# Humboldt-Universität zu Berlin - Mathematisch-Naturwissenschaftliche Fakultät - Institut für Mathematik

FS Stochastische Analysis und Stochastik der Finanzmärkte

## FS Stochastische Analysis und Stochastik der Finanzmärkte

### Bereich für Stochastik

#### P. BANK, Ch. BAYER, Ch. BELAK, D. BECHERER, P. FRIZ, U. HORST, D. KREHER

Das Seminar findet im Sommersemester 2022 in Präsenz an der TU Berlin, Institut für Mathematik, Raum MA 041 statt. Die entsprechende Ankündigung wird per e-mail versendet.

Zeit: Donnerstag, 16.30 Uhr und 17.45 Uhr

 28.04.2022 16.30 Uhr Fausto Gozzi  (LUISS University, Rom) Understanding the Time-Space Evolution of Economic Activities: Recent Mathematical Models and their application Abstract: The goal of this talk is to present some recent models on the time evolution of most important economic variables (e.g. consumption and capital) across different locations, taking into account space heterogeneity. In particular we focus on two recent papers looking at the macro level where there is one planner which, in a spatial Ramsey setting, maximizes utility across space with heterogeneous productivity in a deterministic (paper with R. Boucekkine, G. Fabbri, S. Federico) or in a stochastic setting (paper with M. Leocata). If time allows we will also introduce a mean field game model looking at the micro level where the agents move across space maximizing their own utility which also depends on the choices of the other agents. 28.04.2022 17.45 Uhr Frank Riedel (Universität Bielefeld, IMW) Efficient Allocations under Ambiguous Model Uncertainty Abstract: We investigate consequences of model uncertainty (or ambiguity) on ex ante efficient allocations in an exchange economy. The ambiguity we consider is embodied in the model uncertainty perceived by the decision maker: they are unsure what would be the appropriate probability measure to apply to evaluate contingent consumption contingent plans and keep in consideration a set of alternative probabilistic laws. We study the case where the typical consumer in the economy is ambiguity-averse with smooth ambiguity preferences and the set of priors $\mathcal{P}$ is point identified, i.e., the true law $p\in \mathcal{P}$ can be recovered empirically from observed events. Differently from the literature, we allow for the case where the aggregate risk is ambiguous and agents are heterogeneously ambiguity averse. Our analysis addresses, in particular, the full range of set-ups where under expected utility the Pareto efficient consumption sharing rule is a linear function of the aggregate endowment. We identify systematic differences ambiguity aversion introduces to optimal sharing arrangements in these environments and also characterize the representative consumer. Furthermore, we investigate the implications for the state-price function, in particular, the effect of heterogeneity in ambiguity aversion. 12.05.2022 16.30 Uhr Huilin Zhang  (Shandong University/HU Berlin) Well-posedness of path-dependent semilinear parabolic master equations Abstract: Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in stochastic games and control. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in applications (e.g. option pricing, risk control). We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupire’s vertical derivative, and applying forward-backward stochastic system argument. This talk is based on a joint work with Shanjian Tang 12.05.2022 17.45 Uhr Thomas Kruse (Justus-Liebig-Universität Gießen)  Multilevel Picard approximations for high-dimensional semilinear parabolic PDEs and further applications  Abstract: We present the multilevel Picard approximation method for high-dimensional semilinear parabolic PDEs which in particular appear in the pricing of financial derivatives. A key idea of our method is to combine multilevel approximations with Picard fixed-point approximations. We prove in the case of semilinear heat equations with Lipschitz continuous nonlinearities that the computational effort of the proposed method grows polynomially both in the dimension and in the reciprocal of the required accuracy. Moreover, we present further applications of the multilevel Picard approximation method and illustrate its efficiency by means of numerical simulations. The talk is based on joint works with Weinan E, Martin Hutzenthaler, Arnulf Jentzen, Tuan Nguyen and Philippe von Wurstemberger. 02.06.2022 16.30 Uhr Roxana Dumitrescu (Kings College London)  Control and stopping mean-field games: the linear programming approach Abstract: n this talk, we present recent results on the linear programming approach to mean-field games in a general setting. This relaxed control approach allows to prove existence results under weak assumptions, and lends itself well to numerical implementation. We consider mean-field game problems where the representative agent chooses both the optimal control and the optimal time to exit the game, where the instantaneous reward function and the coefficients of the state process may depend on the distribution of the other agents. Furthermore, we establish the equivalence between mean-field games equilibria obtained by the linear programming approach and the ones obtained via other approaches used in the previous literature. We then present a fictious play algorithm to approximate the mean-field game population dynamics in the context of the linear programming approach. Finally, we give an application of the theoretical and numerical contributions introduced in the first part of the talk to an entry-exit game in electricity markets. The talk is based on several works, joint with R. Aid, G. Bouveret, M. Leutscher and P. Tankov. 