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

A reformulation of the Kelly Criterion is presented. Let $\mathfrak{G}$ be a generic stochastic Bernoulli binary game with outcomes $\mathscr{Z}(I)\in\lbrace -1,1\rbrace$ of N trials for $I=1...N$. The binomial probabilities are $\mathsf{P}(\mathscr{Z}(I)=1)=p$ and ${\mathsf{P}}(\mathscr{Z}(I)=-1)=q$ with $p+q=1$. For a fair game $p=q=\tfrac{1}{2}$ and for a biased game $p>q$. If $\mathscr{W}(0)$ is the initial wealth then at the $I^{th}$ trial one bets a fraction $\mathcal{F}$ so that the bet is $B(I)=\mathcal{F}\mathscr{W}(I-1)$. If one wagers $B(I)$ and wins one recovers the original wager plus $B(I)$ if $\mathscr{Z}(I)=+1$, or a loss of $B(I)$ if $\mathscr{Z}(I)=-1$. The wealth at the $N^{th}$ trial/bet for large $N$ is the random walk $\mathscr{W}(N)=\mathscr{W} (0)+\sum_{I=1}^{N}B(I)\mathscr{Z}(I)=\mathscr{W}(0)\prod_{I=1}^{N}(1+\mathcal{F}\mathscr{Z}(I))$ with expectation $\mathsf{E}[\mathscr{W}(N)]$. Defining a 'utility function' $\mathsf{U}(\mathcal{F},p)=\mathsf{E}[\log(\mathscr{W}(N)/\mathscr{W}(0))^{1/N}]$ 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 $[0,1]=[0,\mathcal{F}_{*})\bigcup [\mathcal{F}_{*}]\bigcup (\mathcal{F}_{*},1]$ with $\mathsf{U}(\mathcal{F}_{*},p)=0$ then $\mathsf{U}(\mathcal{F},p)>0, \forall\mathcal{F}\in[0,\mathcal{F}_{*})$ and $\mathscr{W}(N)$ is a submartingale for $p>1/2$; also $\mathsf{U}(\mathcal{F},p)<0,\forall \mathcal{F}\in(\mathcal{F}_{*},1]$, and $\mathscr{W}(\mathcal{F},p)$ is a supermartingale. Estimates are derived for variance and volatility $\mathsf{VAR}(\mathscr{W}(N))$ and $\sigma(\mathscr{W}(N))=\sqrt{\mathsf{VAR}(\mathscr{W}(N)})$. For large $N$ and $\mathcal{F}=\mathcal{F}_{K}$, $\mathsf{E}[\mathscr{W}(N)]$ grows exponentially.