Itô formula (Rev #4)

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.

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 $(W_1, W_2)$ be a two dimensional Wiener process, and let an two dimensional Itô process be defined via

$d X = a_1 (t, x, y) \; d t + b_{1, 1}(t, x, y) \; d W_1 + b_{1, 2}(t, x, y) \; d W_2$

and

$d Y = a_2 (t, x, y) \; d t + b_{2, 1}(t, x, y) \; d W_1 + b_{2, 2}(t, x, y) \; d W_2$

A coordinate transformation is specified by two $C^2$ functions $U: = g_1(t, x, y)$ and $V := g_2(t, x, y)$, defining two new “coordinates” $U, V$. 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 $U$ and $V$ are again the components of a two dimensional Itô process and satisfy the equations

$d U = \frac{\partial g_1}{\partial t} dt + \frac{\partial g_1}{\partial x} d X + \frac{\partial g_1}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_1}{\partial x \partial y} d X d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial y^2} d Y d Y$

and

$d V = \frac{\partial g_2}{\partial t} dt + \frac{\partial g_2}{\partial x} d X + \frac{\partial g_2}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_2}{\partial x \partial y} d X d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial y^2} d Y d Y$

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

$d W_i \; d W_i = \; d t$

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.

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

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

$x = r \cos(\phi)$

$y = r \sin(\phi)$

The inverse transformation is

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

and

$\phi = \arctan(\frac{y}{x}) =: g_2(x, y)$

for $x \gt 0$.

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

$d X = \lambda \; d W_1$

and

$d Y = \lambda \; d W_2$

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

$d r = \frac{\partial g_1}{\partial x} d X + \frac{\partial g_1}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_1}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_1}{\partial y^2} d Y d Y$

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

$d r = \frac{x}{(x^2+y^2)^{\frac{1}{2}}} \; \lambda \; d W_1 + \frac{y}{(x^2+y^2)^{\frac{1}{2}}} \; \lambda \; d W_2 + \frac{1}{2} (\frac{1}{(x^2 + y^2)^{\frac{1}{2}}}) \; \lambda^2 d t$

Replacing $x, y$ with polar coordinates on the right results in the **SDE for $r$**:

$d r = \frac{\lambda^2}{2 \; r} \; d t + \lambda \; \cos(\phi) \; d W_1 + \lambda \; \sin(\phi) d W_2$

We see that the radius $r$ 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 $\phi$. We keep all nonzero terms in the Itô formula:

$d \phi = \frac{\partial g_2}{\partial x} d X + \frac{\partial g_2}{\partial y} d Y + \frac{1}{2} \frac{\partial^2 g_2}{\partial x^2} d X d X + \frac{1}{2} \frac{\partial^2 g_2}{\partial y^2} d Y d Y$

As before we insert the formulae for the derivatives of $g_2$ and for $d X, d Y$ and get:

$d \phi = \frac{-y}{x^2 + y^2} \; \lambda \; d W_1 + \frac{x}{x^2 + y^2} \; \lambda \; d W_2$

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

$d \phi = \frac{- \sin(\phi)}{r} \; \lambda \; d W_1 + \frac{\cos(\phi)}{r} \; \lambda \; d W_2$

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

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:

$d x = (-y + \beta x - x (x^2 + y^2)) d t + \lambda \; d W_1$

and

$d Y = (x + \beta y - y (x^2 + y^2)) d t + \lambda \; d W_1$

Transforming to polar coordinates we get

$d r = (\beta \; r - r^3 + \frac{\lambda^2}{2 \; r}) \; d t + \lambda \; \cos(\phi) \; d W_1 + \lambda \; \sin(\phi) d W_2$

and

$d \phi = \frac{- \sin(\phi)}{r} \; \lambda \; d W_1 + \frac{\cos(\phi)}{r} \; \lambda \; d W_2$

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

- Ito formula, Wikipedia