Math 6010 • Linear Models

Fix numbers \(p_x,q_x\in (0,1)\), \(x\in \mathbb {N}\), such that \(p_x+q_x\leq 1\). Let \(\{Z_{n,x}:n,x\in \mathbb {N}\}\) be independent random variables that take values in \(\{-1,0,1\}\), with common distribution \[ \mathbb {P}(Z_{n,x}=1)=p_x,\qquad \mathbb {P}(Z_{n,x}=-1)=q_x,\qquad \mathbb {P}(Z_{n,x}=0)=1-p_x-q_x. \] Let \(X_0=1\) and for \(n\in \mathbb {Z}_+\) define inductively \(X_{n+1}=X_n+Z_{n,X_n}\) if \(X_n\in \mathbb {N}\) and \(X_{n+1}=0\) if \(X_n=0\). In particular, \(X_n\in \mathbb {Z}_+\) almost surely. These \(X_n\) satisfy \[ \begin {aligned} \mathbb {P}(X_{n+1}=x+1\mid X_n=x)&=p_x,\\ \mathbb {P}(X_{n+1}=x-1\mid X_n=x)&=q_x,\\ \mathbb {P}(X_{n+1}=x\mid X_n=x)&=1-p_x-q_x, \end {aligned} \] if \(x\in \mathbb {N}\) and \(\mathbb {P}(X_{n+1}=x\mid X_n=x)=1\) if \(x=0\). The process \(X_n\) is called a birth and death process, absorbed at 0. Now define the function \(\phi :\mathbb {Z}_+\to \mathbb {R}\) by \(\phi (0)=0\), \(\phi (1)=1\), and for an integer \(x\geq 2\), \[ \phi (x)=1+\sum _{i=1}^{x-1}\prod _{j=1}^{i}\frac {q_j}{p_j}. \] Prove that \(M_n=\phi (X_n)\) is a martingale in the natural filtration \(\mathcal {F}_n=\sigma (X_0,\ldots ,X_n)\).