欧氏空间和函数空间中的倒向随机方程
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摘要
本文研究倒向随机微分方程,倒向重随机微分方程以及由Brown运动和(?)oisson点过程一起驱动的倒向随机积分偏微分方程.全文分为两部分.
第一部分讨论倒向随机微分方程以及倒向重随机微分方程的Lp解(1 第二部分用概率的方法讨论倒向随机积分偏微分方程的经典解的存在唯一性,这类方程来源于对由Brown运动和Poisson点过程一起驱动的非Markov正倒向随机微分系统的研究.在第四章,我们首先证明了由Brown运动和Poisson点过程一起驱动的半鞅所对应的Ito-Wentzell公式.其次,借助于Galerkin逼近以及压缩映照原理,我们得到了非退化的随机发展方程解的存在唯一性.将这一抽象结果应用到具体的方程并利用逼近的技巧,我们进一步得到了退化的随机积分偏微分方程解的存在唯一性.再结合已建立的Ito-Wentzell公式,我们导出了由Brown运动和Poisson点过程一起驱动的随机微分方程解的逆映照所满足的随机积分偏微分方程.接着,我们对带有Poisson跳的倒向随机微分方程的解进行正则性分析.最后,通过复合倒向随机微分方程的解和正向随机微分方程解的逆映照,我们构造出倒向随机积分偏微分方程的经典解.从而我们得到了随机Feynman-Kac公式.
The thesis is concerned with backward stochastic differential equations (BSDEs, for short), backward doubly stochastic differential equations (BDSDEs, for short), and semi-linear systems of backward stochastic integral partial differential equations driven by both Brownian motions and Poisson point processes. It consists of two parts.
The first part is concerned with the Lp solutions (1< p< 2) to BSDEs and BDSDEs. In Chapter 2, via approximating the generator by smooth functions, we first show that there is unique L2 solution to a one-dimensional BSDE if the generator f is independent of the first unknown variable y, uniformly continuous with respect to the second unknown variable z, and of linear growth. Then, by approximating the generator by monotone functions, we prove that there is unique Lp solution to a one-dimensional BSDE if the generator f is uniformly continuous with respect to the unknown variables (y,z) and is of linear growth, and if the modulus of continuity of f with respect to the first unknown variable y is of certain integrability. In Chapter 3, by means of weak convergence, we first prove that there is unique L2 solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a rather general growth in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than 1. Then we prove an Ito formula which involves both forward Ito integral and backward Ito integral. Finally, by establishing some a prior estimates, we show that there is unique Lp solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a more general integrability in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than (?)
The second part is concerned with, from a probabilistic point of view, semi-linear systems of backward stochastic integral partial differential equations, which arise from the study of non-Markovian forward-backward stochastic differential equations driven by both Brownian motions and Poisson point processes. In Chap-ter 4, we first establish an Ito-Wentzell formula for the semimartingales driven by both Brownian motions and Poisson point processes. Then we obtain the existence and uniqueness of solutions to non-degenerate stochastic evolution equations via Galerkin approximation and the contraction mapping principle. Applying the ab-stract result to concrete equations and using the skill of approximation, we further prove the existence and uniqueness of solutions to degenerate stochastic integral partial differential equations. Combining the Ito-Wentzell formula, we derive the stochastic integral partial differential equation for the inverse of a stochastic flow generated by a stochastic differential equation driven by both Brownian motion and Poisson point process. Then we analyze the regularity of the solutions to the BSDEs with Poisson jumps. Finally, by the composition of the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow stated above, we construct the classical solution to the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.
第一部分讨论倒向随机微分方程以及倒向重随机微分方程的Lp解(1
The thesis is concerned with backward stochastic differential equations (BSDEs, for short), backward doubly stochastic differential equations (BDSDEs, for short), and semi-linear systems of backward stochastic integral partial differential equations driven by both Brownian motions and Poisson point processes. It consists of two parts.
