Hi...I was wondering if anybody could tell me how I could solve this analytically:maximise E[ inetgral frm 0 to T (exp(-at)ln(c(t))dt + K.ln(X(T))]given the wealth dynamics:dX= X[u0(t) + u1(t)]dt -c(t)dt +u1(t)*sigma*XdW where W is a std Brownian motion.Contro0 constraints:c(t)>=0 , for all tu0(t)+u1(t)=1, for all tI am ending up with a highly non-linear HJB equation with a logarithm of the 1st derivative of the value function and have no clue how to solve it analytically. It has ruined my past 2 nights!!

Can you get a copy of Bjoerk's "Arbitrage theory in continuous time" (chapter 14) or Korn's "Optimal Portfolios"? They both describe the solution method in detail. In case of a logarithmic utility function the solution is given by the optimal growth portfolio.By the way, I think you have omitted the risk-free rate in your wealth dynamics...Cheers,Costica.

Hey Costica...Thanks a lot for the reply.The problem I have mentioned here is in Bjork...infact it's problem no. 14.1....and there's no way I can get Korn's book (I have already tried in the library). Yeah, u are right..I forgot to multiply the relative portfolio weights weights with the respective local mean rates (risk-free and stock's local drift)...I figured it must be a optimal growth portfolio but I have to make a guess of the form of the solution (for the value function V(t)) ...i used exp(-at)H(t)logx but the resulting PDE in H(t) contains x explicitly and highly non-linear too. So I am stuck. Can you help me out with this please?