16.06.2022 16.30 Uhr Jörn Sass (TU Kaiserslautern)   Convergence of Optimal Strategies in a Multivariate Black Scholes Type Market with Model Uncertainty on the Drift Abstract: lt is a by now classical observation that in a (realistic) financial market (model) simple portfolio strategies can outperform more sophisticated optimized portfolio strategies. For example, in a one-period setting, the equal weight or 1/N-strategy often provides more stable results than mean-variance-optimal strategies. This is due to the fact that a good estimation of the mean returns is not possible for volatile financial assets. Pflug, Pichler and Wozabel (2012) gave a rigorous explanation of this observation by showing that for increasing uncertainty on the means the equal weight strategy becomes optimal in a mean-variance setting which is due to its robustness. We aim at extending this result to continuous-time strategies in a multivariate Black-Scholes type market. To this end we investigate how optimal trading strategies for maximizing expected utility of terminal wealth under CRRA utility behave when we have Knightian uncertainty on the drift, meaning that the only information is that the drift parameter lies in a so-called uncertainty set. The investor takes this into account by considering that the worst possible drift within this set may occur. In this setting we can show that a minimax theorem holds which enables us to find the worst-case drift and the optimal robust strategy quite explicitly. This again allows us to derive the limits when uncertainty increases and hence to show that a uniform strategy is asymptotically optimal. We also discuss the extension to a financial market with a stochastic drift process, combining the worst-case approach with filtering techniques. This leads to local optimization problems, and the resulting optimal strategy needs to be updated continuously in time. We carry over the minimax theorem for the local optimization problems and derive the optimal strategy. In this setting we show how an ellipsoidal uncertainty set can be defined based on filtering techniques and we demonstrate that investors need to choose a robust strategy to be able to profit from additional information. 16.06.2022 17.45 Uhr Paul Schneider (University of Lugano, USI) Optimal Investment and Equilibrium Pricing under Ambiguity Abstract: We consider portfolio selection under nonparametric alpha-maxmin ambiguity in the neighbourhood of a reference distribution. We show strict concavity of the portfolio problem under ambiguity aversion. Implied demand functions are nondifferentiable, resemble observed bid-ask spreads, and are consistent with existing para- metric limiting participation results under ambiguity. Ambiguity seekers exhibit a discontinuous demand function, implying an empty set of reservation prices. If agents have identical, or sufficiently similar prior beliefs, the first best equilibrium is no trade. Simple conditions yield the existence of a Pareto-efficient second-best equilibrium that reconciles many observed phenomena in financial markets, such as liquidity dry-ups, portfolio inertia, and negative risk premia. Ein joint paper mit Michail Anthropelos. 07.07.2022 16.30 Uhr HU Berlin, Raum 1.115 Ludowig Tangpi (Princeton University)  A probabilistic approach to the convergence of large population games to mean field games: Games in the strong formulation  Abstract: This talk will discuss the convergence problem of mean field games in the strong formulation. The specific example of a price impact model will be presented. If time allows it, an application to stochastic optimal transport will be discussed to showcase the relevance of the method beyond mean field games. 07.07.2022 17.45 Uhr HU Berlin, Raum 1.115 Ludowig Tangpi (Princeton University)  A probabilistic approach to the convergence of large population games to mean field games: Games in the weak formulation Abstract: This talk will discuss the convergence problem of mean field games in the weak formulation. A specific case study will be discussed. Time permitting, we will finish with an outlook on the case of players in non-symmetric interaction. 14.07.2022 16.30 Uhr Ralf Wunderlich (BTU Cottbus).    Achtung! Vortrag entfällt !! Stochastic Optimal Control of Heating Systems with a Geothermal Energy Storage Abstract: Thermal storage facilities help to mitigate and to manage temporal fluctuations of heat supply and demand for heating and cooling systems of single buildings as well as for district heating systems. We focus on a heating system equipped with several heat production units using also renewable energies and an underground thermal storage. The thermal energy is stored by raising the temperature of the soil inside that storage. It is charged and discharged via heat exchanger pipes filled with a moving fluid. Besides the numerous technical challenges and the computation of the spatio-temporal temperature distribution in the storage also economic issues such as the cost-optimal control and management of such systems play a central role. The latter leads to challenging mathematical optimization problems. There we incorporate uncertainties about randomly fluctuating renewable heat production, environmental conditions driving the heat demand and supply. The dynamics of the controlled state process is governed by a PDE, a random ODE, and SDEs modeling energy prices and the difference between supply and demand. Model reduction techniques are adopted to cope with the PDE describing the spatio-temporal temperature distribution in the geothermal storage. Finally, time- discretization leads to a Markov decision process for which we apply numerical methods to determine a cost-optimal control. This is joint work with Paul Honore Takam (BTU Cottbus-Senftenberg) and Olivier Menoukeu Pamen (AIMS Ghana, University of Liverpool). 14.07.2022 17.45 Uhr Julio Backhoff (U Wien) On the martingale projection of a Brownian motion given initial and terminal marginals Abstract: In one of its dynamic formulations, the optimal transport problem asks to determine the stochastic process that interpolates between given initial and terminal marginals and is as close as possible to the constant-speed particle. Typically, the answer to this question is a stochastic process with constant-speed trajectories. We explore the analogue problem in the setting of martingales, and ask: what is the martingale that interpolates between given initial and terminal marginals and is as close as possible to the constant-volatility particle? The answer this time is a process called 'stretched Brownian motion', a generalization of the well-known Bass martingale. After introducing this process and discussing some of its properties, I will present current work in progress (with Mathias Beiglböck, Walter Schachermayer and Bertram Tschiderer) concerning the fine structure of stretched Brownian motions.

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Frau Sabine Bergmann

bergmann@math.hu-berlin.de
Telefon: 2093 45450
Telefax: 2093 45451