Numerical Analysis and Simulation of nonlineare Filtering with Levy noise

In engineering one of the most important areas of application of nonlinear filtering is positioning, navigation and tracking problems. In positioning, one is often interested in estimating its own position, while one is moving around, provided by data from some sensors. To get an idea, imagine a ship on its tour through the ocean having some problems with the GPS signals. First, the signals are corrupted by noise, and secondly, there are some gaps where the ship does not receive any signal. The ship on the way to its harbor is traveling in a certain direction. However, due to the waves the ship does not follow exactly its direction and deviates slightly. Now the problem is to estimate the ship`s position using the historical data of the journey given by the GPS signals. In navigation, beside the position also velocity, acceleration, attitude and heading are included in the filtering problem. Here, the task is to calculate automatically the route, or, to say it better, its direction and its speed in order to navigate the ship to its destination. In target tracking, the position of another object is to be estimated based on measurement of relative range and angles to one`s own position. The target can be again a ship, or a hostile drone in the air. Now, using cheap low quality sensors, the signal may be noisy. Here, the aim in nonlinear filtering is estimating the position of the target from the incomplete and noisy measurements. The aim in stochastic filtering is to reconstruct information about an unobserved (random) process X, called the signal process, given the current available observations of a certain noisy transformation of that process. However, in many applications the noise is often rough, or lives at a time scale which is comparable to the time scale of X and Y quite short. Although, modeling an earthquake one is faced by a noise which has jumps. Or, modeling the change of climate, the time scale of the noise is much faster than the time scale of the dynamic describing the climate. Thereby, the noise appears to have jumps. In finance, high frequency data are modeled successfully by Levy processes. In other words, there are many examples, where the Gaussian noise is replaced by a Levy noise in order to improve the model or to cover features which are not covered by the Gaussian noise. There exist several results where the problem is analyzed, if the observation or signal process is perturbed by Levy noise. Nevertheless, in practice, theoretical results are often not sufficient. Applying the theory means calculating the density on a computer, and this means, performing numerical calculations or simulations. Here, it is necessary to provide strategies, proof of consistency and stability, and error estimates. The aim of the project is to provide some numerical schemes for nonlinear filtering in case the stochastic perturbation comes from a Levy noise and to prove their convergence.

One classical field of applications of Stochastic Partial Differential equations is nonlinear filtering, a topic that 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. In particular, one has a signal (or state) process X described by a stochastic differential equation and an observation process Y, modeled also by a stochastic differential equation. Now, the task is for a given function f to estimate f(Xt) given the trajectory of Y up to time t. Most of the literature devoted to nonlinear filtering assumes that X and Y are driven by a Brownian motion. Although, modelling an earthquake one is faced by a noise which has jumps. Or, modelling the change of climate, the time scale of the noise is much faster than the time scale of the dynamic describing the climate. Thereby, the noise appears to have jumps. In finance, high-frequency data are modelled successfully by Levy processes. In other words, there are many examples, where the Gaussian noise is replaced by a Levy noise in order to improve the model or to cover features that are not covered by the Gaussian noise. Recently, there exist several abstract results about nonlinear filtering, if the observation or signal process is perturbed by Levy noise. Nevertheless, in practice, theoretical results are often not sufficient. Applying the theory means, calculating the density on a computer, and this means, numerical calculations, respective simulations. If the model is linear and Gaussian, the density can be computed exactly using recursive equations, also known as Kalman filters. However, if X is nonlinear or non-Gaussian, the exact filter cannot be calculated by simple means and numerical methods are inevitable. Here, it is necessary to provide strategies, proof of consistency and stability, and error estimates. The aim of the project is to provide some numerical schemes for the measure-valued process n in case the stochastic perturbation comes from a Levy noise and to prove their convergence. But of a few scattered works, there does not exist any articles about this topic. Within this project, we tried to focus on numerical methods relying on particle filters. Here, we designed a particle filter, where we used for the Brownian part standard methods from nonlinear filters, for the jump part, we used control feedback algorithms coming from engineering.