Let X, denote your wealth at time t. Suppose that at any time t you have a choice between two investments: 1) A risky investment where the unit price p = Pict,w) satisfies the equation dp, = 21Pidt +1P1dB.. 2) A safer (less risky) investment where the unit price P2 = Pzt,w) satisfies dpz = 22P2dt + 02P2dB, where di , are constants such that a> az, 01 > 2 and B., B, are independent 1-dimensional Brownian motions. a). Let u (t,w) denote the fraction of the fortune X: (w) which is placed in the riskier investment at time t. Show that dX, = dx") = X(aju + a2(1 - i))dt +X,(g, udB, +0(1 u)dB.). b). Assuming that u is a Markov control, u = u(t, x), find the generator A" of (t, x'). c). Write down the HJB equation for the stochastic control problem D(s,x) = sup ex [(x:"1 where T = min(t, To), To - inf(t> s X, = 0 and 11 is a future time constant), Y E (0,1) is a constant. d). Find the optimal control u for the problem in c). Let X, denote your wealth at time t. Suppose that at any time t you have a choice between two investments: 1) A risky investment where the unit price p = Pict,w) satisfies the equation dp, = 21Pidt +1P1dB.. 2) A safer (less risky) investment where the unit price P2 = Pzt,w) satisfies dpz = 22P2dt + 02P2dB, where di , are constants such that a> az, 01 > 2 and B., B, are independent 1-dimensional Brownian motions. a). Let u (t,w) denote the fraction of the fortune X: (w) which is placed in the riskier investment at time t. Show that dX, = dx") = X(aju + a2(1 - i))dt +X,(g, udB, +0(1 u)dB.). b). Assuming that u is a Markov control, u = u(t, x), find the generator A" of (t, x'). c). Write down the HJB equation for the stochastic control problem D(s,x) = sup ex [(x:"1 where T = min(t, To), To - inf(t> s X, = 0 and 11 is a future time constant), Y E (0,1) is a constant. d). Find the optimal control u for the problem in c)