last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

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11.5 Exercises on Chapter 11

In all the following questions, \(B_t\) denotes standard Brownian motion.

On Brownian motion
  • 11.1 Consider the process \(C_t=\mu t+ \sigma B_t\), for \(t\geq 0\), where \(\mu \in \R \) and \(\sigma >0\) are deterministic constants.

    • (a) Find the mean and variance of \(C_t\).

    • (b) Let \(0\leq u\leq t\). What is the distribution of \(C_t-C_u\)?

    • (c) Is \(C_t\) a random continuous function?

    • (d) Is \(C_t\) a Brownian motion?

  • 11.2 Let \(0\leq u\leq t\). Use the properties of Brownian motion to show that \(\cov (B_t,B_u)=u\).

  • 11.3 Let \(u\geq 0\) and \(t\geq 0\). Show that \(\E [B_t\|\mc {F}_u]=B_{\min (u,t)}\).

  • 11.4

    • (a) Show that \(\E [B_t^n]=t(n-1)\E [B_t^{n-2}]\) for all \(n\geq 2\). (Hint: Integrate by parts!)

    • (b) Deduce that \(\E [B_t^2]=t\) and \(\var (B_t^2)=2t^2\).

    • (c) Write down \(\E [B_t^n]\) for any \(n\in \N \).

    • (d) Show that \(B_t^n\in L^1\) for all \(n\in \N \).

  • 11.5 Let \(Z\sim N(\mu ,\sigma ^2)\). Show that \(\E [e^Z]=\exp (\mu +\frac 12\sigma ^2)\). (Hint: Complete the square!)

  • 11.6 Show that the following processes are martingales.

    • (a) \(X_t=\exp \l (\sigma B_t -\tfrac 12\sigma ^2t\r )\) where \(\sigma >0\) is a deterministic constant.

    • (b) \(Y_t=B_t^3-3tB_t\).

  • 11.7 Fix \(t>0\) and for each \(n\in \N \) let \((t_k)_{k=0}^n\) be such that \(0=t_0<t_1<\ldots <t_n=t\) and \(\max _k|t_{k+1}-t_k|\to 0\) as \(n\to \infty \). (For example: \(t_k=\frac {kt}{n}\)).

    • (a) Show that \(\sum \limits _{k=0}^{n-1} t_{k+1}-t_k=t\) and \(\sum \limits _{k=0}^{n-1} B_{t_{k+1}}-B_{t_k}=B_t\).

    • (b) Show that \(\sum \limits _{k=0}^{n-1} (t_{k+1}-t_k)^2\to 0\) as \(n\to \infty \).

    • (c) Set \(S_n=\sum \limits _{k=0}^{n-1} (B_{t_{k+1}}-B_{t_k})^2\). Show that \(\E [S_n]=t\) and that \(\var (S_n)\to 0\) as \(n\to \infty \).

Challenge Questions
  • 11.8

    • (a) Let \(y\geq 1\). Show that

      \[\P [B_t\geq y]\leq \sqrt {\frac {t}{2\pi }}e^{-\frac {y^2}{2t}}.\]

      Let \(\alpha >\frac 12\). Deduce that \(\P [B_t\geq t^\alpha ]\to 0\) as \(t\to \infty \). What about \(\alpha =\frac 12\)?

    • (b) Let \(y\geq 0\). Show that

      \[\P [B_t\geq y]\leq \frac {t}{\sqrt {2\pi }} \frac {1}{y}e^{-\frac {y^2}{2t}}.\]

      Deduce that \(B_t\to 0\) in probability as \(t\searrow 0\).