(1) Suppose a s and b s are bounded predictable processes with as bounded below by a positive constant. Let W be a one-dimensional Brownian motion. Suppose Y is a one-dimensional semimartingale such...

(1) Suppose a_s
and b_s
are bounded predictable processes with as bounded below by a positive constant. Let W be a one-dimensional Brownian motion. Suppose Y is a one-dimensional semimartingale such that

Prove that if t₀
> 0 and ε > 0, there exists a constant c > 0 depending only on t₀,ε, and the bounds on as and b_s
such that

(2) Now let W be d-dimensional Brownian motion, let
  and let σ be a d × d matrix valued function that is bounded and such that σσ^T(x) is positive definite, uniformly in x. That is, there exists > 0 such that for all x,

Let b be a d × 1 matrix-valued function that is bounded. Let X be the solution to