Weak solutions of mean-field stochastic differential equations

Size: px
Start display at page:

Download "Weak solutions of mean-field stochastic differential equations"

Transcription

1 Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Based on joint works with Hui Min (Beijing University of Technology) Workshop PDE and Probability Methods for Interactions, Inria, Sophia Antipolis, France, March 3-31, / 44

2 Contents 1 Objective of the talk 2 Case 1: The drift coefficient is bounded and measurable. 3 Case 2: The coefficients are bounded, continuous. 2 / 44

3 1 Objective of the talk 2 Case 1: The drift coefficient is bounded and measurable. 3 Case 2: The coefficients are bounded, continuous. 3 / 44

4 Objective of the talk Let T be a fixed time horizon, b, σ measurable mappings defined over appropriate spaces. We are interested in a weak solution of Mean-Field (McKean-Vlasov) SDE : For t [, T ], ξ L 2 (Ω, F, P ; R d ), X t = ξ + b(s, X s, Q X s )ds + σ(s, X s, Q X s )db s, (1.1) where Q is a probability measure with respect to which B is a B.M. Remark: Q X s is the law of X s w.r.t. Q. 3 / 44

5 Brief state of art 1) Such Mean-Field SDEs have been intensively studied: For a longer time as limit equ. for systems with a large number of particles (propagation of chaos)(bossy, Méléard, Sznitman, Talay,...); Mean-Field Games, since (Lasry, Lions,...); 2) Mean-Field SDEs/FBSDEs and associated nonlocal PDEs: Preliminary works in 29 (AP, SPA); Classical solution of non-linear PDE related with the mean-field SDE: Buckdahn, Peng, Li, Rainer (214); Chassagneux, Crisan, Delarue (214); For the case with jumps: Li, Hao (216); Li (216); Weak solution: Oelschläger(1984), Funaki (1984), Gärtner (1988), Lacker (215), Carmona, Lacker (215), Li, Hui (216, 217)... 4 / 44

6 Objective of the talk Our objectives: To prove the existence and the uniqueness in law of the weak solution of mean-field SDE (1.1): * when the coefficient b is bounded, measurable and with a modulus of continuity w.r.t the measure, while σ is independent of the measure and Lipschitz. * when the coefficients (b, σ) are bounded and continuous. 5 / 44

7 Preliminaries We consider + (Ω, F, P ) - complete probability space; + W B.M. over (Ω, F, P ) (for simplicity: all processes 1-dimensional); + F-filtration generated by W, and augmented by F. p-wasserstein metric on P p (R) := {μ μ probab. on (R, B(R)) with R x p μ(x) < + }; W p (μ, ν):= inf { ( x p ρ(dxdy) ) 1 p, ρ( R) = μ, ρ(r ) = ν }. R R (1.2) 6 / 44

8 Preliminaries Generalization of the def. of a weak sol. of a classical SDE (see, e.g., Karatzas and Shreve, 1988) to (1.1): Definition 1.1 A six-tuple ( Ω, F, F, Q, B, X) is a weak solution of SDE (1.1), if (i) ( Ω, F, Q) is a complete probability space, and F = { F t } t T is a filtration on ( Ω, F, Q) satisfying the usual conditions. (ii) X = {X t } t T is a continuous, F-adapted R-valued process; B = {B t } t T is an ( F, Q)-BM. (iii) Q{ T ( b(s, X s, Q X s ) + σ(s, X s, Q X s ) 2 )ds < + } = 1, and equation (1.1) is satisfied, Q-a.s. 7 / 44

9 Preliminaries Definition 1.2 We say that uniqueness in law holds for the mean-field SDE (1.1), if for any two weak solutions (Ω i, F i, F i, Q i, B i, X i ), i = 1, 2, we have Q 1 X = Q 2 1 X, i.e., the two processes X 1 and X 2 have the same law. 2 8 / 44

10 1 Objective of the talk 2 Case 1: The drift coefficient is bounded and measurable. 3 Case 2: The coefficients are bounded, continuous. 9 / 44

11 Case 1: Existence of a weak solution Let b, σ satisfy the following assumption (H1): (i) b : [, T ] C([, T ]; R) P 1 (R) R is bounded and measurable; (ii) σ : [, T ] C([, T ]; R) R is bounded, measurable, and s.t., for all (t, φ) [, T ] C([, T ]; R), 1/σ(t, φ) is bounded in (t, φ); (iii) (Modulus of continuity) ρ : R + R + increasing, continuous, with ρ(+) = s.t., for all t [, T ], φ C([, T ]; R), μ, ν P 1 (R), b(t, φ t, μ) b(t, φ t, ν) ρ(w 1 (μ, ν)); (iv) L s.t., for all t [, T ], φ, ψ C([, T ]; R), σ(t, φ t ) σ(t, ψ t ) L sup φ s ψ s. s t 9 / 44

12 Case 1: Existence of a weak solution We want to study weak solutions of the following mean-field SDE: X t = ξ + σ(s, X s )db s + b(s, X s, Q Xs )ds, t [, T ], (2.1) where (B t ) t [,T ] is a BM under the probability measure Q. Now we can give the main statement of this section. Theorem 2.1 Under assumption (H1) mean-field SDE (2.1) has a weak solution ( Ω, F, F, Q, B, X). Proof: Girsanov s Theorem. Schauder s Fixed Point Theorem. 1 / 44

13 Case 1: Existence of a weak solution Let us give two examples. Example 1. Take diffusion coefficient σ I d and drift coefficient b(s, φ s, μ s ) := b(s, φ s, ψdμ s ), φ C([, T ]), μ P 1 (R), s [, T ]; the function ψ C([, T ]; R) is arbitrarily given but fixed, and Lipschitz. Then our mean-field SDE (2.1) can be written as follows: X t = B t + b(s, X s, E Q [ψ(x s )])ds, t [, T ]. (2.2) Here b : [, T ] C([, T ]) R R is bounded, meas., Lips. in y. Then, the coefficients b and σ satisfy (H1), and from Theorem 2.1, we obtain that the mean-field SDE (2.2) has a weak solution ( Ω, F, F, Q, B, X). 11 / 44

