Stochastic Calculus Research Paper Help

Stochastic Calculus 1. Consider a basket of N assets each following the Geometric Brownian Motion, so the Stochastic Differential Equation for each asset. is given by (153? = Sillidt + SiO?idXi fOI? 1 S1 S [V The price changes are correlated as measured by the linear correlation coefficients pij. Invoke the multi-dimensional Ito Lemma to write down the SDE for F (Sl, SQ, . . . , S N) in the most compact form possible (with clear drift. and diffusion terms). Apply dXide -> pijdt. 2. Construct an SDE for the process Y(t) = e?X(t)?%?2t and show that the process is, in fact, an Exponential Martingale of the form dY(t) = Z (1?) g(t) dX (t). Identify the terms g(t) and Z (t) A diffusion process Y(t) is a martingale if its SDE has no drift term. The SDE can be constructed by evaluating partial derivatives of a function F(t, X) = Y(t) and substituting as follows: oF 1 82F 8F dF= ? ? dt -dXt. (8t+2oX2> +oX Ru Essay writing Help

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