STOCHASTIC CALCULUS 05

Itô's Lemma

Itô's lemma is the stochastic version of the chain rule. It shows how a function of a stochastic process changes when the process contains Brownian motion.

The main idea

In ordinary calculus, suppose

\[ Y=f(X). \]

Then the chain rule gives

\[ dY=f'(X)\,dX. \]

For a stochastic process, this rule must be modified because Brownian motion has nonzero quadratic variation.

\[ \boxed{ (dW_t)^2=dt } \]

This produces an additional second-derivative term in Itô's lemma.

1. Start with an SDE

Suppose

\[ dX_t = a(X_t,t)\,dt + b(X_t,t)\,dW_t. \]

Now define another process

\[ Y_t=f(X_t,t). \]

We want to determine

\[ dY_t. \]

2. Ordinary multivariable calculus first

For a smooth deterministic function

\[ f(x,t), \]

the total differential is

\[ df = \frac{\partial f}{\partial t}\,dt + \frac{\partial f}{\partial x}\,dx. \]

In ordinary calculus, higher-order terms become negligible as the increments shrink.

3. Taylor expansion

For small changes \(dt\) and \(dX\), expand \(f\):

\[ df \approx f_t\,dt + f_x\,dX + \frac12 f_{xx}(dX)^2. \]

Terms involving higher powers are omitted at the relevant stochastic order.

The key difference from ordinary calculus is that

\[ (dX)^2 \]

may contain a term of order \(dt\).

4. Substitute the SDE

Since

\[ dX=a\,dt+b\,dW, \]

we have

\[ (dX)^2 = (a\,dt+b\,dW)^2. \]

Expand:

\[ (dX)^2 = a^2(dt)^2 + 2ab\,dt\,dW + b^2(dW)^2. \]

5. Use the stochastic differential rules

In Itô calculus,

\[ \boxed{ (dt)^2=0 } \] \[ \boxed{ dt\,dW=0 } \] \[ \boxed{ (dW)^2=dt } \]

Therefore,

\[ (dX)^2=b^2dt. \]

6. The extra term

Substitute this into the Taylor expansion:

\[ df = f_t\,dt + f_x\,dX + \frac12f_{xx}b^2dt. \]

Now substitute

\[ dX=a\,dt+b\,dW. \]

Then

\[ df = f_t\,dt + f_x(a\,dt+b\,dW) + \frac12b^2f_{xx}dt. \]

7. Itô's lemma

Collect the \(dt\) and \(dW\) terms:

\[ \boxed{ df(X_t,t) = \left( f_t + af_x + \frac12b^2f_{xx} \right)dt + bf_x\,dW_t } \]

This is Itô's lemma for a one-dimensional stochastic differential equation.

8. Why ordinary chain rule is incomplete

The ordinary chain rule would suggest only

\[ df = f_tdt+f_xdX. \]

But stochastic calculus contains the additional term

\[ \boxed{ \frac12b^2f_{xx}\,dt } \]

because

\[ (dW)^2 \]

contributes at order \(dt\).

9. Simple example

Suppose

\[ dX_t=\mu\,dt+\sigma\,dW_t \]

and define

\[ Y_t=X_t^2. \]

Here

\[ f(x)=x^2. \]

Then

\[ f_x=2x \]

and

\[ f_{xx}=2. \]

10. Apply Itô's lemma

Since \(f\) does not depend explicitly on time,

\[ f_t=0. \]

Therefore,

\[ d(X_t^2) = \left( 2X_t\mu + \frac12\sigma^2(2) \right)dt + 2\sigma X_t\,dW_t. \]

\[ \boxed{ d(X_t^2) = (2\mu X_t+\sigma^2)dt + 2\sigma X_t\,dW_t } \]

11. Compare with ordinary calculus

Ordinary calculus would give

\[ d(X^2)=2X\,dX. \]

Substituting the SDE would produce

\[ 2X\mu\,dt+2X\sigma\,dW. \]

But this misses the additional term

\[ \boxed{ \sigma^2dt } \]

that comes from Brownian quadratic variation.

12. Direct differential calculation

The same result can be seen informally from

\[ d(X^2) = 2X\,dX+(dX)^2. \]

Since

\[ (dX)^2=\sigma^2dt, \]

we obtain

\[ d(X^2) = 2X(\mu\,dt+\sigma\,dW) + \sigma^2dt. \]

Therefore,

\[ d(X^2) = (2\mu X+\sigma^2)dt + 2\sigma X\,dW. \]

13. Example: \(f(X)=\ln X\)

Suppose

\[ dX_t = \mu X_t\,dt + \sigma X_t\,dW_t, \]

with \(X_t>0\).