14 Case 1: Existence of a weak solution Example 2. Take diffusion coefficient σ I d and drift coefficient b(s, φ s, μ s ) := b(s, φ s, y)μ s (dy), φ C([, T ]), μ s P 1 (R), s [, T ], i.e., we consider the following mean-field SDE: X t = B t + R b(s, X s, y)q Xs (dy)ds, t [, T ]. (2.3) Here the coefficient b : [, T ] C([, T ]) R R is bounded, meas. and Lips. in y. Then, the coefficients b and σ satisfy (H1), and from Theorem 2.1 the mean-field SDE (2.3) has a weak solution ( Ω, F, F, Q, B, X). 12 / 44

15 Case 1: Uniqueness in law of weak solutions Let the functions b and σ satisfy the following assumption (H2): (i) b : [, T ] C([, T ]; R) P 1 (R) R is bounded and measurable; (ii) σ : [, T ] C([, T ]; R) R is bounded and measurable, and 1/σ(t, φ) C, (t, φ) [, T ] C([, T ]; R), for some C R + ; (iii) (Modulus of continuity) There exists a continuous and increasing function ρ : R + R + with du ρ(r) >, for all r >, and + ρ(u) = +, such that, for all t [, T ], φ C([, T ]; R), μ, ν P 1 (R), b(t, φ t, μ) b(t, φ t, ν) 2 ρ(w 1 (μ, ν) 2 ); (iv) L such that, for all t [, T ], φ, ψ C([, T ]; R), σ(t, φ t ) σ(t, ψ t ) L sup φ s ψ s. s t 13 / 44

16 Case 1: Uniqueness in law of weak solutions Obviously, under assumption (H2) the coefficients b and σ also satisfy (H1). Thus, due to Theorem 2.1, the following mean-field SDE X t = ξ + b(s, X s, Q Xs )ds + σ(s, X s )db s, t [, T ], (2.1) has a weak solution. Theorem 2.2 Suppose that assumption (H2) holds, and let (Ω i, F i, F i, Q i, B i, X i ), i = 1, 2, be two weak solutions of mean-field SDE (2.1). Then (B 1, X 1 ) and (B 2, X 2 ) have the same law under their respective probability measures, i.e., Q 1 (B 1,X 1 ) = Q2 (B 2,X 2 ). 14 / 44

17 Case 1: Uniqueness in law of weak solutions Sketch of the proof: For φ C([, T ]; R), μ P 1 (R), we define b(s, φ s, μ) = σ 1 (s, φ s )b(s, φ s, μ), and we introduce Wt i = Bt i + L i T = exp{ T b(s, X i s, Q i X )ds, t [, T ], s i b(s, X i s, Q i X i s )dbi s 1 2 T b(s, X i s, Q i X i s ) 2 ds}, (2.4) i = 1, 2. Then from the Girsanov Theorem we know that (W i t ) t [,T ] is an F i -B.M. under the probability measure Q i = L i T Qi, i = 1, 2, respectively. From (H2), for each i, we have a unique strong solution X i of the SDE X i t = X i + σ(s, X i s)dw i s, t [, T ]. (2.5) 15 / 44

18 Case 1: Uniqueness in law of weak solutions It is by now standard that a meas. and non-anticipating function Φ : [, T ] R C([, T ]; R) R not depending on i = 1, 2, s.t. X i t = Φ t (X i, W i ), t [, T ], Qi -a.s. (and, Q i -a.s.), i = 1, 2. (2.6) Then from (2.4) that W i t = B i t + t b(s, Φ s (X i, W i ), Q i X i s)ds, i = 1, 2. Hence, putting f(s, φ s ) = b(s, φ s, Q 1 X s ), (s, φ) [, T ] C([, T ]; R), from (2.4) and (2.6) we have Wt 1 = Bt 1 + f(s, Φ s (X, 1 W 1 ))ds, t [, T ], Wt 2 = B t 2 + f(s, Φ s (X 2, W 2 ))ds, t [, T ], where, t [, T ], B t 2 = Bt 2 + ( b(s, Φ s (X, 2 W 2 ), Q 2 X ) b(s, ) s 2 Φ s (X, 2 W 2 ), Q 1 X ) ds. s 1 16 / 44

19 Case 1: Uniqueness in law of weak solutions Hence, Φ : [, T ] R C([, T ]; R) R meas. s.t., for both B 1, B 2, Bt 1 = Φ t (X, 1 W 1 ) and B t 2 = Φ t (X, 2 W 2 ), t [, T ]. (2.8) Now we define d L 2 t = ( b(s, Φ s (X, 2 W 2 ), Q 2 X ) b(s, s 2 Φ s (X, 2 W 2 ), Q 1 X )) L 2 s 1 t dbt 2, t L 2 = 1. (2.9) From the Girsanov Theorem we know that B 2 is an Brownian motion under the probability measure Q 2 = L 2 T Q2. Moreover, putting 17 / 44

20 Case 1: Uniqueness in law of weak solutions L 2 T = exp{ Q 2 = L 2 T Q 2, T f(s, Φ s (X 2, W 2 ))dw 2 s T f(s, Φ s (X 2, W 2 )) 2 ds}, we have that (W 2 t ) t [,T ] is a B.M. under both Q 2 and Q 2, while (W 1 t ) t [,T ] is a B.M. under Q 1. (2.1) On the other hand, since f is bounded and meas., we can prove that a meas. function Φ : R C([, T ]; R) R, s.t. Φ(X i, W i ) = T f(s, Φ s (X i, W i ))dw i s, Q i -a.s., i = 1, / 44

21 Case 1: Uniqueness in law of weak solutions Therefore, recalling the definition of L 1 T and (2.1), we have L 1 T= exp{ L 2 T= exp{ T T f(s, Φ s (X 1, W 1 ))dw 1 s f(s, Φ s (X 2, W 2 ))dw 2 s T T f(s, Φ s (X 1, W 1 )) 2 ds}, f(s, Φ s (X 2, W 2 )) 2 ds}, (2.11) and we see that a meas. function Φ : R C([, T ]; R) R, s.t. L 1 T = Φ(X, 1 W 1 ), Q 1 -a.s., and L 2 T = Φ(X, 2 W 2 ), Q 2 -a.s. (and, Q 2 -a.s.). (2.12) 19 / 44