The first part is concerned with the Lp solutions (1< p< 2) to BSDEs and BDSDEs. In Chapter 2, via approximating the generator by smooth functions, we first show that there is unique L2 solution to a one-dimensional BSDE if the generator f is independent of the first unknown variable y, uniformly continuous with respect to the second unknown variable z, and of linear growth. Then, by approximating the generator by monotone functions, we prove that there is unique Lp solution to a one-dimensional BSDE if the generator f is uniformly continuous with respect to the unknown variables (y,z) and is of linear growth, and if the modulus of continuity of f with respect to the first unknown variable y is of certain integrability. In Chapter 3, by means of weak convergence, we first prove that there is unique L2 solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a rather general growth in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than 1. Then we prove an Ito formula which involves both forward Ito integral and backward Ito integral. Finally, by establishing some a prior estimates, we show that there is unique Lp solution to a multi-dimensional BDSDE if the drift coefficient f is monotone, continuous, and of a more general integrability in the first unknown variable y, and is Lipschitz continuous in the second unknown variable z, and if the diffusion coefficient g is Lipschitz continuous in the unknown variables (y, z) and moreover, the Lipschitz coefficient with respect to the second unknown variable z is less than (?)
The second part is concerned with, from a probabilistic point of view, semi-linear systems of backward stochastic integral partial differential equations, which arise from the study of non-Markovian forward-backward stochastic differential equations driven by both Brownian motions and Poisson point processes. In Chap-ter 4, we first establish an Ito-Wentzell formula for the semimartingales driven by both Brownian motions and Poisson point processes. Then we obtain the existence and uniqueness of solutions to non-degenerate stochastic evolution equations via Galerkin approximation and the contraction mapping principle. Applying the ab-stract result to concrete equations and using the skill of approximation, we further prove the existence and uniqueness of solutions to degenerate stochastic integral partial differential equations. Combining the Ito-Wentzell formula, we derive the stochastic integral partial differential equation for the inverse of a stochastic flow generated by a stochastic differential equation driven by both Brownian motion and Poisson point process. Then we analyze the regularity of the solutions to the BSDEs with Poisson jumps. Finally, by the composition of the random field generated by the solution of a backward stochastic differential equation with the inverse of the stochastic flow stated above, we construct the classical solution to the system of backward stochastic integral partial differential equations. As a result, we establish a stochastic Feynman-Kac formula.
引文
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[14]R. Buckdahn and Y. Hu. Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res.,23:177-203,1998.
[15]R. Buckdahn and J. Ma. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl.,93:181-204,2001.
[16]R. Buckdahn and E. Pardoux. BSDE's with jumps and associated integral-stochastic differential equations. preprint.
[17]F. Delbaen and S. Tang. Harmonic analysis of stochastic equations and back-ward stochastic differential equations. Probab. Theory Relat. Fields,146:291-336,2010.
[18]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.
[19]N. Englezos and I. Karatzas. Utility maximization with habit formation:dy-namic programming and stochastic PDEs. SI AM J. Control Optim.,48:481-520,2009.
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[22]S. Hamadene. Multidimensional backward stochastic differential equations with uniformly continuous coefficients. Bernoulli,9:517-534,2003.
[23]E. Hausenblas. Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure. Electron. J. Probab.,10:1496-1546,2005.
[24]E. Hausenblas. SPDEs driven by Poisson random measure with non-Lipschitz coefficients:existence results. Probab. Theory Relat. Fields,137:161-200,2007.
[25]Y. Hu, J. Ma, and J. Yong. On semi-linear degenerate backward stochastic partial differential equations. Probab. Theory Related Fields,123:381-411,2002.
[26]Y. Hu and S. Peng. Adapted solution of a backward semilinear stochastic evo-lution equations. Stoch. Anal. Appl.,9:445-459,1991.
