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

Transcription

1 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 of the Center for Financial Mathematics and Actuarial Research UCSB, May 2, 217

2 Introduction Mean-field / McKean-Vlasov SDE: For t [, T ], dxt x = b(t, Xt x, P X x t )dt + σ(t, Xt x, P X x t )db t, X x = x R, where b : R + R P(R) R and σ : R + R P(R) R P(R) space of probability distributions on R P X x t is the law of X x t

3 Introduction Increasing interest, also in Economics and Mathematical Finance. Mean-Field Games: [Lasry, Lions; 7] [Carmona, Delarue; 13, 14, 15], [Carmona, Delarue, Lachapelle; 13], [Carmona, Lacker; 15] Mean-Field BSDEs and PDEs: [Buckdahn, Li, Peng; 9], [Buckdahn, Li, Peng, Rainer; 17] Systemic Risk: [Fouque, Sun; 13], [Fouque, Ichiba; 13], [Garnier, Papanicolaou, Yang; 13], [Kley, Klüppelrg, Reichel; 15] [Carmona, Fouque, Sun; 15], [Carmona, Fouque, Mousavi, Sun; 16]

4 Introduction We here focus on: dx x t = b(t, X x t, E[ϕ(X x t )])dt + db t, X x = x R, (1) with irregular drift coefficient b : [, T ] R R R.

5 Introduction We here focus on: dx x t = b(t, X x t, E[ϕ(X x t )])dt + db t, X x = x R, (1) with irregular drift coefficient b : [, T ] R R R. In the following: ρ x t := E[ϕ(X x t )] ϕ( ) continuously differentiable with bounded Lipschitz derivative

6 Introduction Example (regime-switching mean reversion rate): dx x t = λ(xt x ) ( E[Xt x ] Xt x ) dt + dbt, where λ(x) := λ 1 I {x>r} + λ 2 I {x R} for some levels λ 1, λ 2, R R.

7 Outlook 1) Existence and Uniqueness of Strong Solutions 2) Regularity Properties of Strong Solutions 3) Bismut-Elworthy-Li Formula

8 . 1) Existence and Uniqueness of Strong Solutions

9 Strong Solutions Definition: Mean-field SDE (1) has a weak solution if there exists six-tuple (Ω, F, F, P, B, X x ) such that 1. (Ω, F, F, P) is a complete, filtered probability space, and B = (B t ) t [,T ] is an F-Brownian motion, 2. X x = (X x t ) t [,T ] is a continuous and F-adapted process, 3. X x satisfies P-a.s. dx x t = b(t, X x t, ρ x t )dt + db t, X x = x R, t [, T ],

10 Strong Solutions Definition: A strong solution of Mean-field SDE (1) is a weak solution (Ω, F, F, P, B, X x ) where F = F B is the filtration generated by Brownian motion B and augmented with the P-null sets.

11 Strong Solutions SDE s with irregular drift: Zvonkin (Math. USSR, 74); Veretennikov (Math. USSR, 81); Krylov & Röckner (Prob. Th. Rel.Fields, 5) M.-B., Proske (JFA, 1); M.-B. et al. (Math. Ann., 13); Banos, M.-B., Proske (F.&S., 17) Mean-Field SDE s with irregular drift: Chiang (Sooch. J. Math, 94); Li, Min (SIAM J.C.O., 16); Mishura, Veretenikov (ArXiv,16)

12 Strong Solutions Assumptions: The measurable drift coefficient b(t, y, z) is (i) of at most linear growth in y and z (uniformly in t) (ii) continuous in z (uniformly in t and y) (iii) in the decomposable form b(t, y, z) = ˆb(t, y, z) + b(t, y, z), where ˆb is merely bounded and b is Lipschitz continuous in y (uniformly in t and z)

13 Strong Solutions Theorem [Existence]: Suppose the drift coefficient b(t, y, z) satisfies Assumptions (i) - (iii). Then there exists a strong solution of the mean-field SDE (1). Further, any weak solution (X x t ) t [,T ] of (1) is a strong solution.

