Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions

Size: px
Start display at page:

Download "Two Brownian Particles with Rank-Based Characteristics and Skew-Elastic Collisions"

Transcription

1 Two Brownian Particles with ank-based Characteristics and Skew-Elastic Collisions arxiv:16.435v [math.p] 3 Aug 1 E. OBET FENHOLZ TOMOYUKI ICHIBA IOANNIS KAATZAS August 19, 1 Abstract We construct a two-dimensional diffusion process with rank-dependent local drift and dispersion coëfficients, and with a full range of patterns of behavior upon collision that range from totally frictionless interaction, to elastic collision, to perfect reflection of one particle on the other. These interactions are governed by the left- and right-local times at the origin for the distance between the two particles. We realize this diffusion in terms of appropriate, apparently novel systems of stochastic differential equations involving local times, which we show are well posed. Questions of pathwise uniqueness and strength are also discussed for these systems. The analysis depends crucially on properties of a skew Brownian motion with two-valued drift of the bang-bang type, which we also study in some detail. These properties allow us to compute the transition probabilities of the original planar diffusion, and to study its behavior under time reversal. Key Words and Phrases: Diffusion, Local Time, Skew Brownian Motion, Time eversal, Brownian Motion reflected on Brownian motion. AMS Subject Classifications: Primary 6H1 6G44; secondary 6J55 6J6 1 Introduction We construct a planar diffusion X 1, X according to the following recipe: each of its component particles X 1 and X behaves locally like Brownian motion. The characteristics of these random motions are assigned not by name, but by rank: the leader is assigned drift h INTECH Investment Management LLC, One Palmer Square, Suite 441, Princeton, NJ bob@enhanced.com. Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA ichiba@pstat.ucsb.edu. INTECH Investment Management LLC, One Palmer Square, Suite 441, Princeton, NJ ik@enhanced.com and Department of Mathematics, Columbia University, MailCode 4438, New York, NY 17 ik@math.columbia.edu. esearch supported in part by National Science Foundation Grant DMS

2 and dispersion ρ, whereas the laggard is assigned drift g and dispersion σ. One of the dispersions is allowed to vanish, but not both; similarly for the drifts. In the interest of concreteness and simplicity, we shall set λ := g + h >, ρ + σ = A bit more precisely, we shall construct a complete probability space Ω, F, P endowed with a filtration F = {Ft} t< that satisfies the usual conditions of right continuity and of augmentation by P negligible sets, and on it two pairs B 1, B and X 1, X of continuous, F adapted processes, such that B 1, B is planar Brownian motion and X 1, X a continuous planar semimartingale that starts at some given site X 1, X = x 1, x on the plane and satisfies the dynamics dx 1 t = g 1 {X1 t X t} h 1 {X1 t>x t} dt + ρ 1 {X1 t>x t} + σ 1 {X1 t X t} db 1 t dx t = + 1 ζ 1 g 1 {X1 t>x t} h 1 {X1 t X t} dl X 1 X t + 1 η 1 dl X X 1 t, 1. dt + ρ 1 {X1 t X t} + σ 1 {X1 t>x t} db t + 1 ζ dl X 1 X t + 1 η dl X X 1 t. 1.3 Here and in the sequel we denote by L X L X ; the right-continuous local time accumulated at the origin by a generic continuous semimartingale X, by L X L X ; its left-continuous version, and by L X = L X + L X / its symmetric version; we collect in section the necessary reminders from the theory of semimartingale local time. Each time the particles collide, their trajectories are dragged by amounts proportional to the right local times, L X 1 X t and L X X 1 t, respectively, that have been accumulated up to that instant t at the origin by the differences X 1 X and X X 1 ; this is the significance of the last two terms in each of 1., 1.3. With the notation ζ := 1 + ζ 1 ζ, η := 1 η 1 η, 1.4 the proportionality constants of these interactions, ζ i and η i for X i i = 1,, will be assumed to satisfy the conditions ζ + η, α := η η + ζ We shall discuss in detail the significance of these conditions for the system of equations 1., 1.3; in particular, the fact that they are not only sufficient but also necessary for the wellposedness of the above system of stochastic equations. For the time being, let us note that in the special case ζ 1 = η 1 = 1, the trajectory X 1 of the first particle crosses the trajectory X of the second particle without feeling it, that is, without being subjected to any local time drag; as we shall see in subsection 5., we obtain this same effect under the more general condition

3 5.5. Likewise, the second particle does not feel the first, when ζ = η = 1 or, more generally, under the condition 5.6. When ζ i = η i = 1 i = 1,, the local times vanish completely from 1., 1.3 and we are in the situation studied in detail by FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V In this case, the collisions of the particles are totaly frictionless. At the other extreme ζ = η respectively, η = ζ the trajectory X 1 of the first particle bounces off the trajectory X of the second particle resp., the other way round, as if the latter trajectory were a perfectly reflecting boundary; cf. subsection 5.1. Think of the second resp., the first particle as being heavy, so that in collisions with the light first resp., second particle its motion is unaffected, while the light particle undergoes perfect reflection. In between, for other values of the parameters, we have collisions that are neither totally frictionless without local time drag, nor perfectly reflecting, but elastic : The particles are subjected in general to local time drag, and this kind of friction manifests itself in an asymmetric fashion due to the presence of both right- and left- local times at the origin L Y L Y ; + and L Y L Y ; in 1., 1.3 for the difference Y = X 1 X. We call such collisions skew-elastic. 1.1 Preview Under the conditions of 1.5, the system of equations 1., 1.3 will be shown in section 4 to admit a weak solution, which is unique in the sense of the probability distribution; cf. Theorem 4.1. Using a common terminology: under the conditions of 1.5, the system of equations 1., 1.3 is well posed. A crucial rôle in establishing this result will be played by the properties of the difference Y = X 1 X, for which we show that W t := Y t x 1 x + λ t sgn Y s ds α 1 L Y t, t < is standard Brownian motion W in the notation of 1.1, 1.5. To put a little differently: we identify this difference Y = X 1 X as a so-called Skew Brownian Motion with Bang- Bang drift, a process that we study in detail in Section 6. Similarly, recalling the notation of 1.4 and setting β := η ζ 1 + ζ + ζη 1 + η η + ζ, we identify the process V t := X 1 t + X t x 1 + x g h t 1 β L Y t, t < as another standard Brownian motion, whose cross-variation with the Brownian motion W is V, W = V, Y = γ sgn Y t dt, γ := ρ σ. 3

4 These identifications allow us then to represent the motions X 1, X of the individual particles in the form X 1 t = x 1 + µ t + ρ Y + t y + σ Y t y + 1 β γ L Y t + ρ σ Qt, X t = x + µ t σ Y + t y + + ρ Y t y + 1 β γ L Y t + ρ σ Qt ; here Q is yet another standard Brownian motion, independent of the difference Y = X 1 X, and µ = g ρ h σ. This way we construct a weak solution to the system of equations 1., 1.3, and also show that uniqueness in distribution holds for it. Always under the conditions of 1.5, the system of equations 1., 1.3 is shown in section 4 actually to admit a pathwise unique, strong solution; cf. Theorem 4.. Here we refine the LE GALL LE GALL, J.F LE GALL, J.F methodology, that we used in the recent work FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V. 1 1 to establish pathwise uniqueness for a generalization of the perturbed TANAKA equation of POKAJ POKAJ In fact, the conditions in 1.5 turn out to be not just sufficient but also necessary for the wellposedness of the system 1., 1.3; cf. Proposition 6.1. As we shall see in emarks 3.1 and 3., this system admits no solution in the case η = ζ ; whereas it has lots of solutions, i.e., uniqueness in distribution fails for the system of equations 1. and 1.3, when η = ζ =. Finally, if we do have η + ζ yet 1.5 fails because α / [, 1], it is seen in emark 6.1 that the system 1., 1.3 once again fails to admit a solution. Section 5 discusses some special configurations of the parameters η i, ζ i i = 1, in 1., 1.3 that give rise to some rather interesting structure. We see, in particular in the non-degenerate case ρ σ >, that when β = respectively, β =, the trajectory X 1 X of the leader respectively, X 1 X of the laggard is Brownian motion with drift, with perfect reflection on the trajectory X 1 X of the laggard respectively, X 1 X of the leader, which is then another, independent Brownian motion with drift. Section 6 develops the theory and properties of the Skew Brownian Motion with Bang-Bang drift. Finally, Section 7 uses these properties to compute the transition probabilities and the timereversal of the planar diffusion X 1, X. 1. Extant Work and Open Questions The study of multidimensional stochastic differential equations that involve a local time supported on a smooth hypersurface starts with the work of ANULOVA ANULOVA and POTENKO POTENKO, N.I POTENKO, N.I POTENKO, N.I b 1979 POTENKO, N.I To the best of our knowledge, systems of stochastic equations of the type X i = x i + B i + j i q ij LX i X j, i = 1,, n 1.6 4

5 for a suitable array of real constants q ij 1 i,j n, with B 1,, B n independent standard Brownian motions, were studied first by SZNITMAN & VAADHAN SZNITMAN, A.S. & VAAD- HAN, S..S In fact, these authors consider the more general model Xt = x + Bt + N n q k L k X t, t <, 1.7 k=1 where X := X 1,..., X n, B := B 1,, B n is Brownian motion in n, x n, the unit column vectors n k generate pairwise distinct hyperplanes, and the column vectors q k satisfy the orthogonality conditions q k n k = for k = 1,..., N. When g = h = and σ = ρ, it can be verified using the relationships 3.16 between the symmetric local time and the right local time that the system is equivalent to the model 1.7 with parameters n =, N = 1, n 1 := 1, 1 /, and q 1 := α1 ζ 1 +1 η 1 1 α, α1 ζ +1 η 1 α. The orthogonality conditions amount then to η = ζ. Thus, we can apply the results of SZNITMAN & VAADHAN SZNITMAN, A.S. & VAADHAN, S..S. 1986, if g = h =, σ = ρ, η = ζ in our system There are rather obvious similarities, as well as differences, between the system 1.6 and that of 1., 1.3. In particular, it would be very interesting to extend the results of this paper to systems of stochastic differential equations of the type X i = x i + n k=1 + j i δ k 1 {Xi t=x k t} dt + n k=1 σ k 1 {Xi t=x k t} db i t q + ij LX i X j + q ij LX j X i, i = 1,, n 1.8 for an arbitrary number n N of particles, with x 1,, x n and δ 1,, δ n given real constants, with σ 1,, σ n given positive constants, with suitable arrays q ± ij 1 i,j n of real constants, the descending order statistics notation max X jt =: X 1 t X t X n 1 t X n t := min X jt, 1 j n 1 j n and lexicographic breaking of ties. This system 1.8 exhibits both features of rank-dependent characteristics and skew-elastic collisions that are manifest in 1., 1.3, but involves several particles rather than just two. On Semimartingale Local Time Let us recall the notion of a continuous, real-valued semimartingale X = X + M + C,.1 5