[27]G. Jia. A uniqueness theorem for the solution of backward stochastic differential equations. C. R. Acad. Sci. Paris, Ser Ⅰ,346:439-444,2008.
[28]G. Jia. Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients. Statistics and Probability Letters,79:436-441,2009.
[29]N. El Karoui and S.-J. Huang. A general result of existence and uniqueness of backward stochastic differential equations, Backward Stochastic Differential Equations, volume 364 of Pitman Research Notes in Mathematics Series, pages 27-36. Longman, Harlow, London, UK,1997.
[30]N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez. Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab.,25:702-737,1997.
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[33]N. V. Krylov. On the Ito-wentzell formula for distribution-valued processes and related topics. http://arxiv.org/abs/0904.2752.
[34]N. V. Krylov and B. L. Rozovskii. On the cauchy problem for linear stochastic partial differential equations. Math. USSR Izv.,11:1267-1284,1977.
[35]N. V. Krylov and B. L. Rozovskii. Characteristics of degenerating second-order parabolic Ito equations. Trudy seminara imeni Petrovskogo,8:153-168,1982.
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[38]H. Kunita. Stochastic differential equations based on Levy processes and stochas-tic flows of diffeomorphisms, Real and Stochastic Analysis. Birkhauser, Basel, 2004.
[39]J. P. Lepeltier and J. San Martin. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters,32:425-430,1997.
[40]J. Ma and J. Yong. Adapted solution of a degenerate backward SPDE, with applications. Stochastic Process. Appl.,70:59-84,1997.
[41]J. Ma and J. Yong. On linear, degenerate backward stochastic partial differential equations. Probab. Theory Related Fields,113:135-170,1999.
[42]X. Mao. Adapted solutions of BSDEs with non-Lipschitz coefficients. Stochastic Process. Appl.,58:281-292,1995.
[43]Q. Meng and S. Tang. Backward Stochastic HJB Equations with Jumps. preprint.
[44]L. Mytnik. Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields,123:157-201,2002.
[45]B.(?)ksendal, F. Proske, and T. Zhang. Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields. Stochastics,77:381-399,2005.
[46]B.(?)ksendal and T. Zhang. The Ito-ventzell formula and forward stochastic differential equations driven by Poisson random measures. Osaka J. Math., 44:207-230,2007.
[47]E. Pardoux. Stochastic partial differential equations and filtering of diffusion processes. Stochastics,3:127-167,1979.
[48]E. Pardoux. BSDE's, weak convergence and homogenization of semilinear PDE's, pages 503-549. Nonlinear analysis, Differential Equations and Con-trol. Kluwer Acad. Pub.,1999.
[49]E. Pardoux and S. Peng. Backward Stochastic Differential Equations and Quasi-linear Parabolic Partial Differential Equations, volume 176 of Lecture Notes in Control and Inform. Sci., pages 200-217. Springer, Berlin, Heidelberg, New York.
[50]E. Pardoux and S. Peng. Adapted solution of backward stochastic equation. Systems Control Lett.,14:55-61,1990.
[51]E. Pardoux and S. Peng. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probability Theory and Related Fields,98:209-227,1994.
[52]S. Peng. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics Stochastics Rep.,38:119-134,1992.
[53]S. Peng. Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Op-tim.,30:284-304,1992.
[54]S. Peng. Monotonic limit theorem of BSDE and nonlinear decomposition theo-rem of Doob-Meyer's type. Probab. Theory Related Fields,113:473-499,1999.
[55]P. Protter. Stochastic Integration and Differential Equations (second edition). Springer,2004.
[56]Y. Ren, A. Lin, and L. Hu. Stochastic PDIEs and backward doubly stochas-tic differential equations driven by Levy processes. J. Comput. Appl. Math., 223:901-907,2009.
[57]M. Rockner and T. Zhang. Stochastic evolution equations of jump type:ex-istence, uniqueness and large deviation principles. Potential Analysis,26:255-279,2007.