14 Strong Solutions Ideas in the proof: Consider approximating mean-field SDEs dx n,x t = b n (t, Xt n,x, E[ϕ(Xt n,x )])dt + db t, X n,x = x R, where b n smooth coefficients s.th. b n (t, y, z) b(t, y, z) a.e.

15 Strong Solutions Ideas in the proof: Consider approximating mean-field SDEs dx n,x t = b n (t, Xt n,x, E[ϕ(Xt n,x )])dt + db t, X n,x = x R, where b n smooth coefficients s.th. b n (t, y, z) b(t, y, z) a.e. Show: For all t [, T ] there is Y x t X n,x t L 2 (Ω, F t ) s.th. n Y x t strongly in L 2 (Ω, F t ). Compactness criterion in terms of Malliavin calculus Local space-time calculus: [Eisenbaum, Pot. Anal. ] t 2 b(u, Xu x, E[ϕ(Xt x )])du = b(u, y, E[ϕ(Xt x )])L X x (du, dy) t R

16 Strong Solutions Via Girsanov, construct weak solution X x t to dx x t = b(t, X x t, E[Y x t ])dt + db t, X x = x R,

17 Strong Solutions Via Girsanov, construct weak solution X x t to dx x t = b(t, X x t, E[Y x t ])dt + db t, X x = x R, Show: For all t [, T ] and functions F (x) of at most polynomial growth: F (Xt n,x ) n E[F (Xt x ) F t ] weakly in L 2 (Ω, F t ) X x t is F t -adapted and X x t = Y x t, so X x t is a strong solution.

18 Strong Solutions Assumption: The drift coefficient b(t, y, µ) admits (iv) θ as a modulus of continuity in the third variable, i.e. there exists a continuous function θ : R + R +, with θ(y) > for all y R +, z dy θ(y) = for all z R+, and for all t [, T ], y R, z 1, z 2 R, b(t, y, z 1 ) b(t, y, z 2 ) 2 θ( z 1 z 2 2 ).

19 Strong Solutions Theorem [Uniqueness]: Suppose the drift coefficient b(t, y, z) satisfies Assumptions (i)-(iv). Then there exists a unique strong solution of the mean-field SDE (1).

20 . 2) Regularity Properties of Strong Solutions

21 Regularity Properties Malliavin derivative: Let P be the family of all φ(θ 1,..., θ n ) where φ(x 1,..., x n ) is a polynomial in n variables and for f i ( ) L 2 ([, T ]). θ i = T f i (t)db t Definition: The Malliavin derivative of F P is given by n φ D t (F ) = (θ 1,..., θ n ) f i (t). x i i=1 D 1,2 is the closure of P in L 2 (Ω) w.r.t. the norm F 2 L 2 (Ω) + D (F ) 2 L 2 (Ω [,T ]).

22 Regularity Properties Malliavin derivative: Chain rule: Let F D 1,2 and φ C 1 (R). Then D t φ(f ) = φ (F )D t F.

23 Regularity Properties Malliavin derivative: Chain rule: Let F D 1,2 and φ C 1 (R). Then D t φ(f ) = φ (F )D t F. Duality formula: Let F D 1,2 and u Dom(δ). Then E [ T ] [ T ] F u t δb t = E u t D t Fdt.

24 Regularity Properties Theorem [Malliavin Differentiability]: Suppose the drift coefficient b(t, y, z) satisfies Assumptions (i) - (iii). Then any strong solution X x of the mean-field SDE (1) is Malliavin differentiable, and for s t T, the Malliavin derivative D s Xt x has the following representation D s X x t { t } = exp b(u, y, ρ x t )L X x (du, dy) s R

25 Regularity Properties From now on, instead of continuity in z (Assumption (ii)), we assume b(t, y, z) is continuously differentiable in z with bounded Lipschitz derivative (uniformly in t and y)

26 Regularity Properties Theorem [Hölder Continuity]: For the strong solution X x of the mean-field SDE (1), There exists a continuous version of the random field (t, x) Xt x with Hölder continuous trajectories of order α < 1 2 in t [, T ] and α < 1 in x R.