22 Case 1: Uniqueness in law of weak solutions Consequently, as X i is F i-measurable, i = 1, 2 and Q1 = Q 2, X 1 X 2 from (2.8), (2.1), (2.11) and (2.12) we have that, for all bounded measurable function F : C([, T ]; R d ) 2 R, E Q 1[F (B 1, W 1 1 )] = E Q1[ Φ(X 1, W 1 ) F ( Φ(X, 1 W 1 ), W 1 )] That is, = E Q2[ 1 Φ(X 2, W 2 ) F ( Φ(X, 2 W 2 ), W 2 )] = E Q2[F ( B 2, W 2 )]. Taking into account (2.6), we have Q 1 (B 1,W 1 ) = Q 2 ( B 2,W 2 ). (2.13) Q 1 (B 1,W 1,X 1 ) = Q 2 ( B 2,W 2,X 2 ), (2.14) and, in particular, Q 1 X 1 = Q 2 X 2. 2 / 44

23 Case 1: Uniqueness in law of weak solutions On the other hand, we can prove W 1 (Q 1 X 1 s, Q2 X 2 s )2 = W 1 ( Q 2 X 2 s, Q2 X 2 s )2 C s ρ(w 1(Q 1 X 1 r, Q2 X 2 r )2 )dr; The continuity of s W 1 (Q 1 X 1 s, Q2 X 2 s ). Putting u(s) := W 1 (Q 1 X, s 1 Q2 X ), s [, T ], then we have from above, s 2 u(s) 2 C s ρ(u(r)2 )dr, s t T. From (H2)-(iii), + du ρ(u) = +, it follows from Bihari s inequality that u(s) =, for any s [, T ], that is, Q 1 X = s 1 Q2 X, s [, T ]. Thus, from s 2 (2.7) and (2.9) it follows that B 2 = B 2, L2 T = 1, and, consequently, Q 2 = Q 2. Then, Q 2 ( B = 2,W 2,X 2 ) Q2 (B 2,W 2,X 2 ), and from (2.14) Q 1 (B 1,W 1,X 1 ) = Q2 (B 2,W 2,X 2 ). (2.15) This implies, in particular, Q 1 (B 1,X 1 ) = Q2 (B 2,X 2 ). 21 / 44

24 1 Objective of the talk 2 Case 1: The drift coefficient is bounded and measurable. 3 Case 2: The coefficients are bounded, continuous. 22 / 44

25 Case 2: Preliminaries Definition 3.1 (see, e.g., Karatzas, Shreve, 1988) A probability P on (C([, T ]; R), B(C([, T ]; R))) is a solution to the local martingale problem associated with A, if for every f C 1,2 ([, T ] R; R), M f t := f(t, y(t)) f(, y()) ( s + A )f(s, y(s))ds, t [, T ], (3.1) is a continuous local martingale w.r.t (F y, P ), where y = (y(t)) t [,T ] is the coordinate process on C([, T ]; R), the considered filtration F y = (F y t ) t [,T ] is that generated by y = (y(t)) t [,T ] and augmented by all P -null sets, and A is defined by, y C([, T ]; R), A f(s, y) = b(s, y) x f(s, y(s)) σ2 (s, y) xf(s, 2 y(s)). (3.2) 22 / 44

26 Case 2: Preliminaries Let us first recall a well-known result concerning the equivalence between the weak solution of a functional SDE and the solution to the corresponding local martingale problem (see, e.g., Karatzas, Shreve, 1988). Lemma 3.1 The existence of a weak solution ( Ω, F, F, P, W, X) to the following functional SDE with given initial distribution μ on B(R): X t = ξ + b(s, X s )ds + σ(s, X s )d W s, t [, T ], is equivalent to the existence of a solution P to the local martingale problem (3.1) associated with A defined by (3.2), with P y() = μ. The both solutions are related by P = P X 1, i.e., the probability measure P is the law of the weak solution X on (C([, T ]; R), B(C([, T ]; R))). 23 / 44

27 Case 2: Preliminaries Recall the definition of the derivative of f : P 2 (R) R w.r.t probability measure μ P 2 (R) (in the sense of P.L.Lions)(P.L.Lions lectures at Collège de France, also see the notes of Cardaliaguet). Definition 3.2 (i) f : L 2 (Ω, F, P ; R) R is Fréchet differentiable at ξ L 2 (Ω, F, P ), if a linear continuous mapping D f(ξ)( ) L(L 2 (Ω, F, P ; R); R), s.t. f(ξ + η) f(ξ) = D f(ξ)(η) + o( η L 2), with η L 2 for η L 2 (Ω, F, P ). (ii) f : P 2 (R) R is differentiable at μ P 2 (R), if for f(ξ) := f(p ξ ), ξ L 2 (Ω, F, P ; R), there is some ζ L 2 (Ω, F, P ; R) with P ζ = μ such that f : L 2 (Ω, F, P ; R) R is Fréchet differentiable in ζ. 24 / 44

28 Case 2: Preliminaries From Riesz Representation Theorem there exists a P -a.s. unique variable θ L 2 (Ω, F, P ; R) such that D f(ζ)(η) = (θ, η) L 2 = E[θη], for all η L 2 (Ω, F, P ; R). P.L. Lions proved that there is a Borel function h : R R such that θ = h(ζ), P -a.e., and function h depends on ζ only through its law P ζ. Therefore, f(p ξ ) f(p ζ ) = E[h(ζ) (ξ ζ)] + o( ξ ζ L 2), ξ L 2 (Ω, F, P ; R). Definition 3.3 We call μ f(p ζ, y) := h(y), y R, the derivative of function f : P 2 (R) R at P ζ, ζ L 2 (Ω, F, P ; R). Remark: μ f(p ζ, y) is only P ζ (dy)-a.e. uniquely determined. 25 / 44