[58]M. Royer. Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl.,116:1358-1376,2006.
[59]B. L. Rozovskii. Stochastic evolution systems. Kluwer, Dordrecht,1990.
[60]Y. Shi, Y. Gu, and K. Liu. Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl.,23:97-110,2005.
[61]R. Situ. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl.,66:209-236,1997.
[62]A.-S. Sznitman. Martingales dependant d'un parametre:une formule d'Ito. Z. Wahrscheinlichkeitstheorie verw. Gebiete,60:41-70,1982.
[63]S. Tang. The maximum pinciple for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim.,36:1596-1617,1998.
[64]S. Tang. A New Partially Observed Stochastic Maximum Principle. In Proceed-ings of 37th IEEE Control and Decision Conference, pages 2353-2358, Tampa, Florida,,1998.
[65]S. Tang. Semi-linear systems of backward stochastic partial differential equa-tions in Rn. Chinese Ann. Math. Ser. B,26:437-456,2005.
[66]S. Tang and X. Li. Necessary conditions for optimal control of stochastic systems with random jumps. SIAM J. Control Optim.,32:1447-1475,1994.
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[2]K. Bahlali. Backward stochastic differential equations with locally lipschitz coefficient. C. R. Acad. Sci. Paris, Ser Ⅰ,333:481-486,2001.
[3]K. Bahlali, M. Hassani, B. Mansouri, and N. Mrhardy. One barrier reflected backward doubly stochastic differential equations with continuous generator. C. R. Math. Acad. Sci. Paris,347:1201-1206,2009.
[4]V. Bally and M. Matoussi. Weak solutions for SPDEs and backward doubly stochastic differential equations. J. Theoret. Probab.,14:125-164,2001.
[5]G. Barles, R. Buckdhan, and E. Pardoux. Backward stochastic differential equa-tions and integral-partial differential equations. Stochastics and Stochastics Reports,60:57-83,1997.
[6]A. Bensoussan. Stochastic Control of Partially Observable Systems. Cambridge University Press, Cambridge,1992.
[7]J. M. Bismut. Conjugate convex functions in optimal stochastic control. J. Math. Anal. Appl.,44:384-404,1973.
[8]J. M. Bismut. Linear quadratic optimal stochastic control with random coeffi-cients. SIAM J. Control Optim.,14:419-444,1976.
[9]B. Boufoussi, J. Van Casteren, and N. Mrhardy. Generalized backward doubly stochastic differential equations and SPDEs with nonlinear Neumann boundary conditions. Bernoulli,13:423-446,2007.
[10]Ph. Briand and R. Carmona. BSDEs with polynomial growth generators. J. Appl. Math. Stochastic Anal.,13:207-238,2000.
[11]Ph. Briand, B. Delyon, Y. Hu, E. Pardoux, and L. Stoica. Lp solutions of BSDEs. Stochastic Process. Appl.,108:109-129,2003.
[12]Ph. Briand and Y. Hu. BSDE with quadratic growth and unbounded terminal value. Probab. Theor. Related Fields,136:604-618,2006.
[13]Ph. Briand and Y. Hu. Quadratic BSDEs with convex generators and un-bounded terminal conditions. Probab. Theor. Related Fields,141:543-567,2008.
[14]R. Buckdahn and Y. Hu. Pricing of American contingent claims with jump stock price and constrained portfolios. Math. Oper. Res.,23:177-203,1998.
[15]R. Buckdahn and J. Ma. Stochastic viscosity solutions for nonlinear stochastic partial differential equations. I. Stochastic Process. Appl.,93:181-204,2001.
[16]R. Buckdahn and E. Pardoux. BSDE's with jumps and associated integral-stochastic differential equations. preprint.
[17]F. Delbaen and S. Tang. Harmonic analysis of stochastic equations and back-ward stochastic differential equations. Probab. Theory Relat. Fields,146:291-336,2010.
