Abstract
We consider the minimax impulse control problem in finite horizon, when the cost functions are positive and not bounded from below with a strictly positive constant. We show existence of value function of the problem. Moreover, the value function is characterized as the unique viscosity solution of Hamilton-Jacobi-Bellman-Isaacs equation. This problem is in relation with an application in mathematical finance.
Highlights
In this paper we study a minimax impulse control problem with finite horizon.Minimax impulse control problems appear in many practical situations
For deterministic autonomous systems with infinite horizon, optimal impulse control problems were studied in [2], and optimal control problems with continuous, switching, and impulse controls were studied by the author [24]
We show that the value function of the problem is associated of deterministic functions v which is the unique solution of the following system of HJBI equation:
Summary
In this paper we study a minimax impulse control problem with finite horizon. Minimax impulse control problems appear in many practical situations. Differential games with switching strategies in finite and infinite duration were studied [26, 27]. J. Yong, in [28], studies differential games where one person uses an impulse control and other uses continuous controls. Player I is the Central Bank aims at selecting an admissible impulse control defined by a double sequence t1, ..., tk, ..., ξ1, ..., ξk, ..., k ∈ IN ∗ = IN \{0}, where tk are the strategy, tk ≤ tk+1 and ξk ∈ IR the control at time tk of the jumps in y(tk). Robust control, differential games, quasi-variational inequality, viscosity solution
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