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Stochastic differential equations with reflecting boundary conditions. (English) Zbl 0598.60060

In this paper we solve stochastic differential equations with reflecting boundary conditions by a direct approach based on the Skorokhod problem. Our goal will be two-fold: (i) we give a direct approach to the solution of stochastic differential equations with reflecting boundary conditions (without localization arguments) and (ii) reductions to the case including nonsmooth ones (provided convenient restrictions are assumed).

Let us describe now more precisely the results and methods presented: first, in the case of a normal reflection inside, say, a smooth bounded open set in ${ℝ}^{d}$, we show that for each ${w}_{t}\in C\left(\left[0,\infty \left[;{ℝ}^{d}\right)$ with $w\left(0\right)\in \overline{𝒪}$ there exists a unique solution $\left({x}_{t},{k}_{t}\right)$ of the Skorokhod problem:

$\begin{array}{c}{x}_{t}\in C\left(\left[0,\infty \left[,\overline{𝒪}\right),\phantom{\rule{1.em}{0ex}}{k}_{t}\in C\left(\left[0,\infty \left[;{ℝ}^{d}\right),\phantom{\rule{1.em}{0ex}}{k}_{t}\in BV\left(0,T\right)\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}T<\infty ,\\ {x}_{t}+{k}_{t}={w}_{t}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\ge 0,\\ \phantom{\rule{4pt}{0ex}}\\ {k}_{t}={\int }_{0}^{t}n\left({x}_{s}\right){d|k|}_{s}{,|k|}_{t}={\int }_{0}^{t}{1}_{\left({x}_{s}\in \partial 𝒪\right)}d{|k|}_{s}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}t\ge 0,\end{array}\phantom{\rule{2.em}{0ex}}\left(\mathrm{S}\right)$

where ${k}_{t}$ stands for the total variation of k on [0,t] and n(x) is the unit outward normal to $\partial 𝒪$ at x.

Having solved the deterministic problem (S), we are able to solve stochastic differential equations with reflection along the normal as for example: on some given probability space $\left({\Omega },F,{F}_{t},P,{B}_{t}\right)$ satisfying the usual assumptions, ${B}_{t}$ being an ${F}_{t}$-Brownian motion, find a continuous adapted semimartingale ${X}_{t}$ such that

$\begin{array}{c}{X}_{t}=x+{\int }_{0}^{t}\sigma \left({X}_{s}\right)d{B}_{s}+{\int }_{0}^{t}b\left({X}_{s}\right)ds-{k}_{t},\\ {X}_{t}\in \overline{𝒪}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}t\ge 0,\phantom{\rule{4.pt}{0ex}}\text{a.s.,}\phantom{\rule{4.pt}{0ex}}{k}_{t}\phantom{\rule{4.pt}{0ex}}\text{is}\phantom{\rule{4.pt}{0ex}}\text{a}\phantom{\rule{4.pt}{0ex}}\text{bounded}\phantom{\rule{4.pt}{0ex}}\text{variation}\phantom{\rule{4.pt}{0ex}}\text{process,}\\ {|k|}_{t}={\int }_{0}^{t}{1}_{\left({X}_{s}\in \partial 𝒪\right)}{d|k|}_{s},\phantom{\rule{1.em}{0ex}}{k}_{t}={\int }_{0}^{t}n\left({X}_{s}\right)d{|k|}_{s},\end{array}\phantom{\rule{2.em}{0ex}}\left(\mathrm{SDE}\right)$

for all $t\ge 0$ a.s. We prove that there exists a unique solution $\left({x}_{t},{k}_{t}\right)$ of (SDE) provided, for example, $\sigma$, b are Lipschitz on $\overline{𝒪}$ and $𝒪$ is smooth.

##### MSC:
 60H10 Stochastic ordinary differential equations 60G44 Martingales with continuous parameter 60H25 Random operators and equations