TY - JOUR

T1 - On singularity as a function of time of a conditional distribution of an exit time

AU - Krylov, N. V.

N1 - Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.
Copyright:
Copyright 2016 Elsevier B.V., All rights reserved.

PY - 2016/8/1

Y1 - 2016/8/1

N2 - We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

AB - We establish the singularity with respect to Lebesgue measure as a function of time of the conditional probability distribution that the sum of two one-dimensional Brownian motions will exit from the unit interval before time t, given the trajectory of the second Brownian motion up to the same time. On the way of doing so we show that if one solves the one-dimensional heat equation with zero condition on a trajectory of a one-dimensional Brownian motion, which is the lateral boundary, then for each moment of time with probability one the normal derivative of the solution is zero, provided that the diffusion of the Brownian motion is sufficiently large.

KW - Filtering of partially observable diffusion processes

KW - Heat equation in domains with irregular lateral boundaries

KW - Stochastic partial differential equations

UR - http://www.scopus.com/inward/record.url?scp=84933073560&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84933073560&partnerID=8YFLogxK

U2 - 10.1007/s00440-015-0639-3

DO - 10.1007/s00440-015-0639-3

M3 - Article

AN - SCOPUS:84933073560

VL - 165

SP - 541

EP - 557

JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

SN - 0178-8051

IS - 3-4

ER -