Stochastic Analysis and Stochastic Finance Seminar

Multi-dimensional quadratic BSDEs

Speaker(s): 
Shanjian Tang (Fudan University)
Date: 
Thursday, January 29, 2015 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Quadratic BSDEs refer to those BSDEs whose generators grow quadratically in the second unkown variable. In this talk, I will start with recalling J. M. Bismut's Ph. D work on the linear quadratic optimal stochastic control problem and the introduction of backward stochastic Riccati equations, which motivated the study of general quadratic BSDEs. Then I review the theory of one-dimensional quadratic BSDEs and show the difficulty in a general solution of multi-dimensional quadratic BSDEs even when the terminal value is essentially bounded.

Second order Pontriagin's principle for stochastic control problems - CANCELLED!

Speaker(s): 
F. Frédéric Bonnans (INRIA Saclay, Ecole Polytechnique)
Date: 
Thursday, January 29, 2015 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We discuss stochastic optimal control problems whose volatility does not depend on the control, and which have finitely many equality and inequality constraints on the expected value of function of the final state, as well as control constraints. The main result is a proof of necessity of some second order optimality conditions involving Pontryagin multipliers.

Non-Implementability of Arrow-Debreu Equilibria by Continuous Trading under Knightian Uncertainty

Speaker(s): 
Frank Riedel (Universität Bielefeld)
Date: 
Thursday, January 15, 2015 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

Under risk, Arrow–Debreu equilibria can be implemented as Radner equilibria by continuous trading of few long–lived securities. We show that this result generically fails if there is Knightian uncertainty in the volatility. Implementation is only possible if all discounted net trades of the equilibrium allocation are mean ambiguity–free.

This is a joint work with Patrick Beissner.

Hawkes processes, microstructure and market impact

Speaker(s): 
Mark Hoffmann (Université Paris Dauphine)
Date: 
Thursday, December 18, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

I will first shortly review the issue of obtaining simple lattice price models for assets observed at fine temporal scales that are 1) able to reproduce microstructure effects like variance noise or the Epps effect and 2) behave like continuous semimartingales compatible with the theory of arbitrage on large diffusive scales. The use of mutually exciting point processes enable to track such microstruture effects across scales and I will present some recent (and less recent) models based on Hawkes processes.

Arbitrage-Free Pricing of XVA

Speaker(s): 
Agostino Capponi (John Hopkins University)
Date: 
Thursday, December 18, 2014 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We introduce a framework for computing the total valuation adjustment (XVA) of an European claim accounting for funding spreads, counterparty risk, and collateral mitigation. We use no-arbitrage arguments to derive the nonlinear backward stochastic differential equations (BSDEs) associated with the portfolios which replicate long and short positions in the claim. This leads to defining buyer and sellers? XVAs which in turn identify a no-arbitrage band. When borrowing and lending rates coincide, our framework reduces to a generalized Piterbarg's model.

Dealing with partial hedging or risk management constraints via BSDEs with weak reflections

Speaker(s): 
Romuald Elie (Université Paris-Est Marne-la-Vallée)
Date: 
Thursday, December 4, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In many incomplete markets, the super replication price of a given (eventually non Markovian) claim rewrites in terms of the minimal super-solution of a well chosen Backward stochastic differential equation (BSDE). The price obtained is often numerically very high and hereby useless in practice. In order to lower the price, one must accept to take some risk and this can be formalized via the use of quantile hedging type objectives, where the agent only wishes to upper hedge the claim of interest with a given a priori probability of success p.

Approximations of stochastic partial differential equations and applications in forward markets

Speaker(s): 
Andrea Barth (IANS Stuttgart)
Date: 
Thursday, December 4, 2014 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In this talk I present a model for a yield curve in a forward market using a stochastic partial differential equation driven by an infinite-dimensional Lévy process. This method is well known in interest rate theory. To determine the price of an option one has to calculate the weak error of the solution to the stochastic partial differential equation. The hyperbolic nature of this equation and the non-continuous noise complicate the task of numerical approximation. Furthermore, I make use of a multilevel Monte Carlo method to approximate the said weak error.

Martingale Optimal Transport

Speaker(s): 
H. Mete Soner (ETH Zürich)
Date: 
Thursday, November 20, 2014 - 5:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

In the original transport problem we are given two measures of equal mass and then look for an optimal map that takes one measure to the other one and also minimizes a given cost functional. Kantorovich relaxed this problem by considering a measure whose marginals agree with given two measures instead of a bijection. This generalization linearizes the problem. Hence, allows for an easy existence result and enables one to identify its convex dual. In robust hedging problems, we are also given two measures. Namely, the initial and the final distributions of a stock process.

Convex duality in continuous-time stochastic optimization

Speaker(s): 
Teemu Pennanen (King's College London)
Date: 
Thursday, November 20, 2014 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

We develop a duality framework for convex optimization problems over spaces of predictable stochastic processes. This is done by combining the conjugate duality theory of Rockafellar with some stochastic analysis. Various duality relations in stochastic control and mathematical finance are obtained as special cases. Besides classical models of financial markets, the general framework allows for e.g. illiquidity effects and portfolio constraints.

This is joint work with Ari-Pekka Perkkiö.

BSDEs of Counterparty Risk and Invariant Times

Speaker(s): 
Stéphane Crépey (Evry University)
Date: 
Thursday, November 6, 2014 - 4:00pm
Location: 
HU Berlin, Rudower Chaussee 25, 12489 Berlin, Room 1.115

This work is motivated by the need to generalize the classical credit risk reduced-form modeling approach for counterparty risk applications. We relax the basic immersion conditions of the classical approach by modeling the default time as an invariant time, such that local martingales with respect to a reduced filtration and a possibly changed probability measure, once stopped right before that time, stay local martingales with respect to the original model filtration and probability measure.

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