6 where M is a continuous local martingale and C a continuous process of finite first variation such that M = C =. The local time L X t; ξ accumulated at a given site ξ over the time-interval [, t] by this process, is L X 1 t; ξ := lim ε ε t 1 {ξ Xs<ξ+ε} d X s = Xt ξ + t + X ξ 1 {Xs>ξ} dxs,. where X M. For every fixed ξ this defines a nondecreasing, continuous and adapted process L X ; ξ which is flat off the set {t : Xt = ξ}, namely 1 {Xt ξ} dl X t; ξ = ; and we have also the property 1 {Xt=ξ} d X t =..3 On the other hand, for each fixed T, the mapping ξ L X T ; ξ is almost surely CLL ight-continuous on [,, with Limits from the Left on, paths, and has jumps of size L X T ; ξ L X T ; ξ = We shall employ also the notation T 1 {Xt=ξ} dxt = T 1 {Xt=ξ} dct..4 L X T ; ξ := 1 L X T ; ξ + L X T ; ξ.5 for the so-called symmetric local time" accumulated at the site ξ over the time interval [, T ]. For these local times we prefer to use the simpler notation L X L X ;, L X L X ;, LX L X ;.6 when we evaluate them at the origin ξ =, and note L X = L X 1 = lim ε ε Finally, we recall the occupation time density formulae hxt d X t = L X ; ξ hξ dξ = 1 { Xt> ε} d X t..7 L X ; ξ hξ dξ,.8 valid for every Borel measurable function h : [,, as well as the ITÔ-TANAKA formulae fx = fx + D fxt dxt + L X ; ξ f dξ,.9 fx = fx+ 1 D + fxt+d fxt dxt+ L X ; ξ f dξ..1 Here f : is the difference of two convex functions, D ± f denote its derivatives from left and right, and f denotes its second derivative measure. 6

7 .1 Tanaka Formulae For a continuous, real-valued semimartingale X as in.1, and with the conventions sgn x := 1, x 1, x, sgn x := 1, x 1,] x, x.11 for the symmetric and the left-continuous versions of the signum function, we obtain from.9,.1 the TANAKA formulae X ξ = X ξ + = X ξ + sgn Xt ξ dxt + L X ; ξ.1 sgn Xt ξ dxt + L X ; ξ..13 Applying.1 with ξ = to the continuous, nonnegative semimartingale X, then comparing with the expression of.1 itself, we obtain the companion L X L X = 1 {Xt=} dxt = 1 {Xt=} dct.14 of the property.4. For the theory that undergirds these results we refer, for instance, to KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , section Analysis Let us suppose that such a probability space as stipulated in section 1 has been constructed, and on it a pair B 1, B of independent standard Brownian motions, as well as two continuous, nonnegative semimartingales X 1, X such that the dynamics are satisfied. We import the notation of FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V. 1 1 : in addition to 1.1, we set ν = g h, y = x 1 x, z := x 1 + x >, r 1 = x 1 x, r = x 1 x, 3.1 and introduce the difference and the sum of the two component processes, namely Y := X 1 X, Z := X 1 + X Auxiliary Brownian Motions We introduce also the two planar Brownian motions W 1, W and V 1, V, given respectively by W 1 := W := 1 {Y t>} db 1 t 1 {Y t } db 1 t 1 {Y t } db t, {Y t>} db t 3.4 7

8 and V 1 := V := 1 {Y t>} db 1 t + 1 {Y t } db 1 t + 1 {Y t } db t, {Y t>} db t. 3.6 Finally, we construct the Brownian motions W, V, Q and W, V, U as W := ρ W 1 + σ W, V := ρ V 1 + σ V, Q := σ V 1 + ρ V, 3.7 W := ρ W 1 σ W, V := ρ V 1 σ V, U := σ W 1 ρ W ; 3.8 we note the independence of Q and W, the independence of Q and V, and observe the intertwinements and V j = 1 j+1 sgn Y t dw j t j = 1,, V = V = 3. The Difference and the Sum sgn Y t dw t, Q = sgn Y t dw t 3.9 sgn Y t du t. 3.1 After this preparation, we observe that the difference Y and the sum Z from 3. satisfy, respectively, the stochastic integral equation Y = y λ sgn Y t dt + 1 ζ L Y 1 η L Y + W 3.11 which involves both the right- and the left- local time at the origin of its solution process Y, and the identity Zt = z + ν t + V t + 1 ζ L Y t + 1 η L Y t, t <. 3.1 We have used here the notation of 1.1, 3.1, 1.4, as well as ζ := ζ 1 + ζ, η := η 1 + η We note from.3 and 3.11 that Y is a continuous semimartingale with 1 {Y t=} dt = and that on the strength of.4,.3 we have L Y ; L Y ; = {Y t=} d Y t =, a.s., {Y t=} 1 ζ dl Y t; 1 η dl Y t; 8

9 or equivalently = 1 ζ L Y ; 1 η L Y ; ζ L Y = η L Y From this relationship and.3-.5,.14 we obtain L Y = L Y and L Y = α L Y, L Y = 1 α L Y, 3.16 where we introduce as in 1.5 the skewness parameter" α := η η + ζ With this notation, and recalling 3.14 and 3.16, we see that equation 3.11 takes the form Y = y λ sgn Y t dt + α 1 L Y + W 3.18 of the equation for a Skew Brownian Motion with Bang-Bang drift Skew Bang-Bang Brownian Motion, or SBBBM for short. This is a very close relative of the Skew Brownian motion, that was introduced by ITÔ & MCKEAN ITÔ, K. & MC KEAN, H.P., J , ITÔ, K. & MC KEAN, H.P., J and was furhter studied by WALSH Walsh, J.B , HAISON & SHEPP HAISON, J.M. & SHEPP, L.A ; see LEJAY LEJAY, A. 6 6 for a comprehensive survey. The diffusion process Y of 3.18 is studied in detail in section 6. It is a strong MAKOV and FELLE process, whose transition probabilities can be computed explicitly; see below. In particular, it is shown in section 6 that, for α 1, the stochastic equation 3.18 has a pathwise unique, strong solution and that the filtration identities F Y t = F W t, t [, 3.19 hold. Here and in what follows, given a process Ξ : [, Ω d Euclidean space and CLL paths, we shall use the convention with values in some F Ξ = {F Ξ t} t< for the smallest filtration to which Ξ is adapted that satisfies the usual conditions of right continuity and augmentation by sets of measure zero. Similarly, and with the notation of 3.17, the expression 3.1 takes the form X 1 t + X t = Zt = z + ν t + V t + 1 β L Y t, t < 3. where, as in subsection 1.1, we set β := η ζ + ζ η η + ζ

10 emark 3.1. It is clear from 3.15 that the equation 3.11 can be written as Y = y λ sgn Y t dt + L Y L Y + W. 3. To wit: skew Brownian motion with bang-bang drift solves the equation 3., for any value α [, 1] of its skewness parameter. We conclude that uniqueness in distribution fails for this equation 3., thus also for the equation 3.11 that governs the difference Y = X 1 X when η = ζ =. In particular, uniqueness in distribution cannot possibly hold for the system 1., 1.3 in this case η = ζ =. emark 3.. When η = ζ we get L Y + L Y from 3.15, thus L Y L Y, 3.3 and the equation 3.11 takes the form of Brownian motion with bang-bang drift Y = y λ sgn Y t dt + W. 3.4 This diffusion process was studied in detail by KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , who computed its transition probabilities and the joint distribution of the triple Y t, L Y t, t 1 {Y s>} ds. This diffusion does accumulate local time at the origin: indeed, on the strength of.4,.3, we have L Y L Y = 1 {Y t=} dy t = λ 1 {Y t=} dt = λ 1 {Y t=} d W t = almost surely, but also PL Y t = L Y t > > for every t, ; this contradicts 3.3. In fact, we have PL Y t = L Y t > = 1 for y =. We conclude that the equation 3.11 for the difference Y = X 1 X has no solution in the case η = ζ. Thus, the system 1., 1.3 cannot possibly have a solution in this case. 3.3 Auxiliary Systems From the equations of 3.11, 3.1 and using the notation in , we obtain a system of stochastic differential equations dx 1 t = g 1 {X1 t X t} h 1 {X1 t>x t} dt + ρ 1 {X1 t>x t} dw 1 t + σ 1 {X1 t X t} dw t + 1 ζ 1 dl X 1 X t + 1 η 1 dl X X 1 t, 3.5 dx t = g 1 {X1 t>x t} h 1 {X1 t X t} dt ρ 1 {X1 t X t} dw 1 t σ 1 {X1 t>x t} dw t + 1 ζ dl X 1 X t + 1 η dl X X 1 t, 3.6 1

