Gambler’s ruin with random stopping

Abstract

Let $\{{\mathbf{X}}_j,j\ge 0\}$ denote a Markov process on $[-N-1,N+1]\cup \left\{\mathbf{c}\right\}$. Suppose $\mathbb{P}({\mathbf{X}}_{j+1}=m+1|{\mathbf{X}}_j=m)=\mathit{ph},\mathbb{P}({\mathbf{X}}_{j+1}=m-1|{\mathbf{X}}_j=m)=(1-p)h$, all $j\ge 1$ and $\left|m\right|\le N$, where $p=\frac{1}{2}+\frac{b}{N}$ and $h=1-c_N$ for $c_N=\frac{1}{2}{a}^2/N^2$. Define $\mathbb{P}({\mathbf{X}}_{j+1}=\mathbf{c}|{\mathbf{X}}_j=m)=c_N,j\ge 0,|m|\le N$. $\left\{{\mathbf{X}}_j\right\}$ terminates at the first j such that ${\mathbf{X}}_j\in \{-N-1,N+1,\mathbf{c}\}$. Let $\mathcal{L}=\max \{j\ge 0:{\mathbf{X}}_j=0\}$. On ${\Omega }^{°}=\left\{{\mathbf{X}}_j\text{ terminates at }\mathbf{c}\right\}$, denote by ${\mathcal{R}}^{°}$ and ${\mathcal{L}}^{°}$, respectively, as the numbers of runs and steps from $\mathcal{L}$ until termination. Denote ${\Delta }^{°}={\mathcal{L}}^{°}-2{\mathcal{R}}^{°}$. Then $\underset{N\to \infty }{\lim }\mathbb{E}\left\{e^{\frac{it}{N}{\Delta }^{°}}\right|{\Omega }^{°}\}=C_{a,b}\frac{\sqrt{c^2+t^2}(\cosh \sqrt{c^2+t^2}\text{-}\cosh (2b))}{(a^2+t^2)\sinh \sqrt{c^2+t^2}}$, where $c^2=a^2+4b^2$.

KEYWORDS:

• Catastrophe
• Fibonacci polynomial
• gambler’s ruin
• runs

Acknowledgment

The author wishes to thank an anonymous referee for suggesting improvements to streamline the presentation and emphasize the main extensions of previous work.

Disclosure statement

No potential conflict of interest was reported by the authors.