27 Regularity Properties Theorem [Sobolev Differentiability]: The strong solution X x of the mean-field SDE (1) is Sobolev differentiable in x, and x X t x has the representation x X t x = { exp t t + exp { } b(u, y, ρ x u)l X x (du, dy) R t s R } b(u, y, ρ x u)l X x (du, dy) 3 b(s, Xs x, ρ x s ) x ρx s ds

28 Regularity Properties Corollary: For s, t [, T ], s t, the following relationship holds: x X x t = D s X x t t x X s x + D u Xt x 3 b(u, Xu x, ρ x u) s x ρx udu.

29 . 3) Bismut-Elworthy-Li Formula

30 Bismut-Elworthy-Li Formula We now focus on the Delta for some pay-off function Φ. x E[Φ(X x T )]

31 Bismut-Elworthy-Li Formula We now focus on the Delta for some pay-off function Φ. x E[Φ(X x T )] Consider first a SDE: dx x t = b(t, X x t )ds + db t, X x = x R, t [, T ]

32 Bismut-Elworthy-Li Formula Theorem (Bismut-Elworthy-Li SDEs): [Fournier et al., 99] Let b be continuously differentiable with bounded Lipschitz derivatives, and Φ(X x T ) L2 (Ω). Then the Delta is given by where T x E[Φ(X T [Φ(X x )] = E xt ) a(t) ] x X t x db t, x X x t { t } = exp b (Xs x ) ds, and a(t) is a square integrable deterministic function such that T a(t) dt = 1.

33 Bismut-Elworthy-Li Formula Theorem (Bismut-Elworthy-Li for Mean-Field SDEs): Let the drift b(t, y, z) be as above and Φ L p (R m, w) for all p big enough with w(x) = e x 2 /(2T ). Then u(x) := E [Φ(X x T )] is continuously differentiable in x R and the derivative takes the form ( T x E[Φ(X T [Φ(X x )] = E xt ) a(s) x X s x db s T + 3 b(u, Xu x, ρ x u) u )] x ρx u a(s)dsdb u, where x X s x was given above and a : R R is any bounded function such that T a(s)ds = 1.

34 Bismut-Elworthy-Li Formula Steps in the proof: 1.) For smooth pay-off Φ, show differentiability and x E[Φ(X T x )] = E[Φ (XT x ) x X T x ] by approximation with smooth coefficients.

35 Bismut-Elworthy-Li Formula Steps in the proof: 1.) For smooth pay-off Φ, show differentiability and x E[Φ(X T x )] = E[Φ (XT x ) x X T x ] by approximation with smooth coefficients. 2.) Recall relation Sobolev / Malliavin derivative: T x X T x = = T a(s) x X T x ds ( a(s) D s XT x x X x s + T s D u XT x 3b(u, Xu x, ρ x u) ) x ρx udu ds

36 Bismut-Elworthy-Li Formula F.ex. second summand: by changing order of integration, the chain rule, and the duality formula T T E [Φ (X xt ) a(s)d u XT x 3b(u, Xu x, ρ x u) ] s x ρx ududs [ T = E D u Φ(XT x ) 3b(u, Xu x, ρ x u) u ] a(s)dsdu T = E [Φ(X xt ) 3 b(u, X x u, ρ x u) x ρx u x ρx u u a(s)dsdb u ].

37 Bismut-Elworthy-Li Formula F.ex. second summand: by changing order of integration, the chain rule, and the duality formula T T E [Φ (X xt ) a(s)d u XT x 3b(u, Xu x, ρ x u) ] s x ρx ududs [ T = E D u Φ(XT x ) 3b(u, Xu x, ρ x u) u ] a(s)dsdu T = E [Φ(X xt ) 3 b(u, X x u, ρ x u) x ρx u x ρx u u a(s)dsdb u ]. 4.) Approximation of general pay-offs with smooth pay-offs.

38 THANKS!