Simulation of a Lévy Random Walk

It has not to be mentioned that stochastic analysis is an active area of research and has many applications running from finance mathematics to population dynamics or fatigue analysis in mechanical engineering. In this areas, many models are described by a (partial) differential equation perturbed by a stochastic term, i.e. a Stochastic (Partial) Differential Equation. However, most of the models are based on Gaussian noise, although, in recent years, Lèvy randomness began to draw much attention. Lèvy randomness needs other techniques, is quite intricate and far from amenable to mathematical analysis. As in the deterministic case the numerical methods are also important. In line with the theory of Partial Differential Equations (PDEs), one can prove the existence and uniqueness of solutions; however only in a few cases it is possible to find explicit solutions. Here, it is necessary to simulate the solution of an S(P)DE numerically on a computer. To simulate a stochastic partial differential equation (SPDE) on a computer, not only time and space have to be discretized, also the driving stochastic process has to be simulated. If the driving process is a Wiener process, no real problems arise apart from CPU time and memory. If the driving stochastic process is a general Lèvy process, i.e. is a purely discontinuous process, the increments cannot be a`priori exact simulated. There exist only a few Lèvy processes having a known distribution. In the one dimensional case, some approaches exist to simulate a stochastic differential equation driven by a Lèvy process. However, in higher dimensions these approaches are rather difficult to realize. The aim of this project is to find implementable strategies to simulate a Lèvy walk on a computer and to analyze the convergence of these strategies and their complexity.

Partial Differential Equations (PDEs) play an essential role for mathematical modelling of many physical phenomena, and the literature devoted to their theory and applications is enormous. SPDEs are quite a young research area, the?rst articles appeared in the mid 60s.The presence of noise leads to new and important phenomena. E.g. there exist several examples, like the reaction diffusion equation with white noise forcing, where in the deterministic case, the invariant measure is not unique, and, in the stochastic case the system is uniquely ergodic. This new type of behaviour is often very useful in understanding real processes and leads often to a more realistic description of real systems than their deterministic counterpart.As in the deterministic case the numerical methods are also important. In line with the theory of Partial Differential Equations (PDEs), one can prove the existence and uniqueness of solutions; however only in a few cases it is possible to?nd analytical (i.e. in closed form) solutions. Hence those properties of the solution which cannot be found theoretically have to be studied by numerical simulations. Moreover, many properties cannot be found by experiments and proper measurement cannot be carried out on the physical system itself, e.g.?xing a measurement device on a wing of an aircraft would modify the air?ow and lead to different results. To simulate a stochastic partial differential equation (SPDE) on a computer numerically, not only the time and space have to be discretized, also the driving stochastic process has to be simulated. If the driving process is a Wiener process, apart from CPU time and memory, no real problems arise. If the driving stochastic process is a general Levy process, i.e. is a purely discontinuous process, the increments cannot be apriori exact simulated. There exist only a few Levy processes having a known distribution. In the one dimensional case, some approaches exist to simulate a stochastic differential equation driven by a Levy process.However, in high dimension these approaches are rather difficult to realize. Since discretizing an SPDE in space results in a high dimensional SDE, these approaches are not really useful for approximating an SPDE, which is our aim.Therefore, the aim of this project is to propose and investigate and?nd some schemes to simulate a Levy walk and to give a systematical treatment of the different type of the error induced by the different schemes. Here, it should be taken into account that the aim is to simulate an SPDE, that means a high dimensional system.