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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, D. BECHERER, P. FRIZ, U. HORST, D. KREHER, N. PERKOWSKI

Ort: TU Berlin,
Institut für Mathematik,
Straße des 17. Juni 136, 10623 Berlin
Raum MA 042

Zeit: Donnerstag, 16 Uhr/17 Uhr c.t.

 25.04.2019 (16 Uhr c.t.) Neofytos Rodosthenous (Queen Mary University of London) A two-dimensional bounded-variation stochastic control problem for the management of debt ratio in a regime-switching economy ACHTUNG: Vortrag findet im Raum MA 042 statt   Abstract: This work is motivated by the problem of a government wishing to control the country's debt-to-GDP ratio. The debt-to-GDP ratio evolves stochastically and the interest on debt is affected by an N-state continuous-time Markov chain, representing the country's credit ratings. The debt-to-GDP ratio can be reduced through fiscal interventions policies or increased by public investments. The government aims at choosing a policy minimising the total expected cost of having debt and fiscal interventions counterbalanced by the gain from public investments. Mathematically, this is formulated as a two-dimensional bounded-variation stochastic control problem, that we solve explicitly via its connection to an associated Dynkin game. This is joint work with Giorgio Ferrari. 09.05.2019 (16 Uhr c.t.) David Besslich (TU Berlin) Meyer-\sigma-fields in stochastic control: Frontrunning in optimal liquidation, Irreversible Investment with inventory risk   Abstract: We propose to use Meyer-sigma-elds as a exible tool to model information ow. We illu- strate the possibilities of this approach in a simple limit order book model in which we formulate a rather natural optimal liquidation problem. For such a problem, assessing the short-term order ow is clearly of crucial importance. This is most obvious when a trader (or trading algorithm) seeking to sell a large position receives signals about soon to be posted market sell orders which lead to the decision to forestall by front running the impending order. We show how Meyer--elds can be used to model this continuous- time information ow and how this problem can be reformulated as an irreversible investment problem with inventory risk. We then proceed to explain how such problems can be solved in great generality. To illustrate this by an explicitly worked out case study, we specify an irreversible investment problem with a compound Poisson process, where we nd optimality of ladlag controls whose jumps from the left re ect the agent's reaction to her signals and whose jumps from the right her reaction to the fully revealed shock. As signal quality varies the optimal policies are found to interpolate from the classical predictable controls operating without signals to optional ones that fully account for any exogenous shock as it happens. Joint work with Peter Bank (TU Berlin). 09.05.2019 (17 Uhr c.t.) Luciano Campi (LSE) N-player games and mean-field games with smooth dependence on past absorptions   Abstract: Mean-field games with absorption is a class of games, that have been introduced in Campi and Fischer (2018) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, a direct dependence on past absorptions has been introduced in the drift of players' state dynamics. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth. Therefore, the mean-field interaction among the players takes place in two ways: via the empirical sub-probability measure of the surviving players and through a process representing the fraction of past absorptions over time. Moreover, relaxing the boundedness of the coefficients allows for more realistic dynamics for players' private states. We prove existence of solutions of the mean-field game in strict as well as relaxed feedback form. Finally, we show that such solutions induce approximate Nash equilibria for the N-player game with vanishing error in the mean-field limit as $N \to \infty$. This talk is based on a joint work with M. Ghio and G. Livieri (SNS Pisa). 10.05.2019 (13:30 - 15:00 Uhr c.t.) + 13.05. 2019 (14:30 - 16:00 Uhr c.t.) Julien Guyon (Bloomberg, New York) Außerplanmäßiger Minikurs (Straße des 17. Juni 136, MA 748) The particle method for smile calibration   Abstract: - Single asset models: Calibration of stochastic local volatility (SLV) models; McKean SDEs and the particle method; How to make it fast; Adding stochastic interest rates and stochastic dividend yield; Calibration of path-dependent volatility (PDV) models - Multi-asset models: Calibration of local correlation (LC) models; Calibration of cross-dependent volatility (CDV) models - Theory: McKean SDEs and the propagation of chaos 23.05.2019 (16 Uhr c.t.) Sam Cohen (University of Oxford) Multi-armed bandits with uncertainty   Abstract: Making good decisions based on estimates is difficult, but of clear importance in many applications. This is particularly the case when the decisions made will affect the information available in the future. Formally, this means that the filtration of our probkem is not fixed in advance, but depends on the control used. We will consider the 'simplest' problem of this type, a multi-armed bandit problem, while taking account of uncertainty aversion. We will see that an extension of the classical Gittins' index approach is possible in this framework, despite many dynamic consistency issues. 