Stochastic Analysis and Stochastic Finance Seminar

Numerical Methods for SDEs in Mathematical Finance

Speaker(s): 
Michaela Szoelgyenyi (Vienna University of Economics and Business)
Date: 
Thursday, May 18, 2017 - 5:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

When solving certain stochastic control problems in insurance mathematics or mathematical finance, the optimal control policy sometimes turns out to be of threshold type, meaning that the control depends on the controlled process in a discontinuous way. The stochastic differential equations (SDEs) modeling the underlying process then typically have a discontinuous drift coefficient. This motivates the study of a more general class of such SDEs.

Optimal targeting position and (forward) backward stochastic differential equation

Speaker(s): 
Alexandré Popier (Université Lemans, France)
Date: 
Thursday, May 18, 2017 - 4:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

In this talk we present an optimal stochastic control problem related to portfolio liquidation problems. For the homogeneous case, we give a complete solution using backward stochastic differential equation with singular terminal condition (joint work with T. Kruse (Essen, Germany)). In the Brownian setting, we explain how it can be (partially) solved using forward backward SDE together with the decoupling field method (work in progress with S. Ankirchner, A. Fromm (Jena, Germany) and T. Kruse (Essen, Germany)).

Robust Utility Maximization with Lévy Processes

Speaker(s): 
Ariel Neufeld (ETH Zürich)
Date: 
Thursday, May 4, 2017 - 5:00pm
Location: 
TU Berlin, Straße des 17. Juni 136, 10623 Berlin, Raum MA 043

We present a tractable framework for Knightian uncertainty, the so-called nonlinear Lévy processes, and use it to formulate and solve problems of robust utility maximization for an investor with logarithmic or power utility. The uncertainty is specified by a set of possible Lévy triplets; that is, possible instantaneous drift, volatility and jump characteristics of the price process. Thus, our setup describes uncertainty about drift, volatility and jumps over a class of fairly general models. We show that an optimal investment strategy exists and compute it in semi-closed form.

Optimal Portfolio Choice with Benchmarks

Speaker(s): 
Carole Bernard (Grenoble Ecole de Management)
Date: 
Thursday, February 16, 2017 - 5:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

We construct an algorithm that allows to numerically obtain an investor's optimal portfolio under general preferences. In particular, the objective function and risks constraints may be driven by benchmarks (reflecting state-dependent preferences). We apply the algorithm to various classic optimal portfolio problems for which explicit solutions are available and show that our numerical solutions are compatible with them.

Duality for American options in non-dominated discrete-time models

Speaker(s): 
Xiaolu Tan (Université Paris Dauphine)
Date: 
Thursday, February 16, 2017 - 4:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

The classical pricing-hedging duality for American options with semi-static hedging does not hold in general in the simple formulation inherited from European option set-up. We propose two approaches to recover the duality result. The first approach consists in considering a bigger class of models and rendering an American option a European one. The second way is to relax the static trading and by allowing dynamic trading in the set of vanilla options.

A randomisation approach for the probabilistic representation and approximation of HJB equations

Speaker(s): 
Idris Kharroubi (CEREMADE - Universitè Paris)
Date: 
Thursday, February 2, 2017 - 5:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

We propose a new probabilistic numerical scheme for fully nonlinear equation of Hamilton-Jacobi-Bellman (HJB) type associated to stochastic control problem, which is based on the Feynman-Kac representation by means of control randomization and backward stochastic differential equation with nonpositive jumps. We study a discrete time approximation for the minimal solution to this class of BSDE when the time step goes to zero, which provides both an approximation for the value function and for an optimal control in feedback form.

Stochastic invariance of closed sets with non-Lipschitz coefficients (and applications in finance)

Speaker(s): 
Bruno Bouchard (CEREMADE - Université Paris)
Date: 
Thursday, February 2, 2017 - 4:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

This talk provides a new characterization of the stochastic invariance of a closed subset with respect to a diffusion: we extend the well known inward pointing Stratonovich drift condition to the case where the diffusion matrix can fail to be differentiable (on the boundary). In particular, our result can be directly applied to construct affine diffusions and polynomial preserving diffusions on any arbitrary closed set.

Model-free Ito integration via pathwise super-hedging

Speaker(s): 
David Prömel (ETH Zürich)
Date: 
Thursday, January 12, 2017 - 4:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

Using Vovk’s hedging based approach to mathematical finance, one can determine sample path properties of "typical price paths" belonging to the space of continuous functions or of non-negative càdlàg functions. Interestingly, all results for "typical price paths" hold quasi surely under all martingale measures. We prove that "typical price paths" possess quadratic variation and local limes. This allows us to develop model-free Itô integration as well as pathwise stochastic calculus for local times. This talk is based on joint works with R.M. Lochowski and N. Perkowski.

Risikocontrolling in der Versicherungswirtschaft - Die Solvency II Standardformel, Lineare Algebra und Diversifikation

Speaker(s): 
Joachim Paulusch (R+V Lebensversicherung AG, Risikocontrolling, Wiesbaden)
Date: 
Thursday, December 1, 2016 - 5:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

Seit 01.01.2016 gilt das Aufsichtsregime Solvency II für alle Versicherungsunternehmen in Europa. Die Versicherungsunternehmen müssen eine sogenannte Solvenzkapitalanforderung berechnen und nachweisen, dass sie über Eigenmittel mindestens in Höhe der Solvenzkapitalanforderung verfügen. Dafür verwenden die meisten Versicherungsunternehmen die sogenannte Solvency II Standardformel. Wir untersuchen die Aggregation von Risiken in der Standardformel und beantworten die Fragen:

· Wie kann man Risikokapital in der Standardformel reallokieren, also fair auf Teilrisiken verteilen?

Stability and analytic expansions of local solutions of systems of quadratic BSDEs with applications to a price impact model

Speaker(s): 
Sergio Pulido (LaMME, ENSIIE, Université d'Evry Val d'Essonne)
Date: 
Thursday, December 1, 2016 - 4:15pm
Location: 
HU Berlin, Rudower Chaussee 25, Room 1.115

We obtain stability estimates and derive analytic expansions for local solutions of multi-dimensional quadratic BSDEs. We apply these results to a financial model where the prices of risky assets are quoted by a representative dealer in such a way that it is optimal to meet an exogenous demand. We show that the prices are stable under the demand process and derive their analyticexpansions for small risk aversion coefficients of the dealer. We briefly discuss related results that naturally arise when studying the replication and optimal investment problems under this model of price impact.

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