Introduction to Stochastic Analysis: Integrals and Differential Equations

Solutions, Hints, and Answers

 

 

Chapter 2 exercises

2.1. For s ≥ t, use the independence of B_t and B_s − B_t:

EB_tB_s = EB_t[B_s − B_t) + B_t] = EB_t E(B_S − B_t) + EB²_t = 0 . 0 + t = t, EB²_tB²_s = EB²_t[(B_s− B_t) + B_t]² = EB²_t E(B_S−B_t)² +2EB³_t E(B_S−B_t) + EB⁴_t, and so on.

2.2. The first and second expectations are zeros because of the symmetry of distributions of the random variables. For the second expectation, use the equality

2.3. By part 2 of Theorem 2.4,

2.4.

2.5. where ξ ~ N(0,1).

2.6. A careful reading of the proof of Theorem 2.4 shows that it also works in this case.

2.7. Hint: Check that W satisfies all the properties of a Brownian motion (stated in Theorem 2.3).

2.8. Answer: cov (W_t , B_t) = ρt.

2.9. Answer: cov (Z_t,Z_s) = t ∧ s − ts.

2.10. Using the independence of X and Y, we have:

2.11. Hint: Use Exercises 2.8 and 2.9 to show that the possible limits fill the interval [0,1].

 

Chapter 4 exercises

4.1. Answer: C = 3.

4.2. First, note that, for such that n ≥ t,

and then (correctly) pass ...