Two Brownian Particles with RankBased Characteristics and SkewElastic Collisions


 Edward Waters
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1 Two Brownian Particles with ankbased Characteristics and SkewElastic Collisions arxiv:16.435v [math.p] 3 Aug 1 E. OBET FENHOLZ TOMOYUKI ICHIBA IOANNIS KAATZAS August 19, 1 Abstract We construct a twodimensional diffusion process with rankdependent 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 rightlocal 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 twovalued drift of the bangbang 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 Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA INTECH Investment Management LLC, One Palmer Square, Suite 441, Princeton, NJ and Department of Mathematics, Columbia University, MailCode 4438, New York, NY 17 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 rightcontinuous local time accumulated at the origin by a generic continuous semimartingale X, by L X L X ; its leftcontinuous 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 skewelastic. 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 socalled 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 crossvariation 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 nondegenerate 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 BangBang 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 rankdependent characteristics and skewelastic 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, realvalued 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 timeinterval [, 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 ightcontinuous 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 socalled 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 ITÔTANAKA 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, realvalued 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 leftcontinuous 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 BangBang drift Skew BangBang 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 bangbang 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 bangbang 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 bangbang 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 bangbang 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 bangbang 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 nondegenerate 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 timeinterval [, 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 nondegenerate 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 wellposed, 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 nonisotropic 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 CAUCHYSCHWATZ 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 56 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 bangbang 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 changeofmeasure, 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 91 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 skewelastic 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 14 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 BangBang 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 rightcontinuous 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 leftcontinuous 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 leftcontinuous derivative and its secondderivative 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 ITÔTANAKA 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 occupationtimedensity formula.8, and the property p = α. Now the piecewiselinear 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 ITÔTANAKA rule to obtain dy t = q Zt dzt + d L Z t, z q dz. On the strength of the occupationtimedensity 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 bangbang 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 nonsymmetric reflection principle around the origin. We shall see that even in the presence of the bangbang 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 ADONNIKODÝM 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 nonsymmetric 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 doubleexponential 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 timereversed 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 bangbang 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