Solvability of Forward-Backward Stochastic Partial Differential Equations with Non-Lipschitz Coefficients
Keywords:
Forward-backward stochastic partial differential equations, Yosida Approximation
Abstract
In this paper we study the solvability of a class of fully-coupled forward-backward stochastic partial differential equations (FBSPDEs). Lipschitz conditions are usually required for the well-posedness of such FBSPDEs. We showed that the Lipschitz conditions can actually be removed by the Yosida Approximation Scheme.
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References
\bibitem{Bismut} Bismut, J. M. {\it Conjugate Convex Functions in Optimal Stochastic
Control}, J. Math. Anal. Apl., 44 (1973) 384--404.
\bibitem{Bismut2} Bismut, J. M. {\it Th\'eorie Probabiliste du Contr\^ole des
Diffusions}, Mem. Amer. Math. Soc. 176, Providence, Rhode Island,
1973.
\bibitem{BuckHu} Buckdahn, R. and Hu, Y. {\it Hedging contingent claims for a large investor in an incomplete
market}, Adv. in Appl. Probab. 30 (1998), no. 1, 239--255.
\bibitem{Chow} Chow, P. L. {\it Stochastic Partial Differential Equations}, Taylor \& Francis Group, Boca Raton, 2007.
\bibitem{Confortola} Confortola, F. {\it Dissipative backward stochastic differential equations in infinite dimensions}, Ph.D. thesis, Politecnico di Milano, Milano, Italy, 2005.
\bibitem{CviMa} Cvitanic, J. and Ma, J. {\it Hedging options for a large investor and forward-backward SDE's},
Ann. Appl. Probab. {\bf 6} (1996), no. 2, 370--398.
\bibitem{CviZh} Cvitanic, J. and Zhang, J. {\it Contract Theory in Continuous-Time Models}, Springer, 2012.
Ann. Appl. Probab. {\bf 6} (1996), no. 2, 370--398.
\bibitem{DZ1} Da Prato, G. and Zabczyk, J., {\it Stochastic Equations in Infinite Dimensions}, Encyclopedia of Mathematics and Its Applications. {\bf 44}, Cambridge University Press, 1992.
\bibitem{DZ2} Da Prato, G. and Zabczyk, J., {\it Ergodicity for Infinite Dimensional Systems}, London Mathematical Society Lecture Note Series. {\bf 229}, Cambridge University Press, 1996.
\bibitem{Hu} Hu, Y. {\it On the solution of forward-backward SDEs with monotone and continuous
coefficients}, Nonlinear Anal., 42 (2000) 1--12.
\bibitem{HMY} Hu, Y., Ma, J., and Yong, J. {\it On semi-linear degenerate backward stochastic partial differential equations}, Probab. Theory Related Fields {\bf 123} (2002), no. 3, 381--411.
\bibitem{HuPeng0} Hu, Y. and Peng, S. {\it Adapted solution of a backward semilinear stochastic evolution equation}, Stochastic Analysis and Applications, vol. 9, no. 4, pp. 445-459, 1991.
\bibitem{KallianpurXiong} Kallianpur, G. and Xiong, J. {\it Stochastic Differential
Equations in Infinite Dimensional Spaces}, IMS
Lecture Notes-Monograph Series, 26, 1995.
\bibitem{LiuMa} Liu, Y. and Ma, J. {\it Optimal Reinsurance/Investment for General
Insurance Models}, Ann. Appl. Probab. 19 (2009), no. 4, 1495--1528
\bibitem{MPY} Ma, J., Protter, P. and Yong, J. {\it Solving Forward-Backward
Stochastic Differential Equations Explicitly--A Four Step
Scheme}, Prob. Th. \& Rel. Fields, 98 (1994), 339--359.
\bibitem{PratoZabczyk1} Da Prato, G. and Zabczyk, J. {\it Stochastic Equations in Infinite Dimensions}, Cambridge University Press, Cambridge, 1992.
\bibitem{PratoZabczyk2} Da Prato, G. and Zabczyk, J. {\it Ergodicity for Infinite Dimensional Systems}, Cambridge University Press, Cambridge, 1996.
\bibitem{MYZ} Ma, J., Yin, H., and Zhang, J., {\it On non-Markovian forward-backward SDEs and Backward
SPDEs}, Stochastic Processes and their Applications, 122 (2012) 3980--4004.
\bibitem{MY2} Ma, J. and Yong, J., {\it Adapted solution of a degenerate
backward spde, with applications}, Stochastic Processes and their Applications
70 (1997), 59--84.
\bibitem{MY3} Ma, J. and Yong, J., {\it On linear, degenerate backward stochastic partial differential equations}, Probab. Theory Related Fields {\bf 113} (1999), no. 2, 135--170.
\bibitem{Pardoux} Pardoux, E., {\it Stochastic partial differential equations and filtering of diffusion processes}, {\sl Stochastics}, 3, 127-167 (1979).
\bibitem{PardouxPeng} Pardoux, E. and Peng, S. {\it Adapted Solution of a Backward Stochastic Differential
Equation}, Systems and Control Letters, 14 (1990) 55--66.
\bibitem{PrevotRockner} Pr\'ev\^ot, C. and R\"ockner, M. {\it A concise Course on Stochastic Partial Differential
Equations}, Springer-Verlag, Berlin, 2007.
