2021/01/08 by Giang T. Nguyen, Oscar Peralta, Nguyen, Giang T. +1
Economics, Econometrics and Finance · Mathematics · #41A25 #60J28 #60J65 #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2101.03250
openalex publication_date 2021/01/08 · openalex created_date 2021/10/25 · openalex updated_date 2026/07/28
We construct Wong--Zakai approximations of time--inhomogeneous stochastic differential equations with regime switching (RSSDEs), and provide a convergence rate. %Given a family of finite-variation processes \Fλ\λ≥ 0 that converge strongly to a standard Brownian motion B, we construct pathwise approximations for regime-switching, time-inhomogeneous stochastic differential equations in the Wong-Zakai sense. Moreover, we determine the rate of strong convergence to the solutions of such regime-switching SDEs, showing that this rate is almost as good as that of \Fλ\λ≥ 0 to B. In the proposed approximations, the standard Brownian motion driving the time-inhomogeneous RSSDEs is replaced by a family of finite--variation processes \Fλ\λ> 0. We show that if Fλ strongly converges to B at rate δ(λ), then the Wong--Zakai approximation strongly converges to the original solution of the time--inhomogeneous RSSDE at rate δ(λ) λε, for any ε > 0. This is the first paper on Wong--Zakai approximations for time--inhomogeneous RSSDEs, and significantly extends the counterparts for time--homogeneous SDEs without regime switching in Römisch and Wakolbinger (1985).