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Stochastic Analysis Seminar, 2007
Next seminar:
The seminars aim to share knowledge of the results achieved and open discussion on the ongoing research. The style is informal and the speakers include both students and researchers at CMA and visiting fellows. Student at master level are welcome to take active part in the series.
Time and place
Thursdays 13-14 in room B1036
Preliminary program:
Oct 18 . Rajeev B (Indian Statistical Institute, Bangalore, India)
Title: Probabilistic representations of solutions of the forward equation
Abstract: let L be a second order elliptic differential operator with smooth coefficients. In this talk , we show how we may obtain and represent distribution solutions of the evolution equation corresponding to the adjoint L* using the solutions of the SDE associated with L. The main technique is to represent the flow associated with the SDE as the solution of a stochastic PDE in the space of distributions with compact support.
Aug 30 . 2 talks:
13:15 Josep Vives (Universitat Autonoma de Barcelona)
Title: On the short-time behavior of the implied volatitlity for jump-diffusion models with stochastic volatility
Abstract: In this paper we use the Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a diffusion, nor a Markov process (examples are provided). This expression depends on the derivative of the volatility in the sense of Malliavin calculus.
The presentation is based on a paper in collaboration with: Elisa Alos and Jorge A. Leon.
14:15 Delphine David (Université de La Rochelle, France)
Title: On the optimal control of stochastic delayed systems with jumps
Abstract: We consider the optimal control of stochastic delayed systems with jumps, in which both the state and controls can depend on the past history of the system. In particular we derive necessary and sufficient maximum principles for such problems, and give some applications to financial economics.
Aug 23 . 3 seminars:
13:15 Josep Vives (Universitat Autonoma de Barcelona)
Title: Canonical Levy process and Malliavin Calculus
Abstract: A suitable canonical Levy process is constructed in order to study a Malliavin calculus based on a chaotic representation property of Levy processes proved by Ito using multiple two-parameter integrals.
In this setup, the two-parameter derivative D_{t,x} is studied, depending on whether x is 0 or not. In the first case, we prove a chain rule, in the second case, a formula by trajectories. The presentation is based on a paper in collaboration with: Josep Lluis sole and Frederic Utzet.
14:15 Francesca Biagini (University of Munich, Germany)
Title: Local Risk-Minimization for Defaultable Markets
Abstract: We study the local risk-minimization approach for defaultable markets in a general framework where the asset's behavior and the default time may influence each other. We find the Foellmer-Schweizer decomposition in this general setting, when a recovery payment is due either at time of maturity or at time of default. We compute it explicitly in two particular cases, when default time depends on the risky asset's behavior and when only a dependence of discounted asset price on default time is occurring.
This talk is based on two joint works with Alessandra Cretarola.
15:15 Thilo Meyer-Brandis (CMA)
Title: On generalized solutions of linear stochastic integro-PDE's with applications to non-linear filtering with Gaussian and Cox noise.
Abstract: In this presentation we consider linear parabolic stochastic integro-PDE's driven by Brownian motion and Poisson random measure. We represent solutions of such equations as certain functionals of an associated jump diffusions. Our analysis is done in the framework of white noise analysis for Levy processes and we consider differentiation in the sense of stochastic distributions. This concept of generalized solutions allows for relaxed assumptions on the equation coefficients, identical to those required in problems of similar deterministic integro-PDE's. In particular, uniform non-degeneracy is relaxed. Further, even for equations with terminal consitions we are able to consider forward running time, which is not possible in the classical Ito setting. The study of stochastic integro-PDE's of this type is motivated by its appearance in different applications. In the second part of the talk we show how this type of equations arises in the context of non-linear filtering with Gaussian and Cox noise.
May 24. Zhi-Ming Ma (Chinese Academy of Science, Beijing, China)
Title: Nonlinear Filtering of Semi-Dirichlet Processes
Abstract: The talk is based on my recent joint work with Zhechun Hu and Wei Sun. We consider a nonlinear filtering problem for general right continuous Markov processes associated with semi-Dirichlet forms. We show that in our general setting the fitering processes satisfy also DMZ (Duncan-Mortensen- Zakai) equation. The uniqueness of the solutions of the filtering equations are verified through their Wiener chaos expansions. Our results on the Wiener chaos expansions for nonlinear filters with possibly unbounded observation functions are novel and have their own interests. We investigate further the absolute continuity of the filtering processes with respect to the reference measures and derive the density equations for the filtering processes.
