Approximation of the value function of an impulse control problem of Piecewise Deterministic Markov Processes

We consider an infinite-horizon discounted optimal control problem for piecewise deterministic Markov processes, where a piecewise open-loop control acts continuously on the jump dynamics and on the deterministic flow. For this class of control problems, the value function can in general be characterized as the unique viscosity solution to the corresponding Hamilton−Jacobi−Bellman equation. We prove that the value function can be represented by means of a backward stochastic differential equation (BSDE) on infinite horizon, driven by a random measure and with a sign constraint on its martingale part, for which we give existence and uniqueness results. This probabilistic representation is known as nonlinear Feynman−Kac formula. Finally we show that the constrained BSDE is related to an auxiliary dominated control problem, whose value function coincides with the value function of the original non-dominated control problem.

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OPTIMAL DIVIDEND–REINSURANCE WITH TWO TYPES OF PREMIUM PRINCIPLES

Probability in the Engineering and Informational Sciences ◽

10.1017/s0269964815000352 ◽

2015 ◽

Vol 30 (2) ◽

pp. 224-243 ◽

Cited By ~ 8

Author(s):

Hui Meng ◽

Ming Zhou ◽

Tak Kuen Siu

Keyword(s):

Numerical Analysis ◽

Impulse Control ◽

Value Function ◽

Proportional Transaction Costs ◽

Ruin Time ◽

Closed Form Solutions ◽

Impulse Control Problem ◽

Dividend Payments ◽

Optimal Dividend ◽

The Value Function

A combined optimal dividend/reinsurance problem with two types of insurance claims, namely the expected premium principle and the variance premium principle, is discussed. Dividend payments are considered with both fixed and proportional transaction costs. The objective of an insurer is to determine an optimal dividend–reinsurance policy so as to maximize the expected total value of discounted dividend payments to shareholders up to ruin time. The problem is formulated as an optimal regular-impulse control problem. Closed-form solutions for the value function and optimal dividend–reinsurance strategy are obtained in some particular cases. Finally, some numerical analysis is given to illustrate the effects of safety loading on optimal reinsurance strategy.

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Hereditary Portfolio Optimization with Taxes and Fixed Plus Proportional Transaction Costs—Part I

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2007/82753 ◽

2007 ◽

Vol 2007 ◽

pp. 1-33 ◽

Cited By ~ 1

Author(s):

Mou-Hsiung Chang

Keyword(s):

Portfolio Optimization ◽

Impulse Control ◽

Optimization Problem ◽

Value Function ◽

Infinite Time ◽

Proportional Transaction Costs ◽

Infinite Dimensional ◽

Impulse Control Problem ◽

Portfolio Optimization Problem ◽

The Value Function

This is the first of the two companion papers which treat an infinite time horizon hereditary portfolio optimization problem in a market that consists of one savings account and one stock account. Within the solvency region, the investor is allowed to consume from the savings account and can make transactions between the two assets subject to paying capital gain taxes as well as a fixed plus proportional transaction cost. The investor is to seek an optimal consumption-trading strategy in order to maximize the expected utility from the total discounted consumption. The portfolio optimization problem is formulated as an infinite dimensional stochastic classical-impulse control problem. The quasi-variational HJB inequality (QVHJBI) for the value function is derived in this paper. The second paper contains the verification theorem for the optimal strategy. It is also shown there that the value function is a viscosity solution of the QVHJBI.

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On gradual-impulse control of continuous-time Markov decision processes with exponential utility

Advances in Applied Probability ◽

10.1017/apr.2020.64 ◽

2021 ◽

Vol 53 (2) ◽

pp. 301-334

Author(s):

Xin Guo ◽

Aiko Kurushima ◽

Alexey Piunovskiy ◽

Yi Zhang

Keyword(s):

Control Problem ◽

Markov Decision Processes ◽

Continuous Time ◽

Impulse Control ◽

Decision Processes ◽

Exponential Utility ◽

Impulse Control Problem ◽

Markov Decision ◽

The Value Function ◽

Selection Of

AbstractWe consider a gradual-impulse control problem of continuous-time Markov decision processes, where the system performance is measured by the expectation of the exponential utility of the total cost. We show, under natural conditions on the system primitives, the existence of a deterministic stationary optimal policy out of a more general class of policies that allow multiple simultaneous impulses, randomized selection of impulses with random effects, and accumulation of jumps. After characterizing the value function using the optimality equation, we reduce the gradual-impulse control problem to an equivalent simple discrete-time Markov decision process, whose action space is the union of the sets of gradual and impulsive actions.

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