29 Case 2: Preliminaries Definition 3.4 We say that f C 1 (P 2 (R)), if for all ξ L 2 (Ω, F, P ; R) there exists a P ξ -modification of μ f(p ξ,.), also denoted by μ f(p ξ,.), such that μ f : P 2 (R) R R is continuous w.r.t the product topology generated by the 2-Wasserstein metric over P 2 (R) and the Euclidean norm over R, and we identify this modified function μ f as the derivative of f. The function f is said to belong to C 1,1 b (P 2 (R)), if f C 1 (P 2 (R)) is s.t. μ f : P 2 (R) R R is bounded and Lipschitz continuous, i.e., there exists some constant C such that (i) μ f(μ, x) C, μ P 2 (R), x R; (ii) μ f(μ, x) μ f(μ, x ) C(W 2 (μ, μ )+ x x ), μ, μ P 2 (R), x, x R. 26 / 44

30 Case 2: Preliminaries Definition 3.5 We say that f C 2 (P 2 (R)), if f C 1 (P 2 (R)) and μ f(μ,.) : R R is differentiable, and its derivative y μ f : P 2 (R) R R R is continuous, for every μ P 2 (R). Moreover, f C 2,1 b (P 2 (R)), if f C 2 (P 2 (R)) C 1,1 b (P 2 (R)) and its derivative y μ f : P 2 (R) R R R is bounded and Lipschitzcontinuous. Remark: C 2,1 b (R P 2 (R)), C 1,2,1 b ([, T ] R P 2 (R); R) are similarly defined. 27 / 44

31 Case 2: Preliminaries Now we can give our It^o s formula. Theorem 3.1 Let σ = (σ s ), γ = (γ s ), b = (b s ), β = (β s ) R-valued adapted stochastic processes, such that (i) There exists a constant q > 6 s.t. E[( T ( σ s q + b s q )ds) 3 q ] < + ; (ii) T ( γ s 2 + β s )ds < +, P-a.s. Let F C 1,2,1 b ([, T ] R P 2 (R)). Then, for the It^o processes X t = X + Y t = Y + σ s dw s + γ s dw s + b s ds, t [, T ], X L 2 (Ω, F, P ), β s ds, t [, T ], Y L 2 (Ω, F, P ), 28 / 44

32 Case 2: Preliminaries Theorem 3.1 (continued) we have F (t, Y t, P Xt ) F (, Y, P X ) ( = r F (r, Y r, P Xr ) + y F (r, Y r, P Xr )β r yf (r, Y r, P Xr )γr 2 + E[( μ F )(r, Y r, P Xr, X r ) b r + 1 ) 2 z( μ F )(r, Y r, P Xr, X r ) σ r] 2 dr + y F (r, Y r, P Xr )γ r dw r, t [, T ]. Here ( X, b, σ) denotes an independent copy of (X, b, σ), defined on a P.S. ( Ω, F, P ). The expectation E[ ] on ( Ω, F, P ) concerns only r.v. endowed with the superscript. 29 / 44

33 Case 2: Preliminaries (H3) The coefficients (σ, b) C 1,2,1 b ([, T ] R P 2 (R); R R). Theorem 3.2 (Buckdahn, Li, Peng and Rainer, 214) Let Φ C 2,1 b (R P 2 (R)), then under assumption (H3) the following PDE: = t V (t, x, μ) + x V (t, x, μ)b(x, μ) xv (t, x, μ)σ 2 (x, μ) + ( μ V )(t, x, μ, y)b(y, μ)μ(dy) R + 1 y ( μ V )(t, x, μ, y)σ 2 (y, μ)μ(dy), 2 R V (T, x, μ) = Φ(x, μ), (x, μ) R P 2 (R). (t, x, μ) [, T ) R P 2 (R); has a unique classical solution V (t, x, μ) C 1,2,1 b ([, T ] R P 2 (R); R). 3 / 44

34 Case 2: Existence of a weak solution Let b and σ satisfy the following assumption: (H4) b, σ : [, T ] R P 2 (R) R are continuous and bounded. We want to study weak solution of the following mean-field SDE: X t = ξ + b(s, X s, Q Xs )ds σ(s, X s, Q Xs )db s, t [, T ], (3.3) where ξ L 2 (Ω, F, P ; R) obeys a given distribution law Q ξ = ν P 2 (R) and (B t ) t [,T ] is a B.M. under the probability measure Q. 31 / 44

35 Case 2: Existence of a weak solution Extension of the corresponding local martingale problem: Definition 3.6 A probability measure P on (C([, T ]; R), B(C([, T ]; R))) is a solution to the local martingale problem (resp., martingale problem) associated with A, if for every f C 1,2 ([, T ] R; R) (resp., f C 1,2 b ([, T ] R; R)), the process C f (t, y, μ) := f(t, y(t)) f(, y()) ( ( s + A)f ) (s, y(s), μ(s))ds, is a continuous local (F y, P )-martingale (resp., continuous (F y, P )- martingale), (3.4) 32 / 44

36 Case 2: Existence of a weak solution Definition 3.6 (continued) where μ(t) = P y(t) is the law of the coordinate process y = (y(t)) t [,T ] on C([, T ]; R) at time t, the filtration F y is that generated by y and completed, and A is defined by ( Af)(s, y, ν) := y f(s, y)b(s, y, ν) yf(s, y)σ 2 (s, y, ν), (3.5) (s, y, ν) [, T ] R P 2 (R). Here (( s + A)f)(s, y(s), μ(s)) abbreviates (( s + A)f)(s, y(s), μ(s)) := ( s f)(s, y(s)) + ( Af)(s, y(s), μ(s)). 33 / 44

37 Case 2: Existence of a weak solution Proposition 3.1 The existence of a weak solution ( Ω, F, F, Q, B, X) to equation (3.3) with initial distribution ν on B(R) is equivalent to the existence of a solution P to the local martingale problem (3.4) associated with A defined by (3.5), with P y() = ν. 34 / 44

