# American Institute of Mathematical Sciences

September  2015, 5(3): 401-434. doi: 10.3934/mcrf.2015.5.401

## Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process

 1 Institute of Mathematical Finance and Department of Finance and Control Sciences, School of Mathematical Science, Fudan University, Shanghai 200433, China 2 Institute of Mathematical Finance and Department of Finance and Control Sciences, School of Mathematical Sciences, Fudan University, Shanghai 200433

Received  July 2014 Revised  January 2015 Published  July 2015

In this paper we investigate classical solution of a semi-linear system of backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. By proving an Itô-Wentzell formula for jump diffusions as well as an abstract result of stochastic evolution equations, we obtain the stochastic integral partial differential equation for the inverse of the stochastic flow generated by a stochastic differential equation driven by a Brownian motion and a Poisson point process. By composing the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow, we construct the classical solution of the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.
Citation: Shaokuan Chen, Shanjian Tang. Semi-linear backward stochastic integral partial differential equations driven by a Brownian motion and a Poisson point process. Mathematical Control and Related Fields, 2015, 5 (3) : 401-434. doi: 10.3934/mcrf.2015.5.401
##### References:
 [1] S. Albeverio, J. Wu and T. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 74 (1998), 21-36. doi: 10.1016/S0304-4149(97)00112-9. [2] G. Barles, R. Buckdhan and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastics Reports, 60 (1997), 57-83. doi: 10.1080/17442509708834099. [3] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511526503. [4] J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444. doi: 10.1137/0314028. [5] Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of BSDEs, Stochastic Process. Appl., 108 (2003), 109-129. doi: 10.1016/S0304-4149(03)00089-9. [6] R. Buckdahn and E. Pardoux, BSDE's with jumps and associated integral-stochastic differential equations, preprint. [7] F. Delbaen and S. Tang, Harmonic analysis of stochastic equations and backward stochastic differential equations, Probab. Theory Relat. Fields, 146 (2010), 291-336. doi: 10.1007/s00440-008-0191-5. [8] K. Du and S. Tang, On the dirichlet problem for backward parabolic stochastic partial differential equations in general smooth domains, http://arxiv.org/abs/0910.2289. [9] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in Finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [10] N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs, SIAM J. Control Optim., 48 (2009), 481-520. doi: 10.1137/070686998. [11] T. Fujiwara and H. Kunita, Stochastic differentail equations of jump type and Lévy flows in diffeomorphisms group, J. Math. Kyoto Univ., 25 (1985), 71-106. [12] I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales II, Itô formula in Banach spaces, Stochastics, 6 (1982), 153-173. [13] E. Hausenblas, Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), 1496-1546. doi: 10.1214/EJP.v10-297. [14] E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results, Probab. Theory Relat. Fields, 137 (2007), 161-200. doi: 10.1007/s00440-006-0501-8. [15] Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), 381-411. doi: 10.1007/s004400100193. [16] Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations, Stoch. Anal. Appl., 9 (1991), 445-459. doi: 10.1080/07362999108809250. [17] N. V. Krylov, On the Itô-Wentzell formula for distribution-valued processes and related topics, Probab. Theory Relat. Fields, 150 (2011), 295-319. doi: 10.1007/s00440-010-0275-x. [18] N. V. Krylov and B. L. Rozovskii, The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1329-1347, 1448. [19] N. V. Krylov and B. L. Rozovskii, Characteristics of degenerating second-order parabolic Itô equations, Trudy seminara imeni Petrovskogo, 8 (1982), 153-168 (Russian); English translation: J. Soviet Math., 32 (1986), 336-348. [20] N. V. Krylov and B. L. Rozovskii, Stochastic partial differential equations and diffusion processes, Uspekhi Mat. Nauk, 37 (1982), 75-95 (Russian); English translation: Russian Math. Surveys, 37 (1982), 81-105. [21] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, Vol. 24, Cambridge University Press, Cambridge, 1997. [22] H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in M. Rao (ed.) Real and Stochastic Analysis, Birkhäuser, Basel, (2004), 305-373. [23] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Process. Appl., 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4. [24] J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205. [25] Q. Meng and S. Tang, Backward Stochastic HJB Equations with Jumps, preprint. [26] L. Mytnik, Stochastic partial differential equation driven by stable noise, Probab. Theory Relat. Fields, 123 (2002), 157-201. doi: 10.1007/s004400100180. [27] B, Øksendal, F. Proske and T. Zhang, Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields, Stochastics, 77 (2005), 381-399. doi: 10.1080/17442500500213797. [28] B. Øksendal and T. Zhang, The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures, Osaka J. Math., 44 (2007), 207-230. [29] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142. [30] E. Pardoux and S. Peng, Adapted solution of backward stochastic equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [31] E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, in Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci., B.L.Rozovskii and R.S. Sowers, eds., Springer, Berlin, Heidelberg, New York, 176 (1992), 200-217. doi: 10.1007/BFb0007334. [32] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30 (1992), 284-304. doi: 10.1137/0330018. [33] P. Protter, Stochastic Integration and Differential Equations (second edition), Springer, 2004. [34] M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Analysis, 26 (2007), 255-279. doi: 10.1007/s11118-006-9035-z. [35] B. L. Rozovskii, Stochastic Evolution Systems, Kluwer, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7. [36] A.-S. Sznitman, Martingales dépendant d'un paramètre: Une formule d'Ito, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60 (1982), 41-70. doi: 10.1007/BF01957096. [37] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475. doi: 10.1137/S0363012992233858. [38] S. Tang, The maximum pinciple for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617. doi: 10.1137/S0363012996313100. [39] S. Tang, A New Partially Observed Stochastic Maximum Principle, In Proceedings of 37th IEEE Control and Decision Conference, Tampa, Florida, (1998), 2353-2358. [40] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbb{R}^{N}$, Chinese Ann. Math. Ser. B, 26 (2005), 437-456. doi: 10.1142/S025295990500035X. [41] J. B. Walsh, An introduction to stochastic partial differential equations, In Ecole d'Été de Probabilités de St. Flour XIV, Lecture Notes in Mathematics, Springer, Berlin, 1180 (1986), 265-439. doi: 10.1007/BFb0074920. [42] J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, 1999. doi: 10.1007/978-1-4612-1466-3. [43] X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stochastics and Dynamics, 9 (2009), 549-595. doi: 10.1142/S0219493709002774. [44] X. Zhou, A duality analysis on stochastic partial differential equations, J. Funct. Anal., 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y. [45] X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1992), 1462-1478. doi: 10.1137/0331068.

