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 & 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.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar

[2]

G. Barles, R. Buckdhan and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations,, Stochastics and Stochastics Reports, 60 (1997), 57.  doi: 10.1080/17442509708834099.  Google Scholar

[3]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  Google Scholar

[4]

J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients,, SIAM J. Control Optim., 14 (1976), 419.  doi: 10.1137/0314028.  Google Scholar

[5]

Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of BSDEs,, Stochastic Process. Appl., 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

[6]

R. Buckdahn and E. Pardoux, BSDE's with jumps and associated integral-stochastic differential equations,, preprint., ().   Google Scholar

[7]

F. Delbaen and S. Tang, Harmonic analysis of stochastic equations and backward stochastic differential equations,, Probab. Theory Relat. Fields, 146 (2010), 291.  doi: 10.1007/s00440-008-0191-5.  Google Scholar

[8]

K. Du and S. Tang, On the dirichlet problem for backward parabolic stochastic partial differential equations in general smooth domains,, , ().   Google Scholar

[9]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in Finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[10]

N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs,, SIAM J. Control Optim., 48 (2009), 481.  doi: 10.1137/070686998.  Google Scholar

[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.   Google Scholar

[12]

I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales II,, Itô formula in Banach spaces, 6 (1982), 153.   Google Scholar

[13]

E. Hausenblas, Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure,, Electron. J. Probab., 10 (2005), 1496.  doi: 10.1214/EJP.v10-297.  Google Scholar

[14]

E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results,, Probab. Theory Relat. Fields, 137 (2007), 161.  doi: 10.1007/s00440-006-0501-8.  Google Scholar

[15]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 123 (2002), 381.  doi: 10.1007/s004400100193.  Google Scholar

[16]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar

[17]

N. V. Krylov, On the Itô-Wentzell formula for distribution-valued processes and related topics,, Probab. Theory Relat. Fields, 150 (2011), 295.  doi: 10.1007/s00440-010-0275-x.  Google Scholar

[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.   Google Scholar

[19]

N. V. Krylov and B. L. Rozovskii, Characteristics of degenerating second-order parabolic Itô equations,, Trudy seminara imeni Petrovskogo, 8 (1982), 153.   Google Scholar

[20]

N. V. Krylov and B. L. Rozovskii, Stochastic partial differential equations and diffusion processes,, Uspekhi Mat. Nauk, 37 (1982), 75.   Google Scholar

[21]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge Studies in Advanced Mathematics, (1997).   Google Scholar

[22]

H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms,, in M. Rao (ed.) Real and Stochastic Analysis, (2004), 305.   Google Scholar

[23]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Process. Appl., 70 (1997), 59.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar

[24]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 113 (1999), 135.  doi: 10.1007/s004400050205.  Google Scholar

[25]

Q. Meng and S. Tang, Backward Stochastic HJB Equations with Jumps,, preprint., ().   Google Scholar

[26]

L. Mytnik, Stochastic partial differential equation driven by stable noise,, Probab. Theory Relat. Fields, 123 (2002), 157.  doi: 10.1007/s004400100180.  Google Scholar

[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.  doi: 10.1080/17442500500213797.  Google Scholar

[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.   Google Scholar

[29]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127.  doi: 10.1080/17442507908833142.  Google Scholar

[30]

E. Pardoux and S. Peng, Adapted solution of backward stochastic equation,, Systems Control Lett., 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[31]

E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations,, in Stochastic Partial Differential Equations and Their Applications, 176 (1992), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[32]

S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284.  doi: 10.1137/0330018.  Google Scholar

[33]

P. Protter, Stochastic Integration and Differential Equations (second edition),, Springer, (2004).   Google Scholar

[34]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles,, Potential Analysis, 26 (2007), 255.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[35]

B. L. Rozovskii, Stochastic Evolution Systems, Kluwer,, Dordrecht, (1990).  doi: 10.1007/978-94-011-3830-7.  Google Scholar

[36]

A.-S. Sznitman, Martingales dépendant d'un paramètre: Une formule d'Ito,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60 (1982), 41.  doi: 10.1007/BF01957096.  Google Scholar

[37]

S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps,, SIAM J. Control Optim., 32 (1994), 1447.  doi: 10.1137/S0363012992233858.  Google Scholar

[38]

S. Tang, The maximum pinciple for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596.  doi: 10.1137/S0363012996313100.  Google Scholar

[39]

S. Tang, A New Partially Observed Stochastic Maximum Principle,, In Proceedings of 37th IEEE Control and Decision Conference, (1998), 2353.   Google Scholar

[40]