11 quite similar to that of 1., 1.3, but now driven by the planar Brownian motion W 1, W. In a totally analogous manner, we obtain also the system dx 1 t = g 1 {X1 t X t} h 1 {X1 t>x t} dt + ρ 1 {X1 t>x t} dv 1 t + σ 1 {X1 t X t} dv t + 1 ζ 1 dl X 1 X t + 1 η 1 dl X X 1 t, 3.7 dx t = g 1 {X1 t>x t} h 1 {X1 t X t} dt + ρ 1 {X1 t X t} dv 1 t + σ 1 {X1 t>x t} dv t + 1 ζ now driven by the planar Brownian motion V 1, V. dl X 1 X t + 1 η dl X X 1 t, Skew epresentations In light of the TANAKA formula.13, of the equation 3.18 for the semimartingale Y, and of the last intertwinement in 3.9, we represent now the size of the gap between X 1 t and X t as Y t = y λ t + V t + L Y t 3.9 = y λ t + V t + L Y t, t <. With the help of 3.9, 3.16, let us write the first Brownian motion in 3.8 as W = γ W + δ U, where γ := ρ σ, δ := 1 γ = ρ σ. With this notation, and with the help of 3.1, the Brownian motion V in 3.7 takes the form V t = γ V t + δ Qt = γ Y t y + λ t L Y t + δ Qt, t <. We recall here from 3.1 the standard Brownian motion Q which, being independent of W, is also independent of the process Y in light of In conjunction with X 1 t X t = Y t and the representation 3. for X 1 t + X t, and with the notation µ := 1 ν + λ γ = g ρ h σ, we obtain from this expression the skew representations for the component processes themselves X 1 t = x 1 + µ t + ρ Y + t y + σ Y t y + 1 β γ L Y t + ρσqt 3.3 X t = x + µ t σ Y + t y + + ρ Y t y + 1 β γ L Y t + ρσqt 3.31 in terms of the paths of the skew Brownian motion process Y with bang-bang drift, and of the independent Brownian motion Q. In particular, this shows that uniqueness in distribution holds for the system of stochastic differential equations 1., 1.3. Similar reasoning shows that uniqueness in distribution holds also for each of the systems 3.5, 3.6 and 3.7,

12 emark 3.3. It is clear from 3.9 that the absolute value of the skew Brownian motion with bang-bang drift in 3.18, for any value α [, 1] of the skewness parameter, is Brownian motion with drift λ and reflection at the origin. Arguing as in WALSH Walsh, J.B , Proposition 1, one can conclude that every diffusion process Y, for which Y is Brownian motion with drift λ and reflected at the origin, is a skew Brownian motion with bang-bang drift. 4 Synthesis We reverse now the steps of the analysis in section 3. Let us start with a filtered probability space Ω, F, P, F = {Ft} t< and with two independent, standard Brownian motion W 1, W on it; we shall assume F F W 1,W, i.e., that the F is the smallest filtration satisfying the usual conditions, to which the planar Brownian motion W 1, W is adapted. With given real constants ζ 1, ζ, η 1, η and nonnegative constants g, h, ρ, σ that satisfy 1.1 and 1.5, with a given vector x 1, x, and with the notation of 3.1, we construct the pairs of independent Brownian motions and W : = ρ W 1 + σ W, U : = σ W 1 ρ W 4.1 U : = σ W 1 + ρ W, W : = ρ W 1 σ W 4. as in 3.8, 3.7. Clearly, F W 1,W F W,U F U,W. We construct also the pathwise unique, strong solution Y of the stochastic equation 3.18 driven by the Brownian motion W introduced in 4.1. With the process Y thus in place, we introduce by analogy with 3.9 the independent Brownian motions V 1 = sgn Y t dw 1 t, V t = sgn Y t dw t, 4.3 and by analogy with 3.7, 3.8 the two additional pairs of independent Brownian motions and V : = ρ V 1 + σ V, Q : = σ V 1 ρ V 4.4 Q : = σ V 1 + ρ V, V : = ρ V 1 σ V. 4.5 We introduce also the continuous martingales M 1 := = ρ 1 {Y t>} dw 1 t + σ 1 {Y t } dw t ρ 1 {Y t>} dv 1 t + σ 1 {Y t } dv t, 4.6 1

13 M := = with M 1, M and quadratic variations M 1 = ρ 1 {Y t } dw 1 t + σ 1 {Y t>} dw t ρ 1 {Y t } dv 1 t + σ 1 {Y t>} dv t, 4.7 ρ 1 {Y t>} +σ 1 {Y t } dt, M = ρ 1 {Y t } +σ 1 {Y t>} dt. There exist then independent Brownian motions B 1, B on our filtered probability space Ω, F, P, F = {Ft} t<, so the continuous martingales of 4.6, 4.7 are cast in their DOOB representations as M 1 = ρ 1 {Y t>} +σ 1 {Y t } db 1 t, M = ρ 1 {Y t } +σ 1 {Y t>} db t 4.8 in terms of independent Brownian motions B 1, B ; for instance, by taking B 1 = 1 {Y t>} dw 1 t + 1 {Y t } dw t, 4.9 B = Finally, we introduce the continuous, F adapted processes and X 1 := x 1 + X := x + 1 {Y s } dw 1 t + 1 {Y t>} dw t. 4.1 g 1{Y t } h 1 {Y t>} dt + M1 + 1 ζ 1 g 1{Y t>} h 1 {Y t } dt + M + 1 ζ L Y + 1 η 1 L Y + 1 η L Y 4.11 L Y. 4.1 It is now easy to check X 1 X = Y ; and from this, that the vector process X 1, X solves the system , as well as the systems , It is also straightforward to verify the skew representations of 3.3, emark 4.1. We note that the vector process X 1, X solves also the system of stochastic equations dx 1 t = 1 {X1 t X t} g dt + σ db1 t {X1 t>x t} h dt + ρ db1 t + κ 1 dl X 1 X t, dx t = 1 {X1 t>x t} g dt + σ db t {X1 t X t} h dt + ρ db t + κ dl X 1 X t, where κ j := α 1 ζ j + 1 α 1 η j, j = 1, or equivalently κ 1 = α β/, κ = 1 α β/

14 4.1 anks Let us introduce explicitly the ranked versions leader and laggard, respectively 1 := X 1 X, := X 1 X 4.16 of the components of the vector process X 1, X constructed in 4.11, 4.1. From 3.1 and 3.9, it is rather clear that we have 1 t + t = X 1 t + X t = r 1 + r + ν t + V t + 1 β L Y t, t < 1 t t = X1 t X t = Y t = y λ t + V t + L Y t, 4.17 and these representations lead to the expressions 1 t = r 1 h t + ρ V 1 t + 1 β/ L 1 t, t < 4.18 t = r + g t + σ V t β/ L 1 t, t < A few remarks are in order. The equations 4.18, 4.19 identify the processes V 1 and V of 3.5, 3.6 as the independent Brownian motions associated with the diffusive motion of the ranked particles, the leader 1 and the laggard, respectively; whereas the independent Brownian motions B 1 in 1. and B in 1.3 are associated with the specific names indices, or identities of the individual particles. On the other hand, with the help of the theory of the SKOOKHOD reflection problem e.g., KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , page 1, we obtain from 4.17,.3 the identification of the collision local time L 1 t = L Y t = max s t + y + λ s V s, t <. 4. Let us also observe that, in the non-degenerate case ρ σ >, the equations and the second equation in 4.5 give the filtration comparisons F V 1,V t = F 1, t, t <, 4.1 F V t = F Y t F Y t, < t <, 4. where the inclusion is strict, due to the fact that the process Y changes its sign with positive probability during any time-interval [, t] with t >. 4. Filtration Comparisons, Weak and Strong Solutions We have the following straightforward analogues of Propositions 4.1, 4. and of Theorems 4.1, 4. in FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V

15 Proposition 4.1. In the degenerate case σ =, thus ρ = 1 in light of 1.1, we have the relations F 1, t = F V t = F X 1 X t F X 1 X t = F W t = F X 1,X t 4.3 for every < t <, where the inclusion is strict. In the special case β = 1 of 5. we have in addition σv t = σx 1 t + X t, thus also F V t = F X 1+X t, for every t <. Proposition 4.. In the non-degenerate case ρ σ >, we have for every < t < the filtration relations F V 1,V t = F 1, t = F Y,V t = F Y,Q t 4.4 where the inclusion is strict. F Y,Q t = F Y,V t = F W 1,W t = F X 1,X t, Theorem 4.1. The system of stochastic differential equations 1., 1.3 is well-posed, that is, has a weak solution which is unique in the sense of the probability distribution. The same is true for each of the systems of equations 3.5, 3.6 and 3.7, 3.8. On the other hand, the system of stochastic differential equations 3.5, 3.6 admits a strong solution, which is therefore pathwise unique; whereas the system 3.7, 3.8 admits no strong solution. Theorem 4.. The system of stochastic differential equations 1., 1.3 admits a pathwise unique, strong solution; in particular, the filtration identity F B 1,B t = F X 1,X t holds for all t <. Likewise, the system of equations 4.13, 4.14 has a pathwise unique, strong solution. Proof. epeating almost verbatim the arguments in the proof of Theorem 5.1 in FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V. 1 1, the question boils down to whether the filtration comparison F Y t F B 1,B t, t < 4.5 holds. To decide this issue, we write the equation 3.18 as driven by the pair B 1, B ; in other words, we use 3.7 and 3.3, 3.4 to express the skew Brownian motion Y with bangbang drift as solution of a stochastic differential equation driven by the planar Brownian motion B 1, B. Since this equation does admit a weak solution which is unique in distribution, the issue is whether this solution is also strong, that is, whether 4.5 holds. This question is easy to settle in the isotropic case ρ = σ = 1/ ; then W = B 1 B /, and the comparison 4.5 follows from the strong solvability of the equation 3.18 proved in section 6: F Y t = F W t F B 1,B t holds for all t <, by virtue of In the non-isotropic case ρ σ, we write 3.18 as the extended skew Tanaka equation Y t = y + ρ σ t sgn Y s dβs ρ + σ ϑt + α 1 L Y t, t T

16 with T, arbitrary but fixed. Here β := β 1 + β, ϑ := β 1 β are given standard, independent Brownian motions after an equivalent change of probability measure, and β i t := B i t λ t, t T i = 1,. ρ σ Let us suppose that Y 1 and Y are two solutions of the equation 4.6, defined on the same probability space and with respect to the same, independent standard Brownian motions B 1, B. Following LE GALL 1983, we shall show L Y 1 Y ; we shall then argue that this implies also Y 1 Y, a.s. To this end, we consider the difference D := Y 1 Y, as well as the linear combinations Z u := 1 u Y 1 + u Y for u 1 ; we introduce also a sequence {f k } k N C 1 of continuous and continuously differentiable functions that converge to f := sgn pointwise, and satisfy sup k N f k T V <. Since lim sup k f k T V f T V obviously holds, this is only possible if f is of bounded variation, and in this case an approximating sequence is easily obtained, e.g., by mollifiers. As in the proof of Theorem 8.1 of FENHOLZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V. 1 1, for every δ >, T >, k 1, we establish then [ T E f k Y 1 s f k Y s Y 1 s Y s ] 1 {Y1 s Y s>δ} dt c 1 f k T V sup E L u T, ξ ; ξ,u here L u T, ξ is the symmetric local time of Z u accumulated at the site ξ over the time interval [, T ], and c 1 is a constant chosen independently of k, u, δ. Letting k and δ, we obtain [ T E 1 ] Dt 1 {Dt>} d D t [ T E c 1 f T V sup E ξ,u f Y 1 t f Y t Y 1 t Y t L u T, ξ. ] 1 {Dt>} dt Now the CAUCHY-SCHWATZ inequality, the ITÔ isometry, and the TANAKA formula.13 applied to Z u, allow us to estimate E L u T ; ξ E Z u T Z u + [ E Z u T ] 1/ + α 1 u E LY 1 T + 1 u E LY T [ [E Z u T ] 1/ + α 1 u E LY 1 T + 1 u E LY T ]. 16

