Multilevel Monte Carlo for Stochastic McKean-Vlasov Equations

Transcription

1 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 of Edinburgh) MLMC for MV-SDEs March / 23

2 MKV-SDEs Mckean-Vlasov SDEs on [0, T ], of the form { dx t = R d b(x t, y)µ t (dy)dt + R d σ(x t, y)µ t (dy)dw t, µ t = Law(X t), X 0 R d b C 2,1 b (R d R d, R d ) and σ C 2,1 b (R d R d, R d r ) = For all x 1, x 2, y 1, y 2 R d, there exists a constant L such that b(x 1, y 1 ) b(x 2, y 2 ) + σ(x 1, y 1 ) σ(x 2, y 2 ) L( x 1 x 2 + y 1 y 2 ), = (can be relaxed) b(x 1, y 1) + σ(x 1, y 1) L(1 + x 1 + y 1 ) sup b(x 1, y 1) + sup σ(x 2, y 2) K < (x 1,y 1 ) R d R d (x 2,y 2 ) R d R d The initial law µ 0 satisfies: µ 0 is a Dirac measure at X 0, or p 2, R d x p µ 0 (dx) < Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

3 Motivation MKV-SDEs gives probabilistic interpretation of nonlinear McKean-Vlasov PDEs which weak formulation with f ( ) CK (R d ) is given { t t, f = µ t, 1 d 2 i,j=1 a ij(x, µ t ) 2 f x i x j (x) + d i=1 b i(x, µ t ) f x i (x), µ 0 = P X 1 0 = Law(X 0 ), where a(x, µ t ) = σ(x t, µ t ) T σ(x t, µ t ) and b(x, µ t ) := R d b(x, y)µ t (dy) Example: dy t = R d b(y t, y)µ t (dy)dt + dw t Applications: Lagrangian models (Bossy,Jabir,Talay, 2011) Navier-Stokes equation for the vorticity of a two-dimensional incompressible fluid flow and many more (Bossy,Jourdain,Meleard,Reygner,Talay) Mean-Field Games (Lasry, Lions, 2007, Chassagneux, Crisan, Delarue, 2015) Stochastic Local Volatility Models (Gyongy, 1996; Guyon, Henry-Labordere 2011; Guyon 2014,2015; Jourdain,Zhou 2016) Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

4 Propagation of chaos Stochastic interacting particles (X i,n t ) are solutions to (R d ) N dimensional SDEs { dx i,n t = b(x i,n t µ N t := 1 N N i=1 δ X i,n t, µ N t )dt + σ(x i,n t, t 0,, µ N t )dw i t, i = 1,, N, where {X i,n 0 } i=1,,n are iid samples with the law µ 0 and {W i t } i=1,,n are independent Brownian motions X i,n t X i t when N Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

5 Time discretization - Euler Scheme Consider MV-SDEs dx t = b(x t, y)µ t(dy)dt + σ(x t, y)µ t(dy)dw t, R d R d Euler scheme with time-step h = T /M, i=1,,n, Y i,n k+1 = Y i,n k + 1 N N j=1 b(y i,n k, Y j,n k )h + 1 N N j=1 σ(y i,n k, Y j,n k ) Wk+1 i Due to the particle interactions, its implementation requires N 2 arithmetic operations at each step Euler scheme converges with weak rate of order (( N) 1 + h) Bossy, Talay (1997) Antonelli, Kohatsu-Higa (2002), Bossy, Jourdain (2002) Notice that the same sample is used to approximate MV-SDEs and to evaluate the E[G(X T )] Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

6 Computational cost of the propagation of chaos Consider mean-square-error [( E E[f (X T )] 1 N bias and statistical error are in a nonlinear relationship Consider iid samples Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23 n i=1 f (Y i,n T ) ) 2 ] X k+1 = X k + b( X k, P kh )h + σ( X k, P kh ) W k+1, P kh = P ( X k ) 1 Error decomposition E[f (X T )] 1 N N i=1 f (Y i,n T ) = ( E[f (X T )] E[f ( X T )] ) + ( E[f ( X T )] 1 N + 1 N N i=1 N f (( X T i ) ) i=1 ( f (( X i T ) f (Y i,n T ))

