FS Stochastische Analysis und Stochastik der Finanzmärkte
Bereich für Stochastik
P. BANK, Ch. BAYER, D. BECHERER, P. FRIZ, U. HORST, D. KREHER
 Das Seminar findet an der HU Berlin, Institut für Mathematik, Hörsaal 1.115 (Rudower Chaussee 25, 12489 Berlin) statt.
 Zeit: Donnerstag, 16 Uhr s.t. / 17 Uhr s.t.
26.10.2023

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26.10.2023

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09.11.2023

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23.11.2023 16. Uhr c.t.

John Schoenmakers (WIAS Berlin) Optimal Stopping with Randomly Arriving Opportunities
Abstract: In this talk we develop methods to solve general optimal stopping problems with opportunities to stop that arrive randomly. Such problems occur naturally in applications with market frictions. Pivotal to our approach is that our methods operate on random rather than deterministic time scales. This enables us to convert the original problem into an equivalent discretetime optimal stopping problem with natural number valued stopping times and a possibly infinite horizon. To numerically solve this problem, we design a random times least squares Monte Carlo method. We also analyze an iterative policy improvement procedure in this setting. We illustrate the efficiency of our methods and the relevance of randomly arriving opportunities in a few examples. This is joint work with Josha Dekker, Roger Laeven and Michel Vellekoop from the University of Amsterdam. 

23.11.2023 17 Uhr c.t.

Leandro SanchezBetancourt (University of Oxford) Automated Market Makers Designs beyond Constant Functions
Abstract: Popular automated market makers (AMMs) use constant function markets (CFMs) to clear the demand and supply in the pool of liquidity. A key drawback in the implementation of CFMs is that liquidity providers (LPs) are currently providing liquidity at a loss, on average. In this paper, we propose two new designs for decentralised trading venues, the arithmetic liquidity pool (ALP) and the geometric liquidity pool (GLP). In both pools, LPs choose impact functions that determine how liquidity taking orders impact the marginal exchange rate of the pool, and set the price of liquidity in the form of quotes around the marginal rate. The impact functions and the quotes determine the dynamics of the marginal rate and the price of liquidity. We show that CFMs are a subset of ALP; specifically, given a trading function of a CFM, there are impact functions and quotes in the ALP that replicate the marginal rate dynamics and the execution costs in the CFM. For the ALP and GLP, we propose an optimal liquidity provision strategy where the price of liquidity maximises the LP's expected profit and the strategy depends on the LP's (i) tolerance to inventory risk and (ii) views on the demand for liquidity. Our strategies admit closedform solutions and are computationally efficient. We show that the price of liquidity in CFMs is suboptimal in the ALP. Also, we give conditions on the impact functions and the liquidity provision strategy to prevent arbitrages from rountrip trades. Finally, we use transaction data from Binance and Uniswap v3 to show that liquidity provision is not a lossleading activity in the ALP.


07.12.2023 16 Uhr c.t.

Christoph Czichowski (London School of Economics & Pol. Science) Numeraireinvariance and the law of one price in meanvariance portfolio selection and quadratic hedging
Abstract: Markowitz’s meanvariance portfolio selection is one of the pillars of modern financial economics underpinning large parts of investment practice. Whereas this problem has been considered in one period with or without a riskfree asset, a multiperiod or continuoustime analysis usually involves the existence of a riskfree asset that is taken as the numeraire and/or as the reference asset. In this talk, we provide a numeraireinvariant problem formulation of mean–variance portfolio optimization and quadratic hedging for semimartingale price processes without necessarily assuming a riskfree asset or choosing a numeraire asset. This includes a symmetric definition of admissibility; sufficient conditions for the existence of optimal trading strategies; expressions for the optimal trading strategies that do not depend on the choice of a reference asset and do not require numeraire change; and an equivalence result for hedging with and without numeraire change. Our results complement recent developments in numeraireinvariant modelling in Financial Mathematics. In particular, we provide a new version of the "Fundamental Theorem of Asset Pricing" appropriate in the quadratic context establishing the equivalence of the economic concept of the law of one price with the probabilistic property of the existence of a local Emartingale state price density. The talk is based on joint work with Ales Cerny and Jan Kallsen.


07.12.2023 17 Uhr c.t.

