3 Coursework Exercises 1

• C1.1

  Let $m$ be a positive integer, and consider a simple random walk with $X_0=2m$. Let the random variable $T$ be the time to ruin, and let $N_t$ denote the number of paths which end in ruin at time $t$.

  Using the results in Lemmas 2.3.1 and 2.3.2, calculate $N_t$. Hence write down the probability mass function for $T$.

  [5]

• C1.2

  In the simple random walk starting from $X_0=1$ the probability that $X_t$ eventually returns to 0 is

  $$\begin{array}{c}\hfill 1\text{ if }p\le q\\ \hfill q/p\text{ if }p\ge q\end{array}.$$

  Write down the corresponding result for the case when the random walk starts from $X_0=-1$.

  Now, by conditioning on the outcome of the first step, show that for a simple random walk starting from $X_0=0$, the probability that $X_t$ eventually returns to 0 is

  $$\begin{array}{c}\hfill 2p\text{ if }p\le q\\ \hfill 2q\text{ if }p\ge q\end{array}.$$

  [5]

Full marks may be obtained for correct solutions to the previous questions. A further 5 marks, up to maximum total mark of 10, can be gained by successfully answering the following.

• C1.3

  Consider the random walks defined by

  and

  Define Ruin as the event that $X_t\le 0$. Let $R_k=\text{P}(\text{Ruin}|X_0=k)$. State whether $R_k=R_1^k$ in each case. Explain your answer.

  In general, what property must a random walk have for $R_k=R_1^k$?