On first exit times for homogeneous diffusion processes\thanks{ Theory of Probability and Its Application, 1987, v.31, No 3, pp. 497-498; English translation of the September 1986 issue of the Soviet journal "Teoria Veroyatnostei i ee primneniya"; translated from Russian by Amal Ellis}
math0102099.tex
%% Th Probab Apll 1987, iss 3
\documentstyle[doublespace,leqno,11pt]{article}
\renewcommand{\baselinestretch}{1.2}

%ZDES' numeraciyua sploshnaya
% nazvaniyua Teorema, Lemma i.t.p. v konce (vibor: po russki ili angliiski)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%                         control.sty
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  redefine equation numbering to be in the form SECTIONNUMBER.NUMBER
%
  % \renewcommand{\theequation}{\thesection.\arabic{equation}}
  % \csname @addtoreset\endcsname{equation}{section}
%
%  redefine figure and table numbering to be in the form SECTIONNUMBER.NUMBER
%
   \renewcommand{\thefigure}{\thesection.\arabic{figure}}
   \renewcommand{\thetable}{\thesection.\arabic{table}}
    \csname @addtoreset\endcsname{figure}{section}
    \csname @addtoreset\endcsname{table}{section}

%
% redefine enumerate numbering style
%
    \renewcommand{\theenumi}{(\roman{enumi})}
    \renewcommand{\labelenumi}{(\roman{enumi})}
%
% Use the environment 'eqnabc' instead of 'eqnarray' to label a group of
% equations by letters.  It can be used with or without '\eqnsection.'
\newenvironment{eqnabc}{
   \stepcounter{equation}
   \edef\numericalpart{\theequation}
   \let\originaltheequation=\theequation
   \newcount\equationno
   \equationno=\value{equation}
   \setcounter{equation}{0}
   \renewcommand{\theequation}{\numericalpart\alph{equation}}
   \begin{eqnarray}
   }{
   \end{eqnarray}
   \setcounter{equation}{\number\equationno}
   \renewcommand{\theequation}{\originaltheequation}}

%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%  Command to specify the number of the footnote mark in titlepage

\newcounter{thanksnum}
\def\thanksnumber#1
{\setcounter{thanksnum}{\value{footnote}}\setcounter{footnote}{#1}%
                     \addtocounter{footnote}{-1}\footnotemark
                     \setcounter{footnote}{\value{thanksnum}}}
%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
%  Define newtheoremz command which creates a theorem environment without
%  italics
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%
\def\newtheoremz#1{\@ifnextchar[{\@othmz{#1}}{\@nthmz{#1}}}

\def\@nthmz#1#2{%
\@ifnextchar[{\@xnthmz{#1}{#2}}{\@ynthmz{#1}{#2}}}

\def\@xnthmz#1#2[#3]{\expandafter\@ifdefinable\csname #1\endcsname
{\@definecounter{#1}\@addtoreset{#1}{#3}%
\expandafter\xdef\csname the#1\endcsname{\expandafter\noexpand
  \csname the#3\endcsname \@thmcountersepz \@thmcounterz{#1}}%
\global\@namedef{#1}{\@thmz{#1}{#2}}\global\@namedef{end#1}{\@endtheoremz}}}

\def\@ynthmz#1#2{\expandafter\@ifdefinable\csname #1\endcsname
{\@definecounter{#1}%
\expandafter\xdef\csname the#1\endcsname{\@thmcounterz{#1}}%
\global\@namedef{#1}{\@thm{#1}{#2}}\global\@namedef{end#1}{\@endtheoremz}}}

\def\@othmz#1[#2]#3{\expandafter\@ifdefinable\csname #1\endcsname
  {\global\@namedef{the#1}{\@nameuse{the#2}}%
\global\@namedef{#1}{\@thmz{#2}{#3}}%
\global\@namedef{end#1}{\@endtheoremz}}}

\def\@thmz#1#2{\refstepcounter
    {#1}\@ifnextchar[{\@ythmz{#1}{#2}}{\@xthmz{#1}{#2}}}

\def\@xthmz#1#2{\@begintheoremz{#2}{\csname the#1\endcsname}\ignorespaces}
\def\@ythmz#1#2[#3]{\@opargbegintheoremz{#2}{\csname
       the#1\endcsname}{#3}\ignorespaces}

%DEFAULT VALUES
\def\@thmcounterz#1{\noexpand\arabic{#1}}
\def\@thmcountersepz{.}
%deleted September 2, 1986 MDK
%\def\@makethmnumber#1#2{\bf #1 #2:}
\def\@begintheoremz#1#2{ \trivlist \item[\hskip \labelsep{\bf #1\ #2}]}
\def\@opargbegintheoremz#1#2#3{ \trivlist
      \item[\hskip \labelsep{\bf #1\ #2\ (#3)}]}
\def\@endtheoremz{\endtrivlist}



