Exact asymptotics for the probability of exit from a domain and applications to simulation. (English) Zbl 0856.60033

Baldi, Paolo

Ann. Probab. 23, No. 4, 1644-1670 (1995).

Let \(D\subset \mathbb{R}^n\) be an open set. The diffusion process \(X^\varepsilon\) associated with the stochastic differential equation \[ dX^\varepsilon_t= b(X^\varepsilon_t, t) dt+ \sqrt \varepsilon \sigma(X^\varepsilon_t, t) dB_t,\quad X^\varepsilon_s= x\in D, \] is considered. The problem of exact asymptotics of \(\mathbb{P}^\varepsilon_{x, s}(\tau\leq T)\), \(T> 0\), is investigated, where \(\tau\) is the exit time from \(D\). The case of Brownian bridge is studied. Related problems were considered by W. H. Fleming and M. R. James [ibid. 20, No. 3, 1369-1384 (1992; Zbl 0771.60055)] and R. Azencott [Bull. Sci. Math., II. Sér. 109, 253-308 (1985; Zbl 0591.60023)].

Reviewer: A.Plikusas (Vilnius)

MSC:

60F10	Large deviations
60J60	Diffusion processes
60J65	Brownian motion

Keywords:

stochastic differential equation; asymptotics; Brownian bridge

Citations:

Zbl 0771.60055; Zbl 0591.60023