38 Case 2: Existence of a weak solution Lemma 3.2 Let the probability measure P on (C([, T ]; R), B(C([, T ]; R))) be a solution to the local martingale problem associated with A. Then, for the second order differential operator ( Af)(s, y, ν) := ( Af)(s, y, ν) R R ( μ f)(s, y, ν, z)b(s, z, ν)ν(dz) z ( μ f)(s, y, ν, z)σ 2 (s, z, ν)ν(dz), (3.6) 35 / 44

39 Case 2: Existence of a weak solution Lemma 3.2 (continued) applying to functions f C 1,2 ([, T ] R P 2 (R); R) we have that, for every such f C 1,2 ([, T ] R P 2 (R); R), the process C f (t, y, μ) :=f(t, y(t), μ(t)) f(, y(), μ()) ( s + A)f(s, y(s), μ(s))ds, t [, T ], (3.7) is a continuous local (F y, P )-martingale, where μ(t) = P y(t) is the law of the coordinate process y = (y(t)) t [,T ] on C([, T ]; R) at time t, the filtration F y is that generated by y and completed. Moreover, if f C 1,2,1 b ([, T ] R P 2 (R); R), this process C f is an (F y, P )-martingale. 36 / 44

40 Case 2: Existence of a weak solution Now we can give the main statement of this section. Theorem 3.3 Under assumption (H4) mean-field SDE (3.3) has a weak solution ( Ω, F, F, Q, B, X). Remark 2. If b, σ : [, T ] C([, T ]; R) P 2 (C([, T ]; R)) R are bounded and continuous, then the following mean-field SDE X t = ξ + b(s, X s, Q X s )ds + σ(s, X s, Q X s )db s, t [, T ], (1.1) where ξ L 2 (Ω, F, P ) obeys a given distribution law Q ξ = ν, has a weak solution ( Ω, F, F, Q, X, B). 37 / 44

41 Case 2: Uniqueness in law of weak solutions Now we want to study the uniqueness in law for the weak solution of the mean-field SDE (3.3). Definition 3.7 We call C bb(r) = {φ φ : R R bounded Borel-measurable function} a determining class on R, if for any two finite measures ν 1 and ν 2 on B(R), R φ(x)ν d 1 (dx) = R φ(x)ν d 2 (dx) for all φ C implies ν 1 = ν 2. Remark: The class C (R) is a determining class on R. 38 / 44

42 Case 2: Uniqueness in law of weak solutions Theorem 3.4 For given f C (R), we consider the Cauchy problem t v(t, x, ν) = Av(t, x, ν), (t, x, ν) [, T ] R P 2(R), v(, x, ν) = f(x), x R, (3.8) where Av(t, x, ν) = ( Av)(t, x, ν) + ( μ v)(t, x, ν, u)b(t, u, ν)ν(du) R + 1 z ( μ v)(t, x, ν, u)σ 2 (t, u, ν)ν(du), 2 ( Av)(t, x, ν) = y v(t, x, ν)b(t, x, ν) yv(t, x, ν)σ 2 (t, x, ν), (t, x, ν) [, ) R P 2 (R). R 39 / 44

43 Case 2: Uniqueness in law of weak solutions Theorem 3.4 (continued) We suppose that, for all f C (R), (3.8) has a solution v f C b ([, ) R P 2 (R)) C 1,2,1 b ((, ) R P 2 (R)). Then, the local martingale problem associated with A (Recall Definition 3.6) and with the initial condition δ x has at most one solution. Remark: Theorem 3.4 generalizes a well-known classical uniqueness for weak solutions to the case of mean-field SDE. Corollary 3.1 Under the assumption of Theorem 3.4, we have for the mean-field SDE (3.3) the uniqueness in law, that is, for any weak solutions, i = 1, 2 (Ω i, F i, F i, Q i, B i, X i ), of SDE (3.3), we have Q 1 X 1 = Q 2 X 2. 4 / 44

44 Uniqueness in law of weak solutions Sketch of proof of Theorem 3.4: Let T >, denote by y=(y(t)) t [,T ] the coordinate process on C([, T ]; R). Let P 1 and P 2 be two arbitrary solutions of the local martingale problem associated with A and initial condition x R: P l y() = δ x, l = 1, 2. Consequently, due to Lemma 3.2, for any g C 1,2,1 b ([, T ] R P 2 (R)), C g (t, y, P l y) := g(t, y(t), P l y(t) ) g(, x, δ x) ( s +A)g(s, y(s), P l y(s) )ds, (3.9) is a P l -martingale, l = 1, 2, t [, T ]. For given f C (R), let v f C b ([, T ] R P 2 (R)) C 1,2,1 b ((, T ) R P 2 (R)) be a solution of the Cauchy problem (3.8). 41 / 44

45 Uniqueness in law of weak solutions Then putting g(t, z, ν) := v f (T t, z, ν), t [, T ], z R, ν P 2 (R), defines a function g of class C b ([, T ] R P 2 (R)) C 1,2,1 b ((, T ) R P 2 (R)) which satisfies s g(s, z, ν)+ag(s, z, ν) =, g(t, z, ν) = f(z), (s, z, ν) [, T ] R P 2 (R). From (3.9) we see that {C g (s, y, P l y), s [, T ]} is an (F y, P l )- martingale. Hence, for E l [ ] = Ω l ( )dp l, E l [f(y(t ))] = E l [g(t, y(t ), P l y(t ) )] = g(, x, δ x), x R, l = 1, 2, that is E 1 [f(y(t ))] = E 2 [f(y(t ))], for all f C (R). Combining this with the arbitrariness of T, we have that Py(t) 1 = P y(t) 2, for every t. 42 / 44

46 Uniqueness in law of weak solutions Consequently, P 1, P 2 are solutions of the same classical martingale problem, associated with A = A l, l = 1, 2, A l φ(t, z) = y φ(t, z) b l (t, z) + 2 yφ(t, z)( σ l (t, z)) 2, φ C 1,2 ([, T ] R; R), with the coefficients σ 1 = σ 2, b 1 = b 2 (without mean field term), σ l (t, z) = σ(t, z, Py(t) l ), b l (t, z) = b(t, z, Py(t) l ), (t, z) [, T ] R, and we have seen that Py(t) 1 = P y(t) 2, t [, T ].... P 1 = P 2, i.e., the local martingale problem has at most one solution. 43 / 44