7 Cost Typical mean-square error ( E E[f (X T )] 1 N ) 2 N f (Y i,n T ) C(h N + 1 N ), i=1 Cost C γ = N γ h 1, γ = 1 no-interacting Kernel, γ = 2 interacting Kernel For the root-mean-square-error ϵ the cost is C 1 = O(ϵ 3 ) or C 2 = O(ϵ 5 ) Example of the non-interacting kernel particle system: Y i,n k+1 = Y i,n k + b(y i,n k, 1 N N j=1 f (Y j,n k ))h + σ Wk+1 i Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

8 MLMC for standard SDEs Idea of Giles (2006), Heinrich (2001) was to explore the identity E[P L ] = E[P 0 ] + L E[P l P l 1 ], where P l := P(Y M l ) with P : C([0, T ], R d ) R and {Y M l }, l = 0 L, being discrete time approximation of process X with M l number of time steps l=1 This identity leads to an unbiased estimator of E[P(Y M L )], { N 0 N 1 L P (i,0) 1 l 0 + N 0 i=1 l=1 N l Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23 i=1 (P (i,l) l } P (i,l) l 1 ), where P (i,l) l = P((Y M l ) (i) ) are independent samples at level l But for MLMC variance for particle systems, typically decays as O(N 1 + h) In the case of particle systems, one need to be careful to ensure telescopic sum is preserved

9 Sznitman s iteration proof { Xt = X 0 + W t + t b(x 0 C s, y)µ t (dy)ds, µ t = Law(X t ) 0 t T Step 1: Pick a measure µ P(C[0, T ], R d ) Step 2: Define an operator Φ : P(C[0, T ], R d ) P(C[0, T ], R d ); Φ(µ) = Law(X µ ) t X µ t = X 0 + W t + b(x s µ, y)µ t(dy)ds, 0 t T Iterate Theorem (Sznitman) Let T > 0, and µ P 2 (C[0, T ], R d )) There exists C > 0 st [ W 2 (µ, ν) = inf γ C C 0 C W 2(Φ k+1 (µ), Φ k (µ)) C T k W2(Φ(µ), µ) k! ] u v 2 γ(du, dv); γ( C) = µ, γ(c ) = ν Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

10 Picard s Particle system Consider a sequence of SDEs linear (in a sense of McKean) defined as dxt m = b(xt m, y)µ m 1 t (dy)dt + σ(xt m, y)µ m 1 t (dy)dwt m, Law(X0 m ) = Law(X 0 ), C C where µ m 1 t = Law(Xt m 1 ) Let (Y 0,n,N 0 t ) 1 n N0 be an iid sample with law µ 0 For m 1, we define Key idea: dy m,n,n m t = 1 N m 1 N m 1 j=1 + 1 N m 1 N m 1 j=1 b(y m,n,n m t σ(y m,n,nm t, Y m 1,j,N m 1 t ) dt Use 1 : m 1 steps to approximate R d b(x µ s, y)µ t(dy), Y m 1,j,N m 1 t ) dw m,n t Use the final m Picard step to approximate the quantity of interest Iterated Particle system is less efficient then original Particle system! Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

11 Iterative Particle System Consider a sequence of SDEs linear (in a sense of McKean) defined as dxt m =, y)µ m 1 t (dy)dt + b(xt m R d σ(xt m R d where µ t m 1 = Law(Xt m 1 ) and W m and X0 m are independent Iterative particle system Y i,m,l k+1 = Y i,m,l k + [M t l k ( µ m 1 Y No-interacting Kernels are much easier to analyse, y)µ m 1 t (dy)dwt m, Law(X0 m ) = Law(X 0), )](b(y i,m,l, ))h l + [M t l( µ m 1 k Y )](σ(y i,m,l, )) W i,m,l k+1 Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

12 Iterative Particle System For any borel function G = (G (1),, G (d) ), where G (i) : R d R d R, we define [M t(µ)](g(x, )) := L l=0 where µ l t (G(x, )) = R d G(x, y)µ l t (dy) [ µ l t := ( ) µ l t µ t l 1 (G(x, )), µ P 2, [ t η l (t) h l ]µ l η l (t)+h l + 1 t η l (t) h l ]µ l η l (t), t / Π l, 1 Nl,m N l,m i=1 δ Yt i,m,l, t Π l Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