S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 26 (2005), 437.  doi: 10.1142/S025295990500035X.  Google Scholar

[41]

J. B. Walsh, An introduction to stochastic partial differential equations,, In Ecole d'Été de Probabilités de St. Flour XIV, 1180 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[42]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[43]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients,, Stochastics and Dynamics, 9 (2009), 549.  doi: 10.1142/S0219493709002774.  Google Scholar

[44]

X. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar

[45]

X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1992), 1462.  doi: 10.1137/0331068.  Google Scholar

show all references

References:
[1]

S. Albeverio, J. Wu and T. Zhang, Parabolic SPDEs driven by Poisson white noise,, Stochastic Process. Appl., 74 (1998), 21.  doi: 10.1016/S0304-4149(97)00112-9.  Google Scholar

[2]

G. Barles, R. Buckdhan and E. Pardoux, Backward stochastic differential equations and integral-partial differential equations,, Stochastics and Stochastics Reports, 60 (1997), 57.  doi: 10.1080/17442509708834099.  Google Scholar

[3]

A. Bensoussan, Stochastic Control of Partially Observable Systems,, Cambridge University Press, (1992).  doi: 10.1017/CBO9780511526503.  Google Scholar

[4]

J. M. Bismut, Linear quadratic optimal stochastic control with random coefficients,, SIAM J. Control Optim., 14 (1976), 419.  doi: 10.1137/0314028.  Google Scholar

[5]

Ph. Briand, B. Delyon, Y. Hu, E. Pardoux and L. Stoica, $L^p$ solutions of BSDEs,, Stochastic Process. Appl., 108 (2003), 109.  doi: 10.1016/S0304-4149(03)00089-9.  Google Scholar

[6]

R. Buckdahn and E. Pardoux, BSDE's with jumps and associated integral-stochastic differential equations,, preprint., ().   Google Scholar

[7]

F. Delbaen and S. Tang, Harmonic analysis of stochastic equations and backward stochastic differential equations,, Probab. Theory Relat. Fields, 146 (2010), 291.  doi: 10.1007/s00440-008-0191-5.  Google Scholar

[8]

K. Du and S. Tang, On the dirichlet problem for backward parabolic stochastic partial differential equations in general smooth domains,, , ().   Google Scholar

[9]

N. El Karoui, S. Peng and M. C. Quenez, Backward stochastic differential equations in Finance,, Math. Finance, 7 (1997), 1.  doi: 10.1111/1467-9965.00022.  Google Scholar

[10]

N. Englezos and I. Karatzas, Utility maximization with habit formation: Dynamic programming and stochastic PDEs,, SIAM J. Control Optim., 48 (2009), 481.  doi: 10.1137/070686998.  Google Scholar

[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.   Google Scholar

[12]

I. Gyöngy and N. V. Krylov, On stochastic equations with respect to semimartingales II,, Itô formula in Banach spaces, 6 (1982), 153.   Google Scholar

[13]

E. Hausenblas, Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure,, Electron. J. Probab., 10 (2005), 1496.  doi: 10.1214/EJP.v10-297.  Google Scholar

[14]

E. Hausenblas, SPDEs driven by Poisson random measure with non Lipschitz coefficients: existence results,, Probab. Theory Relat. Fields, 137 (2007), 161.  doi: 10.1007/s00440-006-0501-8.  Google Scholar

[15]

Y. Hu, J. Ma and J. Yong, On semi-linear degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 123 (2002), 381.  doi: 10.1007/s004400100193.  Google Scholar

[16]

Y. Hu and S. Peng, Adapted solution of a backward semilinear stochastic evolution equations,, Stoch. Anal. Appl., 9 (1991), 445.  doi: 10.1080/07362999108809250.  Google Scholar

[17]

N. V. Krylov, On the Itô-Wentzell formula for distribution-valued processes and related topics,, Probab. Theory Relat. Fields, 150 (2011), 295.  doi: 10.1007/s00440-010-0275-x.  Google Scholar

[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.   Google Scholar

[19]

N. V. Krylov and B. L. Rozovskii, Characteristics of degenerating second-order parabolic Itô equations,, Trudy seminara imeni Petrovskogo, 8 (1982), 153.   Google Scholar

[20]

N. V. Krylov and B. L. Rozovskii, Stochastic partial differential equations and diffusion processes,, Uspekhi Mat. Nauk, 37 (1982), 75.   Google Scholar

[21]

H. Kunita, Stochastic Flows and Stochastic Differential Equations,, Cambridge Studies in Advanced Mathematics, (1997).   Google Scholar

[22]