17 The last term is bounded uniformly in ξ, u, since Z u t c t and E L Y i T c 3, for i = 1, and for some constants c, c 3 that do not depend on ξ, u. Thus, we obtain [ T E 1 ] Dt 1 {Dt>} d D t <, < T <. 4.7 Using Lemma 1. of LE GALL LE GALL, J.F see also Exercise 3.7.1, pages 5-6 in KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , we verify that 4.7 gives L D. By exchanging the rôles of Y 1 and Y, we obtain also L D = L Y Y 1, as well as L D. Furthermore, we note that on the strength of Corollary.6 of OUKNINE & UTKOWSKI OUKNINE, Y. & UTKOWSKI, M this implies that the symmetric local time L M of the maximum is given as M := Y 1 Y = Y 1 + Y Y 1 + L M := L Y 1 Y = 1 {Y t } d L Y 1 t + 1 {Y1 t<} d L Y t. We combine now these results with the TANAKA formula, to obtain the dynamics of the maximum M = y + 1 {Y1 t Y t} dy 1 t + 1 {Y1 t<y t} dy t + L Y Y 1 = y + ρ σ sgnmt dβt ρ + σ ϑ + α 1 LM, and observe that these are the same as those of 4.6. But uniqueness in distribution holds for the equation 4.6, so the distribution of the process M is the same as that of Y 1 ; and of course we have M Y 1 a.s. This implies M Y 1, thus Y 1 Y a.s. Therefore, the solution to 4.6 is pathwise unique, hence also strong by the theory of YA- MADA & WATANABE e.g., KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , pages Some Special Cases When α = 1/, that is, η = ζ or equivalently η 1 η = ζ ζ 1, 5.1 the equation 3.18 for the difference Y = X 1 X becomes that of Brownian motion with bang-bang drift Y t = y λ t sgn Y s ds + W t, t < 17

18 as in 3.4. In this special case η = ζ and with σ = ρ, the existence and uniqueness of can be shown also by direct application of Theorem 3.5 of SZNITMAN & VAADHAN SZNITMAN, A.S. & VAADHAN, S..S and a GISANOV s change-of-measure, with the aid of the local time relationships On the other hand, when β = 1 or equivalently η 1 ζ = ζ 1 η, 5. we observe from 3. that the sum X 1 + X is just standard Brownian motion with drift ν = g h. Let us single out now, and study, some more interesting special cases. 5.1 Perfect eflection for Individual Particles Upon Collision Suppose α = 1, or equivalently ζ = and η from 1.5, that is ζ ζ 1 = η 1 η, 5.3 and that x 1 x. We see then from 3.16 that we have L Y L X X 1, and that the equation 3.18 becomes Y t = y λ t + W t + L Y t, t <. From the theory of the SKOOKHOD problem e.g., KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , pages 9-1 we conclude L Y + t = max y + λ s W s, t < s t thus Y, and that strength and pathwise uniqueness hold; for more general results along these lines see CHITASHVILI & LAZIEVA CHITASHVILI,.J. & LAZIEVA, N.L It is also clear from the last two displayed equations, that the filtration identity F Y t = F W t, t < in 3.19 also holds. In this case, then, when the particles collide, the trajectory X 1 of the first particle bounces off the trajectory X of the second particle as if this latter were a perfectly reflecting lower boundary. We can visualize the situation by saying that, under the conditions of 1.5 and 5.3, the second particle is heavy unaffected by collisions, whereas the first particle is light in that it bounces off reflects perfectly when colliding with the heavy particle. The symmetric situation obtains for α =, that is ζ and η = or equivalently ζ ζ 1 = η 1 η ; 5.4 in this case and again with x 1 x, when the two particles collide, the second particle bounces off the first as if this latter were a perfectly reflecting upper boundary; it is the first particle that is now heavy, and the second that is light. 18

19 5. Frictionless Collision It follows also from 3.16 that the local times disappear entirely in 1. when we have the configuration of parameters 1 ζ 1 α + 1 η 1 1 α =, or equivalently 1 ζ 1 η + 1 η 1 ζ = ; 5.5 in this case the trajectory of the first particle crosses that of the second without feeling it, that is, without being subjected to any local time drag. Similarly, the second particle crosses the first in the same frictionless manner, that is, the local times disappear entirely in 1.3, if 1 ζ η + 1 η ζ =. 5.6 If both 5.5 and 5.6 hold, then all such crossings are completely frictionless. We note that 5.5 and 5.6 are both satisfied, if and only if η 1 + ζ 1 = η + ζ = 5.7 holds. This condition implies η = ζ so when this common value is nonzero we are in the case α = 1/ mentioned at the start of the section, and is obviously satisfied in the special case η 1 = ζ 1 = η = ζ = 1 studied by FENHOZ ET AL. FENHOLZ, E.., ICHIBA, T., KAATZAS, I. & POKAJ, V However, 5.7 holds also for other configurations of parameters, for instance ζ 1 = η = 1/, η 1 = ζ = 3/. The condition 5.7 gives the value β = 1 for the parameter of 3.1; back in 4.18, 4.19, this implies that the collision local time L 1 gets apportioned equally to the ranks. 5.3 Elastic Collisions Beyond these two extremes of perfect reflection and frictionless collision that is, for all other configurations of parameters we have collisions that are elastic : neither completely frictionless, nor perfectly reflecting. 5.4 Brownian motion reflected on an independent Brownian motion Finally, let us consider the case β = or equivalently η ζ + ζ η =, that is ζ 1 + ζ + η 1 + η = ζ1 + ζ η1 η η1 + η ζ1 ζ 5.8 in light of 3.1 and 1.4, This happens, for instance, when ζ 1 = 3/4, ζ = 9/4, η 1 = 4/3, η = 8/3 ; in this case we have α = 4/7 and of course β =. Under the condition 5.8, the laggard in 4.19 feels no pressure local time drag from the leader; it just evolves like Brownian motion with variance σ and nonnegative drift. On the other hand, the leader in 4.18 evolves like an independent Brownian motion with variance ρ and nonpositive drift, reflected off the trajectory of the laggard. Such a process has been studied by BUDZY & NUALAT BUDZY, K. & NUALAT, D. see also SOUCALIUC ET AL. SOUCALIUC, F., TÓTH, B. & WENE, W., SOUCALIUC & WENE 19

20 SOUCALIUC, F. & WENE, W. ; here it arises as a special case of the ranked system 4.18, 4.19 for the particles whose motions are governed by the equations 1., 1.3. We have in this case β = an interesting fusion: the perfect reflection we saw in subsection 5.1, and the frictionless motion of subsection 5., are occurring here simultaneously not for the motions of the individual particles, however, but rather for the motions of their ranked versions, the leader 1 and the laggard, respectively. To put it a little differently: starting with two particles that undergo skew-elastic collisions one is able, under the conditions of 1.5 and 5.8, to simulate a heavy particle the laggard and a light particle the leader. The reverse situation obtains when β = or equivalently η ζ + ζ η = η + ζ, that is ζ 1 +ζ +η 1 +η = 4 4+ζ1 ζ η 1 +η + ζ1 +ζ η1 η η1 +η ζ1 ζ ; 5.9 then it is the trajectory of the laggard now the light particle that gets reflected off that of the leader now the heavy particle. This happens, for instance, when ζ 1 = 3/, ζ = 3, η 1 = 7/3, η = 1 ; in this case we have α = 4/7 and β =. 5.5 Some Simulations The pictures Figures 1-4 that follow present simulations of the processes X 1 t in black and X t in red for t [, 1], in black and red, respectively, with drifts g = h = 1 in the degenerate case ρ = Figure 1: ζ 1 = ζ = η 1 = η = 1 ; α = 1/, β = 1.

21 Figure : ζ 1 =, ζ = η 1 = η = 1 ; α = /3, β = / Figure 3: ζ =, ζ 1 = η 1 = η = 1 ; α = /3, β = 4/3. 1

22 Figure 4: ζ 1 =, ζ =, η 1 = η = 1 ; α = 1, β = 1. 6 Skew Brownian Motion with Bang-Bang Drift We study here the stochastic differential equation 3.18 for the skew Brownian motion with bangbang drift by = λ sgny, y 6.1 for some given constant λ >, with the notation of.11, and with skewness parameter α [, 1]. The cases α = and α = 1 have been discussed already in subsection 5.1, so we focus here on the range < α < 1. For this range of values of the skewness parameter, we choose to write the equation 3.18 in terms of the right-continuous local time of the unknown process at the origin, namely Y = y λ sgn Y t dt + W + α 1 α This equation is of the more general form Y = y + τ Y t dw t + with dispersion τ y 1 and measure νdy = by dy + α 1 α L Y. 6. L Y, ξ νdξ, 6.3 δ dy with by = λ sgny, y as in 6.1 above, and with δ the Dirac mass at the origin.

