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Existence of an infinite-horizon optimal impulse consumption of a geometric Brownian motion with variable coefficients. (English) Zbl 1356.49067

Summary: We consider the problem of maximizing expected lifetime utility from consumption of a geometric Brownian motion with variable coefficients, in the presence of controlling costs with a fixed component. Under general assumptions on the utility function and the intervention costs our main result is to show that, if the discount rate is large enough, there always exists an optimal Markovian impulse control for this problem. We compute explicitly the optimal consumption in the case of constant coefficients of the process, linear utility and a two values discount rate. In this illustrative example the value function is not \(C^1\) and the verification theorems commonly used to characterize the optimal control cannot be applied.

MSC:

49N25 Impulsive optimal control problems
49N35 Optimal feedback synthesis
49J55 Existence of optimal solutions to problems involving randomness
49J40 Variational inequalities
49N90 Applications of optimal control and differential games
60J65 Brownian motion
93E20 Optimal stochastic control
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