(a) Let 1 , 2 , ... be independent standard normal r.v.s, S 0 = 0, S t = 1 +...+

(a) Let ξ₁, ξ₂, ... be independent standard normal r.v.’s, S₀ = 0, S_t = ξ₁+...+ξ_t , X₀ = 1, and X_t =C_t exp{S_t} for t = 1, ... . Find a constant C_t for which X_t is a martingale. Find lim_t→∞ X_t .
How will the problem change if we consider X_t =C_t exp{−S_t}?

(b) Do the same if ξ’s are normal with a mean of m and a variance of σ².