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##### References:
 [1] S. Albeverio, J. Wu and T. Zhang, Parabolic SPDEs driven by Poisson white noise, Stochastic Process. Appl., 74 (1998), 21-36. doi: 10.1016/S0304-4149(97)00112-9. [2] G. Barles, R. Buckdhan and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations, Stochastics and Stochastics Reports, 60 (1997), 57-83. doi: 10.1080/17442509708834099. [3] A. Bensoussan, Stochastic Control of Partially Observable Systems, Cambridge University Press, Cambridge, 1992. doi: 10.1017/CBO9780511526503. [4] J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients, SIAM J. Control Optim., 14 (1976), 419-444. doi: 10.1137/0314028. [5] Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of BSDEs, Stochastic Process. Appl., 108 (2003), 109-129. doi: 10.1016/S0304-4149(03)00089-9. [6] R. Buckdahn and E. Pardoux, BSDE's with jumps and associated integral-stochastic differential equations, preprint. [7] F. Delbaen and S. Tang, Harmonic analysis of stochastic equations and backward stochastic differential equations, Probab. Theory Relat. Fields, 146 (2010), 291-336. doi: 10.1007/s00440-008-0191-5. [8] K. Du and S. Tang, On the dirichlet problem for backward parabolic stochastic partial differential equations in general smooth domains, http://arxiv.org/abs/0910.2289. [9] N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in Finance, Math. Finance, 7 (1997), 1-71. doi: 10.1111/1467-9965.00022. [10] N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs, SIAM J. Control Optim., 48 (2009), 481-520. doi: 10.1137/070686998. [11] T. Fujiwara and H. Kunita, Stochastic differentail equations of jump type and Lévy flows in diffeomorphisms group, J. Math. Kyoto Univ., 25 (1985), 71-106. [12] I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales II, Itô formula in Banach spaces, Stochastics, 6 (1982), 153-173. [13] E. Hausenblas, Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure, Electron. J. Probab., 10 (2005), 1496-1546. doi: 10.1214/EJP.v10-297. [14] E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results, Probab. Theory Relat. Fields, 137 (2007), 161-200. doi: 10.1007/s00440-006-0501-8. [15] Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 123 (2002), 381-411. doi: 10.1007/s004400100193. [16] Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations, Stoch. Anal. Appl., 9 (1991), 445-459. doi: 10.1080/07362999108809250. [17] N. V. Krylov, On the Itô-Wentzell formula for distribution-valued processes and related topics, Probab. Theory Relat. Fields, 150 (2011), 295-319. doi: 10.1007/s00440-010-0275-x. [18] N. V. Krylov and B. L. Rozovskii, The Cauchy problem for linear stochastic partial differential equations, Izv. Akad. Nauk SSSR Ser. Mat., 41 (1977), 1329-1347, 1448. [19] N. V. Krylov and B. L. Rozovskii, Characteristics of degenerating second-order parabolic Itô equations, Trudy seminara imeni Petrovskogo, 8 (1982), 153-168 (Russian); English translation: J. Soviet Math., 32 (1986), 336-348. [20] N. V. Krylov and B. L. Rozovskii, Stochastic partial differential equations and diffusion processes, Uspekhi Mat. Nauk, 37 (1982), 75-95 (Russian); English translation: Russian Math. Surveys, 37 (1982), 81-105. [21] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge Studies in Advanced Mathematics, Vol. 24, Cambridge University Press, Cambridge, 1997. [22] H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms, in M. Rao (ed.) Real and Stochastic Analysis, Birkhäuser, Basel, (2004), 