Time reversal and reflected diffusions
Let us consider a reflected diffusion \(X=(Y,Z)\) taking values in \(\mathbb R_{+}\times \mathbb R^{n}\) and described by a stochastic differential equation \[ \begin{aligned} & Z_{t}=Z_{0}+\int ^{t}_{0}\text{sgn}(W_{s}-Z_{0}) ds +l _t(Z_{0},W), \quad \int ^{t}_{0}1_{\{Z_{s}=0\}} ds = 0, \\ &Y_{t}= Y_{0} +\int ^{t}_{0} \chi _{0}(s,X_{s}) ds + \int ^{t}_{0}\chi _{1}(X_{s}) dB_{s} + \int ^{t}_{0} V_{0}(Y_{s}) dl _{s}(Z_{0},W), \end{aligned} \] where \(Z_{0}\geq 0\), \(W\) and \(B\) are independent Wiener processes in \(\mathbb R\) and \(\mathbb R^{n}\), respectively, the coefficients \(\chi _{0}\), \(\chi _{1}\), \(V_{0}\) are assumed to be \(C^\infty \)-smooth and bounded, and \(l _{s}(r,W)\) denotes the local time of the process \(W\) at time \(s\) and level \(r\). Suppose that the process \(X\) has a \(C^\infty \)-smooth density \(q\). Under some assumptions on \(q\) the equation satisfied by the time reversed process \((X_{1-t})_{0\leq t\leq 1}\) is found and the duality equation is derived. To make the main ideas clear, the case of the reflected Brownian motion is discussed separately in the first part of the paper. The methods used by the author extend those employed by \textit{H. Föllmer} in the ``non-reflected'' case [see his contributions in: Stochastic processes -- mathematics and physics. Lect. Notes Math. 1158, 119-129 (1986; Zbl 0582.60078), and in: Calcul des probabilités. Lect. Notes Math. 1362, 101-203 (1988; Zbl 0661.60063)].
scientific article; zbMATH DE number 1240025
scientific article