# Merton's portfolio problem with constraints

Suppose the investor can invest in a Black-Scholes market with one risky asset $$S$$ with drift $$\alpha$$ and volatility $$\sigma$$ and a riskless asset $$B$$ with a riskless rate of return $$r$$, and the investor seeks to solve the problem $$\max_{c,\lambda} E\bigg[\int_0^Tu(c_t)dt+u(X_T)\bigg],$$ where $$c_t>0$$ is a consumption process and $$\lambda_t$$ is the dollar amount invested in the risky asset. u is a CRRA utility function given by $$u(x)=\frac{x^{\gamma}}{\gamma}$$ for $$\gamma$$ $$\in$$ $$(-\infty,1)$$\{0}. $$X_t$$ is the wealth process that solves $$dX_t=[rX_t+\lambda_t(\alpha-r)-c_t]dt+\sigma \lambda_tdW_t, X(0)=x.$$ Using the martingale method (same as Karatzas & Shreve 1998, Methods of Mathematical Finance) , I have arrived at a feedback form solution $$c^*_t=\frac{X_t}{f_t}, \lambda^*_t=\frac{\alpha-r}{\sigma^2(1-\gamma)}X_t$$ for a deterministic function $$f$$. Now I want to impose the constraint $$X_T\geq G$$ for a positive constant $$G$$. The litterature seems to suggest a solution where one divides the initial wealth $$x$$ into 2 parts, $$kx$$ and $$(1-k)x$$, use the amount $$kx$$ to pursue the strategy ($$kc^*_t$$, $$k\lambda^*_t$$) and the remaining ($$1-k)x$$ to buy a European put option on $$kX$$ with maturity $$T$$ and strike $$G$$. k is chosen such that $$x=kx+P_{kX}(0,T,G)$$ where $$P_{kX}(0,T,G)$$ is the price of the put option at time 0.

My question is how we arrive at this kind of solution? The original problem involved static optimization to find the $$c$$ and $$X_T$$ that maximize the total utility and a martingale representation result to find a $$\lambda$$ that replicated that $$X_t$$ How do we go from static optimization to a strategy of this form?

Also, if one wanted to impose the constraint $$c_t\geq C$$ for all $$t$$ some some positive constant $$C$$, would the solution have the same form?

• The Put Option solution is very clever, but in my opinion you are right that it relies on completely different considerations (i.e. on Option Theory) than the Merton Problem stochastic optimization techniques. The Put Option is clearly a feasible strategy, the proponents will have to show that it is also optimal... (not obvious to me). – noob2 Dec 11 '18 at 16:13
• Do you have any references you can point to? – Daneel Olivaw Dec 11 '18 at 23:17