47 Thank you very much! 谢谢! 44 / 44

Mean-field SDE driven by a fractional BM. A related stochastic control problem

Mean-field SDE driven by a fractional BM. A related stochastic control problem Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th,

More information

Strong Solutions and a Bismut-Elworthy-Li Formula for Mean-F. Formula for Mean-Field SDE s with Irregular Drift

Strong Solutions and a Bismut-Elworthy-Li Formula for Mean-F. Formula for Mean-Field SDE s with Irregular Drift Strong Solutions and a Bismut-Elworthy-Li Formula for Mean-Field SDE s with Irregular Drift Thilo Meyer-Brandis University of Munich joint with M. Bauer, University of Munich Conference for the 1th Anniversary

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

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

Large Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm

Large Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm Large Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm Gonçalo dos Reis University of Edinburgh (UK) & CMA/FCT/UNL (PT) jointly with: W. Salkeld, U. of

More information

Prof. Erhan Bayraktar (University of Michigan)

Prof. Erhan Bayraktar (University of Michigan) September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled

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

Backward Stochastic Differential Equations with Infinite Time Horizon

Backward Stochastic Differential Equations with Infinite Time Horizon Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Università di Milano-Bicocca Spring School Stochastic Control in Finance Roscoff, March

More information

The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications

The multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications

More information

LECTURES ON MEAN FIELD GAMES: II. CALCULUS OVER WASSERSTEIN SPACE, CONTROL OF MCKEAN-VLASOV DYNAMICS, AND THE MASTER EQUATION

LECTURES ON MEAN FIELD GAMES: II. CALCULUS OVER WASSERSTEIN SPACE, CONTROL OF MCKEAN-VLASOV DYNAMICS, AND THE MASTER EQUATION LECTURES ON MEAN FIELD GAMES: II. CALCULUS OVER WASSERSTEIN SPACE, CONTROL OF MCKEAN-VLASOV DYNAMICS, AND THE MASTER EQUATION René Carmona Department of Operations Research & Financial Engineering PACM

More information

Multilevel Monte Carlo for Stochastic McKean-Vlasov Equations

Multilevel Monte Carlo for Stochastic McKean-Vlasov Equations Multilevel Monte Carlo for Stochastic McKean-Vlasov Equations Lukasz Szpruch School of Mathemtics University of Edinburgh joint work with Shuren Tan and Alvin Tse (Edinburgh) Lukasz Szpruch (University

More information

On continuous time contract theory

On continuous time contract theory Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem

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

MA8109 Stochastic Processes in Systems Theory Autumn 2013

MA8109 Stochastic Processes in Systems Theory Autumn 2013 Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form

More information

On the Multi-Dimensional Controller and Stopper Games

On the Multi-Dimensional Controller and Stopper Games On the Multi-Dimensional Controller and Stopper Games Joint work with Yu-Jui Huang University of Michigan, Ann Arbor June 7, 2012 Outline Introduction 1 Introduction 2 3 4 5 Consider a zero-sum controller-and-stopper

More information

Information and Credit Risk

Information and Credit Risk Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information

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

Bernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012

Bernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012 1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.

More information

Estimates for the density of functionals of SDE s with irregular drift

Estimates for the density of functionals of SDE s with irregular drift Estimates for the density of functionals of SDE s with irregular drift Arturo KOHATSU-HIGA a, Azmi MAKHLOUF a, a Ritsumeikan University and Japan Science and Technology Agency, Japan Abstract We obtain

More information

Mean-Field optimization problems and non-anticipative optimal transport. Beatrice Acciaio

Mean-Field optimization problems and non-anticipative optimal transport. Beatrice Acciaio Mean-Field optimization problems and non-anticipative optimal transport Beatrice Acciaio London School of Economics based on ongoing projects with J. Backhoff, R. Carmona and P. Wang Robust Methods in

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

On the connection between symmetric N-player games and mean field games

On the connection between symmetric N-player games and mean field games On the connection between symmetric -player games and mean field games Markus Fischer May 14, 214; revised September 7, 215; modified April 15, 216 Abstract Mean field games are limit models for symmetric

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

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

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

A class of globally solvable systems of BSDE and applications

A class of globally solvable systems of BSDE and applications A class of globally solvable systems of BSDE and applications Gordan Žitković Department of Mathematics The University of Texas at Austin Thera Stochastics - Wednesday, May 31st, 2017 joint work with Hao

More information

4th Preparation Sheet - Solutions

4th Preparation Sheet - Solutions Prof. Dr. Rainer Dahlhaus Probability Theory Summer term 017 4th Preparation Sheet - Solutions Remark: Throughout the exercise sheet we use the two equivalent definitions of separability of a metric space

More information

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE

UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.

More information

Some aspects of the mean-field stochastic target problem

Some aspects of the mean-field stochastic target problem Some aspects of the mean-field stochastic target problem Quenched Mass ransport of Particles owards a arget Boualem Djehiche KH Royal Institute of echnology, Stockholm (Joint work with B. Bouchard and

More information

Richard F. Bass Krzysztof Burdzy University of Washington

Richard F. Bass Krzysztof Burdzy University of Washington ON DOMAIN MONOTONICITY OF THE NEUMANN HEAT KERNEL Richard F. Bass Krzysztof Burdzy University of Washington Abstract. Some examples are given of convex domains for which domain monotonicity of the Neumann

More information

Hölder continuity of the solution to the 3-dimensional stochastic wave equation

Hölder continuity of the solution to the 3-dimensional stochastic wave equation Hölder continuity of the solution to the 3-dimensional stochastic wave equation (joint work with Yaozhong Hu and Jingyu Huang) Department of Mathematics Kansas University CBMS Conference: Analysis of Stochastic

