The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we st...
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The present paper considers a stochastic optimal control problem, in which the cost function is defined through a backward stochastic differential equation with infinite horizon driven by G-Brownian motion. Then we study the regularities of the value function and establish the dynamic programming principle. Moreover, we prove that the value function is the unique viscosity solution of the related Hamilton-Jacobi-Bellman-Isaacs (HJBI) equation.
In this paper, we consider an inventory control problem with a nonlinear evolution and a jump-diffusion demand model. This work extends the earlier inventory model proposed by Benkherouf and Johnson [Math. Methods Ope...
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In this paper, we consider an inventory control problem with a nonlinear evolution and a jump-diffusion demand model. This work extends the earlier inventory model proposed by Benkherouf and Johnson [Math. Methods Oper. Res., 76 (2012), pp. 377-393] by including a general jump process. However, as those authors note, their techniques are not applicable to models with demand driven by jump-diffusion processes with drift. Therefore, the combination of diffusion and general compound Poisson demands is not completely solved. From the dynamic programming principle, we transform the problem into a set of quasi-variational inequalities (Q.V.I.). The difficulty arises when solving the Q.V.I. because the second derivative and integration term appear in the same inequality. Our technique is to construct a set of coupled auxiliary functions. Then, an analytical study of the Q.V.I. implies the existence and uniqueness of an (s, S) policy.
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.
This paper is concerned with the singular linear quadratic (SLQ) optimal control problem for stochastic nonregular descriptor systems with time-delay. By means of some reasonable assumptions and a series of equivalent...
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This paper is concerned with the singular linear quadratic (SLQ) optimal control problem for stochastic nonregular descriptor systems with time-delay. By means of some reasonable assumptions and a series of equivalent transformations, the problem is finally transformed into a positive linear quadratic (LQ) problem for standard stochastic systems. Then dynamic programming principle is used to establish the solvability of the original problem, and the desired explicit presentation of the optimal controller is given in terms of matrix iterative form. The results due to Feng etal. are generalized and improved. As an application, a numerical example is presented to demonstrate the efficiency of the proposed approach.
As a functional device, dielectric elastomer balloon with surface electrodes operates under impressed pressure and voltage. The unavoidable pressure disturbance will induce prominent vibration around the equilibrium p...
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As a functional device, dielectric elastomer balloon with surface electrodes operates under impressed pressure and voltage. The unavoidable pressure disturbance will induce prominent vibration around the equilibrium position, which may deteriorate its operating performance and accelerate its failure. This manuscript concentrates on the vibration suppression by slightly adjusting the voltage under certain given constraint. The displacement perturbation is governed by a nonlinear stochastic differential equation which is determined by expanding technique at equilibrium points. The pressure disturbance described by ideal Gaussian white noise is a parametric excitation and the function of control voltage directly multiplies with a function of the displacement perturbation. This comes down to an optimal bounded parametric control problem. The optimal control is first formally determined by the extremum condition in dynamicprogramming equation. The nonlinear dynamicprogramming equation is then approximately solved by the pseudo-inverse algorithm. The good control effectiveness and high robustness to the intensity of pressure disturbance are verified by numerical examples. Particularly, the optimal control strategy with not too small bound can effectively avoid the interwell motion for the cases with bistable potential.
The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The res...
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The authors prove a sufficient stochastic maximum principle for the optimal control of a forward-backward Markov regime switching jump diffusion system and show its connection to dynamic programming principle. The result is applied to a cash flow valuation problem with terminal wealth constraint in a financial market. An explicit optimal strategy is obtained in this example.
We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the...
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We consider a stochastic optimal control problem in a market model with temporary and permanent price impact, which is related to an expected utility maximization problem under finite fuel constraint. We establish the initial condition fulfilled by the corresponding value function and show its first regularity property. Moreover, we can prove the existence and uniqueness of an optimal strategy under rather mild model assumptions. This will then allow us to derive further regularity properties of the corresponding value function, in particular its continuity and partial differentiability. As a consequence of the continuity of the value function, we will prove a dynamic programming principle without appealing to the classical measurable selection arguments. This permits us to establish a tight relation between our value function and a nonlinear parabolic degenerated Hamilton-Jacobi-Bellman (HJB) equation with singularity. To conclude, we show a comparison principle, which allows us to characterize our value function as the unique viscosity solution of the HJB equation.
We establish regularity for functions satisfying a dynamicprogramming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity an...
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We establish regularity for functions satisfying a dynamicprogramming equation, which may arise for example from stochastic games or discretization schemes. Our results can also be utilized in obtaining regularity and existence results for the corresponding partial differential equations. (C) 2018 Elsevier Masson SAS. All rights reserved.
This study develops a numerical method to find optimal ergodic (long-run average) dividend strategies in a regime-switching model. The surplus process is modelled by a regime-switching process subject to liability con...
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This study develops a numerical method to find optimal ergodic (long-run average) dividend strategies in a regime-switching model. The surplus process is modelled by a regime-switching process subject to liability constraints. The regime-switching process is modelled by a finite-time continuous-time Markov chain. Using the dynamic programming principle, the optimal long-term average dividend payment is a solution to the coupled system of Hamilton-Jacobi-Bellman equations. Under suitable conditions, the optimal value of the long-term average dividend payment can be determined by using an invariant measure. However, due to the regime switching, getting the invariant measure is very difficult. The objective is to design a numerical algorithm to approximate the optimal ergodic dividend payment strategy. By using the Markov chain approximation techniques, the authors construct a discrete-time controlled Markov chain for the approximation, and prove the convergence of the approximating sequences. A numerical example is presented to demonstrate the applicability of the algorithm.
The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal...
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The mathematical concept of multiplier robust control is applied to a dam operation problem, which is an urgent issue on river water environment, as a new industrial application of stochastic optimal control. The goal of the problem is to find a fit-for-purpose and environmentally sound operation policy of the flow discharge from a dam so that overgrowth of the harmful algae Cladophora glomerataKutzing in its downstream river is effectively suppressed. A minimal stochastic differential equation for the algae growth dynamics with uncertain growth rate is first presented. The performance index to be maximized by the operator of the dam while minimized by nature is formulated within the framework of differential games. The dynamic programming principle leads to a Hamilton-Jacobi-Bellman-Isaacs equation whose solution determines the worst-case optimal operation policy of the dam, ie, the policy that the operator wants to find. Application of the model to overgrowth suppression of Cladophora glomerataKutzing just downstream of a dam in a Japanese river is then carried out. Values of the model parameters are identified with which the model successfully reproduces the observed population dynamics. A series of numerical experiments are performed to find the most effective operation policy of the dam based on a relaxation of the current policy.
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