Wong-Zakai Approximation of Solutions to Reflecting Stochastic Differential Equations on Domains in Euclidean Spaces II

Abstract

The strong convergence of Wong-Zakai approximations of the solution to the reflecting stochastic differential equations was studied in [2]. We continue the study and prove the strong convergence under weaker assumptions on the domain.

Rough differential equations containing path-dependent bounded variation terms

“…When the driving rough path is a geometric rough path, we can give another proof of the existence of solutions as in [3]. Below we use the notation C θ− = ∩ 0<ε<θ C θ−ε , C 1-var+,θ− = ∩ q>1,0<ε<θ C q-var,θ−ε .…”

“…It is proved in [3] that max 0≤t≤T |Y N t − Y t | converges to 0 in probability under (A) and (B). This and the moment estimate in [4] implies the almost sure convergence in Theorem 6.8 (2).…”

“…The corresponding solution Y N t which is obtained by replacing B t by B N t is called the Wong-Zakai approximation of Y t . It is proved in [4,29,3] that the Wong-Zakai approximations of the solution to a reflected Stratonovich SDE under (A), (B) and (C) converge to the solution in the uniform convergence topology almost surely. Note that a similar statement under the conditions (A) and (B) is proved in [3].…”

“…By using this, we prove a support theorem for path space measures determined by random rough paths. When the driving process is Brownian motion, Wong-Zakai type theorem is proved in [4,29,3,13,25]. Recently, under the conditions (A) and (B) with certain additional natural assumptions on the boundary, support theorem for reflected diffusion processes are proved by Ren and Wu [26] by using the Wong-Zakai type theorem.…”

Reflected rough differential equations

“…For other references of reflected SDEs related with this paper, we refer the readers to [2,25,6,7,8,16,17,18,20,21]. In [1], we study Wong-Zakai approximations ( [23]) in the two cases, (i) the domain is convex, (ii) the conditions (A) and (B) are satisfied which are not contained in the result in [2].…”