### Welcome to Legit Writing  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

Are you interested in this answer? Please click on the order button now to have your task completed by professional writers. Your submission will be unique and customized, so that it is totally plagiarism-free.