David Criens (Universität Freiburg) Nonlinear Diffusions and their Feller Properties
Abstract: Motivated by Knightian uncertainty, S. Peng introduced his celebrated $G$Brownian motion. Intuitively speaking, it corresponds to a dynamic worst case expectation in a model where volatility is uncertain but postulated to take values in a bounded interval. Natural extensions of the $G$Brownian motion are nonlinear diffusions, whose volatility (and drift) takes values in a random set that is allowed to depend on the canonical process in a Markovian way. Nonlinear diffusions satisfy the dynamic programming principle, which entails the semigroup property of a corresponding family of sublinear operators. In this talk, we discuss regularity properties of these semigroups that allow us to relate them to evolution equations. In particular, we explain a novel type of smoothing property and a stochastic representation result for general sublinear semigroups with pointwise generators of HamiltonJacobiBellman type. Latter also implies a unique characterization theorem for such semigroups.


14.12.2023 16 Uhr c.t. 
Olivier M. Pamen (University of Liverpool) A uniqueness and smoothness result of stochastic differential equations on the plane with singular coefficients
Abstract: In this talk, we first discuss the path by path uniqueness for multidimensional SDE on the plane when the drift coefficient verifies a spacial linear growth con dition and is componentwise nondeacreasing. Such equation can also be seeing as quasilinear hyperbolic stochastic differential equations (HSPDEs). The proofs rely on a local timespace representation of Brownian sheet and a type of law of the iterated logarithm for the Brownian sheet. Secondly, when the drift coeffi cient is merely measurable and uniformly bounded, we prove the existence and uniqueness of a Malliavin differentiable solution to the HSPDE. Our approach for proving this result rests on: 1) tools from Malliavin calculus and 2) variational techniques introduced by Davie (2007) non trivially extended to the case of SDEs in the plane by using an algorithm for the selection of certain rectangles. This talk is based on a joint works with A. M. Bogso, M. Dieye and F. Proske.


14.12.2023 17 Uhr c.t. 
Wissal Sabbagh (Le Mans Université) A new Mertens decomposition of Yg,ξsubmartingale systems
Abstract: We first introduce the concept of Y g,ξsubmartingale systems, where the nonlinear operator Y g, corresponds to the first component of the solution of a reflected BSDE with generator g and lower obstacle ξ. We first show that, in the case of a leftlimited rightcontinuous obstacle, any Y g,ξsubmartingale system can be aggregated by a process which is rightlower semicontinuous. We then prove a Mertens decomposition, by using an original approach which does not make use of the standard penalization technique. These results are in particular useful for the treatment of control/stopping game problems and, to the best of our knowledge, they are completely new in the literature. As an application, we introduce a new class of Backward Stochastic Differential Equations (in short BSDEs) with weak constraints at stopping times, which are related to the partial hedging of American options. We study the wellposedness of such equations and, using the Y g,ξ Mertens decomposition, we show that the family of minimal time tvalues Yt, with (Y,Z) a supersolution of the BSDE with weak constraints, admits a representation in terms of a reflected backward stochastic differential equation.


11.01.2024

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25.01.2024 16 Uhr c.t. 
Philipp Jettkant (University of Oxford) On two Formulations of McKeanVlasov Control with Killing
Abstract: We study a McKean–Vlasov control problem with killing and common noise. The particles in this control model live on the real line and are killed at a positive intensity whenever they are in the negative halfline. Accordingly, the interaction between particles occurs through the subprobability distribution of the living particles. We establish the existence of an optimal semiclosedloop control that only depends on the particles’ location and not their cumulative intensity. This problem cannot be addressed through classical mimicking arguments, because the particles’ subprobability distribution cannot be reconstructed from their location alone. Instead, we represent optimal controls in terms of the solutions to semilinear BSPDEs and show those solutions do not depend on the intensity variable. 

25.01.2024 17 Uhr c.t.

Wilfried Kuissi Kamdem (AIMS Ghana / University of Rwanda) Optimal consumption with labor income and borrowing
Abstract: In this talk, we present an optimal consumption and investment problem for an investor with liquidity constraints who has isoelastic recursive EpsteinZin utility preferences and receives a stochastic stream of income. We characterize the optimal consumption strategy as well as the terminal wealth for recursive utility under dynamic liquidity constraints, which prevent the investor to borrow against his stochastic future income. Using duality and backward SDE methods in a possibly nonMarkovian diffusion model for the financial market, this gives rise to an interplay of singular control and optimal stopping problems. Our analysis extends to more general liquidity constraints. (Joint work with Dirk Becherer and Olivier Menoukeu Pamen) 

08.02.2024 16 Uhr c.t.

Sergio Pulido (ENSIIE Paris) Polynomial Volterra processes
Abstract: Recent studies have extended the theory of affine processes to the stochastic Volterra equations framework. In this talk, I will describe how the theory of polynomial processes extends to the Volterra setting. In particular, I will explain the moment formula and an interesting stochastic invariance result in this context. This is joint work with Eduardo Abi Jaber, Christa Cuchiero, Luca Pelizzari and Sara SvalutoFerro.


08.02.2024

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