10-StochDiffEq Page 273 Wednesday, February 4, 2004 12:51 PM
Stochastic Differential Equations 273tYt = t + (^2) ∫B dBs s
0
The nonlinear map g(t,x) = x^2 introduces a second term in dt. Note that
we established the latter formula at the end of Chapter 8 in the form
t
1 2 1
∫ B dBs s= ---Bt – ---t
2 2
0Let’s now generalize Itô’s formula.
Suppose that Xt is an Itô process given by dXt = adt + bdBt. As Xt is
a stochastic process, that is, a function X(t,ω) of both time and the
state, it makes sense to consider another stochastic process Yt, which is
a function of the former, Yt = g(t,Xt). Suppose that g is twice continu-
ously differentiable on [0,∞) × R.
It can then be demonstrated (we omit the detailed proof) that Yt is
another Itô process that admits the representation∂g 1 ∂
2
g
dYt = -----
∂g- (tXt)dt + ------(tXt)dXt + ------------(tXt)(dXt)
2
,
∂t^2
,
∂x,(^2) ∂x
where differentials are computed formally according to the rules^2
dt ⋅ dt = dt ⋅ dBt = dBt ⋅ dt = 0 , dBt ⋅ dBt = dt
Itô’s formula can be written (perhaps more) explicitly as
∂g ∂g 1 ∂^2 g ∂g
dYt = ------+ ------a + ------------b^2
dt + ------bdBt
∂t ∂x^2 ∂x^2 ∂x
This formula reduces to the ordinary formula for the differential of a com-
pound function in the case where b = 0 (that is, when there is no noise).
As a second example of application of Itô’s formula, consider the
geometric Brownian motion:
dXt = μXtdt + σXtdBt
(^2) These rules are known as the Box algebra.