Random Walk - Expected Time Until Absorption

[IniquiTrance]

Homework Statement

Consider the following random walk on the integers:

$$\mathbb{S}=\{0, 1, 2, 3, ... , L\}$$

Let W_n = {the state k $$\in\mathbb{S}$$ you are in after the n'th transition}

$$\mathbb{P}[W_{n+1}= k \pm 1 | W_{n} = k] = \frac{1}_{2}$$

$$For \ 1\leq k\leq L-1$$

Otherwise:

$$\mathbb{P}[W_{n+1}= L | W_{n} = L] = \frac{1}{2} = \mathbb{P}[W_{n+1}= L -1 | W_{n} = L]$$

$$\mathbb{P}[W_{n+1}= 0 | W_{n} = 0] = 1$$

i.e. L is retaining, and 0 is absorbing.

Determine:

$$\mathbb{E}[T | W_{0}=k] \forall \ k \in \mathbb{S}$$

$$Where \ T=min \{n\geq 0 : W_{n}=0 \}$$

Not sure how to proceed...

 

[lippyka]

Homework Equations The Attempt at a Solution I have tried to consider the expected value of T in the case where k=0,1,2,...L-1, but am not sure how to properly combine these values to yield the solution.