Abstract
The notion of weak solution for stochastic differential equation with terminal conditions is introduced. By Girsanov transformation, the equivalence of existence of weak solutions for two-type equations is established. Several sufficient conditions for the existence of the weak solutions for stochastic differential equation with terminal conditions are obtained, and the solution existence condition for this type of equations is relaxed. Finally, an example is given to show that the result is an essential extension of the one under Lipschitz condition ong with respect to (Y,Z).
Similar content being viewed by others
References
Karatzas, I., Shreve, S. E., Brownian Motion and Stochastic Calculus, 2nd ed., New York: Springer-Verlag, 1991.
Huang, Z. Y., Foundations of Stochastic Analysis, Wuhan: Wuhan University Press, 1988.
Skorohod, A. V., Studies in the Theory of Random Procosses, Reading, Mass: Addison Wesley, 1965.
Stroock, D. W., Varadhan, S. R. S., Multidimensional Duffusion Processes, Berlin: Springer, 1979.
Krylov, N. V., Controlled Diffusion Processes, New York: Springer, 1980.
Jeulin, T., Grossisserment d’une filtration et applications, LNM, 1979, 721: 574–609.
Kunita, H., Stochastic Flows and Stochastic Differential Equation, Cambridge: Cambridge Univ. Press, 1990.
Nualart, D., Pardoux, E., Stochastic calculus with anticipating integrands, Probab. Theory and Rel. Fields, 1988, 78: 535–581.
Huang, Z. Y., On the generalized sample solutions of stochastic boundary value problem, Stochastic, 1984, 11: 237–248.
Pardoux, E., Peng, S. G., Adapted solution of a backward stochastic differential equation, System and Control Letters, 1990, 14: 55–61.
Rao, C. R., Linear Statistical Inference and Its Applications, 2nd ed., New York: John Wiley and Sons, Inc., 1973.
Huang, Z. Y., Lin, Q Q., The weak solution for stochastic differential equations with terminal conditions, Applied Math., 1997, 10(3): 56–59.
Peng, S. G., Backward SDE and related g-expectation, in Backward Stochastic Differential Equations (eds. Karoui, E. L., Mazliak, L.), London: Addison-Wesley-Longman, 1997, 141–159.
Karoui, N. E. L., Mazliak, L, Backward Stochastics Differential Equations, Pitman Research Notes in Mathematics, Series 364, London: Addison-Wesley-Longman, 1997.
Ma, J., Yong, J., Forward-Backward Stochastics Differential Equations and Their Applications, LNM, Vol. 1702, New York: Springer, 1999.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Qingquan, L. Weak solution for stochastic differential equations with terminal conditions. Sci. China Ser. A-Math. 45, 1518–1522 (2002). https://doi.org/10.1360/02ys9163
Received:
Issue Date:
DOI: https://doi.org/10.1360/02ys9163