Stochastic differential equations (SDEs) driven by non-Gaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the... Show moreStochastic differential equations (SDEs) driven by non-Gaussian L´evy noises have attracted much attention recently [1, 29]. In [12], the authors studied a scalar SDE driven by a non-Gaussian L´evy motion, and numerically investigate mean exit time and escape probability for arbitrary noise intensity in one dimensional case. In the present thesis, we utilize a different strategy to explore a numerical method for the problem in two dimensional cases. To be specific, we assume the solution u(x) is radially symmetric with respect to the origin, and then represent the equation using radial coordinate, reducing the problem into one dimensional case. Then main difficulty is that, in the integral term, appears a Gauss Hypergeometric function and the unknown function u(r), which makes the error estimates complicated. We exploit some properties of Gauss Hypergeometric function, and finally make out a way for estimating the error [19]. Up to now we are only able to deal with this problem with 0 < α ≤ 1, since our numerical scheme does not converge when 1 < α < 2. Then we compare our numerical solutions with the analytical ones which are given in [3], and they coincide very well. KeyWords: Stochastic dynamical systems; non-Gaussian L´evy motion; L´evy jump measure; First exit time M.S. in Applied Mathematics, December 2013 Show less