% Define Theorem, Lemma, Corollary, Definition and Observation environments

\newtheorem{theorem}{Theorem}%[section]
\newtheorem{lemma}{Lemma}%[section]
\newtheorem{remark}{Remark}%[section]
\newtheorem{assumption}{Assumption}%[section]
\newtheorem{proposition}{Proposition}%[section]
\newtheorem{definition}{Definition}%[section]
\newtheorem{claim}{Claim}%[section]
\newtheorem{corollary}{Corollary}%[section]
\newtheorem{condition}{Condition}%[section]
%\newtheoremz -roman
\newtheorem{observation}{Observation}%[section]
\newtheorem{example}{Example}%[section]
\newtheorem{ruler}{Rule}%[section]
 \def\e{\varepsilon}
\def\tt{\theta}
\def\defi{\stackrel{{\scriptscriptstyle \Delta}}{=}}
\def\bs{{\scriptscriptstyle BS}}
\def\OO{{\scriptscriptstyle O}}
\def\NO{{\scriptscriptstyle NO}}
\def\Cn{C([0,T];\R^n)}
\def\l{l}
\def\N{\mu}
\def\a{\alpha}
\def\d{\delta}
\def\o{\omega}
\def\O{\Omega}
\def\q{q}
\def\Y{{\cal Y}}
\def\F{{\cal F}}
\def\w{\widehat}
\def\Ind{{\,\rm Ind\,}}
\def\col{{\rm col\,}}
\def\sign{{\rm  sign\,}}
\def\mes{{\rm mes\,}}
\def\arctg{{\rm arctg\,}}
\def\diag{{\rm diag\,}}
\def\argmin{\mathop{\rm arg\, min}\,}
\def\argmax{\mathop{\rm arg\, max}\,}
\def\esssup{\mathop{\rm ess\, sup}}
%\def\col{{\rm col \,}}
\def\const{{\rm const\,}}
\def\es{{\rm ess\,}}
\def\Var{{\rm Var\,}}
\def\dist{{\rm dist\,}}
\def\R{{\bf R}}
\def\E{{\bf E}}
\def\P{{\bf P}}
\def\n{{\bf n}}
%\def\S{{\bf S}}
\def\Z{{\cal Z}}
\def\PP{{\cal P}}
\def\J{{\cal J}}
\def\H{{\cal H}}
\def\ttt{{\widehat{\bf  t}}}
\def\h{h}
\def\L{L}
\def\z{{\zeta}}
\def\b{\beta}
\def\s{\delta}
\def\g{\gamma}
\def\C{{\bf C}}
\def\W{{\cal W}^*}
\def\ww{\widetilde}
\def\X{{\cal X}}
\def\t{\theta}
\def\oo{\overline}
\def\s{\sigma}
%def\S{\Sigma}
\def\D{{\Delta}}
\def\p{\partial}
\def\G{\Gamma}
\def\GG{{\cal G}}
\def\U{{\cal U}}
\def\V{{\cal V}}
\def\A{{\cal A}}
\def\M{{\cal M}}
\def\B{{\cal B}}
\def\L{{\cal L}}
\def\I{{\, \cal I}}
\def\h{h}
%\def\YY{{\w Y}}
\def\RO{\stackrel{\circ}{\R}_+}
\def\ROn{\stackrel{\circ\ \ }{\R^n_+}}
\def\LLL{\w{\cal L}}
\def\LL{{\oo{\bf L}}}
\newcommand{\be}{\begin{equation}}
%\newcommand{\l}{\label{#}}
\newcommand{\ee}{\end{equation}}
\newcommand{\bd}{\begin{displaymath}}
\newcommand{\ed}{\end{displaymath}}
%\renewcommand{\theequation}{\thesection.\theneweqn}
\newcommand{\ba}{\bd\begin{array}{rl}}
\newcommand{\ea}{\end{array}\ed}
\font\sm=cmr10
\pagestyle{myheadings}
%\markright{ A new property of absorbed diffusion}
\topmargin=-0.7cm \textheight 245mm \textwidth 16.7cm
\date{\  }
\oddsidemargin=-5mm
\def\L{{\cal L}}
\def\r{\rho}
\def\t{\tau}
\def\c{\zeta}
\def\tt{\theta}
\def\xx{\zeta}
\def\W{{\cal W}}
\def\hh{c}
\def\s{\sigma}
\def\D{{\cal D}}
\def\DD{\oo{\cal D}}
\def\o{\omega}
\def\H{{\cal H}}
\def\Q{{\cal Q}}
\def\CC{{\cal C}}
\def\U{{\cal U}}
\def\R{{\bf R}}
\def\N{{\bf N}}
\def\EE{{\cal E}}
%\renewcommand{\theequation}{\arabic{equation}}
%\csname @addtoreset\endcsname{equation}{section}
%\renewcommand{\theequation}{\arabic{equation}}
%\csname @addtoreset\endcsname{equation}{document}
%\csname @addtoreset\endcsname{equation}{chapter}
\pagestyle{myheadings} \markboth{Theory of Probability and Its
Application, v.31, No 3, pp. 497-498}{Theory of Probability and
Its Application, v.31, No 3, pp. 497-498}
\title{
On first exit times for homogeneous diffusion processes\thanks{
Theory of Probability and Its Application, 1987, v.31, No 3, pp.
497-498; English translation of the September 1986 issue of the
Soviet journal "Teoria Veroyatnostei i ee primneniya"; translated
from Russian by Amal Ellis}}
\author{Nikolai Dokuchaev\thanks{Correspondence address:
The Institute of Mathematics and Mechanics, St.Petersburg State
University, Bibliotechnaya pl.2, Petrodvoretz, St.Petersburg,
198904,  Russia. E-mail: dokuchaev@pobox.spbu.ru; fax
7(812)4286998} }
\begin{document}
\maketitle In this paper we derive an upper bound for the
expression $\E|T_1-T_2|$, first exit times from a region for two
homogeneous diffusion processes. \par Let $(\Omega,{\cal F},\P)$
be a standard probability space and $Q$ be a bounded simply
connected region in $\R^n$. Consider the $n$- dimensional
diffusion processes $y_i(t)$, i=1,2 (see [1, v.3,pp.257-258]) of
the following It\^o equations:
\begin{eqnarray}
\label{(1)}
dy_i(t)=f_i(y(t))dt+\b_i(y(t))dw_t,\\ \label{(2)}
y_i(0)=a_i.
