last updated: October 16, 2024

Stochastic Processes and Financial Mathematics
(part two)

11.5 Exercises on Chapter 11

In all the following questions, Bt denotes standard Brownian motion.

On Brownian motion
  • 11.1 Consider the process Ct=μt+σBt, for t0, where μR and σ>0 are deterministic constants.

    • (a) Find the mean and variance of Ct.

    • (b) Let 0ut. What is the distribution of CtCu?

    • (c) Is Ct a random continuous function?

    • (d) Is Ct a Brownian motion?

  • 11.2 Let 0ut. Use the properties of Brownian motion to show that cov(Bt,Bu)=u.

  • 11.3 Let u0 and t0. Show that E[Bt|Fu]=Bmin(u,t).

  • 11.4

    • (a) Show that E[Btn]=t(n1)E[Btn2] for all n2. (Hint: Integrate by parts!)

    • (b) Deduce that E[Bt2]=t and var(Bt2)=2t2.

    • (c) Write down E[Btn] for any nN.

    • (d) Show that BtnL1 for all nN.

  • 11.5 Let ZN(μ,σ2). Show that E[eZ]=exp(μ+12σ2). (Hint: Complete the square!)

  • 11.6 Show that the following processes are martingales.

    • (a) Xt=exp(σBt12σ2t) where σ>0 is a deterministic constant.

    • (b) Yt=Bt33tBt.

  • 11.7 Fix t>0 and for each nN let (tk)k=0n be such that 0=t0<t1<<tn=t and maxk|tk+1tk|0 as n. (For example: tk=ktn).

    • (a) Show that k=0n1tk+1tk=t and k=0n1Btk+1Btk=Bt.

    • (b) Show that k=0n1(tk+1tk)20 as n.

    • (c) Set Sn=k=0n1(Btk+1Btk)2. Show that E[Sn]=t and that var(Sn)0 as n.

Challenge Questions
  • 11.8

    • (a) Let y1. Show that

      P[Bty]t2πey22t.

      Let α>12. Deduce that P[Bttα]0 as t. What about α=12?

    • (b) Let y0. Show that

      P[Bty]t2π1yey22t.

      Deduce that Bt0 in probability as t0.