The starting point in XXXXXXXXXXcan be random. Suppose Y is a random variable that is measurable with respect to F 0 ,Y is square integrable, and σ and b are bounded and Lipschitz. Prove pathwise...

The starting point in (24.1) can be random. Suppose Y is a random variable that is measurable with respect to F₀,Y is square integrable, and σ and b are bounded and Lipschitz. Prove pathwise existence and uniqueness for the equation

Let W be a one-dimensional Brownian motion and let X x_t
be the solution to

Suppose σ and b are C∞ functions and that σ and b and all their derivatives are bounded. Show that for each t the map
  is continuous in x with probability one. Show that the map is differentiable in x.

Suppose A(t) and B(t) are deterministic functions of t. Find an explicit solution to the one dimensional SDE

Chapter 25