This text is divided into two parts. In the first part we present a methodology for approximating complex stochastic processes. Furthermore, we present an application to finance to calculate the price of American or European options when the price of the underlying equity obeys these complex processes. In the second part we investigate the exponential behavior of the solution of the parabolic Anderson model when the time goes to infinity. We show that the relevant quantity (the Lyapunov exponent) exists, and we provide tight lower and upper bounds for it.

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