Discrete approximations of reflected backward stochastic differential equations with random terminal time

• Autorzy: Jańczak K.
• Języki publikacji: EN

Abstrakty

EN

We study convergence of discrete approximations of reflected backward stochastic differential equations with random terminal time in a general convex domain. Applications to investigation of the viability property for backward stochastic differential equations and to obstacle problem for partial differential equations are given.

Słowa kluczowe

EN

stochastic differential equations; reflected backward stochastic differential equations; random terminal time; discrete approximation methods

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2008
• Tom: Vol. 28, Fasc. 1
• Strony: 41--74
• Opis fizyczny: Bibliogr. 23 poz.

Twórcy

autor

Jańczak K.

• Institute of Mathematics and Physics, University of Technology and Life Sciences, ul. Kaliskiego 7, 85-225 Bydgoszcz, Poland

Bibliografia

• [1] V. Bally and G. Pages, Error analysis of the optimal quantization algorithm for obstacle problems, Stochastic Process. Appl. 106 (2003), pp. 1-40.
• [2] P. Briand, B. Delyon and J. Mémin, Donsker-type theorem for BSDEs, Electron. Comm. Probab. 6 (2001), pp. 1-14.
• [3] P. Briand, B. Delyon and J. Mémin, On the robustness of backward stochastic differential equations, Stochastic Process. Appl. 97 (2002), pp. 229-253.
• [4] R. Buckdahn, M. Quincampoix and A. Rascanu, Viability property for a backward stochastic differential equations and applications to partial differential equations, Probab. Theory Related Fields 116 (2000), pp. 485-504.
• [5] K.L. Chung and Z.X. Zhao, From Brownian motion to Schrödinger’s equation, Grundlehren Math. Wiss. 312 (1995).
• [6] F. Coquet, J. Mémin and L. Słomiński, On weak convergence of filtrations, Lecture Notes in Math. No 1755, Springer, 2001, pp. 306-328.
• [7] M. G. Crandall, H. Ishii and P. L. Lions, User’s guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. 27 (1992), pp. 1-67.
• [8] A. Gegout-Petit and É. Pardoux, Equations différentielles stochastiques retrogrades réfléchies dans un convexe, Stoch. Stoch. Rep. 57 (1996), pp. 111-128.
• [9] J. Jacod, Convergence en loi de semimartingales et variation quadratique, Lecture Notes in Math. No 850, Springer, 1981, pp. 547-560.
• [10] J. Jacod and A.N. Shiryayev, Limit Theorems for Stochastic Processes, Springer, Berlin 1987.
• [11] A. Jakubowski, J. Mémin and G. Pagès, Convergence en loi des suites d’intégrales stochastiques sur l’espace D1 de Skorokhod, Probab. Theory Related Fields 81 (1989), pp. 111-137.
• [12] N. Kazamaki, Continuous expotential martingales and BMO, Lecture Notes in Math. No 1579, Springer, 1994.
• [13] J. Ma, P. Protter, J. San Martin and S. Torres, Numerical method for backward stochastic differential equations, Ann. Appl. Probab. 12 (2002), pp. 302-316.
• [14] J. Mémin, S. Peng and M. Xu, Convergence of solutions of discrete reflected backward SDE’s with numerical simulations, preprint.
• [15] É. Pardoux, Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in: Stochastic Analysis and Related Topics, VI (Geilo, 1996), pp. 79-127.
• [16] É. Pardoux and A. Răşcanu, Backward stochastic differential equations with subdifferential operator and related variational inequalities, Stochastic Process. Appl. 76 (1998), pp. 191-215.
• [17] L. Słomiński, Stability of strong solutions of stochastic differential equations, Stochastic Process. Appl. 31 (1989), pp. 173-202.
• [18] L. Słomiński, Euler’s approximations of solutions of SDE’s with reflecting boundary, Stochastic Process. Appl. 94 (2001), pp. 317-337.
• [19] D. Stroock and S. Varadhan, Multidimensional Diffusion Processes, Springer, Berlin 1979.
• [20] S. Toldo, Convergence de filtrations; application `a la discrétisation de processus et à la stability ´e de temps d’arrêt, PhD Thesis, University of Rennes I, 2005.
• [21] S. Toldo, Stability of solutions of BSDEs with random terminal time, ESAIM Probab. Stat. 10 (2006), pp. 141-163.
• [22] D. Williams, Probability with Martingales, Cambridge University Press, 1991.
• [23] Z. Zheng, Reflected BSDEs with random terminal time and applications. Part II: Variational inequalities, preprint.