[18]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.
[19]N. Englezos and I. Karatzas. Utility maximization with habit formation:dy-namic programming and stochastic PDEs. SI AM J. Control Optim.,48:481-520,2009.
[20]T. Fujiwara and H. Kunita. Stochastic differentail equations of jump type and Levy flows in diffeomorphisms group. J. Math. Kyoto Univ.,25:71-106,1985.
[21]I. Gyongy and N. V. Krylov. On stochastic equations with respect to semi-martingales II, Ito formula in Banach spaces. Stochastics,6:153-173,1982.
[22]S. Hamadene. Multidimensional backward stochastic differential equations with uniformly continuous coefficients. Bernoulli,9:517-534,2003.
[23]E. Hausenblas. Existence, Uniqueness and Regularity of Parabolic SPDEs driven by Poisson random measure. Electron. J. Probab.,10:1496-1546,2005.
[24]E. Hausenblas. SPDEs driven by Poisson random measure with non-Lipschitz coefficients:existence results. Probab. Theory Relat. Fields,137:161-200,2007.
[25]Y. Hu, J. Ma, and J. Yong. On semi-linear degenerate backward stochastic partial differential equations. Probab. Theory Related Fields,123:381-411,2002.
[26]Y. Hu and S. Peng. Adapted solution of a backward semilinear stochastic evo-lution equations. Stoch. Anal. Appl.,9:445-459,1991.
[27]G. Jia. A uniqueness theorem for the solution of backward stochastic differential equations. C. R. Acad. Sci. Paris, Ser Ⅰ,346:439-444,2008.
[28]G. Jia. Some uniqueness results for one-dimensional BSDEs with uniformly continuous coefficients. Statistics and Probability Letters,79:436-441,2009.
[29]N. El Karoui and S.-J. Huang. A general result of existence and uniqueness of backward stochastic differential equations, Backward Stochastic Differential Equations, volume 364 of Pitman Research Notes in Mathematics Series, pages 27-36. Longman, Harlow, London, UK,1997.
[30]N. El Karoui, C. Kapoudjian, E. Pardoux, S. Peng, and M. C. Quenez. Reflected solutions of backward SDE's, and related obstacle problems for PDE's. Ann. Probab.,25:702-737,1997.
[31]N. El Karoui, S. Peng, and M. C. Quenez. Backward stochastic differential equations in Finance. Math. Finance,7:1-71,1997.
[32]M. Kobylanski. Backward stochastic differential equations and partial differen-tial equations with quadratic growth. Ann. Prob.,28:558-602,2000.
[33]N. V. Krylov. On the Ito-wentzell formula for distribution-valued processes and related topics. http://arxiv.org/abs/0904.2752.
[34]N. V. Krylov and B. L. Rozovskii. On the cauchy problem for linear stochastic partial differential equations. Math. USSR Izv.,11:1267-1284,1977.
[35]N. V. Krylov and B. L. Rozovskii. Characteristics of degenerating second-order parabolic Ito equations. Trudy seminara imeni Petrovskogo,8:153-168,1982.
[36]N. V. Krylov and B. L. Rozovskii. Stochastic partial differential equations and diffusion processes. Uspekhi Mat. Nauk 37:6,37:75-95 (Russian),1982.
[37]H. Kunita. Stochastic flows and stochastic differential equations, volume 24 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge,1997.
[38]H. Kunita. Stochastic differential equations based on Levy processes and stochas-tic flows of diffeomorphisms, Real and Stochastic Analysis. Birkhauser, Basel, 2004.
[39]J. P. Lepeltier and J. San Martin. Backward stochastic differential equations with continuous coefficient. Statistics and Probability Letters,32:425-430,1997.
[40]J. Ma and J. Yong. Adapted solution of a degenerate backward SPDE, with applications. Stochastic Process. Appl.,70:59-84,1997.