More information

Robust Markowitz portfolio selection. ambiguous covariance matrix

Robust Markowitz portfolio selection. ambiguous covariance matrix under ambiguous covariance matrix University Paris Diderot, LPMA Sorbonne Paris Cité Based on joint work with A. Ismail, Natixis MFO March 2, 2017 Outline Introduction 1 Introduction 2 3 and Sharpe ratio

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

Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Was

Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Was Dynamic and Stochastic Brenier Transport via Hopf-Lax formulae on Wasserstein Space With many discussions with Yann Brenier and Wilfrid Gangbo Brenierfest, IHP, January 9-13, 2017 ain points of the

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

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

BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009

BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009 BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, 16-20 March 2009 1 1) L p viscosity solution to 2nd order semilinear parabolic PDEs

More information

Question 1. The correct answers are: (a) (2) (b) (1) (c) (2) (d) (3) (e) (2) (f) (1) (g) (2) (h) (1)

Question 1. The correct answers are: (a) (2) (b) (1) (c) (2) (d) (3) (e) (2) (f) (1) (g) (2) (h) (1) Question 1 The correct answers are: a 2 b 1 c 2 d 3 e 2 f 1 g 2 h 1 Question 2 a Any probability measure Q equivalent to P on F 2 can be described by Q[{x 1, x 2 }] := q x1 q x1,x 2, 1 where q x1, q x1,x

More information

Harnack Inequalities and Applications for Stochastic Equations

Harnack Inequalities and Applications for Stochastic Equations p. 1/32 Harnack Inequalities and Applications for Stochastic Equations PhD Thesis Defense Shun-Xiang Ouyang Under the Supervision of Prof. Michael Röckner & Prof. Feng-Yu Wang March 6, 29 p. 2/32 Outline

More information

Long time behavior of Mean Field Games

Long time behavior of Mean Field Games Long time behavior of Mean Field Games P. Cardaliaguet (Paris-Dauphine) Joint work with A. Porretta (Roma Tor Vergata) PDE and Probability Methods for Interactions Sophia Antipolis (France) - March 30-31,

More information

Lecture 12: Diffusion Processes and Stochastic Differential Equations

Lecture 12: Diffusion Processes and Stochastic Differential Equations Lecture 12: Diffusion Processes and Stochastic Differential Equations 1. Diffusion Processes 1.1 Definition of a diffusion process 1.2 Examples 2. Stochastic Differential Equations SDE) 2.1 Stochastic

More information

Variational approach to mean field games with density constraints

Variational approach to mean field games with density constraints 1 / 18 Variational approach to mean field games with density constraints Alpár Richárd Mészáros LMO, Université Paris-Sud (based on ongoing joint works with F. Santambrogio, P. Cardaliaguet and F. J. Silva)

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

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

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University

Lecture 4: Ito s Stochastic Calculus and SDE. Seung Yeal Ha Dept of Mathematical Sciences Seoul National University Lecture 4: Ito s Stochastic Calculus and SDE Seung Yeal Ha Dept of Mathematical Sciences Seoul National University 1 Preliminaries What is Calculus? Integral, Differentiation. Differentiation 2 Integral

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

Differential Games II. Marc Quincampoix Université de Bretagne Occidentale ( Brest-France) SADCO, London, September 2011

Differential Games II. Marc Quincampoix Université de Bretagne Occidentale ( Brest-France) SADCO, London, September 2011 Differential Games II Marc Quincampoix Université de Bretagne Occidentale ( Brest-France) SADCO, London, September 2011 Contents 1. I Introduction: A Pursuit Game and Isaacs Theory 2. II Strategies 3.

More information

Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review

Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review L. Gawarecki Kettering University NSF/CBMS Conference Analysis of Stochastic Partial

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

Approximation of BSDEs using least-squares regression and Malliavin weights

Approximation of BSDEs using least-squares regression and Malliavin weights Approximation of BSDEs using least-squares regression and Malliavin weights Plamen Turkedjiev (turkedji@math.hu-berlin.de) 3rd July, 2012 Joint work with Prof. Emmanuel Gobet (E cole Polytechnique) Plamen

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

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

Continuous Time Finance

Continuous Time Finance Continuous Time Finance Lisbon 2013 Tomas Björk Stockholm School of Economics Tomas Björk, 2013 Contents Stochastic Calculus (Ch 4-5). Black-Scholes (Ch 6-7. Completeness and hedging (Ch 8-9. The martingale

More information

Non-Zero-Sum Stochastic Differential Games of Controls and St

Non-Zero-Sum Stochastic Differential Games of Controls and St Non-Zero-Sum Stochastic Differential Games of Controls and Stoppings October 1, 2009 Based on two preprints: to a Non-Zero-Sum Stochastic Differential Game of Controls and Stoppings I. Karatzas, Q. Li,

More information

{σ x >t}p x. (σ x >t)=e at.

{σ x >t}p x. (σ x >t)=e at. 3.11. EXERCISES 121 3.11 Exercises Exercise 3.1 Consider the Ornstein Uhlenbeck process in example 3.1.7(B). Show that the defined process is a Markov process which converges in distribution to an N(0,σ

More information

Stochastic optimal control with rough paths

Stochastic optimal control with rough paths Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction

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

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

Large Deviations for Perturbed Reflected Diffusion Processes

Large Deviations for Perturbed Reflected Diffusion Processes Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The

More information

An Overview of the Martingale Representation Theorem

An Overview of the Martingale Representation Theorem An Overview of the Martingale Representation Theorem Nuno Azevedo CEMAPRE - ISEG - UTL nazevedo@iseg.utl.pt September 3, 21 Nuno Azevedo (CEMAPRE - ISEG - UTL) LXDS Seminar September 3, 21 1 / 25 Background

More information

Particle models for Wasserstein type diffusion

Particle models for Wasserstein type diffusion Particle models for Wasserstein type diffusion Vitalii Konarovskyi University of Leipzig Bonn, 216 Joint work with Max von Renesse konarovskyi@gmail.com Wasserstein diffusion and Varadhan s formula (von

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

Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term

Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term 1 Strong uniqueness for stochastic evolution equations with possibly unbounded measurable drift term Enrico Priola Torino (Italy) Joint work with G. Da Prato, F. Flandoli and M. Röckner Stochastic Processes

