# Stochastic impulse control with discounted and ergodic optimization criteria: A comparative study for the control of risky holdings

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## Abstract

We consider a single-asset investment fund that in the absence of transactions costs would hold a constant amount of wealth in the risky asset. In the presence of market frictions wealth is allowed to fluctuate within a control band: Its upper (lower) boundary is chosen so that gains (losses) from adjustments to the target minus (plus) fixed plus proportional transaction costs maximize (minimize) a power utility function. We compare stochastic impulse control policies derived via ergodic and discounted optimization criteria. For the solution of the ergodic problem we use basic tools from the theory of diffusions whereas the discounted problem is solved after being characterized as a system of quasi-variational inequalities. For both versions of the problem, derivation of the control bands pertains to the numerical solution of a system of nonlinear equations. We solve numerous such systems and present an extensive comparative sensitivity analysis with respect to the parameters that characterize investor’s preferences and market behavior.

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