Game theory, economics, finance, and other social and behavioral sciences Portfolio theory Problem

Merton's portfolio problem

Builds on Stochastic optimal control Geometric Brownian motion Hamilton–Jacobi–Bellman equation

Problem

Consider a market with a riskless asset with interest rate $r$ and a stock. The stock price follows the geometric Brownian motion $dS_t = \mu S_t \, dt + \sigma S_t \, dW_t$ , where $\mu > r$ , $\sigma > 0$ and $W$ is a Wiener process. An investor with initial wealth $x > 0$ chooses, at each time $t$ , the fraction $\pi_t$ of wealth held in the stock and a consumption rate $c_t \ge 0$ . Both are adapted to the information generated by $W$ . The wealth then evolves by $dX_t = \left( r X_t + \pi_t (\mu - r) X_t - c_t \right) dt + \pi_t \sigma X_t \, dW_t, \qquad X_0 = x.$ Let $\rho > 0$ be a discount rate and let $U(c) = \dfrac{c^{1 - \gamma}}{1 - \gamma}$ with $\gamma > 0$ and $\gamma \ne 1$ . The problem is to find the strategies $(\pi, c)$ with $X_t \ge 0$ for all $t$ that maximise the expected discounted utility of consumption $E \int_0^\infty e^{-\rho t} U(c_t) \, dt.$

Remarks

The problem is solved, and it is a standard example of stochastic optimal control. Write $\theta = (\mu - r) / \sigma$ and assume $\nu = \dfrac{\rho - (1 - \gamma)\left( r + \theta^2 / (2 \gamma) \right)}{\gamma} > 0$ . The value function is $V(x) = \nu^{-\gamma} U(x)$ . The optimal strategy holds the constant fraction $\pi^* = \dfrac{\mu - r}{\gamma \sigma^2}$ in the stock and consumes at the rate $c_t = \nu X_t$ . These follow from the Hamilton–Jacobi–Bellman equation $\rho V = \max_{\pi, c} \left\{ U(c) + V'(x) \left( r x + \pi (\mu - r) x - c \right) + \frac{1}{2} \sigma^2 \pi^2 x^2 V''(x) \right\}$ with the guess $V = A U$ for a constant $A > 0$ . Merton also treated a finite horizon with utility of final wealth, and the logarithmic utility $U = \ln$ , which is the limit case $\gamma = 1$ and gives $\pi^* = (\mu - r) / \sigma^2$ and $c_t = \rho X_t$ .

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Geometric Brownian motion concept
Hamilton–Jacobi–Bellman equation concept
Stochastic optimal control concept
Merton's portfolio problem problem

Tag	`DZKBF576W5`
URI	`m2://DZKBF576W5`
Kind	problem
Writing	W0 stub
MSC	91G10, 93E20, 49L20
Status	This copy of the site has no build data. Run `m2 status DZKBF576W5`.
About	Stochastic optimal control
Uses	Stochastic optimal control, Geometric Brownian motion, Hamilton–Jacobi–Bellman equation
Same as	Wikidata Q3467003 Wikipedia Merton's portfolio problem

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