Stochastic Processes and Financial Mathematics
(part one)

4.5 Exercises on Chapter 4

On stochastic processes

• 4.1 Let $S_n=\sum _{i=1}^n{X}_i$ be the symmetric random walk from Section 4.1 and let $Z_n=e^{S_n}$. Show that $(Z_n)$ is a submartingale and that

  $$M_n={\left(\frac{2}{e+\frac{1}{e}}\right)}^n{Z}_n$$

  is a martingale.

• 4.2 Let $S_n=\sum _{i=1}^n{X}_i$ be the asymmetric random walk from Section 4.1, where $\mathbb{P}[X_i=1]=p$, $\mathbb{P}[X_i=-1]=q$ and with $p>q$ and $p+q=1$. Show that $(S_n)$ is a submartingale and that

  $$M_n={\left(\frac{q}{p}\right)}^{S_n}$$

  is a martingale.

• 4.3 Let $(X_i)$ be a sequence of identically distributed random variables with common distribution

  $$X_i=\{\begin{array}{ll}a& \text{ with probability }{p}_a\\ -b& \text{ with probability }{p}_b=1-p_a.\end{array}$$

  where $0\le a,b$. Let $S_n=\sum _{i=1}^n{X}_i$. Under what conditions on $a,b,p_a,p_b$ is $(S_n)$ a martingale?

• 4.4 Let $S_n=\sum _{i=1}^n{X}_i$ be the symmetric random walk from Section 4.1. Show that $S_n^2$ is a submartingale and that $M_n=S_n^2-n$ is a martingale.

• 4.5 Let $(X_i)$ be an i.i.d. sequence of random variables such that $\mathbb{P}[X_i=1]=\mathbb{P}[X_i=-1]=\frac{1}{2}$. Define a stochastic process $S_n$ by setting $S_0=1$ and

  $$S_{n+1}=\{\begin{array}{ll}{S}_n+X_{n+1}& \text{ if }{S}_n>0,\\ 1& \text{ if }{S}_n=0.\end{array}$$

  That is, $S_n$ behaves like a symmetric random walk but, whenever it becomes zero, on the next time step it is ‘reflected’ back to $1$. Let

  $$L_n=\sum _{i=0}^{n-1}\text{𝟙}\{S_i=0\}$$

  be the number of time steps, before time $n$, at which $S_n$ is zero. Show that

  $$\mathbb{E}[S_{n+1}|{\mathcal{F}}_n]=S_n+\text{𝟙}\{S_n=0\}$$

  and hence show that $S_n-L_n$ is a martingale.

• 4.6 Consider an urn that may contain balls of three colours: red, blue and green. Initially the urn contains one ball of each colour. Then, at each step of time $n=1,2,\dots$ we draw a ball from the urn. We place the drawn ball back into the urn and add an additional ball of the same colour.

  Let $(M_n)$ be the proportion of balls that are red. Show that $(M_n)$ is a martingale.

• 4.7 Let $S_n=\sum _{i=1}^n{X}_i$ be the symmetric random walk from Section 4.1. State, with proof, which of the following processes are martingales:

  $$

  $$\text{(i) }{S}_n^2+n\text{(ii) }{S}_n^2+S_n-n\text{(iii) }\frac{S_n}{n}$$

  Which of the above are submartingales?

Challenge questions

• 4.8 Let $(S_n)$ be the symmetric random walk from Section 4.1. Prove that there is no deterministic function $f:\mathbb{N}\to \mathbb{R}$ such that $S_n^3-f(n)$ is a martingale.