ABSTRACT
Suppose Xt satisfies the SDE dXt = a(t,Xt) dt + b(t,Xt) dWt , (59.27)
or equivalently, the integral equation,
Xt = Xs + ∫ t s
a(θ,Xθ) dθ+ ∫ t s
b(θ,Xθ) dWθ. (59.28)
If Yt = g(Xt), where g is twice continuously differentiable, we show that Yt satisfies the SDE
dYt = g ′(Xt) dXt + 1 2 g ′′(Xt)b(t,Xt)2 dt. (59.29)
Using the Taylor expansion of g as we did in the preceding section, write
{ g ′(Xti )(Xti+1 −Xti )+
2 g ′′(Xti )(Xti+1 −Xti )2
} .
