Theory of Stochastic Differential

We consider Itô’s stochastic differential equation (SDE). First, we would review a standard theory under standard assumptions. Then we would see how such a standard theory should be modified under different and more general assumptions and how and what new notions need be introduced to discuss such modifications. Consider the following SDE on R dX (t) = σ k(X(t))dB (t) + b(X(t))dt, X(0) = x (1) where X(t) = (X(t), · · · , X(t)) is a d-dimensional continuous process and x = (x, · · · , x) ∈ R. σ k(x) and b(x), i = 1, · · · , d, k = 1, · · · , r, are real Borel functions on R. We allow the case r = ∞ and then we always assume that ||σ(x)|| := √∑d i=1 ∑r k=1 σ i k(x) 2 < ∞ for each x. Set |b(x)| = √∑d i=1 b i(x)2. B(t) = (B(t)) is an r-dimensional Wiener process with B(0) = 0.