May 10. Thomas Liebmann (University of Ulm, Germany)
Title: Martingale measures minimizing generalized relative entropy
Abstract: In an incomplete market the number of equivalent martingale maesures is infinite and an additional criterion is required in order to choose the pricing measure. Here, minimization of generalized relative entropy with respect to a prior measure in a market modeled by exponential Lévy processes is investigated.
Mar 8. Atle Hahn (University of Bonn, Germany)
Title: From path integrals to topological invariants of 3-manifolds
Abstract: The study of the heuristic Chern-Simons path integral by E. Witten inspired (at least) two general approaches to the theory of "quantum" 3-manifold invariants. Firstly, the perturbative approach based on the CS path integral in the Lorentz gauge and, secondly, the "quantum group approach" by Reshetikhin/Turaev. While for the first approach the relation to the CS path integral is obvious for the second approach it is not. In my talk, which summarizes the results of a recent preprint, I will sketch a strategy that should lead to a clarification of this issue in the special case where the base manifold is of product form. This strategy is based on the "torus gauge fixing" procedure introduced by Blau/Thompson for the study of the partition function of CS models. I will show that the formulas of Blau/Thompson can be generalized to Wilson lines and that the evaluation of the expectation values of these Wilson lines leads to the same state sum expressions in terms of which Turaev's shadow invariant is defined. Finally, I will sketch how, using White Noise Analysis, one can obtain a rigorous realization of the path integral expressions appearing in this treatment.
Mar 8. Christina Erlwein (Brunel University West London, UK)
Title: HMM filtering and parameter estimation of an electricity spot price model
Abstract: We develop a hidden Markov model for electricity spot price dynamics, where the spot price is assumed to follow an exponential Ornstein-Uhlenbeck (OU) process with an added compound Poisson process. A sinusoidal factor is introduced to capture the seasonality component of prices. The mean-reverting level, speed of adjustment and volatility of the OU process as well as the mean and variance of the normally distributed jump sizes of the compound Poisson process are all modulated by a hidden Markov chain in discrete time. The parameters are able to switch between different economic regimes representing various levels of supply and demand. Through the application of reference probability technique, adaptive filters are derived, which in turn, provide optimal estimates for the state of the Markov chain and related quantities of the observation process. The EM algorithm is applied to find optimal estimates of the model parameters in terms of the recursive filters. We implement this self-calibrating model on a deseasonalized series of daily spot electricity prices from the Nordic exchange Nord Pool. On the basis of one-step ahead forecasts, we found that the model is able to capture the stylised features of Nord Pool spot prices.
Feb. 22 . Bernt Øksendal (CMA)
Title: Risk indifference pricing
Abstract: We study a jump diffusion market and present a new pricing principle, based on the concept of a convex risk measure (introduced by Föllmer and Schied), as follows: The (seller´s) risk indifference price p of a European option with payoff G, say, is the value of p which makes the seller of the option risk indifferent to the two choices: (i) either selling the option and receiving the initial payment p and being committed to pay out G to the buyer at the terminal time T or (ii) not selling the option, and hence not receiving any initial payment and not paying out anything at time T. The problem of finding p leads to the problem of solving two stochastic differential games. We are able to find a (relatively) explicit formula for the risk indifference price p, for a large class of given convex risk measures. The talk is based on recent joint work (in progress) with Agnès Sulem.
Feb. 6 . Annika Niehage (University of Mannheim, Germany)
(NB time! This week Tuesday at 13.15)
Title: Gaussian Random Fields
Abstract: In this talk I will present fast algorithms for the generation of Gaussian random fields (GRF) with given space correlation. I will especially focus on two dimensional random fields. Further I will show the statistic behaviour of the simulated GRFs and I will present different correlation functions with certain properties, e.g. exponential decay and symmetry under rotation. Images of the created random fields should give a feeling of how these GRFs look like and what kind of correlation functions might be interesting for different applications. Finally I will show the use of the generated GRFs in the context of different SPDEs. These examples shall emphasize the opportunities that these random fields with different correlation functions give for the simulation of SDEs.
Organizers:
Fred Espen Benth, CMA
Giulia Di Nunno, CMA
Bernt Øksendal, CMA
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