# mean time to return random walk continuous whit drift to origen

Let $$S_t$$ a random walk with rate $$a>0$$ on $$Z$$ that such has a drift in direction of $$0$$, this is, if $$(0,0)\in Z\times[0,\infty)$$, and defined a sequence $$(S_n, T_n)_{n\in N}$$ such that $$(S_0, T_0)=(0,0)$$ and, for $$n\geq1$$

$$T_n-T_{n-1}=\tau_n\sim Exp(a), \ \ \ \ \ \ S_n-S_{n-1}= \left\{ \begin{array}{lcc} 1, & if & S_n \leq 0 \\ -1, & if & S_n <0 \\ \end{array} \right.$$

What is $$E(TR_o)$$ where $$TR_0=\inf\{t>0: S_t=0\}$$?

• I don't get what your $T_n$ does in here. Can you be more precise on this point ? – Joseph Budin Oct 1 '19 at 13:28
• $T_n$ are the renewal times of random walk – Joan Jesus Amaya Triana Oct 1 '19 at 22:04
• And where is the stochasticity of $S_n$ ? As far as I understand, its behaviour is deterministic. – Joseph Budin Oct 2 '19 at 7:23