23 We shall deal with the equation 6. using a direct methodology that removes the parts of finite variation, that is, both the drift and the local time, and reduces 6. to a stochastic differential equation in natural scale Z = py + s Zt dw t 6.4 for appropriate functions p and s. This approach was pioneered for the skew Brownian motion itself i.e., with λ = by HAISON & SHEPP HAISON, J.M. & SHEPP, L.A , and for more general equations of the form 6.3 for suitable measurable functions τ and measures ν on B, by LE GALL LE GALL, J.F LE GALL, J.F and ENGELBET & SCHMIDT ENGELBET, H.J. & SCHMIDT, W The results in these works do not seem to cover the equation 6., but those in BASS & CHEN. BASS, & CHEN, Z.Q. 5 5 do; we have preferred to detail a direct construction which is, in our opinion at least, quite simpler. In this spirit, let us introduce the scale function py = 1 α λ This has left-continuous derivative e λy 1, y > ; p = ; py = α 1 e λy, y <. λ p y = 1 α e λy 1, y + α e λy 1,] y, y which is bounded away from zero, and second derivative measure Likewise, we introduce the inverse qz = 1 λ log 1 + λ z 1 α p dy = by p y dy + 1 α δ dy., z > ; q = ; qz = 1 λ log 1 λ z, z < α of the function p, as well as its left-continuous derivative and its second-derivative measure q z = 1 α + λ z 1 1, z + α λ z 1 1,] z, q dz = bz q z dz + α 1 α 1 α δ dz. Analysis: Assume that a solution to 6. has been constructed; in particular, the process Y is then a continuous semimartingale for which 3.14 holds a.s. We look at the process Z := py and apply the ITÔ-TANAKA rule dzt = p Y t dy t + d L Y t, y p dy of.9, to obtain [ dzt = p Y t by t dt + dw t + α 1 α 3 ] dl Y t

24 d by p y L Y t, y dy + 1 α dl Y t = p qy t dw t. We have used here the occupation-time-density formula.8, and the property p = α. Now the piecewise-linear function sz := p qz = 1 q z = 1 α+ λ z 1, z+ α λ z 1,] z, z 6.5 is bounded away from the origin, so the process Z is the pathwise unique, strong solution of the stochastic differential equation 6.4 for this new dispersion function NAKAO NAKAO, S ; and because Y and Z are bijections of each other, we have again the filtration identities F Z t = F Y t = F W t, t <. 6.6 Synthesis: Consider the strong solution Z of the stochastic differential equation 6.4 with the new dispersion function of 6.5, and define the process Y := qz. Since dzt = szt dw t and d Z t = s Zt dt, this process satisfies almost surely 1 {Zt=} dt = 1 {Zt=} d Z t s Zt minα, 1 α 1 {Zt=} d Z t =, and we apply the ITÔ-TANAKA rule to obtain dy t = q Zt dzt + d L Z t, z q dz. On the strength of the occupation-time-density formula and s q 1, this gives dy t = dw t + d by q y L Y t, y dy + α 1 α 1 α dlz t = dw t + by t q Y t s Y t dt + α 1 α 1 α dlz t = dw t + by t dt + α 1 α dl Y t, that is, the equation 6. of the skew Brownian motion with bang-bang drift for the process Y. We have used here the comparison of the local times at the origin for these two processes: L Z = 1 α L Y. 6.7 This last identity 6.7 can be justified as follows: We start by noting L Z 1 = lim ε ε 1 1 {<Zt<ε} d Z t = lim ε ε 1 {<Y t<qε} p Y t dt = 1 α lim ε 1 α ε 1 {<Y t<qε} p Y t 1 α 4 dt.

25 On the event { < Y t < qε} we have 1 p Y t qε 1 α e λ qε ; and since lim ε ε we deduce the claimed identity of 6.7, namely L Z = 1 α lim ε 1 qε = 1 1 α, 1 {<Y t<qε} dt = 1 α L Y. Taken together, the Analysis and Synthesis parts of this argument establish the following result. Theorem 6.1. The equation 3.18 admits a pathwise unique, strong solution for all values of its skewness parameter α [, 1], and we have the filtration identity F Y t = F W t, t < in emark 6.1. As shown towards the end of section 3 in HAISON & SHEPP HAISON, J.M. & SHEPP, L.A , the stochastic equation 3.18 admits no solution for α / [, 1]. Consequently, when η + ζ holds but the condition 1.5 fails because α = η/η + ζ / [, 1], the system of equations 1., 1.3 admits no solution. emark 6.. We compute in the next subsection the transition probabilities of the diffusion Y. It follows from these computations, and in conjunction with the theory developed in POTENKO POTENKO, N.I POTENKO, N.I b 1979 POTENKO, N.I , that this process has the strong MAKOV and FELLE properties. From the emarks 3.1, 3., 6.1 and in conjunction with Theorem 4.1, we obtain now the following result. Proposition 6.1. The conditions of 1.5 are not just sufficient but also necessary for the wellposedness of the system of equations 1., Joint Distribution of SBBBM and its Local Time Let us recall a construction of the skew Brownian motion ITÔ & MC KEAN ITÔ, K. & MC KEAN, H.P., J , ITÔ, K. & MC KEAN, H.P., J , WALSH Walsh, J.B We take a Brownian motion starting from y, reflect it at the origin, and consider its excursions away from the origin. Then we change the sign of each excursion independently with probability 1 α, 1. The resulting process is positive with probability α, negative with probability 1 α. This implies a non-symmetric reflection principle around the origin. We shall see that even in the presence of the bang-bang drifts by = λ sgny, y as in 6.1, this principle continues to hold for the skew Brownian motion. Thus, the joint distribution of SBBBM and its local time are derived here. By GISANOV s theorem e.g., KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , section 3.5, we consider the reference probability measure P, under which the 5

26 process by t dt + W becomes standard Brownian motion. For every given t [, the ADON-NIKODÝM derivative on F Y t of the original measure with respect to the reference measure, is dp { t dp = exp by sdw s + 1 t } b Y sds F Y t = exp {λ y Y t + L Y t } λ t ; we have used in this last equation the relationships 3.9, 3.9. Under the reference probability measure P, the process Y is skew Brownian motion which starting at y. As shown in WALSH Walsh, J.B see also LANG LANG, , the transition probability density function p t; y, ξ = P Y t dξ / dξ for this process is given by p t; y, ξ = 1 { exp y ξ } 1 { + α 1 sgnξ exp y + ξ } πt t πt t for ξ, y, t >. Moreover, by the method of elastic Brownian motion e.g., KAATZAS & SHEVE KAATZAS, I. & SHEVE, S.E , APPUHAMILLAGE ET AL. AP- PUHAMILLAGE, TH., VUSHALI, B., THOMANN, E., WAYMIE, E. & WOOD, B the joint distribution of the skew Brownian motion and its symmetric local time is computed as P Y t dξ, L Y t db = = { 1 + α 1 sgnξ } ξ + b + y πt 3 { exp ξ + b + y } dξ db ; b > t and P Y t dξ, LY t = = 1 [ { exp y ξ } { exp y + ξ }] dξ πt t t for ξ. Note that we have 1 + α 1 sgnξ = α if ξ >, and 1 + α 1 sgnξ = 1 α if ξ. Thus, the non-symmetric reflection principle around the origin works intuitively even for the joint distribution. As expected, when there is no accumulation of local time at the origin, the skewness parameter α does not affect the transition probabilities. We bring the above formulae from the reference measure P back to the original measure P, by means of P Y ± t A, Y t =, L Y t B = 6.8 = exp { λ y λ t/ } [ E P exp { L Y t Y ± t } ] 1 {Y ± t A,Y t =, L Y t B} 6

27 for A, B B B, t >. With ξ, b [,,, the joint density functions are PY + t dξ, Y t =, L Y t db = 6.9 as well as = α e λξ ξ + b + y πt 3 { exp ξ + b + y λt t } dξ db, PY t dξ, Y + t =, L Y t db = 6.1 = 1 α e λξ ξ + b + y { exp ξ + b + y λt } dξ db. πt 3 t Whereas, when there is no accumulation of local time, we have = P Y ± t dξ, Y t =, L Y t = = { exp ξ y + λt } { e λ ξ exp ξ + y + λt } dξ, ξ >. πt t t The marginal density pt; y, ξ dξ = PY t dξ of Y t under the original probability measure P is obtained from PY t dξ = PY t dξ, L Y t > + PY t dξ, L Y t = 1 {ξ y >}, 6.1 where the second term takes care of the case when the local time is absent. If ξ > and y >, the marginal density becomes pt; y, ξ = α 1 e λξ 1 { exp ξ + y λt } 6.13 πt t + 1 { exp ξ y + λt } πt t + α λ e λξ πt whereas, if ξ < and y <, this expression becomes pt; y, ξ = 1 αe λξ 1 { exp ξ y λt } πt t e u λt ξ+y t du ; { exp ξ + y + λt } + 1 α λ eλξ e u λt t du. πt t πt ξ y If ξ y, then this expression becomes pt; y, ξ = { 1 + α 1 sgnξ } e λ ξ πt [ { exp ξ + y λt } + λ t ξ + y where the term PY t dξ, L Y t = in 6.1 is now equal to zero. ] e u λt t du,

28 emark 6.3. Letting t in , we derive for the process Y the stationary measure m = p ξ dξ with the double-exponential probability density function p ξ := lim t pt; y, ξ = α λ e λ ξ 1 {ξ>} + 1 α λ e λ ξ 1 {ξ } It can be verified that 6.16 is the invariant distribution. Furthermore, it follows from the transition density and the stationary distribution 6.16 that the following duality holds: gy fξpt; y, ξdξ p ydy = fξ gypt; ξ, ydy p ξdξ, for arbitrary bounded, measurable functions f, g, or equivalently [ [ gy E y fy t] p y dy = fξ E ξ gy t] p ξ dξ ; t > Here E y stands for the expectation under the measure P y induced by Y which starts from y. Thus, under the probability measure P := P y mdy, the process Y is stationary; and moreover, given a fixed time T,, the time reversal Ŷ t := Y T t, t T 6.18 satisfies E P [ f Y t f n Y t n ] = E P [ f Ŷ t n f n Ŷ t ] 6.19 for every integer n N, collection of time points = t < t 1 < < t n = T, and bounded, measurable functions f,..., f n. emark 6.4. The infinitesimal generator of the process Y may be defined formally by [Lf]ξ := 1 f ξ λ sgnξf ξ + α 1f ξ δ ξ, ξ 6. for f D := C, where δ is the Dirac delta function at the origin. Here we use the parametrization for the symmetric local time L Y. Let us denote formally the symmetric version of the density of m by p ξ = α λ e λξ 1 {ξ>} + 1 α λ e λξ 1 {ξ<} + λ 1 {ξ = }. Then by direct calculation fξ [Lg]ξ mdξ = gξ [Lf]ξ mdξ, [Lf] mdξ = ; f, g D. 6.1 Applying Theorem.3 of FUKUSHIMA & STOOCK FUKUSHIMA, M. & STOOCK, D , we arrive at the same conclusion