[41]J. Ma and J. Yong. On linear, degenerate backward stochastic partial differential equations. Probab. Theory Related Fields,113:135-170,1999.
[42]X. Mao. Adapted solutions of BSDEs with non-Lipschitz coefficients. Stochastic Process. Appl.,58:281-292,1995.
[43]Q. Meng and S. Tang. Backward Stochastic HJB Equations with Jumps. preprint.
[44]L. Mytnik. Stochastic partial differential equation driven by stable noise. Probab. Theory Relat. Fields,123:157-201,2002.
[45]B.(?)ksendal, F. Proske, and T. Zhang. Backward stochastic partial differential equations with jumps and application to optimal control of random jump fields. Stochastics,77:381-399,2005.
[46]B.(?)ksendal and T. Zhang. The Ito-ventzell formula and forward stochastic differential equations driven by Poisson random measures. Osaka J. Math., 44:207-230,2007.
[47]E. Pardoux. Stochastic partial differential equations and filtering of diffusion processes. Stochastics,3:127-167,1979.
[48]E. Pardoux. BSDE's, weak convergence and homogenization of semilinear PDE's, pages 503-549. Nonlinear analysis, Differential Equations and Con-trol. Kluwer Acad. Pub.,1999.
[49]E. Pardoux and S. Peng. Backward Stochastic Differential Equations and Quasi-linear Parabolic Partial Differential Equations, volume 176 of Lecture Notes in Control and Inform. Sci., pages 200-217. Springer, Berlin, Heidelberg, New York.
[50]E. Pardoux and S. Peng. Adapted solution of backward stochastic equation. Systems Control Lett.,14:55-61,1990.
[51]E. Pardoux and S. Peng. Backward doubly stochastic differential equations and systems of quasilinear SPDEs. Probability Theory and Related Fields,98:209-227,1994.
[52]S. Peng. A generalized dynamic programming principle and Hamilton-Jacobi-Bellman equation. Stochastics Stochastics Rep.,38:119-134,1992.
[53]S. Peng. Stochastic Hamilton-Jacobi-Bellman equations. SIAM J. Control Op-tim.,30:284-304,1992.
[54]S. Peng. Monotonic limit theorem of BSDE and nonlinear decomposition theo-rem of Doob-Meyer's type. Probab. Theory Related Fields,113:473-499,1999.
[55]P. Protter. Stochastic Integration and Differential Equations (second edition). Springer,2004.
[56]Y. Ren, A. Lin, and L. Hu. Stochastic PDIEs and backward doubly stochas-tic differential equations driven by Levy processes. J. Comput. Appl. Math., 223:901-907,2009.
[57]M. Rockner and T. Zhang. Stochastic evolution equations of jump type:ex-istence, uniqueness and large deviation principles. Potential Analysis,26:255-279,2007.
[58]M. Royer. Backward stochastic differential equations with jumps and related non-linear expectations. Stochastic Process. Appl.,116:1358-1376,2006.
[59]B. L. Rozovskii. Stochastic evolution systems. Kluwer, Dordrecht,1990.
[60]Y. Shi, Y. Gu, and K. Liu. Comparison theorems of backward doubly stochastic differential equations and applications. Stoch. Anal. Appl.,23:97-110,2005.
[61]R. Situ. On solutions of backward stochastic differential equations with jumps and applications. Stochastic Process. Appl.,66:209-236,1997.
[62]A.-S. Sznitman. Martingales dependant d'un parametre:une formule d'Ito. Z. Wahrscheinlichkeitstheorie verw. Gebiete,60:41-70,1982.
[63]S. Tang. The maximum pinciple for partially observed optimal control of stochastic differential equations. SIAM J. Control Optim.,36:1596-1617,1998.
[64]S. Tang. A New Partially Observed Stochastic Maximum Principle. In Proceed-ings of 37th IEEE Control and Decision Conference, pages 2353-2358, Tampa, Florida,,1998.
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