An algorithm for numerical solution of discrete Hamilton-Jacobi-Bellman equations is proposed. The method begins with a suitable initial guess value of the solution,then finds a suitable matrix to linearize the system...An algorithm for numerical solution of discrete Hamilton-Jacobi-Bellman equations is proposed. The method begins with a suitable initial guess value of the solution,then finds a suitable matrix to linearize the system and constructs an iteration algorithm to generate the monotone sequence. The convergence of the algorithm for nonlinear discrete Hamilton-Jacobi-Bellman equations is proved. Some numerical examples are presented to confirm the effciency of this algorithm.展开更多
The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for ...The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.展开更多
Abstract--In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman （GHJB） equation through successively approximate it. Firstly, the GHJB equation is converted to an algebra...Abstract--In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman （GHJB） equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems. Index Terms--Generalized Hamilton-Jacobi-Bellman （HJB） equation, iterative method, nonlinear dynamic system, optimal control.展开更多
In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the sch...In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.展开更多
This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergenc...This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.展开更多
We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, wher...We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.展开更多
文摘An algorithm for numerical solution of discrete Hamilton-Jacobi-Bellman equations is proposed. The method begins with a suitable initial guess value of the solution,then finds a suitable matrix to linearize the system and constructs an iteration algorithm to generate the monotone sequence. The convergence of the algorithm for nonlinear discrete Hamilton-Jacobi-Bellman equations is proved. Some numerical examples are presented to confirm the effciency of this algorithm.
基金supported in partby National Science Foundation of China (10371088,10671144)National Basic Research Program of China(2007CB814903)+3 种基金Development Funds of Shanghai Higher Education (05D210)the Special Funds for Major Specialties of Shanghai Education Committee (T0401)Supported by Special Fund for the Excellent Young Teachers of Shanghai Higher Learning Institutions (ssd08029)the Research Program of Shanghai Normal University (SK200812)
文摘The passport option is a call option on the balance of a trading account. The option holder retains the gain from trading, while the issuer is liable for the net loss. In this article, the mathematical foundation for pricing the European passport option is established. The pricing equation which is a fully nonlinear equation is derived using the dynamic programming principle. The comparison principle, uniqueness and convexity preserving of the viscosity solutions of related H J13 equation are proved. A relationship between the passport and lookback options is discussed.
基金supported by the National Natural Science Foundation of China(U1601202,U1134004,91648108)the Natural Science Foundation of Guangdong Province(2015A030313497,2015A030312008)the Project of Science and Technology of Guangdong Province(2015B010102014,2015B010124001,2015B010104006,2018A030313505)
文摘Abstract--In this paper, a new iterative method is proposed to solve the generalized Hamilton-Jacobi-Bellman （GHJB） equation through successively approximate it. Firstly, the GHJB equation is converted to an algebraic equation with the vector norm, which is essentially a set of simultaneous nonlinear equations in the case of dynamic systems. Then, the proposed algorithm solves GHJB equation numerically for points near the origin by considering the linearization of the non-linear equations under a good initial control guess. Finally, the procedure is proved to converge to the optimal stabilizing solution with respect to the iteration variable. In addition, it is shown that the result is a closed-loop control based on this iterative approach. Illustrative examples show that the update control laws will converge to optimal control for nonlinear systems. Index Terms--Generalized Hamilton-Jacobi-Bellman （HJB） equation, iterative method, nonlinear dynamic system, optimal control.
文摘In this paper we propose a relaxation scheme for solving discrete HJB equations based on scheme II [1] of Lions and Mercier. The convergence of the new scheme has been established. Numerical example shows that the scheme is efficient.
文摘This paper presents a modified domain decomposition method for the numerical solution of discrete Hamilton-Jacobi-Bellman equations arising from a class of optimal control problems using diffusion models. A convergence theorem is established. Numerical results indicate the effectiveness and accuracy of the method.
基金supported by the NSF of China(11071144,11171187,11222110 and 71671104)Shandong Province(BS2011SF010,JQ201202)+4 种基金SRF for ROCS(SEM)Program for New Century Excellent Talents in University(NCET-12-0331)111 Project(B12023)the Ministry of Education of Humanities and Social Science Project(16YJA910003)Incubation Group Project of Financial Statistics and Risk Management of SDUFE
文摘We establish a new type of backward stochastic differential equations(BSDEs)connected with stochastic differential games(SDGs), namely, BSDEs strongly coupled with the lower and the upper value functions of SDGs, where the lower and the upper value functions are defined through this BSDE. The existence and the uniqueness theorem and comparison theorem are proved for such equations with the help of an iteration method. We also show that the lower and the upper value functions satisfy the dynamic programming principle. Moreover, we study the associated Hamilton-Jacobi-Bellman-Isaacs(HJB-Isaacs)equations, which are nonlocal, and strongly coupled with the lower and the upper value functions. Using a new method, we characterize the pair(W, U) consisting of the lower and the upper value functions as the unique viscosity solution of our nonlocal HJB-Isaacs equation. Furthermore, the game has a value under the Isaacs’ condition.