29 emark 6.5. Let us define the time reversal of Y as in Following PADOUX PADOUX, E and PETIT PETIT, F , we may show that the time reversal is a solution of the stochastic equation Ŷ t = Ŷ + W t + 1 α L Ŷ t + t + λ sgn Ŷ s + ξ log p T s; y, Ŷ s ds 6. for t T, where W is a standard Brownian motion with respect to the backwards filtration F Ŷ generated by the time-reversed process Ŷ of 6.18, and L Ŷ t := L Y T L Y T t, t T. 6.3 In the special case y = = Ŷ T, the logarithmic derivative of the transition probability density function is ξ log p t;, ξ = λ sgnξ ξ t C 1 t, ξ C 1 t, ξ + C t, ξ, where C 1 t, ξ := exp ξ + λt t, C t, ξ := λ e λ ξ Thus, the time reversal is a skew Brownian bridge with bang-bang drift Ŷ = Ŷ + W + 1 α L Ŷ ξ u λt exp du. t [ λ sgn Ŷ t + Ŷ t T t C1 T t, Ŷ t ] dt. C 1 + C This result suggests that the time reversal of SBBBM in general looks like a skew Brownian bridge drifted towards the target point Ŷ T = y. 7 Applications of the Skew epresentations With the skew representations in section 3.4 and the joint distribution of Y, L Y in section 6.1 it is now straightforward to compute the transition density of the system as well as its time reversal. 7.1 Transition Density Let us discuss only some special cases, since the other cases are quite similar. For example, in the degenerate case with σ =, thus ρ = 1, γ = 1 and with x 1 x, become X 1 t = x 1 + g t + Y + t y + β L Y t, X t = x + g t + Y t y β L Y t, 9

30 for t <, where y = x 1 x, and hence the transition density of X 1, X is P X 1 t dξ 1, X t dξ = α β e λξ 1 ξ where + β c 1 := ξ 1 β c { 1 exp π t 3 β ξ + x 1 + x + β β g t, if β >, ξ 1 ξ and ξ < x + g t. Similarly, by skew symmetry: P X 1 t dξ 1, X t dξ = 1 α β e λξ ξ 1 where + β c := ξ β c { exp π t 3 β ξ 1 + x 1 + x + β β g t, c 1 λ t } dξ 1 dξ, t c λ t } dξ 1 dξ, t if β >, ξ ξ 1 and ξ 1 < x + g t. If β >, ξ 1 > ξ = x + g t, then the local time L Y is absent, and the transition density is easily obtained from 6.11: = P X 1 t dξ 1, X t = x + g t = 1 exp { a x 1 + x + λt } λ e a exp { a + x 1 x + λt } a dξ 1. πt t t = ξ1 x g t For another extreme example, in the degenerate case with σ = 1, thus ρ =, γ = 1 and with x 1 x, become X 1 t = x 1 h t Y t y + β L Y t, X t = x h t Y + t y + + β L Y t, for t <. If β <, ξ 1 ξ and ξ 1 > x 1 h t, then P c { X 1 t dξ 1, X t dξ = α β e λξ 1 ξ 3 exp π t 3 where c 3 := If β <, ξ ξ 1 and ξ > x 1 h t, then P X 1 t dξ 1, X t dξ = 1 α e λξ ξ 1 where c 4 := c 3 λ t t 4 β β 4 β ξ 1 ξ x 1 x + h t β β β β } dξ 1 dξ, c { 4 exp c 4 λ t } dξ 1 dξ, π t 3 t 4 β β 4 β ξ ξ 1 x 1 x + h t. β β β If β <, ξ 1 = x 1 h t > ξ, then the local time L Y does not accumulate, that is, the transitions density is obtained from 6.11 : PX 1 t = x 1 h t, X t dξ = 3

EQUITY MARKET STABILITY

EQUITY MARKET STABILITY EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference,

More information

1 Brownian Local Time

1 Brownian Local Time 1 Brownian Local Time We first begin by defining the space and variables for Brownian local time. Let W t be a standard 1-D Wiener process. We know that for the set, {t : W t = } P (µ{t : W t = } = ) =

More information

The Pedestrian s Guide to Local Time

The Pedestrian s Guide to Local Time The Pedestrian s Guide to Local Time Tomas Björk, Department of Finance, Stockholm School of Economics, Box 651, SE-113 83 Stockholm, SWEDEN tomas.bjork@hhs.se November 19, 213 Preliminary version Comments

More information

Hybrid Atlas Model. of financial equity market. Tomoyuki Ichiba 1 Ioannis Karatzas 2,3 Adrian Banner 3 Vassilios Papathanakos 3 Robert Fernholz 3

Hybrid Atlas Model. of financial equity market. Tomoyuki Ichiba 1 Ioannis Karatzas 2,3 Adrian Banner 3 Vassilios Papathanakos 3 Robert Fernholz 3 Hybrid Atlas Model of financial equity market Tomoyuki Ichiba 1 Ioannis Karatzas 2,3 Adrian Banner 3 Vassilios Papathanakos 3 Robert Fernholz 3 1 University of California, Santa Barbara 2 Columbia University,

More information

Reflected Brownian Motion

Reflected Brownian Motion Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please

More information

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem

On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem On the martingales obtained by an extension due to Saisho, Tanemura and Yor of Pitman s theorem Koichiro TAKAOKA Dept of Applied Physics, Tokyo Institute of Technology Abstract M Yor constructed a family

More information

Stochastic Differential Equations.

Stochastic Differential Equations. Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)

More information

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen

Some SDEs with distributional drift Part I : General calculus. Flandoli, Franco; Russo, Francesco; Wolf, Jochen Title Author(s) Some SDEs with distributional drift Part I : General calculus Flandoli, Franco; Russo, Francesco; Wolf, Jochen Citation Osaka Journal of Mathematics. 4() P.493-P.54 Issue Date 3-6 Text

More information

On a class of stochastic differential equations in a financial network model

On a class of stochastic differential equations in a financial network model 1 On a class of stochastic differential equations in a financial network model Tomoyuki Ichiba Department of Statistics & Applied Probability, Center for Financial Mathematics and Actuarial Research, University

More information

LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES

LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES LOCAL TIMES OF RANKED CONTINUOUS SEMIMARTINGALES ADRIAN D. BANNER INTECH One Palmer Square Princeton, NJ 8542, USA adrian@enhanced.com RAOUF GHOMRASNI Fakultät II, Institut für Mathematik Sekr. MA 7-5,

More information

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3

Brownian Motion. 1 Definition Brownian Motion Wiener measure... 3 Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................

More information

Gaussian Processes. 1. Basic Notions

Gaussian Processes. 1. Basic Notions Gaussian Processes 1. Basic Notions Let T be a set, and X : {X } T a stochastic process, defined on a suitable probability space (Ω P), that is indexed by T. Definition 1.1. We say that X is a Gaussian

More information

Some Aspects of Universal Portfolio

Some Aspects of Universal Portfolio 1 Some Aspects of Universal Portfolio Tomoyuki Ichiba (UC Santa Barbara) joint work with Marcel Brod (ETH Zurich) Conference on Stochastic Asymptotics & Applications Sixth Western Conference on Mathematical

More information

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS

PROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please

More information

Stochastic integration. P.J.C. Spreij

Stochastic integration. P.J.C. Spreij Stochastic integration P.J.C. Spreij this version: April 22, 29 Contents 1 Stochastic processes 1 1.1 General theory............................... 1 1.2 Stopping times...............................

More information

Stable Diffusions Interacting through Their Ranks, as Models of Large Equity Markets

Stable Diffusions Interacting through Their Ranks, as Models of Large Equity Markets 1 Stable Diffusions Interacting through Their Ranks, as Models of Large Equity Markets IOANNIS KARATZAS INTECH Investment Management, Princeton Department of Mathematics, Columbia University Joint work

More information

Weak solutions of mean-field stochastic differential equations

Weak solutions of mean-field stochastic differential equations Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Email: juanli@sdu.edu.cn Based on joint works

More information

Exercises in stochastic analysis

Exercises in stochastic analysis Exercises in stochastic analysis Franco Flandoli, Mario Maurelli, Dario Trevisan The exercises with a P are those which have been done totally or partially) in the previous lectures; the exercises with

More information

Lecture 17 Brownian motion as a Markov process

Lecture 17 Brownian motion as a Markov process Lecture 17: Brownian motion as a Markov process 1 of 14 Course: Theory of Probability II Term: Spring 2015 Instructor: Gordan Zitkovic Lecture 17 Brownian motion as a Markov process Brownian motion is

More information

Generalized Gaussian Bridges of Prediction-Invertible Processes

Generalized Gaussian Bridges of Prediction-Invertible Processes Generalized Gaussian Bridges of Prediction-Invertible Processes Tommi Sottinen 1 and Adil Yazigi University of Vaasa, Finland Modern Stochastics: Theory and Applications III September 1, 212, Kyiv, Ukraine

More information

Wiener Measure and Brownian Motion

Wiener Measure and Brownian Motion Chapter 16 Wiener Measure and Brownian Motion Diffusion of particles is a product of their apparently random motion. The density u(t, x) of diffusing particles satisfies the diffusion equation (16.1) u

More information

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p

p 1 ( Y p dp) 1/p ( X p dp) 1 1 p Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p

More information

Kolmogorov Equations and Markov Processes

Kolmogorov Equations and Markov Processes Kolmogorov Equations and Markov Processes May 3, 013 1 Transition measures and functions Consider a stochastic process {X(t)} t 0 whose state space is a product of intervals contained in R n. We define

More information

Applications of Ito s Formula

Applications of Ito s Formula CHAPTER 4 Applications of Ito s Formula In this chapter, we discuss several basic theorems in stochastic analysis. Their proofs are good examples of applications of Itô s formula. 1. Lévy s martingale

More information

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have

Solution for Problem 7.1. We argue by contradiction. If the limit were not infinite, then since τ M (ω) is nondecreasing we would have 362 Problem Hints and Solutions sup g n (ω, t) g(ω, t) sup g(ω, s) g(ω, t) µ n (ω). t T s,t: s t 1/n By the uniform continuity of t g(ω, t) on [, T], one has for each ω that µ n (ω) as n. Two applications