\end{eqnarray}
Here $t\ge 0$, $w_t$ is a standard $d$-dimensional Wiener process,
coordinated as usual with some right-continuous non-decreasing
flow of $\s$-algebras $\F_t\subset \F$; $f_i$ and $\b_i$ are
nonrandom functions with respective values in $\R^n$ and
$\R^{n\times d}$, (here and elsewhere $i=1,2$). The random vectors
$a_i$ are measurable with respect to the $\s$-algebra $\F_0$ and
$a_i\in\bar Q$ with probability 1($\bar Q$ denotes the closure of
the region $Q$ ). All vectors and matrices are real with Euclidean
norm $|\cdot|$.
\par
Consider the r.v.'s $T_i=\inf\{t:\ y_i(t)\notin Q\}$. These are
the first exit times of the processes $y_i(t)$ from the region
$Q$.
\par
Consider in $Q$ the Dirichlet problem
\be
\label{(3)} L_iv_i=-1,\quad v_i|_{\p Q}=0. \ee Here $\p Q$ is the
boundary of $Q$ and the differential operators \be\label{(4)}
L_i=\sum_{j=1}^nf_i^{(j)}\frac{\p}{\p
y^{(j)}}+\frac{1}{2}\sum_{j,k=1}^nb_i^{(j,k)}\frac{\p^2}{\p
y^{(j)}\p y^{(k)}}, \ee where $f_i^{(j)}$, $y^{(j)}$,
$b_i^{(j,k)}$ are the components of the vectors $f_i$, $y$ and the
matrices $b_i(\cdot)=\b_i(\cdot)\b_i(\cdot)^\top$.
\par
We assume that all the components of the function $f$, $\b_i$ are
continuously differentiable, the eigenvalues of the matrices $b_i$
are isolated from zero uniformly in $Q$, and the boundary $\p Q$
is smooth. Problem (3)-(4) then has  a (unique) solution that is
twice continuously differentiable up to the boundary. \par Denote
by $dv_i/dy$ the vector with components $\p v_i/\p y^{(j)}$. \par
{\bf Theorem}.{\it \be \label{(5)} \E |T_1-T_2|\le \max_{i=1,2} \,
\sup_{x\in Q}\left|\frac{d v_i}{d y}(y)\right| \E|y_1(\ww
T)-y_2(\ww T)|, \ee
 where $\ww T=T_1\land T_2=\min(T_1,T_2)$. }
 \par
 {\it Proof}.
 Let $e_1$ and $e_2$ be the indicator functions of the events
 $\{T_1>T_2\}$ and $\{T_2>T_1\}$ respectively.
 Obviously,
 \be
 \label{(6)}
\E|T_1-T_2|=\E\{e_1(T_1-T_2)\}+\E\{e_2(T_2-T_1)\}.
 \ee
 The r.v.'s $e_i$ are measurable with respect to the $\s$-algebras
 $\F_{\ww T}$, $\F_{T_j}$, associated with the Markov times (with
 respect to the flow $\F_t$) $\ww T$, $T_j$, $j=1,2$ (see [1, v.2,
 Chap.4, \S 2]). Using It\^ o's formula, we obtain the equality
 \be
 \label{(7)}
 \begin{array}{ll}
\E\{e_1\{v_1[y_1(T_2)]-v_1[y_1(T_2)]\}\}&=-
\E\{e_1\{v_1[y_1(T_1)]-v_1[y_1(T_2)]\}\}\\
&=-\E\left\{e_1\int_{\ww T}^{T_1}L_1v_1[y_1(t)]dt\right\}
=\E\{e_1(T_1-T_2)\}.
 \end{array}
 \ee
 If we replaced the indices $1,2$ in (7) by 2,1, we get an
 analogous expression for $\E\{e_1(T_1-T_2)\}$. Now the assertion of
 the theorem follows at once from (6).
 \par
 {\bf Example.} In equations (1)-(2) let $n=d=1$,
 $y_i(t)=a_i+w_t$,
 $Q=(0,1)$. Then, obviously, $v_i(y)=y(1-y)$, $dv_i(y)/dy=1-2y$ and
 $\E|T_1-T_2|\le |a_1-a_2|$.
 \par
 The behavior of the first exit times of the process when the
 drift and diffusion coefficients vary has been studied in details
 in the book [2]. However, it solves problems of different sort
 than the one considered above.
 \par
 For non-homogeneous processes and $T_1$, $T_2$ bounded times an
 estimate similar to (5)
 was derived in [3] and used to prove the "stochastic maximum
 principle" established in [4]
 for a controlled diffusion process in a bounded region. For
 strong Markov processes of arbitrary form, an estimate of type
 (5) was derived in [5] (however, the assumptions made there are
 restrictive and not easy to check for nondiffusional processes).
\begin{thebibliography}{10} \bibitem{1}
% [1]
 Gihman I.I. and Skorohod, A.V.
{\it The Theory of Stochastic Processes.} Vol. 2, 1975; vol. 3,
1979.  Springer-Verlag, New York, 1975.\bibitem{2} A.D.Vent-tsel'
and M.I.Freidlin,  {\it Fluctuations in Dynamic Systems under the
Action of Small Random Perturbations,} Nauka, Moscow, 1979 (in
Russian).
\bibitem{3}
Dokuchaev, N. On moment's of first emergency for diffusion-type
processes. Leningrad Univ. Mechanical Bull. (1984), N 1, 58-61.
\par
\bibitem{4}  Dokuchaev, N. Optimal program control of a stochastic process
cutting off on the boundary of a region. Working paper, Dep.
VINITI No. 4705-83, 46 pp., 1983 (in Russian). \par
\bibitem{5}  Dokuchaev, N. On the closeness of the first exit times for diffusion
 processes. Working paper, Dep.
VINITI No. 5398-84, 8 pp., 1984 (in Russian).
\end{thebibliography}
\end{document}\\

