Christoph Belak

Non-Smooth Verification for Impulse Control Problems

Stochastic impulse control problems are continuous-time optimization problems in which a stochastic system is controlled through finitely many impulses causing a discontinuous displacement of the state process. The objective is to choose the impulses optimally so as to maximize or minimize a reward or cost functional of the state process. This type of optimization problem arises in many branches of applied probability and economics such as optimal portfolio management under transaction costs, optimal forest harvesting, inventory control, and real options analysis.

In this talk, I will give an introduction to optimal impulse control and discuss classical solution techniques. I will then introduce a new method to solve impulse control problems based on superharmonic functions and a stochastic analogue of Perron's method, which allows to construct optimal impulse controls under a very general set of assumptions. Finally, I will show how the general results can be applied to a problem of optimal investment in the presence of constant and proportional transaction costs.

This talk is based on joint work with Sören Christensen (University of Hamburg) and Frank T. Seifried (University of Trier).

References

Belak, C., Christensen, S., Seifried, F.T. (2017). A General verification result for stochastic impulse control problems. SIAM Journal on Control and Optimization 55(2), pp. 627-649.

Belak, C., Christensen, C. (2017). Utility maximization in a factor model with constant and proportional costs. Preprint available at https://ssrn.com/abstract=2774697.

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