23.05.2019 (17 Uhr c.t.) Daniel Lacker (Columbia University) Beyond mean field limits: Local dynamics for large sparse networks of interacting processes   Abstract:  We study large systems of stochastic processes (particles), in discrete or continuous time, in which each particle is associated with a vertex in a graph and interacts only with its neighbors. It is known that when the graph is complete and the number of particles grows to innity, the system is well-approximated by a nonlinear (McKean-Vlasov) process, which describes the behavior of one typical particle. For general (sparse) graphs, however, the system is no longer exchangeable, and the mean eld approximation fails. Nevertheless, for a broad class of 􀀀locallytree - like graphs, we show that a single particle and its nearest neighbors are characterized by an autonomous evolution we call the 􀀀localdynamics Joint work with Kavita Ramanan and Ruoyu Wu. 06.06.2019 (17 Uhr c.t.) Giorgio Ferrari (Universität Bielefeld) On an Optimal Dividend Problem with Stochastic Discount   Abstract:  We study an optimal dividend problem with stochastic discount. Dividends can be paid to shareholder at unrestricted rates, are discounted at a stochastic time-varying rate, and the aim is to maxi-mize the total expected ow of discounted dividends, until a possible default time. From the mathematical point of view, we model the problem as a two-dimensional singular stochastic control problem, in which the state variable consists of the surplus process, and of a stochastic interest rate driving the discount factor. The rst evolves as a linearly controlled drifted Brownian motion, while the second is a CIR process. We solve the problem by relying it to a fully two-dimensional optimal stopping problem where the state process is a reected Brownian motion with drift and a CIR process. By using almost exclusively probabilistic arguments, we show that the optimal dividend strategy is triggered by a monotone free boundary and that the value function is a classical solution (in the a.e. sense) to the associated dynamic programming equation. This talk is based on a joint work with E. Bandini, T. De Angelis, and Fausto Gozzi. 20.06.2019 (16 Uhr c.t.) Mykhaylo Shkolnivov (Princeton University) From systemic risk to supercooling and back   Abstract:  I will explain how structural models of default cascades in the systemic risk literature naturally lead to the supercooled Stefan problem of mathematical physics. On the one hand, this connection allows us to uncover a notion of global solutions to the supercooled Stefan problem, which we analyze in detail. On the other hand, the supercooled Stefan problem formulation allows to provide a truly intrinsic denition of systemic crises and to characterize the fragile states of the economy. Time permitting, I will also explain the network and game extensions of the problem. Based on a series of works with Francois Delarue and Sergey Nadtochiy. 20.06.2019 (17 Uhr c.t.) David Prömel (University of Oxford) MOT Duality and Robust Finance   Abstract:  Without assuming any probabilistic price dynamics, we consider a frictionless nancial market given by the Skorokhod space, on which some nancial options are liquidly traded. In this model-free setting we show various pricing-hedging dualities and the analogue of the fundamental theorem of asset pricing. For this purpose we study the corresponding martingale optimal transport (MOT) problem: We obtain a dual representation of the Kantorovich functional (super-replication functional) dened for functions (nancial derivatives) on the Skorokhod space using quotient sets (hedging sets). Our representation takes the form of a Choquet capacity generated by martingale measures satisfying additional constraints to ensure compatibility with the quotient sets. The talk is based on a joint work with Patrick Cheridito, Matti Kiiski and H. Mete Soner. 04.07.2019 (16 Uhr c.t.) Torben Koch (Universität Bielefeld) Optimal Installation of Solar Panels with Price Impact: a Solvable Singular Stochastic Control Problem   Abstract: In this talk we consider a price-maker company which generates electricity and sells it in the spot market. The company can increase its level of installed power by irreversible installations of solar panels. In absence of any actions of the company, the electricity's spot price evolves as an {Ornstein-Uhlenbeck process}, and therefore it has a mean-reverting behavior. The current level of the company's installed power has a permanent impact on the electricity's price and affects its mean-reversion level. The company aims at maximizing the total expected profits from selling electricity in the market, net of the total expected proportional costs of installation. This problem is modeled as a {two-dimensional degenerate singular stochastic control problem} in which the installation strategy is identified as the company's control variable. We follow a {guess-and-verify approach} to solve the problem. We find that the optimal installation strategy is triggered by a curve which separates the {waiting region}, where it is not optimal to install additional panels, and the {installation region,} where it is. Such a curve depends on the current level of the company's installed power, and is the unique strictly increasing function which solves a first-order {ordinary differential equation} (ODE). While studying the ODE, we obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. This is a joint work with Tiziano Vargiolu. 04.07.2019 (17 Uhr c.t.) Sören Christensen (Universität Kiel) ENTFÄLLT !!!! Nonparametric learning in stochastic control – exploration vs. exploitation   Abstract:  One of the fundamental assumptions in stochastic control of continuous time processes is that the dynamics of the underlying (diusion) process is known. This is, however, usually obviously not fullled in practice. On the other hand, over the last decades, a rich theory for nonparametric estimation of the drift (and volatility) for continuous time processes has been developed. The aim of this talk is to make a first (small) step to bringing together techniques from stochastic control with methods from statistics for stochastic processes to nd a way to both learn the dynamics of the underlying process and control good at the same time. To this end, we study a toy example motivated from optimal harvesting, mathematically described as an impulse control problem. One of the problems that immediately arises is an Exploration vs. Exploitation behavior as is well known in from bandit problems and reinforcement learning. We propose a way to deal with this issue and analyse the proposed strategy asymptotically.

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