The Kelly Criterion And Utility Function Optimisation For Stochastic Binary Games: Submartingale And Supermartingale Regimes

Abstract

A reformulation of the Kelly Criterion is presented. Let $\mathfrak{G}$ be a generic stochastic Bernoulli binary game with outcomes $\mathcal{Z}(I)\in \{-1,1\}$ of N trials for $I=\mathrm{1...}N$. The binomial probabilities are $\mathsf{P}(\mathcal{Z}(I)=1)=p$ and $\mathsf{P}(\mathcal{Z}(I)=-1)=q$ with $p+q=1$. For a fair game $p=q=\frac{1}{2}$ and for a biased game $p>q$. If $\mathcal{W}(0)$ is the initial wealth then at the $I^{th}$ trial one bets a fraction $\mathcal{F}$ so that the bet is $B(I)=\mathcal{F}\mathcal{W}(I-1)$. If one wagers $B(I)$ and wins one recovers the original wager plus $B(I)$ if $\mathcal{Z}(I)=+1$, or a loss of $B(I)$ if $\mathcal{Z}(I)=-1$. The wealth at the $N^{th}$ trial/bet for large $N$ is the random walk $\mathcal{W}(N)=\mathcal{W}(0)+{\sum }_{I=1}^{N}B(I)\mathcal{Z}(I)=\mathcal{W}(0){\prod }_{I=1}^N(1+\mathcal{F}\mathcal{Z}(I))$ with expectation $\mathsf{E}[\mathcal{W}(N)]$. Defining a 'utility function' $\mathsf{U}(\mathcal{F},p)=\mathsf{E}[\log (\mathcal{W}(N)/\mathcal{W}(0){)}^{1/N}]$ then $\mathsf{U}(\mathcal{F},p)$ is optimised by the Kelly fraction $\mathcal{F}={\mathcal{F}}_K=p-q=2p-1$, which is essentially a critical point of $\mathsf{U}(\mathcal{F},p)$. Also $\mathsf{U}({\mathcal{F}}_K,p)$ can be related to the Shannon entropy. If $[0,1]=[0,{\mathcal{F}}_{\ast })\bigcup [{\mathcal{F}}_{\ast }]\bigcup ({\mathcal{F}}_{\ast },1]$ with $\mathsf{U}({\mathcal{F}}_{\ast },p)=0$ then $\mathsf{U}(\mathcal{F},p)>0,\forall \mathcal{F}\in [0,{\mathcal{F}}_{\ast })$ and $\mathcal{W}(N)$ is a submartingale for $p>1/2$; also $\mathsf{U}(\mathcal{F},p)<0,\forall \mathcal{F}\in ({\mathcal{F}}_{\ast },1]$, and $\mathcal{W}(\mathcal{F},p)$ is a supermartingale. Estimates are derived for variance and volatility $\mathsf{VAR}(\mathcal{W}(N))$ and $\ddot{I}ƒ(\mathcal{W}(N))=\sqrt{\mathsf{VAR}(\mathcal{W}(N)})$. For large $N$ and $\mathcal{F}={\mathcal{F}}_K$, $\mathsf{E}[\mathcal{W}(N)]$ grows exponentially.

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