Abstract It is given an effective upper estimate of expectation of
|T_1-T_2|, where T_1 and T_2 are the first exit times  from a
region for two vector diffusion processes.

math0102099.extracted_bib
\begin{thebibliography}{10} \bibitem{1}
% [1]
 Gihman I.I. and Skorohod, A.V.
{\it The Theory of Stochastic Processes.} Vol. 2, 1975; vol. 3,
1979.  Springer-Verlag, New York, 1975.\bibitem{2} A.D.Vent-tsel'
and M.I.Freidlin,  {\it Fluctuations in Dynamic Systems under the
Action of Small Random Perturbations,} Nauka, Moscow, 1979 (in
Russian).
\bibitem{3}
Dokuchaev, N. On moment's of first emergency for diffusion-type
processes. Leningrad Univ. Mechanical Bull. (1984), N 1, 58-61.
\par
\bibitem{4}  Dokuchaev, N. Optimal program control of a stochastic process
cutting off on the boundary of a region. Working paper, Dep.
VINITI No. 4705-83, 46 pp., 1983 (in Russian). \par
\bibitem{5}  Dokuchaev, N. On the closeness of the first exit times for diffusion
 processes. Working paper, Dep.
VINITI No. 5398-84, 8 pp., 1984 (in Russian).
\end{thebibliography}