TU Berlin University

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 construct impulses which optimize a given performance 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 valuation of real options.

In this talk, I will give an introduction to stochastic 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 optimal investment problems in the presence of transaction costs.

This talk is based on joint work with Sören Christensen (Christian-Albrechts-University Kiel), Lukas Mich (Trier University), and Frank T. Seifried (Trier University).

References:
C. Belak, S. Christensen, F. T. Seifried: A General Verification Result for Stochastic Impulse Control Problems. SIAM Journal on Control and Optimization, Vol. 55, No. 2, pp. 627--649, 2017.
C. Belak, S. Christensen: Utility Maximisation in a Factor Model with Constant and Proportional Transaction Costs. Finance and Stochastics, Vol. 23, No. 1, pp. 29--96, 2019.
C. Belak, L. Mich, F. T. Seifried: Optimal Investment for Retail Investors with Floored and Capped Costs. Preprint, available at http://ssrn.com/abstract=3447346, 2019.

We aim to study the long time behaviour of the solution to a rough differential equation (in the sense of Lyons) driven by a random rough path. To do so, we use the theory of random dynamical systems. In a first step, we show that rough differential equations naturally induce random dynamical systems, provided the driving rough path has stationary increments. If the equation satisfies a strong form of stability, we can show that the solution admits an invariant measure.

This is joint work with I. Bailleul (Rennes) and M. Scheutzow (Berlin).    

Abstract: Kolmogorov backward equations related to stochastic evolution equations (SEE) in Hilbert space, driven by trace class Gaussian noise have been intensively studied in the literature. In this talk I discuss the extension to non trace class Gaussian noise in the particular case when the leading linear operator generates an analytic semigroup. This natural generalization leads to several complications, requiring new existence and uniqueness results for SEE with initial singularities and a new notion of an extended transition semigroup. This is joint work with Arnulf Jentzen and Ryan Kurniawan (ETH).

We consider a broker who has to place a large order which consumes a sizable part of average daily trading volume. By contrast to the previous literature, we allow the liquidity parameters of market depth and resilience to vary deterministically over the course of the trading period. The resulting singular optimal control problem is shown to be tractable by methods from convex analysis and, under

minimal assumptions, we construct an explicit solution to the scheduling problem in terms of some concave envelope of the resilience adjusted market depth.