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SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

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Chinese Annals of mathematics,Series B 2005年第4期26卷 651-658页

作者： WANG MENG department of mathematics, Zhejiang university, Hangzhou 310027, china.school of mathematical sciences, Pudan university, shanghai 200433, china. department of mathematics Zhejiang University Hangzhou 310027 China. school of mathematical sciences Fudan University Shanghai 200433 China

Let M be an n dimensional complete Riemannian manifold satisfying the doublingvolume property and an on-diagonal heat kernel estimate. The necessary-sufficientcondition for the Sobolev inequality ‖f‖q ≤ Cn,,v,p,q(... 详细信息

关键词： Sobolev inequality Complete manifold Riesz transform Potential Heat kernel

Jensen's Inequality for Backward Stochastic Differential Equations

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Chinese Annals of mathematics,Series B 2006年第5期27卷 553-564页

作者： Long JIANG department of mathematics, china.university of Mining and Technology, Xuzhou 221008, Jiangsu, china.school of mathematical sciences, fudan university, shanghai 200433, china.school of mathematics and System sciences, Shandong university, Jinan 250100, china. department of mathematics China University of Mining and Technology Xuzhou 221008 Jiangsu China school of mathematical sciences Fudan University Shanghai 200433 China school of mathematics and System sciences Shandong University Jinan 250100 China

Under the Lipschitz assumption and square integrable assumption on g, the author proves that Jensen＇s inequality holds for backward stochastic differential equations with generator g if and only if g is independent of y, g（t, 0）≡ 0 and g is super homogeneous with respect to z. This result generalizes the known results on Jensen＇s inequality for gexpectation in [4, 7-9].

关键词： Backward stochastic differential equation g-Expectation Jensen's inequality for g-expectation Jensen's inequality for BSDEs

Approximate solutions to infinite dimensional LQ problems over infinite time horizon

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Science china.mathematics 2006年第7期49卷 865-876页

作者： PAN Liping, ZHANG Xu & CHEN Qihong school of mathematical sciences, fudan university, shanghai 200433, china.school of mathematics, Sichuan university, Chengdu 610064, china.department of Applied mathematics, shanghai university of Finance & Economics, shanghai 200433, china school of mathematical sciences Fudan University Shanghai China school of mathematics Sichuan University Chengdu China department of Applied mathematics Shanghai University of Finance & Economics Shanghai China

This paper is addressed to develop an approximate method to solve a class of infinite dimensional LQ optimal regulator problems over infinite time horizon. Our algorithm is based on a construction of approximate solutions which solve some finite dimensional LQ optimal regulator problems over finite time horizon, and it is shown that these approximate solutions converge strongly to the desired solution in the double limit sense.

关键词： approximate solution LQ problem infinite dimensional infinite time hori-

GLOBAL EXISTENCE OF WEAKLY DISCONTINUOUS SOLUTIONS TO A KIND OF MIXED INITIAL-BOUNDARY VALUE PROBLEM FOR QUASILINEAR HYPERBOLIC SYSTEMS

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Applied mathematics(A Journal of Chinese Universities) 2007年第2期22卷 181-200页

作者： Guo Fei school of mathematical sciences, fudan university, shanghai 200433, china.department of mathematics, Qufu Normal university, Shandong, 273165, china. school of mathematical sciences Fudan University Shanghai China department of mathematics Qufu Normal University Shandong China

In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {（t, x）｜ t ≥ 0, x ≥0} is considered. A suffic... 详细信息

关键词： quasilinear hyperbolic system mixed initial-boundary value problem global weakly discontinu-ous solution weak linear degeneracy

Optimal variational principle for backward stochastic control systems associated with Lévy processes

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Science china.mathematics 2012年第4期55卷 745-761页

作者： TANG MaoNing 1 & ZHANG Qi 2,1 department of mathematical sciences,Huzhou university,Huzhou 313000,china.2 school of mathematical sciences,fudan university,shanghai 200433,china department of mathematical sciences Huzhou University Huzhou China school of mathematical sciences Fudan University Shanghai China

The paper is concerned with optimal control of backward stochastic differentiM equation （BSDE） driven by Teugel＇s martingales and an independent multi-dimensional Brownian motion, where Teugel＇s martingales are a family of pairwise strongly orthonormal martingales associated with L6vy processes （see e.g., Nualart and Schoutens＇ paper in 2000）. We derive the necessary and sufficient conditions for the existence of the optimal control by means of convex variation methods and duality techniques. As an application, the optimal control problem of linear backward stochastic differential equation with a quadratic cost criteria （or backward linear-quadratic problem, or BLQ problem for short） is discussed and characterized by a stochastic Hamilton system.

