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A new existence result for quadratic BSDEs with jumps with application to the utility maximization problem

STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2010年第10期120卷 1966-1995页

作者： Morlais, Marie-Amelie Univ Maine Lab Manceau Math F-72085 Le Mans France

In this study, we consider the exponential utility maximization problem in the context of a jump-diffusion model. To solve this problem, we rely on the dynamic programming principle to express the value process of this problem in terms of the solution of a quadratic BSDE with jumps. Since the quadratic BSDE1 under study is driven by both a Wiener process and a Poisson random measure having a Levy measure with infinite mass, our main task is therefore to establish a new existence result for the specific BSDE introduced. (C) 2010 Elsevier B.V. All rights reserved.

关键词： Backward stochastic differential equations Levy measure Utility maximization dynamic programming principle

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OPTIMAL INVESTMENT POLICY AND DIVIDEND PAYMENT STRATEGY IN AN INSURANCE COMPANY

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ANNALS OF APPLIED PROBABILITY 2010年第4期20卷 1253-1302页

作者： Azcue, Pablo Muler, Nora Univ Torcuato Tella Dept Matemat Buenos Aires DF Argentina

We consider in this paper the optimal dividend problem for an insurance company whose uncontrolled reserve process evolves as a classical Cramer-Lundberg process. The firm has the option of investing part of the surplus in a Black-Scholes financial market. The objective is to find a strategy consisting of both investment and dividend payment policies which maximizes the cumulative expected discounted dividend pay-outs until the time of bankruptcy. We show that the optimal value function is the smallest viscosity solution of the associated second-order integro-differential Hamilton-Jacobi-Bellman equation. We study the regularity of the optimal value function. We show that the optimal dividend payment strategy has a band structure. We find a method to construct a candidate solution and obtain a verification result to check optimality. Finally, we give an example where the optimal dividend strategy is not barrier and the optimal value function is not twice continuously differentiable.

关键词： Cramer-Lundberg process insurance dividend payment strategy optimal investment policy Hamilton-Jacobi-Bellman equation viscosity solution risk control dynamic programming principle band strategy barrier strategy

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Mean-field backward stochastic differential equations and related partial differential equations

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STOCHASTIC PROCESSES AND THEIR APPLICATIONS 2009年第10期119卷 3133-3154页

作者： Buckdahn, Rainer Li, Juan Peng, Shige Shandong Univ Sch Math & Stat Weihai 264200 Peoples R China Univ Bretagne Occidentale Dept Math F-29285 Brest France Shandong Univ Sch Math Jinan 250100 Peoples R China

In [R. Buckdahn, B. Djehiche, J. Li, S. Peng, Mean-field backward stochastic differential equations. A limit approach. Arm. Probab. (2007) (in press). Available online: http://***/aop/future_papers. h(m] the authors obtained mean-field Backward Stochastic Differential Equations (BSDE) associated with a mean-field Stochastic Differential Equation (SDE) in a natural way as a limit of a high dimensional system of forward and backward SDEs, corresponding to a large number of "particles" (or "agents"). The objective of the present paper is to deepen the investigation of such mean-field BSDEs by studying them in a more general framework, with general coefficient, and to discuss comparison results for them. In a second step we are interested in Partial Differential Equations (PDE) whose solutions can be stochastically interpreted in terms of mean-field BSDEs. For this we Study a mean-field BSDE in a Markovian framework, associated with a McKean-Vlasov forward equation. By combining classical BSDE methods, in particular that of "backward semigroups" introduced by Peng [S. Peng, J. Yan, S. Peng, S. Fang, L. Wu (Eds.), in: BSDE and Stochastic Optimizations;Topics in Stochastic Analysis, Science Press, Beijing (1997) (Chapter 2) (ill Chinese)], with specific arguments for mean-field BSDEs, we prove that this mean-field BSDE gives the viscosity solution of a nonlocal PDE. The uniqueness of this viscosity solution is obtained for the space of continuous functions with polynomial growth. With the help of an example it is shown that for the nonlocal PDEs associated with mean-field BSDEs one cannot expect to have uniqueness in a larger space of continuous functions. (C) 2009 Elsevier B.V. All rights reserved.

