g-期望的无穷小生成元,g-上鞅与g-上调和函数
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摘要
本文主要讨论了BSDE及相关g-期望在非线性分析中的几个新应用,给出的新结果有两个:分别是g-期望情形下的随机无穷小生成元及g-上鞅与g-上调和函数的对应关系。它们分别在文章的第一节与第三节中进行详细讨论。笔者认为本文最大贡献在于将粘性解理论应用于第二个问题的研究,将g-上调和函数的范围由光滑函数扩大到连续函数,得到了较满意的结果,请见§3.4。另外,本文在第二节中讨论了一类由BSDE导出的非线性半群,并利用非线性Feynman-Kac公式得出了该半群的微分算子与无穷小生成元的关系,虽然本节中的结果不算新颖,但其中对g-期望系统的半群性质即马氏性质的探讨也是比较有趣的,以期望读者能从中受到启发,得到更好的结果。
In this paper, we study several problems about the application of BSDE and the related g-expectation in nonlinear analysis. In §1 and §3 we give two new results: The stochastic Infinitesimal Generator under g-expectation and the Corresponding relation between g-supermartingale and g-superharmonic function. Particularly, in §3.4 we expand the range of the g-superharmonic functions by applicating the concept of viscosity solution, however the correspondence still hold. In §2 we use BSDE to construct a nonlinear semi-group and discuss its infitesimal generator and its differential operator, and we expect more new development in this direction.
In this paper, we study several problems about the application of BSDE and the related g-expectation in nonlinear analysis. In §1 and §3 we give two new results: The stochastic Infinitesimal Generator under g-expectation and the Corresponding relation between g-supermartingale and g-superharmonic function. Particularly, in §3.4 we expand the range of the g-superharmonic functions by applicating the concept of viscosity solution, however the correspondence still hold. In §2 we use BSDE to construct a nonlinear semi-group and discuss its infitesimal generator and its differential operator, and we expect more new development in this direction.
引文
[1] E.Pardoux and S.Peng, Adapted solution of a backward stochastic differential equation, Systems and Cdntrol letters, 14, 55-61, 1990.
[2] S.Peng, Probabilistic Interpretation for Systems of Quasi linear Parabolic Differential Equations, Stochastics, vol37, 61-74, 1991.
[3] S.Peng, BSDE and related g-expectation, Pitman research note series 364:141-159, 1997.
[4] S.Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Prob. Theory Rel. Fields 113, no 4, 473-499, 1999.
[5] Z.Chen and S.Peng, Continous Properties of g-martingale, Chin. Ann. of Math, 22B:1, 115-128, 2001.
[6] Z.Chen and S.Peng, A general down crossing inequality for g-martingales, Statist. Probab. Lett. 56, no 2, 169-175, 2000.
[7] S.Peng, Nonlinear expectations and nonlinear Markov chains, in the proceedings of the 3rd Colloquium on "Backward Stochastic Differential Equations and Applications", Chin. Ann. Math. 26B:2(2005), 159-184.
[8] S.Peng, Nonlinear expectations nonlinear evaluations and risk measures, Stochastic Methods in Finance Lectures, 143-217, LNM 1856, Springer-Verlag (2004).
[9] B. ф ksendal, Stochastic Differential Equations, 5th editon, Springer-Verlag, pp115-118, 1998.
[10] P.Briand, F.Coquet, Y.Hu, J.Memin and Peng, A converse comparison theorem for BSDEs and related properties of g-expectations, Electron. Comm. Probab, 5 (2000).
[11] J.L.Doob, Semimartingales and subharmonic functions, Trans. Amer. math. Soc. 77, 86-121, 1954.
[12] Kakutani,S., Two-dimentional Brownian motion and harmonic functions,Proc. Imp. Acad. Tokyo, 20, 706-714, 1944.
[13] 王梓坤,布朗运动与位势,科学出版社,1983.
[14] E.Hill, Functional Analysis and Semi-groups, Colloq. Pubt. Amer. Math. Soc., 1948.
[15] K.Yosida,On the differentiability and the representation of one-parameter semi-groups of linear operators,J. Math. Soc. Japan 1, 15-21, 1948.
[16] K.Yosida, Functional Analysis, 5th version, Springer-Verlag, 1977.
[17] S.Peng,Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Prob. Theory Rel. Feilds 113, no4, 473- 499, 1999.
[18] M.C.Crandall and P.L.Lions, Viscosity Solutions of Hamilton-Bellman- Jacobi Equations, Trans. Amer. Math. Soc., 1983.
