Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 30K (Stochastic Financial Models)

A process $(X_n{)}_{n\ge 0}$ is a martingale with respect to a filtration $({\mathcal{F}}_n{)}_{n\ge 0}$ when:

Equivalently, every increment has conditional mean zero given the past.

Because $X$ is a martingale,Writingand using ${\xi }_n\in \{-1,1\}$ givesThus

For signs $e_1,\dots ,e_m\in \{-1,1\}$, repeated use of the tower property of conditional expectation givesHence the increments are IID symmetric signs, proving the symmetric signs forced by the martingale property result. Thus $X_n-X_0$ is a simple symmetric random walk.

Because $H_k$ is previsible, $H_k$ is ${\mathcal{F}}_{k-1}$-measurable. The finite sum ${\hat{X}}_n$ is therefore ${\mathcal{F}}_n$-measurable. Boundedness of $H$ and integrability of the martingale increments implyFinally,Thusso $\hat{X}$ is a martingale. It is the martingale transform of $X$ by $H$.

DefineThe event $\{n\le T_a\}$ means that level $a$ has not been reached by time $n-1$, so it belongs to ${\mathcal{F}}_{n-1}$. Hence $H$ is a bounded predictable process. A direct check across the hitting time givesand thereforePart (c) shows that $\hat{X}$ is a martingale. Its increments still take values in $\{-1,1\}$, so part (b) shows that they are IID symmetric signs. Consequently $\hat{X}$ is a simple symmetric random walk. This is the path transformation behind the reflection principle for simple symmetric random walk.

The reflection map from part (d) bijects paths that hit $a$ and finish below $a$ with paths whose reflected endpoint lies strictly above $a$. Since the reflected walk has the same distribution,Subtracting the corresponding identity at level $a+1$ gives

For a simple symmetric random walk,when $|k|\le n$ and $n+k$ is even, and it is zero otherwise. Therefore the explicit answer iswhere a binomial coefficient is interpreted as zero when its lower argument is not an integer in $\{0,\dots ,n\}$. This is the point probability for the maximum of simple symmetric random walk.