Constrained Merton Portfolio Optimization with a Terminal Wealth Floor

# Merton's portfolio problem with constraints


# 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?