×

zbMATH — the first resource for mathematics

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 \({\mathbb{R}}^ d\), we show that for each \(w_ t\in C([0,\infty [;{\mathbb{R}}^ d)\) with \(w(0)\in \bar {\mathcal O}\) there exists a unique solution \((x_ t,k_ t)\) of the Skorokhod problem: \[ \begin{gathered} x_ t\in C([0,\infty[, \bar{\mathcal O}),\quad k_ t\in C([0,\infty[; {\mathbb{R}}^ d), \quad k_ t\in BV(0,T) \text{ for all } T<\infty, \\ x_ t+k_ t=w_ t \text{ for all } t\geq 0, \\ \;\\ k_ t=\int^{t}_{0}n(x_ s)d| k|_ s, | k|_ t=\int^{t}_{0}1_{(x_ s\in \partial{\mathcal O})}d| k|_ s \text{ for }t\geq 0, \end{gathered} \tag{S} \] where \(k_ t\) stands for the total variation of k on [0,t] and n(x) is the unit outward normal to \(\partial {\mathcal O}\) 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 \((\Omega,F,F_ t,P,B_ t)\) satisfying the usual assumptions, \(B_ t\) being an \(F_ t\)-Brownian motion, find a continuous adapted semimartingale \(X_ t\) such that \[ \begin{gathered} X_ t=x+\int^{t}_{0}\sigma (X_ s)dB_ s+\int^{t}_{0}b(X_ s)ds-k_ t, \\ X_ t\in \bar {\mathcal O}\text{ for all }t\geq 0,\text{ a.s., \(k_ t\) is a bounded variation process,} \\ | k|_ t=\int^{t}_{0} 1_{(X_ s\in \partial {\mathcal O})} d| k|_ s,\quad k_ t=\int^{t}_{0}n(X_ s)d| k|_ s, \end{gathered}\tag{SDE} \] for all \(t\geq 0\) a.s. We prove that there exists a unique solution \((x_ t,k_ t)\) of (SDE) provided, for example, \(\sigma\), b are Lipschitz on \(\bar {\mathcal O}\) and \({\mathcal O}\) is smooth.

MSC:
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60G44 Martingales with continuous parameter
60H25 Random operators and equations (aspects of stochastic analysis)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Anderson, Nagoya Math. J. 60 pp 189– (1976) · Zbl 0324.60063
[2] and , Controle Impulsionnel et Inéquations Quasi-Variationnelles, Dunod, Paris, 1982.
[3] and , Equations différentielles stochastiques, In Séminaire de probabilité XI, Lecture Notes in Math., Springer, Berlin, 1977.
[4] Doss, Z für W. Theorie 61 3 pp 327– (1982)
[5] and , Petites perturbations de systèmes dynamiques avec réflexion, Séminaire de probabilité XVII, Lecture Notes in Math. 986, Springer, Berlin.
[6] Processus de réflexion sur RN, Séminaire de Probabilité IX, Lecture Notes in Math. n{\(\deg\)} 465, Springer, Berlin, 1975.
[7] and , Un probléme de réflexion et ses applications au temps local et aux équations différentielles stochastiques sur R, Cas continu, Temps Locaux, Astérisque 52–53, 1978, pp. 117–144.
[8] El Karoui, Ann. of Prob. 8 pp 1049– (1980)
[9] and , Elliptic Partial Differential Equations of Second-Order, Springer, Berlin, 1977.
[10] and , Stochastic Differential Equations and Diffusion Processes, North-Holland, Amsterdam, 1981.
[11] Krylov, Izv. Acad. Nauk. SSSR 37 pp 691– (1973)
[12] Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 1: The dynamic programming principle and applications, preprint.
[13] Optimal control of diffusion processes and Hamilton-Jacobi-Bellman equations. Part 3: Regularity of the optimal cost function, in Nonlinear Partial Differential Equations and their applications, Collège de France Seminar, Vol. IV, Pitman, London, 1983.
[14] Lions, Comptes-Rendus Paris 292 pp 559– (1981)
[15] McKean, J. Math. Kyoto Univ. 3 pp 86– (1963)
[16] Stochastic Integrals, Academic, New York, 1969. · Zbl 0191.46603
[17] Serrin, Phil. Trans. Roy. Soc. London 265 pp 413– (1969)
[18] Skorokhod, Theor. Veroyatnost. i Primenen 6 pp 264– (1961)
[19] Theor. Veroyatnost. i Primenen 7 pp 3– (1962)
[20] Stroock, Comm. Pure Appl. Math. 24 pp 147– (1971)
[21] and , Multidimensional Diffusion Processes, Springer, Berlin, 1979. · Zbl 0426.60069
[22] Tanaka, Hiroshima Math. J. 9 pp 163– (1979)
[23] Watanabe, J. Math. Kyoto Univ. 11 pp 169– (1971)
[24] Watanabe, J. Math. Kyoto Univ. 11 pp 155– (1971)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.