Suppose (X,W, P) is a weak solution to where W is a one-dimensional Brownian motion and σ and b are bounded and continuous, but we do not assume that σ is bounded below by a positive constant....

Suppose (X,W, P) is a weak solution to

where W is a one-dimensional Brownian motion and σ and b are bounded and continuous, but we do not assume that σ is bounded below by a positive constant. Suppose the solution to (32.8) is unique in law. Suppose σ_n
and b_n
are Lipschitz functions which are uniformly bounded and which converge uniformly to σ and b, respectively. Let X_t(n) be the unique path wise solution to

the probability measure here is P. Prove that X (n) converges weakly to X with respect to C[0, 1].

Let W be a d-dimensional Brownian motion and let

be the solution to (24.22). If x ∈ R^d, prove that the support of P_x
is all of C[0, 1].

Chapter 33