关键词： Mean-field models McKean-Vlasov equation Backward stochastic differential equations Comparison theorem dynamic programming principle Viscosity solution

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Stochastic optimization theory of backward stochastic differential equations with jumps and viscosity solutions of Hamilton-Jacobi-Bellman equations

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NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS 2009年第4期70卷 1776-1796页

作者： Li, Juan Peng, Shige Shandong Univ Dept Math Weihai 264200 Peoples R China Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Shandong Univ Sch Math & Syst Sci Jinan 250100 Peoples R China

In this paper we Study stochastic optimal control problems with jumps with the help of the theory of Backward Stochastic Differential Equations (BSDEs) with jumps. We generalize the results of Peng [S. Peng, BSDE and stochastic optimizations, in: J. Yan, S. Peng, S. Fang, L, Wu. Topics in Stochastic Analysis, Science Press, Beijing, 1997 (Chapter 2) (in Chinese)] by considering cost functionals defined by controlled BSDEs with jumps. The application of BSDE methods, in particular, the use of the notion of stochastic backward semigroups introduced by Peng in the above-mentioned work allows a straightforward proof of a dynamic programming principle for Value functions associated with stochastic optimal control problems with jumps. We prove that the value functions are the viscosity solutions of the associated generalized Hamilton-Jacobi-Bellman equations with integral-differential operators. For this proof, we adapt Peng's BSDE approach, given in the above-mentioned reference, developed in the framework of stochastic control problems driven by Brownian motion to that of stochastic control problems driven by Brownian motion and Poisson random measure. (C) 2008 Elsevier Ltd. All rights reserved.

关键词： Poisson random measure Value function Backward stochastic differential equations dynamic programming principle Viscosity solution Stochastic backward semigroup

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OPTIMAL STOPPING PROBLEM FOR STOCHASTIC DIFFERENTIAL EQUATIONS WITH RANDOM COEFFICIENTS

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2009年第2期48卷 941-971页

作者： Chang, Mou-Hsiung Pang, Tao Yong, Jiongmin USA Res Off Div Math Res Triangle Pk NC 27709 USA N Carolina State Univ Dept Math Ctr Res Sci Computat Raleigh NC 27695 USA Univ Cent Florida Dept Math Orlando FL 32816 USA

An optimal stopping problem for stochastic differential equations with random coefficients is considered. The dynamic programming principle leads to a Hamiltion-Jacobi-Bellman equation, which, for the current case, is a backward stochastic partial differential variational inequality (BSPDVI, for short) for the value function. Well-posedness of such a BSPDVI is established, and a veri. cation theorem is proved.

关键词： optimal stopping random coefficients dynamic programming principle backward stochastic partial differential variational inequality verification theorem

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Probabilistic interpretation for systems of Isaacs equations with two reflecting barriers

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NODEA-NONLINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS 2009年第3期16卷 381-420页

作者： Buckdahn, Rainer Li, Juan Univ Bretagne Occidentale Dept Math F-29238 Brest 3 France Shandong Univ Weihai Sch Math & Stat Weihai 264209 Peoples R China

In this paper we investigate zero-sum two-player stochastic differential games whose cost functionals are given by doubly controlled reflected backward stochastic differential equations (RBSDEs) with two barriers. For admissible controls which can depend on the whole past and so include, in particular, information occurring before the beginning of the game, the games are interpreted as games of the type "admissible strategy" against "admissible control", and the associated lower and upper value functions are studied. A priori random, they are shown to be deterministic, and it is proved that they are the unique viscosity solutions of the associated upper and the lower Bellman-Isaacs equations with two barriers, respectively. For the proofs we make full use of the penalization method for RBSDEs with one barrier and RBSDEs with two barriers. For this end we also prove new estimates for RBSDEs with two barriers, which are sharper than those in Hamadene, Hassani (Probab Theory Relat Fields 132: 237-264, 2005). Furthermore, we show that the viscosity solution of the Isaacs equation with two reflecting barriers not only can be approximated by the viscosity solutions of penalized Isaacs equations with one barrier, but also directly by the viscosity solutions of penalized Isaacs equations without barrier.

关键词： Stochastic differential games Zero-sum games Value function Reflected backward stochastic differential equations dynamic programming principle Isaacs equations with obstacles Viscosity solution

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A note on negative dynamic programming for risk-sensitive control

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OPERATIONS RESEARCH LETTERS 2008年第5期36卷 531-534页

作者： Jaskiewicz, Anna Polish Acad Sci Inst Math PL-00956 Warsaw Poland

Negative dynamic programming for risk-sensitive control is studied. Under some compactness and semicontinuity assumptions the following results are proved: the convergence of the value iteration algorithm to the optimal expected total reward, the Borel measurability or upper semicontinuity of the optimal value functions, and the existence of an optimal stationary policy. (c) 2008 Elsevier B.V. All rights reserved.

