Approximating irregular SDEs via iterative Skorokhod embeddings

Transcription

1 Approximating irregular SDEs via iterative Skorokhod embeddings Universität Duisburg-Essen, Essen, Germany Joint work with Stefan Ankirchner and Thomas Kruse Stochastic Analysis, Controlled Dynamical Systems and Applications Jena, March 9 13, 2015

2 Introduction

3 Random walk

4 Shrink time N times and space N times

5 Limit as N

6 The Donsker Prokhorov invariance principle Input: µ δ 0 centered probability measure on R with x 2 µ(dx) = 1 Theorem Let X 1, X 2,... be iid random variables with distribution µ, let 1 Y0 N = 0 and Yk N = Yk 1 N + N X k. Set Yt N = Y t N + (t t )(Y t +1 N Y t N ). Then ( ) YNt N t R + converges in distribution to a BM as N. Main question: Prove something of this kind for a 1-dim diffusion dm t = η(m t ) dw t

7 1-dim diffusion M Interior of the state space: I = (l, r) with l < r SDE: dm t = η(m t ) dw t, M 0 = m I η : I R Borel measurable with (a) (b) (c) η(x) 0 for x I 1 η 2 L1 loc(i) η(x) = 0 for x R \ I Theorem (Engelbert & Schmidt 1985) There is a weak solution of the SDE and we have uniqueness in law.

8 Main question Inputs: µ δ 0 centered probability measure on R dm t = η(m t ) dw t, M 0 = m I Can we find a scale factor a N : I (0, ) such that we have: Theorem Let X 1, X 2,... be iid random variables with distribution µ, let Y N 0 = m and Y N k = Y N k 1 + a N (Y N k 1)X k. Set Yt N = Y t N + (t t )(Y t +1 N Y t N ). Then ( ) YNt N t R + converges in distribution to M as N. One possibility: the (weak) Euler scheme a Eu N (y) = η(y) N x 2 µ(dx)

9 Answer and views Answer: Yes We will construct scale factors a N via iterative Skorokhod embeddings of shifted and scaled µ into M This method works even for irregular or quickly growing η View 1: A generalization of the DP invariance principle Different from [Stone 1963],..., [Étoré & Lejay 2007] View 2: A weak approximation scheme for SDEs Compare with the Euler scheme

10 Main results

11 Skorokhod s proof of the Donsker Prokhorov invariance principle µ δ 0 centered probability measure on R with x 2 µ(dx) = 1 X k iid µ, Y0 N = 0, Y k N = Y k 1 N + 1 N X k Skorokhod embeds Y N into a BM B with stopping times 0 = τ N (0) < τ N (1) <, i.e. (B τ N (k); k 0) d = (Y N k ; k 0), where τ N (k) τ N (k 1) are iid with E[τ N (k) τ N (k 1)] <. Then, by Wald s identity, ( ) 2 1 E[τ N (k) τ N (k 1)] = E(B τ N (k) B τ N (k 1)) 2 = E N X k = 1 N. One can show that (B τ N (k)) k 0 converges to B in probability, hence (Y N k ) k 0 converges to B in distribution.

12 Following Skorokhod s approach dm t = η(m t ) dw t, M 0 = m, Y N k+1 = Y N k + a N (Y N k ) X k+1, Y N 0 = m 1. Fix N N. Find stopping times 0 = τ0 N < τ 1 N a N : I (0, ) such that < and a function (M τ N k ; k 0) d = (Y N k ; k 0) and E[τ N k+1 τ N k F τ N k ] = 1 N. Problem (P) 2. (M τ N k ) k 0 converges to M in probability.

13 The Skorokhod embedding problem for M dm t = η(m t ) dw t, M 0 = y x ν(dx) = y SEP: find τ s.t. M τ ν By Itô s formula, q(y, x) = x u y y 2 η 2 dz du, (z) y q(y, M t ) t, t 0, I, x R is a local martingale starting from 0. If it is a true martingale, and the optional sampling theorem applies for a solution τ of the SEP for M, then Eτ = Eq(y, M τ ) = q(y, x) ν(dx). In general, the latter integral is the minimal possible Eτ. Source: [Ankirchner, Hobson, Strack 2013]

14 Solving Problem (P) Fix N N. Let M 0 = y I and X µ. Find a stopping time τ and a number a N (y) s.t. M τ d = y + an (y)x and Eτ = 1 N, ( ) i.e. solve the SEP with ν( ) = µ y a N (y) in expected time 1/N. Define ( ) dx y G y (a) := q(y, x) µ = q(y, y + ax) µ(dx), a 0. a Recall: G y (a) is expected time needed to embed µ ( ) y a into M. Solution method: For each y I find a solution a N (y) (0, ) of the equation G y (a) = 1 N. Problem: There are situations when this equation has no solution, while Problem (P) can be successfully solved.

