# Continous-time portfolio allocation optimization for a given consumption rate

I have the following PDE $0 = V_t - c(t)V_x - \lambda^2 V_x^2/V_{xx} + rxV_x + 1/2\lambda^2x^2V_{xx}$ where $t\mapsto c(t)$ is some given function and $r,\lambda$ are given constants. If necessary, one can impose $V(T,x) = \log(x)$ for some finite $T$.

What I would like to know is whether the quantity $V_x/(xV_{xx})$ depends on $t$.

The short background to this question is that there is an agent who wants to maximize his terminal wealth utility at $T$ and this agent consumes at a rate $c(t)$. Dependence of the quantity above on $t$ would imply dependence of optimal allocation strategy (stocks versus bonds) on $t$ and that is what I would like to find out.

The PDE above comes from the HJB equation. Namely, $$0 = V_t +\sup_{\delta}\left((\delta(\mu-r)+r)xV_x - cV_x + \frac{1}{2}\delta^2x^2\sigma^2V_{xx}\right)$$ with $V(T,x) = U_2(x)$ where $\delta$ is the allocation to stock and $c$ is the consumption rate.

The underlying model is the Merton model and the agent wants to maximize $E[U_2(X_T)]$ where $U_2$ is some utility function and $(X_t)_{t\in[0,T]}$ is the wealth process. I could write more details but this is a very standard problem. The only thing that makes it non-standard is that there is a prescribed consumption. If $c(t) = 0$ for all $t \in [0,T]$, then the optimal $\delta = \frac{\mu -r}{\gamma\sigma^2}$ where $\gamma$ is the risk-aversion coefficient in power utility.

• Do you use stochastic control theory to solve your problem? Could you edit your post and write down how you get this pde (in details)? – Leon Apr 2 '16 at 16:39
• @Leon I made some changes. – Calculon Apr 2 '16 at 16:50
• Anything new in this subject? – Leon Apr 11 '16 at 6:30
• @Leon Unfortunately no. – Calculon Apr 11 '16 at 6:32
• Maybe you should write your question (only PDE with boundary condition $V(T,x)=\log(x)$, without financial mathematics background) on this site: math.stackexchange.com/?newreg=7eefc143c8c7428ab8c1a2b0b58896df . More interesting question is whether your PDE has unique solution or not. If yes, we can look for it and then compute $V_{x}/ xV_{xx}$. To solve your PDE we have to substitute some function for $V(t,x)$ to eleminate non-linear part $V_{x}^{2}/V_{xx}$. I hope that somebody will help us with this task. – Leon Apr 11 '16 at 6:53