A functional limit theorem for irregular SDEs

Abstract

Let X 1 , X 2 , . . . be a sequence of i.i.d. real-valued random variables with mean zero, and consider the scaled random walk of the formWe show, under mild assumptions on the law of X i , that one can choose the scale factor a N in such a way that the process (Y N ⌊N t⌋ ) t∈R + converges in distribution to a given diffusion (M t ) t∈R + solving a stochastic differential equation with possibly irregular coefficients, as N → ∞. To this end we embed the scaled random walks into the diffusion M with a sequence of stopping times with expected time step 1/N. 2010 MSC : 60F17, 60J60, 65C30. Keywords : stochastic differential equations, irregular diffusion coefficient, weak law of large numbers for u.i. arrays, weak convergence of processes, Skorokhod embedding problem.

Numerical approximation of irregular SDEs via Skorokhod embeddings

“…For the remainder of this section we suppose that for all h ∈ (0, ∞) we can find a unique scale factor a such that the random walk (Y k ) with scale factor a is embeddable into M with expected time step h. In Section 2 we recall a sufficient condition from [2] guaranteeing that this assumption is satisfied (see Proposition 2.1). We next explain why the embeddable scaled random walks can be used for approximating the law of the diffusion M .…”

“…Since M has continuous sample paths, this further indicates that lim N →∞ M N = M in probability uniformly in the space C([0, T ]) of continuous functions [0, T ] → R; in other words, we have convergence of the distributions of M N on the path space. The previous line of argument is made precise in [2]. Theorem 1.1 (Theorem 3.6 in [2]).…”

“…The previous line of argument is made precise in [2]. Theorem 1.1 (Theorem 3.6 in [2]). Suppose that |η| and 1 |η| are locally bounded on I, µ has a compact support and that the following implications hold true:…”

“…One can extend (Y k ) to a continuoustime process by setting Y t = Y t + (t − t )(Y t +1 − Y t ) for all t ∈ R + . A functional limit theorem implies that the distribution of (Y t/h ) t∈[0,T ] converges to the distribution of (M t ) t∈[0,T ] as h ↓ 0 (see [2]). One can thus use (Y k ) for approximating the law of M .…”

“…Our idea works, because there is a simple integral formula for the minimal expected time needed for embedding a distribution into M (see [1]). The integral formula allows to determine the scale factor a in (2) such that the transition probabilities of (Y k ) can be embedded into M with stopping times having expectation h.…”