Discrete approximations of strong solutions of reflecting SDEs with discontinuous coefficients

• Autorzy: Semrau A.
• Języki publikacji: EN

Abstrakty

EN

We study Lp convergence for the Euler scheme for stochastic differential equations reflecting on the boundary of a general convex domain D ⊆ Rd. We assume that the equation has the pathwise uniqueness property and its coefficients are measurable and continuous almost everywhere with respect to the Lebesgue measure. In the case D = [0, ∞) new sufficient conditions ensuring pathwise uniqueness for equations with possibly discontinuous coefficients are given.

Słowa kluczowe

EN

stochastic differential equation; reflecting boundary condition; Skorokhod problem; pathwise uniqueness

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2009
• Tom: Vol. 57, no 2
• Strony: 169--180
• Opis fizyczny: Bibliogr. 19 poz.

Twórcy

autor

Semrau A.

• Institute of Mathematics and Physics, University of Technology and Life Sciences, Kaliskiego 7, 85-796 Bydgoszcz, Poland,

Bibliografia

• D. J. Aldous (1978), Stopping time and tightness, Ann. Probab. 6, 335-340.
• P. Billingsley (1999), Convergence of Probability Measures, Wiley, New York.
• R. J. Chitashvili and N. L. Lazrieva (1981), Strong solutions of stochastic differential equations with boundary conditions, Stochastics 5, 225-309.
• I. Gyöngy and N. Krylov (1996), Existence of strong solutions for Itô’s stochastic equations via approximations, Probab. Theory Related Fields 105, 143-158.
• J. Jacod and A. N. Shiryaev (2003), Limit Theorems for Stochastic Processes, Springer, Berlin.
• A. Jakubowski, J. Mémin et G. Pages (1989), Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod, Probab. Theory Related Fields 81, 111-137.
• N. V. Krylov (1982), Controlled Diffusion Processes, Springer, New York.
• J. F. Le Gall (1983), Applications du temps local aux equations différentielles stochastiques unidimensionnelles, in: Sém. de Probab. XVII, Lecture Notes in Math. 986, Springer, Berlin, 15-31.
• D. Lépingle (1995), Euler scheme for reflected stochastic differential equations, Math. Comput. Simulation 38, 119-126.
• S. Nakao (1972), On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations, Osaka J. Math. 9, 513-518.
• R. Pettersson (1995), Approximations for stochastic differential equations with reflecting convex boundaries, Stochastic Process. Appl. 59, 295-308.
• S. Rong (2000), Reflecting stochastic differential equations with jumps and applications, Chapman & Hall/CRC Res. Notes in Math. 408, Chapman &: Hall/CRC.
• A. Rozkosz and L. Słomiński (1997), On stability and existence of solutions of SDEs with reflection at the boundary, Stochastic Process. Appl. 68, 285-302.
• A. Semrau (2007), Euler’s approximations of weak solutions of reflecting SDEs with discontinuous coefficients, Bull. Polish Acad. Sci. Math. 55, 171-182.
• M. A. Shashiashvili (1988), On the variation of the difference of singular components in the Skorokhod problem and on stochastic differential systems in a half-space, Stochastics 24, 151-169.
• L. Słomiński (1994), On approximation of solutions of multidimensional SDEs with reflecting boundary conditions, Stochastic Process. Appl. 50, 197-219.
• L. Słomiński (2001), Euler’s approximations of solutions of SDEs with reflecting boundary, Stochastic Process. Appl. 94, 317-337.
• H. Tanaka (1979), Stochastic differential equations with reflecting boundary condition in convex regions, Hiroshima Math. J. 9, 163-177.
• T. S. Zhang (1994), On the strong solutions of one-dimensional stochastic differential equations with reflecting boundary, Stochastic Process. Appl. 50, 135-147.