[19] P.L.Lions, Optimal control of diffusion processes and Hamilton-Jacobi- Bellman equations. Part 1: The dynamic programming Principle and applications and Part2: Viscosity solutions and uniqueness, Comm. P.D.E, 8(1983), 1101-1174 and 1229-1276.
[20] S.Peng, A Generalized Dynamic Programming Principle and Hamilton- Jacobi-Bellman Equation, Stochastics. 38. 119-134, 1992
[21] G.Barles and E.Lesigne, SDE, BSDE and PDE, Pitman Research Notes Series 364: 47-80, 1997
[22] 朱学红,关于受控带跳扩散过程生存性质的研究及应用,山大硕士学位论文,2003。
[23] C.Dellacherie, Une version non lineaire du theoreme de representation de Riesz, CRAS 324, 1997, p.1011-1014.
[24] M.H. Protter, Weinberger H.F., Maximum principles in differential equations, Springer 1984.
[25] W. Walter, Ordinary differential equations, Springer 1998.
[2] S.Peng, Probabilistic Interpretation for Systems of Quasi linear Parabolic Differential Equations, Stochastics, vol37, 61-74, 1991.
[3] S.Peng, BSDE and related g-expectation, Pitman research note series 364:141-159, 1997.
[4] S.Peng, Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Prob. Theory Rel. Fields 113, no 4, 473-499, 1999.
[5] Z.Chen and S.Peng, Continous Properties of g-martingale, Chin. Ann. of Math, 22B:1, 115-128, 2001.
[6] Z.Chen and S.Peng, A general down crossing inequality for g-martingales, Statist. Probab. Lett. 56, no 2, 169-175, 2000.
[7] S.Peng, Nonlinear expectations and nonlinear Markov chains, in the proceedings of the 3rd Colloquium on "Backward Stochastic Differential Equations and Applications", Chin. Ann. Math. 26B:2(2005), 159-184.
[8] S.Peng, Nonlinear expectations nonlinear evaluations and risk measures, Stochastic Methods in Finance Lectures, 143-217, LNM 1856, Springer-Verlag (2004).
[9] B. ф ksendal, Stochastic Differential Equations, 5th editon, Springer-Verlag, pp115-118, 1998.
[10] P.Briand, F.Coquet, Y.Hu, J.Memin and Peng, A converse comparison theorem for BSDEs and related properties of g-expectations, Electron. Comm. Probab, 5 (2000).
[11] J.L.Doob, Semimartingales and subharmonic functions, Trans. Amer. math. Soc. 77, 86-121, 1954.
[12] Kakutani,S., Two-dimentional Brownian motion and harmonic functions,Proc. Imp. Acad. Tokyo, 20, 706-714, 1944.
[13] 王梓坤,布朗运动与位势,科学出版社,1983.
[14] E.Hill, Functional Analysis and Semi-groups, Colloq. Pubt. Amer. Math. Soc., 1948.
[15] K.Yosida,On the differentiability and the representation of one-parameter semi-groups of linear operators,J. Math. Soc. Japan 1, 15-21, 1948.
[16] K.Yosida, Functional Analysis, 5th version, Springer-Verlag, 1977.
[17] S.Peng,Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob-Meyer's type, Prob. Theory Rel. Feilds 113, no4, 473- 499, 1999.
[18] M.C.Crandall and P.L.Lions, Viscosity Solutions of Hamilton-Bellman- Jacobi Equations, Trans. Amer. Math. Soc., 1983.
[19] P.L.Lions, Optimal control of diffusion processes and Hamilton-Jacobi- Bellman equations. Part 1: The dynamic programming Principle and applications and Part2: Viscosity solutions and uniqueness, Comm. P.D.E, 8(1983), 1101-1174 and 1229-1276.
[20] S.Peng, A Generalized Dynamic Programming Principle and Hamilton- Jacobi-Bellman Equation, Stochastics. 38. 119-134, 1992
[21] G.Barles and E.Lesigne, SDE, BSDE and PDE, Pitman Research Notes Series 364: 47-80, 1997
[22] 朱学红,关于受控带跳扩散过程生存性质的研究及应用,山大硕士学位论文,2003。
[23] C.Dellacherie, Une version non lineaire du theoreme de representation de Riesz, CRAS 324, 1997, p.1011-1014.
[24] M.H. Protter, Weinberger H.F., Maximum principles in differential equations, Springer 1984.
[25] W. Walter, Ordinary differential equations, Springer 1998.