Questions on Kelly criterion

Question

I am new to asset allocation problems and have some concerns regarding the derivation of the continuous-time Kelly criterion (i.e. not the original version destined to discrete sports betting/Casino).

I am following the derivation of Martin & Schöneborn, see page 4 and following.

As far as I understand it, the assumptions are:

1. Economy with 2 investment vehicles: a risky asset $X$ and a risk-free one $B$
2. $X$ is driven by a time-homogeneous diffusion process: $$dX_t = \mu(X_t) dt + \sigma (X_t) dW_t $$ while $B$ follows the usual: $dB_t = rB_t dt$
3. The investment strategy consists in holding $\{\theta_t\}_{t\geq 0}$ units of the risk-asset at any time $t \geq 0$. It is assumed self-financing, so that the P&L over an infinitesimal time interval $[t,t+dt[$ writes: $$ d\Pi_t = \theta_t dX_t + dB_t $$
4. The investor aims at maximizing the total discounted expectation of the utility of her $\theta$-controlled P&L over a forward-looking horizon stretching from today $t$ up to infinity: $$ V_t = \int_{s=t}^\infty e^{-r(s-t)} \mathbb{E}_t\left[ \mathcal{U}(\theta_s dX_s) \right] $$
5. Since $X_t$ is a diffusive process with bounded quadratic variation, a second order definition of the utility function $\mathcal{U}(.)$ is enough. Typically, we take: $$\mathcal{U}(0)=0,\ \ \mathcal{U}'(0)=1,\ \ \mathcal{U}''(0)=-1/G$$
6. Continuous trading is possible and there are no market frictions.

And the result is: $$ \theta_t = \frac{\mu(X_t)}{\sigma^2(X_t)}G $$

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What I don't understand is:

1. In the paper, they look for $\{\theta_t\}$ which maximises $V_t$. This leads to the following ODE $$ \frac{\partial f}{\partial x}(X_t) \mu(X_t) + \frac{1}{2} \frac{\partial^2 f}{\partial x^2}(X_t) \sigma^2(X_t) - rf(X_t) = -\dot{U}(X_t, \theta_t) $$ Now they jump to the "intuitively clear conclusion" that this is equivalent to finding $\theta_t$ which maximises $\dot{U}(X_t, \theta_t)$. Unfortunately, it is not clear to me.
2. The fact that the optimal allocation strategy does not involve the risk-free rate $r$ seems weird to me. In my opinion, this comes from the fact that we should rather define the value function $V_t$ to be maximised as: $$ V_t = \int_{s=t}^\infty e^{-r(s-t)} \mathbb{E}_t\left[ \mathcal{U}(\theta_s dX_s + dB_s) \right] $$
3. How do you use this Kelly allocation criterion in practice? I mean, suppose time $t$ corresponds to the market close of day $D$. It seems natural to want to know the optimal allocation (or position) one should take at $t+\Delta t$ (e.g. day $D+1$). So what we need is something like: $$ \theta_{t+\Delta t} = f(\text{information contained in the filtration } \mathcal{F}_t) $$ and not $\theta_t$ on the LHS, since it is already known at time $t$ (in the demonstration we always assume $\theta_t$ to be adapted, hence measurable at any $t$). Should I assume a particular continuity assumption for $\theta_t$?
4. I've read somewhere that $\forall G$ the Kelly strategy is optimal in the following sense: $$\mathbb{E} \left[ \frac{ \Pi_T(\{\theta_t^{\text{Kelly}(G)}\}_{t\in[0,T]}) }{ \Pi_T( \{\theta_t\}_{t\in[0,T]}) } \right] \geq 1$$ for all $\{\theta_t\}_{t\in[0,T]}$ adapted to the natural filtration of $W_t$ and where $\Pi_T( \{\theta_t\}_{t\in[0,T]}$ denotes the final wealth of an investment strategy based on the risky asset position sizing $\{\theta_t\}$. But some simulations show me that this is not true, especially not $\forall G$. Even the growth rate $\dot{U}(\theta_t,X_t)$ mentioned in the paper does not seem optimal (sometimes I have basic long only or short only strategy that perform better).

Could someone please help me out? If the answer to all these questions is that this derivation is wrong and not the one I should follow, I would be glad if someone could provide a nice reference.

Cheers

Answer 1

As wrt. to (1): ODE and Maximization:
The paper's transition from an ODE to maximizing $\dot{U}(X_t, \theta_t)$ can seem non-intuitive. The key here is recognizing that the original problem involves maximizing expected utility. The dynamic programming principle involves finding a control strategy (i.e., $\theta_t$) that maximizes the Hamilton-Jacobi-Bellman (HJB) equation associated with this utility maximization problem. The ODE forms the basis of this optimization, where we seek to choose $\theta_t$ such that the effect of adding more risky exposure is optimized.

Intuition:

• HJB Equation: The ODE is derived from the HJB equation which is a necessary condition for optimality in continuous-time optimization problems.
• Utility Maximization: By maximizing $\dot{U}(X_t, \theta_t)$, you're directly influencing the value function $V_t$, thus aligning with the optimization goal.

The derivation uses the idea of infinitesimal time increments where maximizing the utility over a short period gives rise to the expression for the optimal allocation $\theta_t$.

Answer 2

A continuous-time strategy that aims at maximizing the long-term growth rate of wealth is the Kelly criterion, which optimizes the proportion of wealth to be invested in risky assets. One has to solve a Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) and find out the optimal allocation. It maximizes growth rate of wealth and also considers risk and return of assets. Nevertheless, actual application of this strategy needs accurate estimation of parameters as well as updating allocation at every decision point of rebalancing. On the other hand, deviation from expected outcomes may be due to model mis-specification, estimation issues or utility function choice.