15 Summary to Problem (P) Messages: In many cases there exists a solution to Problem (P). We have sufficient conditions in terms of η and µ. Some µ always work. For instance, µ = 1 2 (δ 1 + δ 1 ). For some η no restrictions on µ (except minimal natural restrictions ). For instance, GBM.

16 Convergence results Assume sufficient conditions for solvability of Problem (P) discrete-time Markov chain (Yk N ) k Z+ extension to continuous-time process (Yt N ) t 0 via linear interpolation (C1) η and 1 η are bounded on I. Theorem Assume (C1). Then the processes (Y N Nt ) converge to (M t) in distribution. (C2) η and 1 η are locally bounded on I. Theorem Suppose (C2) and that µ has a compact support. Then the processes (Y N Nt ) converge to (M t) in distribution.

17 Convergence rate Under (C1) and x 4 µ(dx) < the order of convergence is 1/4 (η is just Borel measurable) Under (C2) this is no longer true (a counterexample)

18 EXAMPLES

19 Summary of the algorithm Aim: Approximate distributional properties of M, dm t = η(m t ) dw t, M 0 = m. 1. Determine q(y, x) = x y u y 2 dz du, y I, x R. η 2 (z) 2. Choose the discretization parameter N N. 3. Choose a reference measure µ such that Problem (P) has a solution. Below we always take µ = 1 2 (δ 1 + δ 1 ). 4. Solve in a the equation q(y, y + ax) µ(dx) = 1/N for all y I solution a N (y). 5. Simulate Y N k = Y N k 1 + a N(Y N k 1 )X k, Y N 0 = m, where X k iid µ.

20 Exponentially growing η I = R, η(x) = cosh(x)

21 Exponentially growing η dm t = cosh(m t ) dw t, M 0 = 0. [ ( ) ] q(y, x) = 2 log cosh(x) cosh(y) tanh(y)(x y), for y, x R. Choose µ = 1 2 (δ 1 + δ 1 ) Then for N N and y R a N (y) = 1 ) (2(exp(1/N) 2 arcosh 1) cosh 2 (y) + 1. a N has linear growth Euler scheme a Eu N (y) = 1 N cosh(y) General property: our scale factors always have linear growth

22 A realization of the Euler scheme for dm t = cosh(m t ) dw t : saw effect Saw effect diverging oscillations

23 Scale factor for the Euler scheme a Eu N (y) = 1 N cosh(y) N = y y + an Eu (y) and y y aeu N (y)

24 Scale factor for our scheme N = y y + a N (y) and y y a N (y) Saw effect is impossible (comparison principle)

25 Comparing realizations for dm t = cosh(m t ) dw t 5 Euler Our method 0 5 5

26 Approximating an expectation of M 1 Aim: For α (0, 1) approximate E[ M 1 α ] numerically. Euler scheme Y N,Eu converges a.s. to M ([Gyöngy 1998], for Gaussian increments). But it follows from [Hutzenthaler, Jentzen, Kloeden 2010] that E[ Y N,Eu N α ] as N. Proposition The family ( Y N N α ) N N is uniformly integrable. Hence, as N. E[ Y N N α ] E[ M 1 α ]

27 Not locally bounded η I = R, η(x) = 1 x, η(0) = 1

28 Not locally bounded η dm t = η(m t ) dw t, M 0 = 0. η(x) = 1 x for x 0 and η(0) = 1. q(y, x) = 1 6 x xy y 4, for y, x R. Choose µ = 1 2 (δ 1 + δ 1 ) Then for N N and y R a N (y) = 9y N 3y 2. Euler scheme an Eu (y) = 1 1 for y 0, N y aeu N (0) = 1. N

29 Scale factors N = 10000, y a N (y) and y a Eu N (y) a N (0) = 4 6 N and lim N NaN (y) = 1 y. General property: smoothing if η has irregularities

30 Convergence The Euler approximation Y N,Eu does not converge in law to M. Indeed, for every N N we have Y N,Eu 2 = X 1 N + X 2 X 1. Proposition The sequence of continuous processes (Y N Nt ) t 0 converges in law to the process M, as N. Moreover, we have E[f (Y N N )] E[f (M 1 )] as N for every continuous function f : R R with f (x) c(1 + x α ), x R, for some c R + and α (0, 4).

31 Dear Hans-Jürgen, happy birthday and many happy returns! Thank you!

32 Conclusion We constructed Markov chains that can be embedded into a driftless diffusion with a fixed mean time lag 1 N and a non-local, implicit numerical scheme to approximate diffusions with irregular coefficients and superlinear growth The scale factors may differ significantly from their counterparts in the Euler scheme smoothing if η has discontinuities tempered growth behavior