Nonlinear Filtering with Respect to Lévy Noise

One classical field of applications of stochastic partial differential equations is nonlinear filtering, a topic which belongs originally to Statistics. The aim in nonlinear filtering is to reconstruct information about an unobserved random process, called the signal process, given the current available observations of a certain noisy transformation of that process. To illustrate the idea, let us give the following example: The estimation of the position of a satellite in a geostationary transfer orbit --GTO-- The geostationary transfer orbit of a satellite is an elliptic orbit used to transfer the satellite from an initial orbit to the geostationary one. The movement of the satellite, denoted by X, is described by a stochastic differential equation perturbed by a random noise representing the modeling error. A radar follows the movement of the satellite and registers the information about. This observed data Y can be represented as the solution of a stochastic differential equation perturbed by a random noise coming from the measurement. The objective of nonlinear filtering is to study the properties of the process X using the data generated by the process Y. The process X is treated as a signal process and the process Y is the available observation process. The aim in "filtering" is to filter out the noise Y. By this we mean reconstructing V; the best mean-square estimate of X for a given time t on the basis of the observations generated by Y. The main task in filtering theory is to study the well posedness and the properties of the unnormalized filter V as well as to calculate it numerically. In the classical theory, the signal and the observable processes satisfy stochastic differential equations perturbed by Gaussian noises. The aim of this project is to extend the filtering theory, known for the Wiener process to the framework of the Lévy process. A topic which is treated partially for special cases such as for the degenerate case and for the Poisson random noise, but still far to reach the whole general theory. Here, the key tools to get the unnormalized filter will be completely different. In fact, the leading operator in the Zakai equation will be a pseudo differential operator instead of the second order operator known for the Gaussian case and the equation will be perturbed by a general Lévy noise.

One classical field of applications of Stochastic Partial Differential equations is nonlinear filtering, a topic which belongs to Statistic. The objective in stochastic filtering is to reconstruct information about an unobserved (random) process, called the signal process, given the current available observations of a certain noisy transformation of that process. To illustrate the idea, let us give the following example see e.g. [1]: In our example, the process X is treated as a signal process and the process Y is the available observation process perturbed by the noise Y . In many problems arising from physics, engineering, finance and many other applied sciences the state X of a dynamical system cannot be measured directly and has to be estimated from observations Y. In general, observations made on a dynamical system are corrupted by random errors. To extract from them the most precise information about the underlying system, it is important and necessary to filter out the noise in the observations. Roughly speaking, the aim of the project is was to filter out the noise Y in case Y is a jump noise, and to investigate noise, and to investigate the best estimate of [1] Pardoux E. Filtrage non lineaire et equations aux derive`es partielles stochastiques associe`es, Lecture Notes in Mathematics, Ecole de probabilite`s de Saint-Flour XIX-1989. 2359