Ito’s Lemma and the Kelly Fraction for Log-Utility Investors
Summary
The document derives how an investor’s wealth changes when a fraction is allocated to a risky asset following geometric Brownian motion and the remainder earns a risk-free rate. Applying Ito’s lemma transforms the wealth process into a stochastic process for utility. For logarithmic utility, the drift includes the risk-free return, the risky asset’s excess return weighted by the allocation, and a volatility-related penalty; the random component scales with the allocation and volatility.
Taking expectations over time removes the Brownian increment, leaving an objective that can be maximized with respect to the risky allocation. Under the stated constant-parameter model, the maximizing fraction is the excess expected return divided by variance, commonly identified as the Kelly fraction. This result applies to log utility and idealized continuous trading under the assumed dynamics. The answer does not discuss constraints, estimation error, changing market parameters, or practical position limits, all of which can alter a real allocation.
Key ideas
- Wealth growth combines the risky asset’s excess return with the risk-free rate.
- Ito’s lemma adds a second-derivative term when converting wealth dynamics into utility dynamics.
- For log utility, expected growth includes a penalty proportional to allocation squared and variance.
- Maximizing expected log utility under the stated model gives the Kelly allocation.
- The result depends on the model assumptions and does not account for estimation error or investment constraints.
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Full text
# Application of Ito's Lemma in expected utility theory
# 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.
## Answer by RRL (score 6)
https://quant.stackexchange.com/a/55039
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.