\bibitem{Yin} Yin, H. {\it Solvability of Forward-Backward Stochastic Partial Differential Equations}, Stochastic Processes and their Applications, 124 (2014) 2583-2604.
Control}, J. Math. Anal. Apl., 44 (1973) 384--404.
\bibitem{Bismut2} Bismut, J. M. {\it Th\'eorie Probabiliste du Contr\^ole des
Diffusions}, Mem. Amer. Math. Soc. 176, Providence, Rhode Island,
1973.
\bibitem{BuckHu} Buckdahn, R. and Hu, Y. {\it Hedging contingent claims for a large investor in an incomplete
market}, Adv. in Appl. Probab. 30 (1998), no. 1, 239--255.
\bibitem{Chow} Chow, P. L. {\it Stochastic Partial Differential Equations}, Taylor \& Francis Group, Boca Raton, 2007.
\bibitem{Confortola} Confortola, F. {\it Dissipative backward stochastic differential equations in infinite dimensions}, Ph.D. thesis, Politecnico di Milano, Milano, Italy, 2005.
\bibitem{CviMa} Cvitanic, J. and Ma, J. {\it Hedging options for a large investor and forward-backward SDE's},
Ann. Appl. Probab. {\bf 6} (1996), no. 2, 370--398.
\bibitem{CviZh} Cvitanic, J. and Zhang, J. {\it Contract Theory in Continuous-Time Models}, Springer, 2012.
Ann. Appl. Probab. {\bf 6} (1996), no. 2, 370--398.
\bibitem{DZ1} Da Prato, G. and Zabczyk, J., {\it Stochastic Equations in Infinite Dimensions}, Encyclopedia of Mathematics and Its Applications. {\bf 44}, Cambridge University Press, 1992.
\bibitem{DZ2} Da Prato, G. and Zabczyk, J., {\it Ergodicity for Infinite Dimensional Systems}, London Mathematical Society Lecture Note Series. {\bf 229}, Cambridge University Press, 1996.
\bibitem{Hu} Hu, Y. {\it On the solution of forward-backward SDEs with monotone and continuous
coefficients}, Nonlinear Anal., 42 (2000) 1--12.
\bibitem{HMY} Hu, Y., Ma, J., and Yong, J. {\it On semi-linear degenerate backward stochastic partial differential equations}, Probab. Theory Related Fields {\bf 123} (2002), no. 3, 381--411.
\bibitem{HuPeng0} Hu, Y. and Peng, S. {\it Adapted solution of a backward semilinear stochastic evolution equation}, Stochastic Analysis and Applications, vol. 9, no. 4, pp. 445-459, 1991.
\bibitem{KallianpurXiong} Kallianpur, G. and Xiong, J. {\it Stochastic Differential
Equations in Infinite Dimensional Spaces}, IMS
Lecture Notes-Monograph Series, 26, 1995.
\bibitem{LiuMa} Liu, Y. and Ma, J. {\it Optimal Reinsurance/Investment for General
Insurance Models}, Ann. Appl. Probab. 19 (2009), no. 4, 1495--1528
\bibitem{MPY} Ma, J., Protter, P. and Yong, J. {\it Solving Forward-Backward
Stochastic Differential Equations Explicitly--A Four Step
Scheme}, Prob. Th. \& Rel. Fields, 98 (1994), 339--359.
\bibitem{PratoZabczyk1} Da Prato, G. and Zabczyk, J. {\it Stochastic Equations in Infinite Dimensions}, Cambridge University Press, Cambridge, 1992.
\bibitem{PratoZabczyk2} Da Prato, G. and Zabczyk, J. {\it Ergodicity for Infinite Dimensional Systems}, Cambridge University Press, Cambridge, 1996.
\bibitem{MYZ} Ma, J., Yin, H., and Zhang, J., {\it On non-Markovian forward-backward SDEs and Backward
SPDEs}, Stochastic Processes and their Applications, 122 (2012) 3980--4004.
\bibitem{MY2} Ma, J. and Yong, J., {\it Adapted solution of a degenerate
backward spde, with applications}, Stochastic Processes and their Applications
70 (1997), 59--84.
\bibitem{MY3} Ma, J. and Yong, J., {\it On linear, degenerate backward stochastic partial differential equations}, Probab. Theory Related Fields {\bf 113} (1999), no. 2, 135--170.
\bibitem{Pardoux} Pardoux, E., {\it Stochastic partial differential equations and filtering of diffusion processes}, {\sl Stochastics}, 3, 127-167 (1979).
\bibitem{PardouxPeng} Pardoux, E. and Peng, S. {\it Adapted Solution of a Backward Stochastic Differential
Equation}, Systems and Control Letters, 14 (1990) 55--66.
\bibitem{PrevotRockner} Pr\'ev\^ot, C. and R\"ockner, M. {\it A concise Course on Stochastic Partial Differential
Equations}, Springer-Verlag, Berlin, 2007.
\bibitem{Yin} Yin, H. {\it Solvability of Forward-Backward Stochastic Partial Differential Equations}, Stochastic Processes and their Applications, 124 (2014) 2583-2604.
Published
2021-04-29
How to Cite
Yin, H. (2021). Solvability of Forward-Backward Stochastic Partial Differential Equations with Non-Lipschitz Coefficients. Journal of Progressive Research in Mathematics, 18(1), 87-98. Retrieved from http://www.scitecresearch.com/journals/index.php/jprm/article/view/2040
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