More information

Lecture 19 L 2 -Stochastic integration

Lecture 19 L 2 -Stochastic integration Lecture 19: L 2 -Stochastic integration 1 of 12 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 19 L 2 -Stochastic integration The stochastic integral for processes

More information

The Cameron-Martin-Girsanov (CMG) Theorem

The Cameron-Martin-Girsanov (CMG) Theorem The Cameron-Martin-Girsanov (CMG) Theorem There are many versions of the CMG Theorem. In some sense, there are many CMG Theorems. The first version appeared in ] in 944. Here we present a standard version,

More information

A Concise Course on Stochastic Partial Differential Equations

A Concise Course on Stochastic Partial Differential Equations A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original

More information

I forgot to mention last time: in the Ito formula for two standard processes, putting

I forgot to mention last time: in the Ito formula for two standard processes, putting I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy

More information

Lecture 21 Representations of Martingales

Lecture 21 Representations of Martingales Lecture 21: Representations of Martingales 1 of 11 Course: Theory of Probability II Term: Spring 215 Instructor: Gordan Zitkovic Lecture 21 Representations of Martingales Right-continuous inverses Let

More information

Walsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1

Walsh Diffusions. Andrey Sarantsev. March 27, University of California, Santa Barbara. Andrey Sarantsev University of Washington, Seattle 1 / 1 Walsh Diffusions Andrey Sarantsev University of California, Santa Barbara March 27, 2017 Andrey Sarantsev University of Washington, Seattle 1 / 1 Walsh Brownian Motion on R d Spinning measure µ: probability

More information

OPTIONAL DECOMPOSITION FOR CONTINUOUS SEMIMARTINGALES UNDER ARBITRARY FILTRATIONS. Introduction

OPTIONAL DECOMPOSITION FOR CONTINUOUS SEMIMARTINGALES UNDER ARBITRARY FILTRATIONS. Introduction OPTIONAL DECOMPOSITION FOR CONTINUOUS SEMIMARTINGALES UNDER ARBITRARY FILTRATIONS IOANNIS KARATZAS AND CONSTANTINOS KARDARAS Abstract. We present an elementary treatment of the Optional Decomposition Theorem

More information

Stability of Stochastic Differential Equations

Stability of Stochastic Differential Equations Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010

More information

An Introduction to Malliavin calculus and its applications

An Introduction to Malliavin calculus and its applications An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart

More information

Lecture 22 Girsanov s Theorem

Lecture 22 Girsanov s Theorem Lecture 22: Girsanov s Theorem of 8 Course: Theory of Probability II Term: Spring 25 Instructor: Gordan Zitkovic Lecture 22 Girsanov s Theorem An example Consider a finite Gaussian random walk X n = n

More information

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )

Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( ) Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical

More information

Exercises. T 2T. e ita φ(t)dt.

Exercises. T 2T. e ita φ(t)dt. Exercises. Set #. Construct an example of a sequence of probability measures P n on R which converge weakly to a probability measure P but so that the first moments m,n = xdp n do not converge to m = xdp.

More information

An essay on the general theory of stochastic processes

An essay on the general theory of stochastic processes Probability Surveys Vol. 3 (26) 345 412 ISSN: 1549-5787 DOI: 1.1214/1549578614 An essay on the general theory of stochastic processes Ashkan Nikeghbali ETHZ Departement Mathematik, Rämistrasse 11, HG G16

More information

LogFeller et Ray Knight

LogFeller et Ray Knight LogFeller et Ray Knight Etienne Pardoux joint work with V. Le and A. Wakolbinger Etienne Pardoux (Marseille) MANEGE, 18/1/1 1 / 16 Feller s branching diffusion with logistic growth We consider the diffusion

More information

The Skorokhod reflection problem for functions with discontinuities (contractive case)

The Skorokhod reflection problem for functions with discontinuities (contractive case) The Skorokhod reflection problem for functions with discontinuities (contractive case) TAKIS KONSTANTOPOULOS Univ. of Texas at Austin Revised March 1999 Abstract Basic properties of the Skorokhod reflection

More information

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt.

The concentration of a drug in blood. Exponential decay. Different realizations. Exponential decay with noise. dc(t) dt. The concentration of a drug in blood Exponential decay C12 concentration 2 4 6 8 1 C12 concentration 2 4 6 8 1 dc(t) dt = µc(t) C(t) = C()e µt 2 4 6 8 1 12 time in minutes 2 4 6 8 1 12 time in minutes

More information

On the submartingale / supermartingale property of diffusions in natural scale

On the submartingale / supermartingale property of diffusions in natural scale On the submartingale / supermartingale property of diffusions in natural scale Alexander Gushchin Mikhail Urusov Mihail Zervos November 13, 214 Abstract Kotani 5 has characterised the martingale property

More information

Some Properties of NSFDEs

Some Properties of NSFDEs Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline

More information

1 Introduction. 2 Measure theoretic definitions

1 Introduction. 2 Measure theoretic definitions 1 Introduction These notes aim to recall some basic definitions needed for dealing with random variables. Sections to 5 follow mostly the presentation given in chapter two of [1]. Measure theoretic definitions

More information

Risk-Minimality and Orthogonality of Martingales

Risk-Minimality and Orthogonality of Martingales Risk-Minimality and Orthogonality of Martingales Martin Schweizer Universität Bonn Institut für Angewandte Mathematik Wegelerstraße 6 D 53 Bonn 1 (Stochastics and Stochastics Reports 3 (199, 123 131 2

More information

1. Stochastic Processes and filtrations

1. Stochastic Processes and filtrations 1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S

More information

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS

OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze

More information

The Azéma-Yor Embedding in Non-Singular Diffusions

The Azéma-Yor Embedding in Non-Singular Diffusions Stochastic Process. Appl. Vol. 96, No. 2, 2001, 305-312 Research Report No. 406, 1999, Dept. Theoret. Statist. Aarhus The Azéma-Yor Embedding in Non-Singular Diffusions J. L. Pedersen and G. Peskir Let

More information

An introduction to Mathematical Theory of Control

An introduction to Mathematical Theory of Control An introduction to Mathematical Theory of Control Vasile Staicu University of Aveiro UNICA, May 2018 Vasile Staicu (University of Aveiro) An introduction to Mathematical Theory of Control UNICA, May 2018

More information

Brownian Motion and Stochastic Calculus

Brownian Motion and Stochastic Calculus ETHZ, Spring 17 D-MATH Prof Dr Martin Larsson Coordinator A Sepúlveda Brownian Motion and Stochastic Calculus Exercise sheet 6 Please hand in your solutions during exercise class or in your assistant s

More information

6. Brownian Motion. Q(A) = P [ ω : x(, ω) A )

6. Brownian Motion. Q(A) = P [ ω : x(, ω) A ) 6. Brownian Motion. stochastic process can be thought of in one of many equivalent ways. We can begin with an underlying probability space (Ω, Σ, P) and a real valued stochastic process can be defined

More information

Nonlinear representation, backward SDEs, and application to the Principal-Agent problem

Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem

More information

1 Euclidean geometry. 1.1 The metric on R n

1 Euclidean geometry. 1.1 The metric on R n 1 Euclidean geometry This chapter discusses the geometry of n-dimensional Euclidean space E n, together with its distance function. The distance gives rise to other notions such as angles and congruent

More information

Squared Bessel Process with Delay

Squared Bessel Process with Delay Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 216 Squared Bessel Process with Delay Harry Randolph Hughes Southern Illinois University Carbondale, hrhughes@siu.edu

More information

GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1. Hungarian Academy of Sciences

GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1. Hungarian Academy of Sciences The Annals of Probability 1996, Vol. 24, No. 3, 1324 1367 GENERALIZED RAY KNIGHT THEORY AND LIMIT THEOREMS FOR SELF-INTERACTING RANDOM WALKS ON Z 1 By Bálint Tóth Hungarian Academy of Sciences We consider

More information

ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME

ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems

More information

Exercises: Brunn, Minkowski and convex pie

Exercises: Brunn, Minkowski and convex pie Lecture 1 Exercises: Brunn, Minkowski and convex pie Consider the following problem: 1.1 Playing a convex pie Consider the following game with two players - you and me. I am cooking a pie, which should

More information

Skew Brownian Motion and Applications in Fluid Dispersion

Skew Brownian Motion and Applications in Fluid Dispersion Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 *Based on joint work with Thilanka Appuhamillage, Vrushali Bokil,

More information

Tools of stochastic calculus

Tools of stochastic calculus slides for the course Interest rate theory, University of Ljubljana, 212-13/I, part III József Gáll University of Debrecen Nov. 212 Jan. 213, Ljubljana Itô integral, summary of main facts Notations, basic

More information

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1)

n [ F (b j ) F (a j ) ], n j=1(a j, b j ] E (4.1) 1.4. CONSTRUCTION OF LEBESGUE-STIELTJES MEASURES In this section we shall put to use the Carathéodory-Hahn theory, in order to construct measures with certain desirable properties first on the real line

More information

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games

Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Stability of Feedback Solutions for Infinite Horizon Noncooperative Differential Games Alberto Bressan ) and Khai T. Nguyen ) *) Department of Mathematics, Penn State University **) Department of Mathematics,

More information

CHAPTER V. Brownian motion. V.1 Langevin dynamics

CHAPTER V. Brownian motion. V.1 Langevin dynamics CHAPTER V Brownian motion In this chapter, we study the very general paradigm provided by Brownian motion. Originally, this motion is that a heavy particle, called Brownian particle, immersed in a fluid

More information

Introduction to self-similar growth-fragmentations

Introduction to self-similar growth-fragmentations Introduction to self-similar growth-fragmentations Quan Shi CIMAT, 11-15 December, 2017 Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 1 / 34 Literature Jean Bertoin, Compensated fragmentation

More information

STIT Tessellations via Martingales

STIT Tessellations via Martingales STIT Tessellations via Martingales Christoph Thäle Department of Mathematics, Osnabrück University (Germany) Stochastic Geometry Days, Lille, 30.3.-1.4.2011 Christoph Thäle (Uni Osnabrück) STIT Tessellations

More information

Filtrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition

Filtrations, Markov Processes and Martingales. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition Filtrations, Markov Processes and Martingales Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 3: The Lévy-Itô Decomposition David pplebaum Probability and Statistics Department,

