Solutions for Chapter 6

Summary

In the solutions developed below, (ℱ_t) will often denote the “obvious” filtration involved in the question, and F = F(X_u,u ≤ t) a functional on the canonical path space C(IR₊, IR^d), which is measurable with respect to the past up to time t.

Solution to Exercise 6.1

1. (a) By assumption, (R_t, t ≥ 0) is a positive process, so it follows from Fubini theorem that

Then we derive from the assumptions (i) and (ii) that E[R_t] = E [R₁] < ∞, so that if, then and hence, a.s.

(b) Applying Fubini theorem, we obtain that for any measurable set B,

Then the inequality follows from the hint.

(c) By definition of the set B_N, we have. Moreover, the assumption (i) implies that

Then we conclude thanks to the inequality of question 1(a).

(d) If, a.s., then lim_N→∞ P(B_N) = 1. Let us exclude the trivial case where R₁ = 0, a.s. So we can ch∞se N sufficiently large and ε > 0 sufficiently small to have, for all, from which we derive that

Then we deduce that μ(0,1] < ∞ from the inequality of question 1 (c).

2. (a) Let us compute, for s < t.

First we note that

The process β is clearly a continuous centered Gaussian process.

Moreover, as we have just proved, its covariance function is given by E(β_tβ_s) = s Λ t, s, t ≥ 0, hence β is a standard Brownian motion

This identity may be expressed as

Then considering the above equation as a linear differential equation whose solution exists and is (B_u, u < t), we see that B can be written as

for some measurable functional Φ, hence.