Optimal stopping rules for directionally reinforced processes (Q2749134)

The authors consider optimal single and multiple stopping problems for a class of correlated random walks \(\{S_n\}_{n\in\mathbb{Z}^+}\) with \(S_0=0\) and \(S_n=\sum^n_{i=1}X_i\), \(n\geq 1\), where \(X_n\), \(n\geq 1\), are \(\{-1,1\}\)-valued random variables such that \(P(X_n=1)=1/2\) and satisfy NEWLINE\[NEWLINE\begin{aligned} g^+(j) & = P(X_{n+j+1}=1\mid X_{n+j}=1,\dots,X_{n+1}=1,\;X_n=-1)\\ & = P(X_{n+j+1}=1\mid X_{n+j}=1,\dots,X_{n+1}=1,\;X_n=-1,\;X_{n-1}=\varepsilon_{n-1},\dots,X_1=\varepsilon_1)\end{aligned}NEWLINE\]NEWLINE and NEWLINE\[NEWLINE\begin{aligned} g^-(j) & =P(X_{n+j+1}=-1\mid X_{n+j}=-1,\dots,X_{n+1}=-1,\;X_n=1)\\ & = P(X_{n+j+1}=-1\mid X_{n+j}=-1,\dots,X_{n+1}=-1,\;X_n=1,\;X_{n-1}=\varepsilon_{n-1},\dots,X_1=\varepsilon_1)\end{aligned}NEWLINE\]NEWLINE for \(n\geq 0\), \(j\geq 1\) and \(\varepsilon_1,\dots,\varepsilon_{n-1}\in \{-1,1\}\). It is assumed that the sequences \(\{(g^+(n)\}\) and \(\{g^-(n)\}\) are nondecreasing and \(g^+(1)\), \(g^-(1)\geq 1/2\). Given a finite time horizon \(N\), the objective of a single stopping problem is to find an optimal stopping time \(\tau^*\) such that \(ES_{\tau^*} =\sup_{\tau\in{\mathcal T}^N}S_\tau\) where \({\mathcal T}^N\) is the set of stopping times \(\tau\) such that \(1 \leq\tau \leq N\) almost surely. They establish the general form for the optimal stopping time. For the special case of \(g^+(n) = q\geq 1/2\), a simple optimal stopping time is given. For a multiple stopping problem, the player stops at times \(0\leq \tau_1 <\cdots< \tau_2 < \tau_{2m}\leq N\) and the total return is given by \(\sum^m_{j=1}(S_{\tau_{2j}}-S_{\tau_{2j-1}})-2mc\). For the case without transaction costs, that is, \(c = 0\), they prove that the optimal stopping time has a particularly nice form. For the case with transaction costs some reasonably straightforward stopping times are obtained under various assumptions for \(g^+(n)\) or \(g^-(n)\). Finally numerical examples are presented comparing optimal stopping times to simpler buy and hold strategies.