More information

Brownian Motion on Manifold

Brownian Motion on Manifold Brownian Motion on Manifold QI FENG Purdue University feng71@purdue.edu August 31, 2014 QI FENG (Purdue University) Brownian Motion on Manifold August 31, 2014 1 / 26 Overview 1 Extrinsic construction

More information

ASYMPTOTIC BEHAVIOR OF A TAGGED PARTICLE IN SIMPLE EXCLUSION PROCESSES

ASYMPTOTIC BEHAVIOR OF A TAGGED PARTICLE IN SIMPLE EXCLUSION PROCESSES ASYMPTOTIC BEHAVIOR OF A TAGGED PARTICLE IN SIMPLE EXCLUSION PROCESSES C. LANDIM, S. OLLA, S. R. S. VARADHAN Abstract. We review in this article central limit theorems for a tagged particle in the simple

More information

MAXIMAL COUPLING OF EUCLIDEAN BROWNIAN MOTIONS

MAXIMAL COUPLING OF EUCLIDEAN BROWNIAN MOTIONS MAXIMAL COUPLING OF EUCLIDEAN BOWNIAN MOTIONS ELTON P. HSU AND KAL-THEODO STUM ABSTACT. We prove that the mirror coupling is the unique maximal Markovian coupling of two Euclidean Brownian motions starting

More information

Convergence at first and second order of some approximations of stochastic integrals

Convergence at first and second order of some approximations of stochastic integrals Convergence at first and second order of some approximations of stochastic integrals Bérard Bergery Blandine, Vallois Pierre IECN, Nancy-Université, CNRS, INRIA, Boulevard des Aiguillettes B.P. 239 F-5456

More information

Deterministic Dynamic Programming

Deterministic Dynamic Programming Deterministic Dynamic Programming 1 Value Function Consider the following optimal control problem in Mayer s form: V (t 0, x 0 ) = inf u U J(t 1, x(t 1 )) (1) subject to ẋ(t) = f(t, x(t), u(t)), x(t 0

More information

Simulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan

Simulation of diffusion. processes with discontinuous coefficients. Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan Simulation of diffusion. processes with discontinuous coefficients Antoine Lejay Projet TOSCA, INRIA Nancy Grand-Est, Institut Élie Cartan From collaborations with Pierre Étoré and Miguel Martinez . Divergence

More information

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION

LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in

More information

LECTURE 15: COMPLETENESS AND CONVEXITY

LECTURE 15: COMPLETENESS AND CONVEXITY LECTURE 15: COMPLETENESS AND CONVEXITY 1. The Hopf-Rinow Theorem Recall that a Riemannian manifold (M, g) is called geodesically complete if the maximal defining interval of any geodesic is R. On the other

More information

9 Radon-Nikodym theorem and conditioning

9 Radon-Nikodym theorem and conditioning Tel Aviv University, 2015 Functions of real variables 93 9 Radon-Nikodym theorem and conditioning 9a Borel-Kolmogorov paradox............. 93 9b Radon-Nikodym theorem.............. 94 9c Conditioning.....................

More information

A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES

A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES A NOTE ON STOCHASTIC INTEGRALS AS L 2 -CURVES STEFAN TAPPE Abstract. In a work of van Gaans (25a) stochastic integrals are regarded as L 2 -curves. In Filipović and Tappe (28) we have shown the connection

More information

Stochastic Volatility and Correction to the Heat Equation

Stochastic Volatility and Correction to the Heat Equation Stochastic Volatility and Correction to the Heat Equation Jean-Pierre Fouque, George Papanicolaou and Ronnie Sircar Abstract. From a probabilist s point of view the Twentieth Century has been a century

More information

WEYL S LEMMA, ONE OF MANY. Daniel W. Stroock

WEYL S LEMMA, ONE OF MANY. Daniel W. Stroock WEYL S LEMMA, ONE OF MANY Daniel W Stroock Abstract This note is a brief, and somewhat biased, account of the evolution of what people working in PDE s call Weyl s Lemma about the regularity of solutions

More information

The Asymptotic Theory of Transaction Costs

The Asymptotic Theory of Transaction Costs The Asymptotic Theory of Transaction Costs Lecture Notes by Walter Schachermayer Nachdiplom-Vorlesung, ETH Zürich, WS 15/16 1 Models on Finite Probability Spaces In this section we consider a stock price

More information

A MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT

A MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields

More information

µ X (A) = P ( X 1 (A) )

µ X (A) = P ( X 1 (A) ) 1 STOCHASTIC PROCESSES This appendix provides a very basic introduction to the language of probability theory and stochastic processes. We assume the reader is familiar with the general measure and integration

More information

On Reflecting Brownian Motion with Drift

On Reflecting Brownian Motion with Drift Proc. Symp. Stoch. Syst. Osaka, 25), ISCIE Kyoto, 26, 1-5) On Reflecting Brownian Motion with Drift Goran Peskir This version: 12 June 26 First version: 1 September 25 Research Report No. 3, 25, Probability

More information

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS

(2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS (2m)-TH MEAN BEHAVIOR OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS UNDER PARAMETRIC PERTURBATIONS Svetlana Janković and Miljana Jovanović Faculty of Science, Department of Mathematics, University

More information

Martingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi

Martingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi s Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi Lectures on Probability and Stochastic Processes III Indian Statistical Institute, Kolkata 20 24 November

More information

INVARIANT MANIFOLDS WITH BOUNDARY FOR JUMP-DIFFUSIONS

INVARIANT MANIFOLDS WITH BOUNDARY FOR JUMP-DIFFUSIONS INVARIANT MANIFOLDS WITH BOUNDARY FOR JUMP-DIFFUSIONS DAMIR FILIPOVIĆ, STFAN TAPP, AND JOSF TICHMANN Abstract. We provide necessary and sufficient conditions for stochastic invariance of finite dimensional

More information

Newtonian Mechanics. Chapter Classical space-time

Newtonian Mechanics. Chapter Classical space-time Chapter 1 Newtonian Mechanics In these notes classical mechanics will be viewed as a mathematical model for the description of physical systems consisting of a certain (generally finite) number of particles

More information

Stochastic Processes II/ Wahrscheinlichkeitstheorie III. Lecture Notes

Stochastic Processes II/ Wahrscheinlichkeitstheorie III. Lecture Notes BMS Basic Course Stochastic Processes II/ Wahrscheinlichkeitstheorie III Michael Scheutzow Lecture Notes Technische Universität Berlin Sommersemester 218 preliminary version October 12th 218 Contents

More information

THE SKOROKHOD OBLIQUE REFLECTION PROBLEM IN A CONVEX POLYHEDRON

THE SKOROKHOD OBLIQUE REFLECTION PROBLEM IN A CONVEX POLYHEDRON GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 2, 1996, 153-176 THE SKOROKHOD OBLIQUE REFLECTION PROBLEM IN A CONVEX POLYHEDRON M. SHASHIASHVILI Abstract. The Skorokhod oblique reflection problem is studied

More information

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009

A new approach for investment performance measurement. 3rd WCMF, Santa Barbara November 2009 A new approach for investment performance measurement 3rd WCMF, Santa Barbara November 2009 Thaleia Zariphopoulou University of Oxford, Oxford-Man Institute and The University of Texas at Austin 1 Performance

More information

Homework #6 : final examination Due on March 22nd : individual work

Homework #6 : final examination Due on March 22nd : individual work Université de ennes Année 28-29 Master 2ème Mathématiques Modèles stochastiques continus ou à sauts Homework #6 : final examination Due on March 22nd : individual work Exercise Warm-up : behaviour of characteristic

More information

The Skorokhod problem in a time-dependent interval

The Skorokhod problem in a time-dependent interval The Skorokhod problem in a time-dependent interval Krzysztof Burdzy, Weining Kang and Kavita Ramanan University of Washington and Carnegie Mellon University Abstract: We consider the Skorokhod problem

More information

The Wiener Itô Chaos Expansion

The Wiener Itô Chaos Expansion 1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in

More information

Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of. F s F t

Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of. F s F t 2.2 Filtrations Let (Ω, F) be a measureable space. A filtration in discrete time is a sequence of σ algebras {F t } such that F t F and F t F t+1 for all t = 0, 1,.... In continuous time, the second condition

More information

Skorokhod Embeddings In Bounded Time

Skorokhod Embeddings In Bounded Time Skorokhod Embeddings In Bounded Time Stefan Ankirchner Institute for Applied Mathematics University of Bonn Endenicher Allee 6 D-535 Bonn ankirchner@hcm.uni-bonn.de Philipp Strack Bonn Graduate School

More information

for all f satisfying E[ f(x) ] <.

for all f satisfying E[ f(x) ] <. . Let (Ω, F, P ) be a probability space and D be a sub-σ-algebra of F. An (H, H)-valued random variable X is independent of D if and only if P ({X Γ} D) = P {X Γ}P (D) for all Γ H and D D. Prove that if

More information

Lecture 12. F o s, (1.1) F t := s>t

Lecture 12. F o s, (1.1) F t := s>t Lecture 12 1 Brownian motion: the Markov property Let C := C(0, ), R) be the space of continuous functions mapping from 0, ) to R, in which a Brownian motion (B t ) t 0 almost surely takes its value. Let

More information

Mathematics for Economists

Mathematics for Economists Mathematics for Economists Victor Filipe Sao Paulo School of Economics FGV Metric Spaces: Basic Definitions Victor Filipe (EESP/FGV) Mathematics for Economists Jan.-Feb. 2017 1 / 34 Definitions and Examples

More information

Stochastic Processes

Stochastic Processes Stochastic Processes A very simple introduction Péter Medvegyev 2009, January Medvegyev (CEU) Stochastic Processes 2009, January 1 / 54 Summary from measure theory De nition (X, A) is a measurable space

More information

ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT

ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT Elect. Comm. in Probab. 8 23122 134 ELECTRONIC COMMUNICATIONS in PROBABILITY ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WIT MONOTONE DRIFT BRAIM BOUFOUSSI 1 Cadi Ayyad University FSSM, Department

More information

A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1

A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1 Chapter 3 A Change of Variable Formula with Local Time-Space for Bounded Variation Lévy Processes with Application to Solving the American Put Option Problem 1 Abstract We establish a change of variable

More information