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Itô formula

Contents

Idea

The Itô formula is the chain rule for Itô stochastic calculus. The involvement of the white noise process leads to an inclusion of second order derivatives, instead of first order derivatives as in usual calculus.

Details

We will formulate the formula for a two dimensional process, the general formula as well as the one dimensional formula are simple to get from that.

Let (W 1,W 2)(W_1, W_2) be a two dimensional Wiener process, and let an two dimensional Itô process be defined via

dX=a 1(t,x,y)dt+b 1,1(t,x,y)dW 1+b 1,2(t,x,y)dW 2 d X = a_1 (t, x, y) \; d t + b_{1, 1}(t, x, y) \; d W_1 + b_{1, 2}(t, x, y) \; d W_2

and

dY=a 2(t,x,y)dt+b 2,1(t,x,y)dW 1+b 2,2(t,x,y)dW 2 d Y = a_2 (t, x, y) \; d t + b_{2, 1}(t, x, y) \; d W_1 + b_{2, 2}(t, x, y) \; d W_2

A coordinate transformation is specified by two C 2C^2 functions U:=g 1(t,x,y)U: = g_1(t, x, y) and V:=g 2(t,x,y)V := g_2(t, x, y), defining two new “coordinates” U,VU, V. Note that we need the functions to be twice continuous differentiable, since the transformation formula includes derivatives of the second order.

The Itô formula says that UU and VV are again the components of a two dimensional Itô process and satisfy the equations

dU=g 1tdt+g 1xdX+g 1ydY+12 2g 1x 2dXdX+12 2g 1xydXdY+12 2g 1y 2dYdY d U = \frac{\partial g_1}{\partial t} dt + \frac{\partial g_1}{\partial x} d X + \frac{\partial g_1}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_1}{\partial x \partial y} d X d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial y^2} d Y d Y

and

dV=g 2tdt+g 2xdX+g 2ydY+12 2g 2x 2dXdX+12 2g 2xydXdY+12 2g 2y 2dYdY d V = \frac{\partial g_2}{\partial t} dt + \frac{\partial g_2}{\partial x} d X + \frac{\partial g_2}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_2}{\partial x \partial y} d X d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial y^2} d Y d Y

Further, the Itô formula states that when one inserts the equations for dXd X and dYd Y in the above formula, products of “differentials” are all zero with the notable exception

dW idW i=dt d W_i \; d W_i = \; d t

So we see that, for example, a process without a drift (a is zero) could get transformed in a process with drift by a coordinate transformation.

Examples

The Hopf-Bifurcation Example from TWF 308

In this example we will take a look at the two dimensional system of SDE explained in This Weeks Finds 308.

Polar Coordinates

We start with the Euclidean plane 2\mathbb{R}^2 and Cartesian coordinates x,yx, y, we would like to do a transformation to polar coordinates r,ϕr, \phi:

x=rcos(ϕ) x = r \cos(\phi)
y=rsin(ϕ) y = r \sin(\phi)

The inverse transformation is

r=(x 2+y 2) 12=:g 1(x,y) r = (x^2 + y^2)^{\frac{1}{2}} =: g_1(x, y)

and

ϕ=arctan(yx)=:g 2(x,y) \phi = \arctan(\frac{y}{x}) =: g_2(x, y)

for x>0x \gt 0.

Polar Coordinates, Two Dimensional Random Walk

A simple two dimensional random walk with independent but equally strong noise terms is given by

dX=λdW 1 d X = \lambda \; d W_1

and

dY=λdW 2 d Y = \lambda \; d W_2

We calculate the SDE in polar coordinates according to the Itô formula. For rr we get:

dr=g 1xdX+g 1ydY+12 2g 1x 2dXdX+12 2g 1y 2dYdY d r = \frac{\partial g_1}{\partial x} d X + \frac{\partial g_1}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_1}{\partial y^2} d Y d Y

All the other terms in the Itô formula are zero. Inserting the formulae for the differentials of g 1g_1 and the SDE we get:

dr=x(x 2+y 2) 12λdW 1+y(x 2+y 2) 12λdW 2+12(1(x 2+y 2) 12)λ 2dt d r = \frac{x}{(x^2+y^2)^{\frac{1}{2}}} \; \lambda \; d W_1 + \frac{y}{(x^2+y^2)^{\frac{1}{2}}} \; \lambda \; d W_2 + \frac{1}{2} (\frac{1}{(x^2 + y^2)^{\frac{1}{2}}}) \; \lambda^2 d t

Replacing x,yx, y with polar coordinates on the right results in the SDE for rr:

dr=λ 22rdt+λcos(ϕ)dW 1+λsin(ϕ)dW 2 d r = \frac{\lambda^2}{2 \; r} \; d t + \lambda \; \cos(\phi) \; d W_1 + \lambda \; \sin(\phi) d W_2

We see that the radius rr picks up a drift term that pushes the process away from the origin, the nearer it comes the stronger. This accounts for the effect that the influence of noise tends to move the process away from any concentrically confined area.

Now we turn to the other coordinate ϕ\phi. We keep all nonzero terms in the Itô formula:

dϕ=g 2xdX+g 2ydY+12 2g 2x 2dXdX+12 2g 2y 2dYdY d \phi = \frac{\partial g_2}{\partial x} d X + \frac{\partial g_2}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_2}{\partial y^2} d Y d Y

As before we insert the formulae for the derivatives of g 2g_2 and for dX,dYd X, d Y and get:

dϕ=yx 2+y 2λdW 1+xx 2+y 2λdW 2 d \phi = \frac{-y}{x^2 + y^2} \; \lambda \; d W_1 + \frac{x}{x^2 + y^2} \; \lambda \; d W_2

The second derivatives cancel. Replacing x,yx, y with polar coordinates on the right results in the SDE for ϕ\phi:

dϕ=sin(ϕ)rλdW 1+cos(ϕ)rλdW 2 d \phi = \frac{- \sin(\phi)}{r} \; \lambda \; d W_1 + \frac{\cos(\phi)}{r} \; \lambda \; d W_2

So the angle ϕ\phi does not pick up a drift term, but the effects of the noise decrease with the distance from the origin, as one would expect.

The Hopf Bifurcation Example System

There is no complication in applying our results for the “free” two dimensional random walk to the example of week 308, the system of equations in cartesian coordinates is:

dx=(y+βxx(x 2+y 2))dt+λdW 1 d x = (-y + \beta x - x (x^2 + y^2)) d t + \lambda \; d W_1

and

dY=(x+βyy(x 2+y 2))dt+λdW 1 d Y = (x + \beta y - y (x^2 + y^2)) d t + \lambda \; d W_1

Transforming to polar coordinates we get

dr=(βrr 3+λ 22r)dt+λcos(ϕ)dW 1+λsin(ϕ)dW 2 d r = (\beta \; r - r^3 + \frac{\lambda^2}{2 \; r}) \; d t + \lambda \; \cos(\phi) \; d W_1 + \lambda \; \sin(\phi) d W_2

and

dϕ=sin(ϕ)rλdW 1+cos(ϕ)rλdW 2 d \phi = \frac{- \sin(\phi)}{r} \; \lambda \; d W_1 + \frac{\cos(\phi)}{r} \; \lambda \; d W_2

Now we can calculate the Fokker-Planck operator for this system in polar coordinates, see over there.

References