305-373. [23] J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications, Stochastic Process. Appl., 70 (1997), 59-84. doi: 10.1016/S0304-4149(97)00057-4. [24] J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations, Probab. Theory Related Fields, 113 (1999), 135-170. doi: 10.1007/s004400050205. [25] Q. Meng and S. Tang, Backward Stochastic HJB Equations with Jumps, preprint. [26] L. Mytnik, Stochastic partial differential equation driven by stable noise, Probab. Theory Relat. Fields, 123 (2002), 157-201. doi: 10.1007/s004400100180. [27] B, Øksendal, F. Proske and T. Zhang, Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields, Stochastics, 77 (2005), 381-399. doi: 10.1080/17442500500213797. [28] B. Øksendal and T. Zhang, The Itô-Ventzell formula and forward stochastic differential equations driven by Poisson random measures, Osaka J. Math., 44 (2007), 207-230. [29] E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes, Stochastics, 3 (1979), 127-167. doi: 10.1080/17442507908833142. [30] E. Pardoux and S. Peng, Adapted solution of backward stochastic equation, Systems Control Lett., 14 (1990), 55-61. doi: 10.1016/0167-6911(90)90082-6. [31] E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations, in Stochastic Partial Differential Equations and Their Applications, Lecture Notes in Control and Inform. Sci., B.L.Rozovskii and R.S. Sowers, eds., Springer, Berlin, Heidelberg, New York, 176 (1992), 200-217. doi: 10.1007/BFb0007334. [32] S. Peng, Stochastic Hamilton-Jacobi-Bellman equations, SIAM J. Control Optim., 30 (1992), 284-304. doi: 10.1137/0330018. [33] P. Protter, Stochastic Integration and Differential Equations (second edition), Springer, 2004. [34] M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles, Potential Analysis, 26 (2007), 255-279. doi: 10.1007/s11118-006-9035-z. [35] B. L. Rozovskii, Stochastic Evolution Systems, Kluwer, Dordrecht, 1990. doi: 10.1007/978-94-011-3830-7. [36] A.-S. Sznitman, Martingales dépendant d'un paramètre: Une formule d'Ito, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60 (1982), 41-70. doi: 10.1007/BF01957096. [37] S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps, SIAM J. Control Optim., 32 (1994), 1447-1475. doi: 10.1137/S0363012992233858. [38] S. Tang, The maximum pinciple for partially observed optimal control of stochastic differential equations, SIAM J. Control Optim., 36 (1998), 1596-1617. doi: 10.1137/S0363012996313100. [39] S. Tang, A New Partially Observed Stochastic Maximum Principle, In Proceedings of 37th IEEE Control and Decision Conference, Tampa, Florida, (1998), 2353-2358. [40] S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbb{R}^{N}$, Chinese Ann. Math. Ser. B, 26 (2005), 437-456. doi: 10.1142/S025295990500035X. [41] J. B. Walsh, An introduction to stochastic partial differential equations, In Ecole d'Été de Probabilités de St. Flour XIV, Lecture Notes in Mathematics, Springer, Berlin, 1180 (1986), 265-439. doi: 10.1007/BFb0074920. [42] J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations, Springer, 1999. doi: 10.1007/978-1-4612-1466-3. [43] X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients, Stochastics and Dynamics, 9 (2009), 549-595. doi: 10.1142/S0219493709002774. [44] X. Zhou, A duality analysis on stochastic partial differential equations, J. Funct. Anal., 103 (1992), 275-293. doi: 10.1016/0022-1236(92)90122-Y. [45] X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations, SIAM J. Control Optim., 31 (1992), 1462-1478. doi: 10.1137/0331068.
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