关键词： Risk-sensitive control Borel state space dynamic programming principle

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Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations

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SIAM JOURNAL ON CONTROL AND OPTIMIZATION 2008年第1期47卷 444-475页

作者： Buckdahn, Rainer Li, Juan Univ Bretagne Occidentale Dept Math F-29285 Brest France Fudan Univ Sch Math Sci Shanghai 200433 Peoples R China Shandong Univ Weihai Dept Math Weihai 264200 Peoples R China

In this paper we study zero-sum two-player stochastic differential games with the help of the theory of backward stochastic differential equations (BSDEs). More precisely, we generalize the results of the pioneering work of Fleming and Souganidis [Indiana Univ. Math. J., 38 (1989), pp. 293-314] by considering cost functionals defined by controlled BSDEs and by allowing the admissible control processes to depend on events occurring before the beginning of the game. This extension of the class of admissible control processes has the consequence that the cost functionals become random variables. However, by making use of a Girsanov transformation argument, which is new in this context, we prove that the upper and the lower value functions of the game remain deterministic. Apart from the fact that this extension of the class of admissible control processes is quite natural and reflects the behavior of the players who always use the maximum of available information, its combination with BSDE methods, in particular that of the notion of stochastic "backward semigroups" introduced by Peng [BSDE and stochastic optimizations, in Topics in Stochastic Analysis, Science Press, Beijing, 1997], allows us then to prove a dynamic programming principle for both the upper and the lower value functions of the game in a straightforward way. The upper and the lower value functions are then shown to be the unique viscosity solutions of the upper and the lower Hamilton-Jacobi-Bellman-Isaacs equations, respectively. For this Peng's BSDE method is extended from the framework of stochastic control theory into that of stochastic differential games.

关键词： stochastic differential games value function backward stochastic differential equations dynamic programming principle viscosity solution

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One Kind of Corporate International Optimal Investment and Consumption Choice Problem

One Kind of Corporate International Optimal Investment and C...

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27th Chinese Control Conference

作者： Huang Zongyuan Wu Zhen Shandong Univ Sch Math Jinan 250100 Peoples R China

ISBN: (纸本)9787900719706

In this paper, we study a the specific Hyperbolic Absolute Risk Aversion (HARA) case of corporate international optimal Portfolio and consumption choice problem. The investor can invest his wealth bond (bank account). On the other hand, he can invest his money to a real project with production in a foreign country. Using the celebrated dynamical programming principle method we provide the explicit optimal investment and consumption solution and give some simulation results to illustrates the influence of the volatility parameters on the optimal choice.

关键词： Investment Choice dynamic programming principle Hamilton-Jacobi-Bellman Equations

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A d-person differential game with state space constraints

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APPLIED MATHEMATICS AND OPTIMIZATION 2007年第3期56卷 312-342页

作者： Ramasubramanian, S. Indian Stat Inst Stat Math Unit Bangalore 560059 Karnataka India

We consider a network of d companies ( insurance companies, for example) operating under a treaty to diversify risk. Internal and external borrowing are allowed to avert ruin of any member of the network. The amount borrowed to prevent ruin is viewed upon as control. Repayment of these loans entails a control cost in addition to the usual costs. Each company tries to minimize its repayment liability. This leads to a d-person differential game with state space constraints. If the companies are also in possible competition a Nash equilibrium is sought. Otherwise a utopian equilibrium is more appropriate. The corresponding systems of HJB equations and boundary conditions are derived. In the case of Nash equilibrium, the Hamiltonian can be discontinuous;there are d interlinked control problems with state constraints;each value function is a constrained viscosity solution to the appropriate discontinuous HJB equation. Uniqueness does not hold in general in this case. In the case of utopian equilibrium, each value function turns out to be the unique constrained viscosity solution to the appropriate HJB equation. Connection with Skorokhod problem is briefly discussed.

关键词： d-person differential game state space constraints nash equilibrium Utopian equilibrium dynamic programming principle system of HJB equations constrained viscosity solution semicontinuous envelope deterministic Skorokhod problem drift reflection

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