Stochastic Analysis Seminar
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Mon, 09/05/2011 14:15 |
Mauro Mariani (Université Aix-Marseille III - Paul Cézanne) |
Stochastic Analysis Seminar  |
Oxford-Man Institute |
| We consider parabolic scalar conservation laws perturbed by a (conservative) noise. Large deviations are investigated in the singular limit of jointly vanishing viscosity and noise. The model is supposed to feature the same behavior of "asymmetric" particles systems (e.g. TASEP) under Euler scaling. A first large deviations principle is obtained in a space of Young measures. A "second order" large deviations principle is then discussed, including connections with the Jensen and Varadhan functional. As time allows, more recent "long correlation" models will be treated. | |||
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Mon, 09/05/2011 15:45 |
Lukas Szpruch |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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Numerical Approximations of Non-linear Stochastic Systems. Abstract: The explicit solution of stochastic differential equations (SDEs can be found only in a few cases. Therefore, there is a need fo accurate numerical approximations that could, for example, enabl Monte Carlo Simulations. Convergence and stability of these methods are well understood for SDEs with Lipschit continuous coefficients. Our research focuses on those situations wher the coefficients of the underlying SDEs are non-Lipschitzian It was demonstrated in the literature, that in this case using the classical methods we may fail t obtain numerically computed paths that are accurate for small step-sizes, or to obtain qualitative information about the behaviour of numerical methods over long time intervals. Our work addresses both of these issues, giving a customized analysis of the most widely used numerical methods. |
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Mon, 16/05/2011 14:15 |
Monique Pontier (Inst. Math. De Toulouse (IMT)) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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The paper analyses structural models for the evaluation of risky debt following H.E. LELAND [2], with an approach of optimal stopping problem (for instance cf. N. EL KAROUI [1]) and within a more general context: a dividend is paid to equity holders, moreover a different tax schedule is introduced, depending on the firm current value. Actually, an endogenous default boundary is introduced and a nonlinear convex tax schedule allowing for a possible switching in tax benefits. The aim is to find optimal capital structure such that the failure is delayed, meaning how to decrease the failure level VB, anyway preserving D debtholders and E equity holders’interests: for the firm VB is needed as low as possible, for the equity holder, an optimal equity is requested, finally an optimal coupon C is asked for the total value. Keywords: corporate debt, optimal capital structure, default, |
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Mon, 16/05/2011 15:45 |
Jean-Francois Chassagneux (Université d'Evry-Val-d 'Essonne) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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Mon, 23/05/2011 14:15 |
Vassili Kolokoltsov (ETH Zurich) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| I will introduce the notion of a nonlinear Levy process, discuss basic well-posednes, SDE links and the connection with interacting particles. The talk is aimed to be an introduction to the topic of my recent CUP monograph 'Nonllinear Markov processes and kinetic equations'. | |||
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Mon, 23/05/2011 15:45 |
Matteo Casserini (joint work with Gechun Liang) (ETH Zurich) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| Recently, Liang, Lyons and Qian developed a new methodology for the study of backward stochastic differential equations (BSDEs) on general filtered probability spaces. Their approach is based on the analysis of a particular class of functional differential equations, where the driver of the equation does not depend only on the present, but also on the terminal value of the solution. The purpose of this work is to study fully coupled systems of forward functional differential equations, which are related to a broad class of fully coupled forward-backward stochastic dynamics with respect to general filtrations. In particular, these systems of functional differential equations have a more homogeneous structure with respect to the underlying forward-backward problems, allowing to partly avoid the conflicting nature between the forward and backward components. Another advantage of the approach is that its generality allows to consider many other types of forward-backward equations not treated in the classical literature: this is shown with the help of several examples, which have interesting applications to mathematical finance and are related to parabolic integro-partial differential equations. In the second part of the talk, we introduce a numerical scheme for the approximation of decoupled systems, based on a time discretization combined with a local iteration approach. | |||
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Mon, 06/06/2011 14:15 |
Konstantinos Zygalakis (University of Oxford) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| : Backward error analysis is a technique that has been extremely successful in understanding the behaviour of numerical methods for ordinary differential equations. It is possible to fit an ODE (the so called modified equation) to a numerical method to very high accuracy. Backward error analysis has been of particular importance in the numerical study of Hamiltonian problems, since it allows to approximate symplectic numerical methods by a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. Such a systematic theory in the case of numerical methods for stochastic differential equations (SDEs) is currently lacking. In this talk we will describe a general framework for deriving modified equations for SDEs with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed, as well as the use of modified equations as a tool for constructing higher order methods for stiff stochastic differential equations. This is joint work with A. Abdulle (EPFL). D. Cohen (Basel), G. Vilmart (EPFL). | |||
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Mon, 06/06/2011 15:45 |
Herbert Spohn |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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In 1986 Kardar, Parisi, and Zhang
proposed a stochastic PDE for the motion of driven interfaces, |
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Mon, 06/06/2011 17:00 |
Sasha Grigoryan (Bielefeld University) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
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Mon, 13/06/2011 14:15 |
Nizar Touzi (London) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| This problem is classically addressed by the so-called Skorohod Embedding problem. We instead develop a stochastic control approach. Unlike the previous literature, our formulation seeks the optimal no arbitrage bounds given the knowledge of the distribution at some (or various) point in time. This problem is converted into a classical stochastic control problem by means of convex duality. We obtain a general characterization, and provide explicit optimal bounds in some examples beyond the known classical ones. In particular, we solve completely the case of finitely many given marginals. | |||
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Mon, 13/06/2011 15:45 |
Keith Ball (University of Edinburgh) |
Stochastic Analysis Seminar |
Oxford-Man Institute |
| The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics. | |||
