MATHEMATICAL TRIPOS Part II PAPER 3 Before you begin read ...

1629J Stochastic Financial ModelsSuppose that (ε t ) t=0,1,...,T is a sequence of independent and identically distributedrandom variables such that E exp(zε 1 ) < ∞ for all z ∈ R. Each day, an agent receives anincome, the income on day t being ε t . After receiving this income, his wealth is w t . Fromthis wealth, he chooses to consume c t , and invests the remainder w t − c t in a bank accountwhich pays a daily interest rate of r > 0. Write down the equation for the evolution of w t .Suppose we are given constants β ∈ (0, 1), A T , γ > 0, and define the functionsU(x) = − exp(−γx), U T (x) = −A T exp(−νx) ,where ν := γr/(1 + r). The agent’s objective is to attainV 0 (w) := sup E{ T∑ −1β t U(c t ) + β T U T (w T )t=0}∣ w 0 = w ,where the supremum is taken over all adapted sequences (c t ). If the value function isdefined for 0 n < T by{ T −1}∑V n (w) = sup E β t−n U(c t ) + β T −n U T (w T )∣ w n = w ,t=nwith V T = U T , explain briefly why you expect the V n to satisfy[ {V n (w) = sup U(c) + βE Vn+1 ((1 + r)(w − c) + ε n+1 ) } ] . (∗)cShow that the solution to (∗) has the formV n (w) = −A n exp(−νw) ,for constants A n to be identified. What is the form of the consumption choices that achievethe supremum in (∗) ?Part II, Paper 3