Time reversal and reflected diffusions
type: Item
instance of: scholarly article
title: Time reversal and reflected diffusions
zbMATH Open document ID: 0911.60059
DOI: 10.1016/S0304-4149(97)00035-5
author: F. Petit
published in: Stochastic Processes and their Applications
publication date: 1999-01-14T00:00:00Z
review text: 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)].
Mathematics Subject Classification ID: 60J60
Mathematics Subject Classification ID: 60J55
Mathematics Subject Classification ID: 60H10
zbMATH DE Number: 1240025
zbMATH Keywords: reflected diffusions
zbMATH Keywords: time reversal
zbMATH Keywords: entropy
zbMATH Keywords: reflected Brownian motion
reviewed by: Jan Seidler
MaRDI profile type: Publication
cites work: Q3038325
cites work: Conservative diffusions
cites work: Q3799448
cites work: Time reversal of diffusion processes with a boundary condition
cites work: Correction to: Minimization of the Kullback information of diffusion processes
cites work: Large deviations and Nelson processes
cites work: The Representation of Functionals of Brownian Motion by Stochastic Integrals
cites work: Q3704700
cites work: Q3810626
cites work: A transformation of a symmetric Markov process and the Donsker-Varadhan theory
cites work: Q3959169
cites work: Calcul stochastique et problèmes de martingales
cites work: Q3678418
cites work: Q5552132
cites work: Q3881642
cites work: Q3746599
cites work: Symmetric reflected diffusions
cites work: Q5528072
cites work: Tightness results for laws of diffusion processes application to stochastic mechanics
Recommended article: Reflection and coalescence between independent one-dimensional Brownian paths
Recommended article: On the local time of the Brownian motion
Recommended article: Reflection between two conjugate diffusions
Recommended article: A Dirichlet process characterization of a class of reflected diffusions
Recommended article: A note on reflecting Brownian motions
Recommended article: Brownian motion reflected on Brownian motion
Recommended article: Reflected solutions of backward stochastic differential equations with continuous coefficient
description: scientific article; zbMATH DE number 1240025