# Application of Ito's Lemma in expected utility theory

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.

• You should post what have you tried to answer this question. – phdstudent Jun 19 '20 at 8:16

## 1 Answer

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,

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

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.