Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since ...
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作者：
Khlopin, DmitryRussian Acad Sci
Krasovskii Inst Math & Mech Ural Branch 16 S Kovalevskaja St Ekaterinburg 620990 Russia Ural Fed Univ
Inst Math & Comp Sci Chair Appl Math 4 Turgeneva St Ekaterinburg 620083 Russia
For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and t...
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For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long-run average cost as the time horizon tends to infinity and the limit of value functions of -discounted games as the discount tends to zero. We prove that the dynamic programming principle for value functions directly leads to the Tauberian theorem-that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and superoptimality principles (the dynamic programming principle). With their aid, we obtain certain inequalities on asymptotics of sub- and supersolutions, which lead to the Tauberian theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point;a very simple stochastic game model is also considered.
We study optimal control of the general stochastic McKean-Vlasov equation. Such a problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players...
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We study optimal control of the general stochastic McKean-Vlasov equation. Such a problem is motivated originally from the asymptotic formulation of cooperative equilibrium for a large population of particles (players) in mean-field interaction under common noise. Our first main result is to state a dynamic programming principle for the value function in the Wasserstein space of probability measures, which is proved from a flow property of the conditional law of the controlled state process. Next, by relying on the notion of differentiability with respect to probability measures due to [P.L. Lions, Cours au College de France : Theorie des jeux a champ moyens, (2012), pp. 2006-2012] and Ito's formula along a flow of conditional measures, we derive the dynamicprogramming Hamilton-Jacobi-Bellman equation and prove the viscosity property together with a uniqueness result for the value function. Finally, we solve explicitly the linear-quadratic stochastic McKean-Vlasov control problem and give an application to an interbank systemic risk model with common noise.
A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascad...
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A new kind of multiple stochastic optimal stopping problem is formulated and its associated recursive variational inequalities are derived. We show that these variational inequalities can be solved exactly in a cascading manner. The relevance of the present problem in analyzing animal migration, which is an ecologically important problem, is also briefly discussed.
Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since ...
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Dealing with high-dimensional feedback control problems is a difficult task when the classical dynamic programming principle is applied. Existing techniques restrict the application to relatively low dimensions since the discretizations typically suffer from the curse of dimensionality. In this paper we introduce a novel approximation technique for the value function of an infinite horizon optimal control. The method is based on solving optimal open loop control problems on a finite horizon with a sampling of the global value function along the generated trajectories. For the interpolation we choose a kernel orthogonal greedy strategy, because these methods are able to produce extreme sparse surrogates and enable rapid evaluations in high dimensions. Two numerical examples prove the performance of the approach and show that the method is able to deal with high-dimensional feedback control problems, where the dimensionality prevents the approximation by most existing methods. (C) 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.
This thesis consists of three parts which deal with quasi linear parabolic PDE on a junction, stochastic diffusion on a junction and stochastic control on a junction with control at the junction point. We begin in the...
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This thesis consists of three parts which deal with quasi linear parabolic PDE on a junction, stochastic diffusion on a junction and stochastic control on a junction with control at the junction point. We begin in the first Chapter by introducing and studying a new class of non degenerate quasi linear parabolic PDE on a junction, satisfying a Neumann (or Kirchoff) non linear and non dynamical condition at the junction point. We prove the existence and the uniqueness of a classical solution. The main motivation of studying this new mathematical object is the analysis of stochastic control problems with control at the junction point, and the characterization of the value function of the problem in terms of Hamilton Jacobi Bellman equations. For this end, in the second Chapter we give a proof of the existence of a diffusion on a junction. The process is characterized by its local time at the junction point, whose quadratic approximation is centrally related to the ellipticty assumption of the second order terms around the junction point. We then provide an It 's formula for this process. Thanks to the previous results, in the last Chapter we study a problem of stochastic control on a junction, with control at the junction point. The set of controls is the set of the probability measures (admissible rules) satisfying a martingale problem. We prove the compactness of the admissible rules and the dynamic programming principle.
This paper is concerned with a kind of optimal portfolio and consumption choice problem, where an investor can invest his wealth in a trade project and foreign exchange deposit. The trade project earns profit by buyin...
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This paper is concerned with a kind of optimal portfolio and consumption choice problem, where an investor can invest his wealth in a trade project and foreign exchange deposit. The trade project earns profit by buying the merchandise and selling it with a higher price. The bank pays at an interest rate for any deposit, and vice takes at a large rate for any loan. The optimal strategy is obtained by Hamilton-Jacobi-Bellman(HJB) equation, which is derived from dynamic programming principle. For the specific Hyperbolic Absolute Risk Aversion(HARA) case, we get the explicit form of optimal portfolio and consumption solution, and we give some simulation results.
This paper is concerned with the optimal control problem, where the recursive cost functional is defined as one of the solution to a controlled fully coupled forward-backward stochastic differential equation (FBSDE), ...
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ISBN:
(纸本)9789881563958
This paper is concerned with the optimal control problem, where the recursive cost functional is defined as one of the solution to a controlled fully coupled forward-backward stochastic differential equation (FBSDE), and the control domain is convex. Two different approaches-dynamic programming principle (DPP) and maximum principle (MP)-are applied to solve the problem and the relationship between them are studied. Under some differentiable assumptions, relations among the adjoint processes, the value function and the generalized Hamiltonian function are proved, whereas the diffusion term of the forward equation is independent of the state variable z. The general case for the problem is open. A linear example is discussed as the illustration of our main result.
In this paper, we study zero-sum two-player stochastic differential games in which the state equations are competing Brownian particles and the cost functional is defined by generalized backward stochastic differentia...
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In this paper, we study zero-sum two-player stochastic differential games in which the state equations are competing Brownian particles and the cost functional is defined by generalized backward stochastic differential equations with more than one increasing process. After we study the regularity of competing Brownian particles, we establish the dynamic programming principle for the upper and lower value functions and show that these are the unique viscosity solution of the associated upper and lower Isaacs' equations, which are fully nonlinear parabolic partial differential equations with nonlinear Neumann boundary conditions.
The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financi...
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The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the continuous-time martingale transport on the Skorokhod space of cadlag paths. Similar to the classical setting of optimal transport, we introduce different dual problems and establish the corresponding dualities by a crucial use of the S-topology and the dynamic programming principle. (C) 2016 Elsevier B.V. All rights reserved.
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