3 An individual solves an optimal portfolio problem as follows: max f()g E Z0 1 e-tW1 -t1- dt where his wealth evolves according to dWt = Wt(t(dt + dZt) + (1 - t)rdt) Notice that ct does not appear in the utility function or on the right hand side of the wealth equation.

what is the answer for 3.6 3.7 3.8

3.6 Substitute the conjectured form of the value function V into the HJB equation, along with the actual utility function and simplify (but again you do not need to solve for the abstract constant(s)). [15 points] 3.7 Write down the Wt process under the assumption that = 1. Note that no measure change has been imposed. [15 points] 3.8 In the lectures we discovered that the state price t is equal to the first derivative of the value function Vw; set t Vw. Using Ito's lemma, state the stochastic process for dt, under the assumption that = 1. [Hint: Because Vw is a function of Wt use the dWt process from your answer to the previous subquestion. Note again that no measure change has been imposed.]

3.1 Write down the consolidated wealth process dWt. 3.2 Write down the HJB equation.

[10 points]

3.3 State the optimality condition for the portfolio holding . [10 points] 3.4 Conjecture a form for V , and also state the derivatives Vw and Vww for the conjectured form. Your conjecture might have one or more constants; because this is an exam I do not expect you to solve for the constant(s)|just carry them forward to this and later questions in abstract form. [10 points] V (w) = Vw = Vww = 3.5 State the solution for the optimal holding of the risky asset using the conjectured form (leaving the unknown constant(s) undetermined for now). [15 points] 3.6 Substitute the conjectured form of the value function V into the HJB equation, along with the actual utility function and simplify (but again you do not need to solve for the abstract constant(s)). [15 points] 3.7 Write down the Wt process under the assumption that = 1. Note that no measure change has been imposed. [15 points] 3.8 In the lectures we discovered that the state price t is equal to the first derivative of the value function Vw; set t Vw. Using Ito's lemma, state the stochastic process for dt, under the assumption that = 1. [Hint: Because Vw is a function of Wt use the dWt process from your answer to the previous subquestion. Note again that no measure change has been imposed.]