The linear growth of the coefficients is essential for Corollary 21.31.

a) Consider the case where \(d=n=1, b(x)=-e^{x}\) and \(\sigma(x)=0\). Find the solution of this deterministic ODE and compare your findings for \(x \rightarrow+\infty\) with Corollary 21.31

b) Assume that \(|b(x)|+|\sigma(x)| \leqslant M\) for all \(x\). Find a simpler proof of Corollary 21.31. Hint. Find an estimate for \(\mathbb{P}\left(\left|\int_{0}^{t} \sigma\left(X_{s}^{x}\right) d B_{s}\right|>r\right)\).

c) Assume that \(|b(x)|+|\sigma(x)|\) grow, as \(|x| \rightarrow \infty\), like \(|x|^{1-\epsilon}\) for some \(\epsilon>0\). Show that one can still find a (relatively) simple proof of Corollary 21.31.

Data From Corollary 21.31

Transcribed Image Text:

21.31 Corollary. Let (Bt, Ft)to be a d-dimensional Brownian motion and assume that the coefficients b: [0, 0) x R" R" and : [0,00) x R" Rxd of the SDE (21.1) satisfy the Lipschitz condition (21.17) for all x, y = R" with the global Lipschitz constant L. Denote by (X) to the solution of the SDE (21.1) with initial condition X = x R". Then lim_X(w)] = co_almost surely for every t 0. X-00 Proof. For x = R \{0} we set x = x/1x1 e IR", and define for every t > 0 (21.41) (1+X) If Y 0, if x = 0. Observe that |x| 0 if, and only if, |x|co. If x, y 0 we find for all p > 2 - -