New Algorithm for Stochastic Problems with Random Fields of Non-finite Variances

A new algorithm is developed to solve stochastic problems with random fields of non-finite variances. Developing this algorithm motives from an attempt of representing random fields following the Lévy distribution. The first step of current algorithm is deriving moving least square reproducing kernel (MLSRK) approximations of random fields. These MLSRK approximations are derived over local support domains in the probability space. Thus, equating such approximations is still possible, even if the variance of random fields to be studied is infinite. The stochastic problem is next solved with respect to these MLSRK approximations. Testing the succeeding algorithm finds that it doesn’t require many samples and any empirical coefficient to represent accurately random fields following such as Lévy, Cauchy, and multivariate Cauchy distributions. It also provides accurate computation of means and variances of the option price with the stochastic volatility following two empirical Pareto- Lévy and non-stable Lévy distributions. Except for MLSRK approximations of the option price and stochastic volatility, such computation is tested with a deterministic meshless collocation formulation of the Black-Scholes equation.

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