(2 marks) Let M (t) = e(w (t)-t), where W(t) is a standard Brownian motion. Determine...

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(2 marks) Let M (t) = e(w (t)-t), where W(t) is a standard Brownian motion. Determine the value of A such that M(t) is a martingale with respect to the information available up to time s, where s < t. (3 marks) Consider the process X(t) satisfying the following stochastic differential equa- tion (SDE): dX(t) = K(0-X(t))dt + odW(t), X (0) = 0.5, where K, and o are positive numbers, and W(t) is a standard Brownian motion. By first deriving the SDE for et X(t) and solving for X(t), find E(X(t)) and Var(X(t)). (2 marks) Let M (t) = e(w (t)-t), where W(t) is a standard Brownian motion. Determine the value of A such that M(t) is a martingale with respect to the information available up to time s, where s < t. (3 marks) Consider the process X(t) satisfying the following stochastic differential equa- tion (SDE): dX(t) = K(0-X(t))dt + odW(t), X (0) = 0.5, where K, and o are positive numbers, and W(t) is a standard Brownian motion. By first deriving the SDE for et X(t) and solving for X(t), find E(X(t)) and Var(X(t)).

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Solution Part 1 We need to find the value of A such that Mt is a martingale with respect to the information available up to time s where s t First not... View the full answer