Stochastic Analysis & Mathematical Finance Seminars

Please note that the list below only shows forthcoming events, which may not include regular events that have not yet been entered for the forthcoming term. Please see the past events page for a list of all seminar series that the department has on offer.

Past events in this series
18 November 2019
14:15
JOSE BLANCHET
Abstract

We revisit portfolio selection models by considering a distributionally robust version, where the region of distributional uncertainty is around the empirical measure and the discrepancy between probability measures is dictated by optimal transport costs. In many cases, this problem can be simplified into an empirical risk minimization problem with a regularization term. Moreover, we extend a recently developed inference methodology in order to select the size of the distributional uncertainty in a data-driven way. Our formulations allow us to inform the distributional uncertainty region using market information (e.g. via implied volatilities). We provide substantial empirical tests that validate our approach.
(This presentation is based on the following papers: https://arxiv.org/pdf/1802.04885.pdf and https://arxiv.org/abs/1810.02403.)

  • Stochastic Analysis & Mathematical Finance Seminars
18 November 2019
15:45
WALTER SCHACHERMAYER
Abstract

We reconsider the approximations of the Black-Scholes model by discrete time models such as the binominal or the trinominal model.

We show that for continuous and bounded claims one may approximate the replication in the Black-Scholes model by trading in the discrete time models. The approximations holds true in measure as well as "with bounded risk", the latter assertion being the delicate issue. The remarkable aspect is that this result does not apply to the well-known binominal model, but to a much wider class of discrete approximating models, including, eg.,the trinominal model. by an example we show that we cannot do the approximation with "vanishing risk".

We apply this result to portfolio optimization and show that, for utility functions with "reasonable asymptotic elasticity" the solution to the discrete time portfolio optimization converge to their continuous limit, again in a wide class of discretizations including the trinominal model. In the absence of "reasonable asymptotic elasticity", however, surprising pathologies may occur.

Joint work with David Kreps (Stanford University)

  • Stochastic Analysis & Mathematical Finance Seminars
25 November 2019
14:15
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 talk, 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 goes to infinity. This is based on a joint work with Maddalena Ghio and Giulia Livieri (SNS Pisa). 

  • Stochastic Analysis & Mathematical Finance Seminars
25 November 2019
15:45
CHRISTOPH BELAK
Abstract


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.

  • Stochastic Analysis & Mathematical Finance Seminars
2 December 2019
14:15
RICHARD NICKL
Abstract

Bayes methods for inverse problems have become very popular in applied mathematics in the last decade after seminal work by Persi Diaconis and Andrew Stuart. They provide reconstruction algorithms as well as in-built “uncertainty quantification” via Bayesian credible sets, and particularly for Gaussian process priors can be efficiently implemented by MCMC methodology. For linear inverse problems, they are closely related to classical penalised least squares methods and thus not fundamentally new, but for non-linear and non-convex problems, they give genuinely distinct and computable algorithmic alternatives that cannot be studied by variational analysis or convex optimisation techniques. In this talk we will discuss recent progress in Bayesian non-parametric statistics that allows to give rigorous statistical guarantees for posterior consistency in such models, and illustrate the theory in a variety of concrete non-linear inverse problems arising with partial differential equations.

  • Stochastic Analysis & Mathematical Finance Seminars
2 December 2019
15:45
PAOLO GUASONI
Abstract

This paper evaluates the effect of market integration on prices and welfare, in a model where two Lucas trees grow in separate regions with similar investors. We find equilibrium asset price dynamics and welfare both in segmentation, when each region holds its own asset and consumes its dividend, and in integration, when both regions trade both assets and consume both dividends. Integration always increases welfare. Asset prices may increase or decrease, depending on the time of integration, but decrease on average. Correlation in assets' returns is zero or negative before integration, but significantly positive afterwards, explaining some effects commonly associated with financialization.

  • Stochastic Analysis & Mathematical Finance Seminars
9 December 2019
14:15
to
15:45
ILYA CHEVYREV
Abstract

Yang-Mills theory plays an important role in the Standard Model and is behind many mathematical developments in geometric analysis. In this talk, I will present several recent results on the problem of constructing quantum Yang-Mills measures in 2 and 3 dimensions. I will particularly speak about a representation of the 2D measure as a random distributional connection and as the invariant measure of a Markov process arising from stochastic quantisation. I will also discuss the relationship with previous constructions of Driver, Sengupta, and Lévy based on random holonomies, and the difficulties in passing from 2 to 3 dimensions. Partly based on joint work with Ajay Chandra, Martin Hairer, and Hao Shen.

  • Stochastic Analysis & Mathematical Finance Seminars
9 December 2019
15:45
GONCALO DOS REIS
Abstract


We present several Itô-Wentzell formulae on Wiener spaces for real-valued functionals random field of Itô type depending on measures. We distinguish the full- and marginal-measure flow cases. Derivatives with respect to the measure components are understood in the sense of Lions.
This talk is based on joint work with V. Platonov (U. of Edinburgh), see https://arxiv.org/abs/1910.01892.
 

  • Stochastic Analysis & Mathematical Finance Seminars
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