Exact Asymptotics for the Exit Time Probabilities of Scalar Ornstein-Uhlenbeck Bridges

Abstract

This paper aims to derive accurate asymptotic estimates for the exit time probabilities of scalar Ornstein-Uhlenbeck (OU) bridges. The exit time probabilities are expressed as an asymptotic series in powers of a small parameter that characterizes the intensity of the noise inputs. It is shown that the series is valid in certain regions where all its terms are smooth functions. The results enable an accurate evaluation of the probability for a corresponding OU process to escape from a domain before a specified time, provided its initial and terminal states are known.

Figures (2)

• Figure 1: The vector field $(b(x,t),1)^{\top}$ and the intersection between $\{(x^{0}_{x,s}(t),t):t\in[s,T]\}$ and $\partial{D}\times[0,T)$ for Case $\text{A}_{VII}$. Subfigures (a) and (b) correspond to Cases $\text{A}_{VII,I}$ and $\text{A}_{VII,II}$, respectively, while (c) and (d) illustrate an example for Case $\text{A}_{VII,III}$. The parameters are $a_0=0$, $a_1=1$, $T=1$, $x_T=2$ with (a) $d_1=0.2$, $d_2=1.2$; (b) $d_1=1.2$, $d_2=1.8$; (c)(d) $d_1=1.7$, $d_2=1.9$.
• Figure 2: Strongly regular regions for Case $\text{A}_{VII}$. Subfigures (a) and (b) correspond to Cases $\text{A}_{VII,I}$ and $\text{A}_{VII,II}$, respectively, while (c) illustrates an example for Case $\text{A}_{VII,III}$. In each subfigure, the solid green line represents a segment of the trajectory $\{(x^{0}_{x,s}(t),t):t\in[0,T]\}$ that satisfies $x^{0}_{x,s}(T')=d_2$, and the solid magenta line denotes a segment of the critical trajectory $\{(x^{0,*}(t),t):t\in[0,T]\}$. The parameters are $a_0=0$, $a_1=1$, $T=1$, $x_T=2$, $T'=0.9$ with (a) $d_1=0.2$, $d_2=1.2$; (b) $d_1=1.2$, $d_2=1.8$; (c) $d_1=1.7$, $d_2=1.9$.

Theorems & Definitions (44)

• Lemma 2.1
• proof
• Remark 2.1
• Corollary 2.2
• Corollary 2.3
• proof
• Remark 2.2
• Corollary 2.4
• proof
• Proposition 2.5