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
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
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]
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.
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
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.
Outlook 1) Existence and Uniqueness of Strong Solutions 2) Regularity Properties of Strong Solutions 3) Bismut-Elworthy-Li Formula
. 1) Existence and Uniqueness of Strong Solutions
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 ],
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.
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)
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)
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.
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.
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
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,
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.
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 ).
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).
. 2) Regularity Properties of Strong Solutions
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 ]).
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.
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.
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
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)
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.
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
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.
. 3) Bismut-Elworthy-Li Formula
Bismut-Elworthy-Li Formula We now focus on the Delta for some pay-off function Φ. x E[Φ(X x T )]
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 ]
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.
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.
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.
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
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 ].
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.
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