关键词： stochastic control stochastic maximum principle Ldvy processes Teugel's martingales backwardstochastic differential equations

THE CODIMENSION FORMULA ON AF-COSUBMODULES

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Chinese Annals of mathematics,Series B 2002年第3期23卷 419-424页

作者： GUO KUNYU department of mathematics, fudan university, shanghai 200433, china department of mathematics Fudan University Shanghai 200433 China

Let M1, M2 be submodules of analytic Hilbert module X on ?(? Cn) such that M1 ? M2 and dimM1/M2 = k < '. If M2 is an AF-cosubmodule, then the codimension dimM1/M2 of M2 in M1 equals the cardinality of zeros of M2 rel... 详细信息

Let M1, M2 be submodules of analytic Hilbert module X on ?(? Cn) such that M1 ? M2 and dimM1/M2 = k 2 is an AF-cosubmodule, then the codimension dimM1/M2 of M2 in M1 equals the cardinality of zeros of M2 related to MI by counting multiplicities. The codimension formula has some interesting applications. In particular, the author calculates out the dimension of Rudin quotient module, which is raised in [14].

关键词： AF-cosubmodule Characteristic space Analytic Hilbert module

On Galois Extension of Hopf Algebras

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Chinese Annals of mathematics,Series B 2008年第1期29卷 65-70页

作者： Guohua LIU Shenglin ZHU department of mathematics Southeast University Nanjing 210096 China Key Laboratory of mathematics for Nonlinear sciences School of Mathematical Sciences Pudan University Shanghai 200433 China

Let H be a cosemisimple Hopf algebra over a field k, and π ： A→ H be a surjective cocentral bialgebra homomorphism of bialgebras. The authors prove that if A is Galois over its coinvariants B=LH Ker π and B is a sub-Hopf algebra of A, then A is itself a Hopf algebra. This generalizes a result of Cegarra [3] on group-graded algebras.

关键词： Hopf algebra Galois extension

Limit theorem and uniqueness theorem of backward stochastic differential equations

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Science china.mathematics 2006年第10期49卷 1353-1362页

作者： JIANG Long department of mathematics, china.university of Mining and Technology, Xuzhou 221008, china.Institute of mathematics, fudan university, shanghai 200433, china.school of mathematics and System sciences, Shandong university, Jinan 250100, china department of mathematics China University of Mining and Technology Xuzhou China Institute of mathematics Fudan University Shanghai China school of mathematics and System sciences Shandong University Jinan China

This paper establishes a limit theorem for solutions of backward stochastic differential equations (BSDEs). By this limit theorem, this paper proves that, under the standard assumption g(t,y,0) = 0, the generator g of a BSDE can be uniquely determined by the corresponding g-expectationεg;this paper also proves that if a filtration consistent expectation S can be represented as a g-expectationεg, then the corresponding generator g must be unique.

关键词： backward stochastic differential equation, g-expectation, generator.

Parametric estimation of discretely sampled Gamma-OU processes

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Science china.mathematics 2006年第9期49卷 1231-1257页

作者： ZHANG Shibin,ZHANG Xinsheng & SUN Shuguang school of Management, fudan university, shanghai 200433, china.department of mathematics, shanghai Maritime university, shanghai 200135, china school of Management Fudan University Shanghai China department of mathematics Shanghai Maritime University Shanghai China

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.

关键词： Gamma-OU process transition function Lévy process Lévy density stochastic volatility background driving Lévy process Laplace transformation maximum likelihood estimation