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

Penalization of the Wiener Measure and Principal Values

Penalization of the Wiener Measure and Principal Values Penalization of the Wiener Measure and Principal Values by Catherine DONATI-MARTIN and Yueyun HU Summary. Let Q µ be absolutely continuous with respect to the Wiener measure with density D t exp hb sdb

More information

Linear Ordinary Differential Equations

Linear Ordinary Differential Equations MTH.B402; Sect. 1 20180703) 2 Linear Ordinary Differential Equations Preliminaries: Matrix Norms. Denote by M n R) the set of n n matrix with real components, which can be identified the vector space R

More information

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals

Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Functional Limit theorems for the quadratic variation of a continuous time random walk and for certain stochastic integrals Noèlia Viles Cuadros BCAM- Basque Center of Applied Mathematics with Prof. Enrico

More information

13 The martingale problem

13 The martingale problem 19-3-2012 Notations Ω complete metric space of all continuous functions from [0, + ) to R d endowed with the distance d(ω 1, ω 2 ) = k=1 ω 1 ω 2 C([0,k];H) 2 k (1 + ω 1 ω 2 C([0,k];H) ), ω 1, ω 2 Ω. F

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

Course 212: Academic Year Section 1: Metric Spaces

Course 212: Academic Year Section 1: Metric Spaces Course 212: Academic Year 1991-2 Section 1: Metric Spaces D. R. Wilkins Contents 1 Metric Spaces 3 1.1 Distance Functions and Metric Spaces............. 3 1.2 Convergence and Continuity in Metric Spaces.........

More information

On a class of optimal stopping problems for diffusions with discontinuous coefficients

On a class of optimal stopping problems for diffusions with discontinuous coefficients On a class of optimal stopping problems for diffusions with discontinuous coefficients Ludger Rüschendorf and Mikhail A. Urusov Abstract In this paper we introduce a modification of the free boundary problem

More information

ON NON-UNIQUENESS AND UNIQUENESS OF SOLUTIONS IN FINITE-HORIZON MEAN FIELD GAMES

ON NON-UNIQUENESS AND UNIQUENESS OF SOLUTIONS IN FINITE-HORIZON MEAN FIELD GAMES ON NON-UNIQUENESS AND UNIQUENESS OF SOLUTIONS IN FINITE-HOIZON MEAN FIELD GAMES MATINO BADI AND MAKUS FISCHE Abstract. This paper presents a class of evolutive Mean Field Games with multiple solutions

More information

Bernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010

Bernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010 1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes

More information

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS

ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary

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

Hardy-Stein identity and Square functions

Hardy-Stein identity and Square functions Hardy-Stein identity and Square functions Daesung Kim (joint work with Rodrigo Bañuelos) Department of Mathematics Purdue University March 28, 217 Daesung Kim (Purdue) Hardy-Stein identity UIUC 217 1 /

More information

Verona Course April Lecture 1. Review of probability

Verona Course April Lecture 1. Review of probability Verona Course April 215. Lecture 1. Review of probability Viorel Barbu Al.I. Cuza University of Iaşi and the Romanian Academy A probability space is a triple (Ω, F, P) where Ω is an abstract set, F is

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

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

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

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

Theoretical Tutorial Session 2

Theoretical Tutorial Session 2 1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations

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

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

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

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

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

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

arxiv: v1 [math.pr] 24 Sep 2018

arxiv: v1 [math.pr] 24 Sep 2018 A short note on Anticipative portfolio optimization B. D Auria a,b,1,, J.-A. Salmerón a,1 a Dpto. Estadística, Universidad Carlos III de Madrid. Avda. de la Universidad 3, 8911, Leganés (Madrid Spain b

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

EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS

EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS Qiao, H. Osaka J. Math. 51 (14), 47 66 EULER MARUYAMA APPROXIMATION FOR SDES WITH JUMPS AND NON-LIPSCHITZ COEFFICIENTS HUIJIE QIAO (Received May 6, 11, revised May 1, 1) Abstract In this paper we show

More information

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539

Brownian motion. Samy Tindel. Purdue University. Probability Theory 2 - MA 539 Brownian motion Samy Tindel Purdue University Probability Theory 2 - MA 539 Mostly taken from Brownian Motion and Stochastic Calculus by I. Karatzas and S. Shreve Samy T. Brownian motion Probability Theory

More information

Weakly interacting particle systems on graphs: from dense to sparse

Weakly interacting particle systems on graphs: from dense to sparse Weakly interacting particle systems on graphs: from dense to sparse Ruoyu Wu University of Michigan (Based on joint works with Daniel Lacker and Kavita Ramanan) USC Math Finance Colloquium October 29,

More information

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 22 12/09/2013. Skorokhod Mapping Theorem. Reflected Brownian Motion

MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 22 12/09/2013. Skorokhod Mapping Theorem. Reflected Brownian Motion MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.7J Fall 213 Lecture 22 12/9/213 Skorokhod Mapping Theorem. Reflected Brownian Motion Content. 1. G/G/1 queueing system 2. One dimensional reflection mapping

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

LAN property for sde s with additive fractional noise and continuous time observation

LAN property for sde s with additive fractional noise and continuous time observation LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,

More information

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents

MATH MEASURE THEORY AND FOURIER ANALYSIS. Contents MATH 3969 - MEASURE THEORY AND FOURIER ANALYSIS ANDREW TULLOCH Contents 1. Measure Theory 2 1.1. Properties of Measures 3 1.2. Constructing σ-algebras and measures 3 1.3. Properties of the Lebesgue measure

More information

Solvability of G-SDE with Integral-Lipschitz Coefficients

Solvability of G-SDE with Integral-Lipschitz Coefficients Solvability of G-SDE with Integral-Lipschitz Coefficients Yiqing LIN joint work with Xuepeng Bai IRMAR, Université de Rennes 1, FRANCE http://perso.univ-rennes1.fr/yiqing.lin/ ITN Marie Curie Workshop

More information