13 Further assumptions Define B(t, x) = E[b(x, X t )] and Σ(t, x) = E[σ(x, X t )] ( B(, ) C 1,2 and Σ(, ) C 1,2 ) Consider stochastic flow ( X s,x t, t [s, T ] ) We consider X s,x t = x + t s B ( θ, X s,x θ ) dθ + t s Σ ( θ, X s,x ) θ dwθ, t [s, T ] v y (s, x) := E[G(y, X s,x t )], y R d and (s, x) [0, t] R d, and associated (family of) PDEs reads d v y s (s, x) i,j=1 v y (t, x) = G(y, x), A ij (s, x) 2 v y x i x j (s, x) + where A = Σ(s, x)σ(s, x) T We have d j=1 B j (s, x) vy x j (s, x) = 0, (s, x) [0, t] R d sup v y y R d x (t, x ) L v and sup 2 v y y R d x (t, x ) 2 L v (t, x ) [0, T ] R d Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

14 Main Result Theorem Assume regularity of the coefficients and the initial law Then ] [([M ηl(t)(µ MY )](G(x, )) E[G(x, X ηl(t))]) 2 sup x R d sup E 0 t T c { h 2 L + M m=1 c M m (M m)! L l=0 } h l + cm, N m,l M! We can reduce computational cost of approximating expectation by the order of magnitude Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

15 Glimpse into the analysis SDEs with random coefficients du t = B(U t, V ηl (t))dt + Σ(U t, V ηl (t))dw t, Law(U 0) = Law(X 0), where V ηl (t) P(R d ) is random measure Euler scheme is given dzt l = B(Z η(t), l V η(t) )dt + Σ(Z η(t), l V η(t) )dw t, Law(Z0 l ) = Law(X 0 ), (Conditional) Independence: The random measure V is independent of W and Z 0 Integrability: Let G be Lipschitz continuous, then sup sup E[ G(x, y)) p V s(dy)] c p 1 0 s T R d x R d Regularity: Let G be Lipschitz continuous, then sup sup E[ G(x, y)(v t V s )(dy) 2 ] c(t s) x R d 0 s t T R d Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

16 Glimpse into the analysis Theorem Assume regularity of the coefficients and the initial law There exists a constant c such that Theorem sup sup E V Z l η L (t)(g(x, )) V Z l 1 η L (t) (G(x, )) 2 ch l x R d 0 t T Assume regularity of the coefficients and the initial law Then there exists a constant c such that t [0, T ], sup x R d sup E[G(x, Zs L )] E[G(x, X s )] 0 s t ( t c h L + 0 sup E B(x, V ηl (s)) E[b(x, X ηl (s))] ds x R d + t 0 sup E Σ(x, V ηl (s)) E[σ(x, X ηl (s))] ds x R d ) Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

17 New idea The idea is to use previous Iteration as Control Variates ie E[G(x, X M )] = E[G(x, X 0 )] + M m=1 E[G(x, X m ) G(x, X m 1 )] Coupling is obtained by simulating both iterations with the same BM Inner expected value can be further expanded into a telescopic sum First numerical examples are very promising Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

18 Numerical Experiment: Non-interacting kernel The target stochastic differential equation is dx t = sin(x t E[X t ])dt + σdw t, X 0 = 0 The testing payoff function is P(x) = max(x K, 0), where strike K is set to 01 Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

19 Non-interacting kernel Epsilon level (ǫ) Computation cost particle system revised MC+picard MC+picard Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

20 Non-interacting case MLMC with ǫ 1 >ǫ 2 >+ picard MLMC + picard particle systems 25 2 Computational cost Epsilon level (ǫ) Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

21 Interacting kernel The target stochastic differential equation is dx t = E[sin(x X t)] x=xt dt + σdw t, X 0 = 0, where Y t is an independent copy of X t The payoff P(x) = 1 + x 2 Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

22 Interacting kernel particle system revised MC+picard MC+picard MLMC+picard 2 Computational cost Epsilon level (ǫ) Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23

23 Interacting kernel MLMC with ǫ 1 >ǫ 2 >+ picard MLMC + picard particle systems 4 35 Computational cost Epsilon level (ǫ) Lukasz Szpruch (University of Edinburgh) MLMC for MV-SDEs March / 23