H. Kunita, Stochastic differential equations based on Lévy processes and stochastic flows of diffeomorphisms,, in M. Rao (ed.) Real and Stochastic Analysis, (2004), 305.   Google Scholar

[23]

J. Ma and J. Yong, Adapted solution of a degenerate backward SPDE, with applications,, Stochastic Process. Appl., 70 (1997), 59.  doi: 10.1016/S0304-4149(97)00057-4.  Google Scholar

[24]

J. Ma and J. Yong, On linear, degenerate backward stochastic partial differential equations,, Probab. Theory Related Fields, 113 (1999), 135.  doi: 10.1007/s004400050205.  Google Scholar

[25]

Q. Meng and S. Tang, Backward Stochastic HJB Equations with Jumps,, preprint., ().   Google Scholar

[26]

L. Mytnik, Stochastic partial differential equation driven by stable noise,, Probab. Theory Relat. Fields, 123 (2002), 157.  doi: 10.1007/s004400100180.  Google Scholar

[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.  doi: 10.1080/17442500500213797.  Google Scholar

[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.   Google Scholar

[29]

E. Pardoux, Stochastic partial differential equations and filtering of diffusion processes,, Stochastics, 3 (1979), 127.  doi: 10.1080/17442507908833142.  Google Scholar

[30]

E. Pardoux and S. Peng, Adapted solution of backward stochastic equation,, Systems Control Lett., 14 (1990), 55.  doi: 10.1016/0167-6911(90)90082-6.  Google Scholar

[31]

E. Pardoux and S. Peng, Backward Stochastic Differential Equations and Quasilinear Parabolic Partial Differential Equations,, in Stochastic Partial Differential Equations and Their Applications, 176 (1992), 200.  doi: 10.1007/BFb0007334.  Google Scholar

[32]

S. Peng, Stochastic Hamilton-Jacobi-Bellman equations,, SIAM J. Control Optim., 30 (1992), 284.  doi: 10.1137/0330018.  Google Scholar

[33]

P. Protter, Stochastic Integration and Differential Equations (second edition),, Springer, (2004).   Google Scholar

[34]

M. Röckner and T. Zhang, Stochastic evolution equations of jump type: Existence, uniqueness and large deviation principles,, Potential Analysis, 26 (2007), 255.  doi: 10.1007/s11118-006-9035-z.  Google Scholar

[35]

B. L. Rozovskii, Stochastic Evolution Systems, Kluwer,, Dordrecht, (1990).  doi: 10.1007/978-94-011-3830-7.  Google Scholar

[36]

A.-S. Sznitman, Martingales dépendant d'un paramètre: Une formule d'Ito,, Z. Wahrscheinlichkeitstheorie verw. Gebiete, 60 (1982), 41.  doi: 10.1007/BF01957096.  Google Scholar

[37]

S. Tang and X. Li, Necessary conditions for optimal control of stochastic systems with random jumps,, SIAM J. Control Optim., 32 (1994), 1447.  doi: 10.1137/S0363012992233858.  Google Scholar

[38]

S. Tang, The maximum pinciple for partially observed optimal control of stochastic differential equations,, SIAM J. Control Optim., 36 (1998), 1596.  doi: 10.1137/S0363012996313100.  Google Scholar

[39]

S. Tang, A New Partially Observed Stochastic Maximum Principle,, In Proceedings of 37th IEEE Control and Decision Conference, (1998), 2353.   Google Scholar

[40]

S. Tang, Semi-linear systems of backward stochastic partial differential equations in $\mathbbR^n$,, Chinese Ann. Math. Ser. B, 26 (2005), 437.  doi: 10.1142/S025295990500035X.  Google Scholar

[41]

J. B. Walsh, An introduction to stochastic partial differential equations,, In Ecole d'Été de Probabilités de St. Flour XIV, 1180 (1986), 265.  doi: 10.1007/BFb0074920.  Google Scholar

[42]

J. Yong and X. Zhou, Stochastic Controls: Hamiltonian Systems and HJB Equations,, Springer, (1999).  doi: 10.1007/978-1-4612-1466-3.  Google Scholar

[43]

X. Zhang, On stochastic evolution equations with non-Lipschitz coefficients,, Stochastics and Dynamics, 9 (2009), 549.  doi: 10.1142/S0219493709002774.  Google Scholar

[44]

X. Zhou, A duality analysis on stochastic partial differential equations,, J. Funct. Anal., 103 (1992), 275.  doi: 10.1016/0022-1236(92)90122-Y.  Google Scholar

[45]

X. Zhou, On the necessary conditions of optimal controls for stochastic partial differential equations,, SIAM J. Control Optim., 31 (1992), 1462.  doi: 10.1137/0331068.  Google Scholar

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