Application of Ito's Lemma in expected utility theory

Question

An investor with utility curve $U(.)$ has wealth $X_t$ at time t. He invests

• A proportion $p$ of his wealth in a risky asset that follows a geometric Brownian motion, with parameters $\mu$ and $\sigma$, and the remaining proportion $1-p$ in the risk-free asset, with continuous rate r.

Q1.) How would one find the stochastic differential equation for $U(X_t)$, where $dX_t$ can be written by combining the proportions of the different assets ?

Q2.) By considering $U(X_{t+dt}) | F^X_t$, where $F^X_t$ is the filtration generated by the wealth process, how would one derive the optimal proportion p that the investor should use for the time period $[t; t + dt]$ based on Expected Utility Theory.

Answer 1

The risky and riskless assets follow processes,

$$\frac{dS_t}{S_t}= \mu \, dt + \sigma \, dB_t, \,\,\, \frac{dM_t}{M_t}= r \, dt$$

If the proportion invested in the risky asset at time $t$ is $p_t$, then the wealth process is

$$\frac{dX_t}{X_t}= p_t \frac{dS_t}{S_t}+ (1-p_t)\frac{dM_t}{M_t}= (r + p_t(\mu -r)) dt + p_t \sigma dB_t$$

Finding the process for a utility function $x \mapsto U(x)$ requires an application of Ito's lemma,

$$dU(X_t) = \left(\mu \frac{dU}{dx} + \frac{1}{2} \sigma^2\frac{d^2U}{dx^2}\right) \, dt + \sigma \frac{dU}{dx} \, dB_t$$

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To illustrate, suppose we have a log utility function, $U(X_t) = \log X_t$. Using Ito's lemma we obtain the process

$$dU(X_t) = d\log X_t= (r + p_t(\mu -r)- \frac{1}{2} p_t^2 \sigma^2) dt + p_t \sigma dB_t$$

Integrating over $[0,T]$ we get

$$\log X_T = \log X_0 + \int_0^T(r + p_t(\mu -r)- \frac{1}{2} p_t^2 \sigma^2)\, dt+ \int_0^T \sigma p_t \, dB_t,$$

with the expected terminal utility of wealth

$$\mathbb{E}(\log W_T) = \log X_0 + \int_0^T(r + p_t(\mu -r)- \frac{1}{2} p_t^2 \sigma^2)\, dt$$

In this case the optimal allocation is the constant proportion $p^*$ given by

$$p^* = \text{argmax}_p(r + p(\mu -r)- \frac{1}{2} p^2 \sigma^2)T = \frac{\mu-r}{\sigma^2}$$

This, by the way, is the famous Kelly optimal fraction that maximizes the geometric growth rate of the portfolio.