Let

\[ f(X)=\ln X. \]

Then

\[ f_x=\frac1X \]

and

\[ f_{xx} = -\frac1{X^2}. \]

14. Apply Itô's lemma to \(\ln X_t\)

Here,

\[ a=\mu X, \qquad b=\sigma X. \]

Therefore,

\[ d(\ln X_t) = \left[ \mu X\left(\frac1X\right) + \frac12\sigma^2X^2 \left( -\frac1{X^2} \right) \right]dt + \sigma X\left(\frac1X\right)dW_t. \]

Simplifying,

\[ \boxed{ d(\ln X_t) = \left( \mu-\frac{\sigma^2}{2} \right)dt + \sigma\,dW_t } \]

15. Why this result is important

The logarithm transforms geometric Brownian motion into an SDE with constant coefficients.

Integrating gives

\[ \ln X_t = \ln X_0 + \left( \mu-\frac{\sigma^2}{2} \right)t + \sigma W_t. \]

Exponentiating,

\[ \boxed{ X_t = X_0 \exp \left[ \left( \mu-\frac{\sigma^2}{2} \right)t + \sigma W_t \right] } \]

16. The \(-\sigma^2/2\) correction

The term

\[ -\frac{\sigma^2}{2} \]

is not present in ordinary chain-rule calculations.

It appears because of the Itô correction term

\[ \frac12b^2f_{xx}. \]

This is a clear example of why stochastic calculus differs from ordinary calculus.

17. Time-dependent function

\[ Y_t=f(X_t,t), \]

then the explicit time derivative must also be included:

\[ \boxed{ df = \left( f_t + af_x + \frac12b^2f_{xx} \right)dt + bf_xdW_t } \]

18. Example with explicit time dependence

Suppose

\[ f(X,t)=e^tX. \]

Then

\[ f_t=e^tX, \] \[ f_x=e^t, \]

and

\[ f_{xx}=0. \]

\[ dX=a\,dt+b\,dW, \]

then

\[ df = (e^tX+ae^t)dt + be^t\,dW. \]

There is no Itô correction here because

\[ f_{xx}=0. \]

19. When does the correction vanish?

The Itô correction is

\[ \frac12b^2f_{xx}dt. \]

It vanishes when

\[ b=0, \]

which removes the stochastic component, or when

\[ f_{xx}=0, \]

meaning \(f\) is linear in \(x\).

20. Diagram: why second-order change survives

21. Itô's lemma in several dimensions

Suppose

\[ \mathbf{X}_t = (X_t^{(1)},\ldots,X_t^{(n)}) \]

is a multidimensional stochastic process.

Then Itô's formula contains first derivatives and second partial derivatives.

In compact form,

\[ \boxed{ df = f_tdt + \nabla f^{\,T}d\mathbf{X} + \frac12 \sum_{i,j} \frac{\partial^2f}{\partial x_i\partial x_j} \,dX_i\,dX_j } \]

The covariance structure of the Brownian components determines the \(dX_i\,dX_j\) terms.

22. Why the Hessian appears

The matrix of second partial derivatives,

\[ H_f, \]

is called the Hessian.

In multidimensional Itô calculus, the second-order correction depends on this Hessian and the diffusion covariance matrix.

So stochastic calculus naturally connects with multivariable calculus and linear algebra.

23. Itô's lemma and expected values

After applying Itô's lemma, the stochastic integral term often has expectation zero.

This can allow us to derive differential equations for expectations.

For example, from

\[ d(X_t^2) = (2\mu X_t+\sigma^2)dt + 2\sigma X_t\,dW_t, \]

taking expectations gives

\[ \frac{d}{dt}E[X_t^2] = 2\mu E[X_t]+\sigma^2, \]

under suitable conditions.

24. This leads to moment equations

Quantities such as

\[ E[X_t], \qquad E[X_t^2], \qquad \operatorname{Var}(X_t) \]

are called moments or are built from moments.

Itô's lemma provides one of the main tools for deriving equations governing these quantities.

25. The practical procedure

\[ \boxed{ 1.\; \text{Identify }a\text{ and }b } \] \[ \boxed{ 2.\; \text{Calculate }f_t,\ f_x,\ f_{xx} } \] \[ \boxed{ 3.\; \text{Use } df= \left( f_t+af_x+\frac12b^2f_{xx} \right)dt + bf_xdW } \] \[ \boxed{ 4.\; \text{Simplify the }dt\text{ and }dW\text{ terms} } \]

26. The central idea

\[ \boxed{ \text{ordinary chain rule} + \text{Brownian quadratic variation} = \text{Itô's lemma} } \]

The defining difference is the extra second-order contribution:

\[ \boxed{ \frac12b^2f_{xx}\,dt } \]

It appears because Brownian increments are of order \(\sqrt{dt}\), so their squares remain of order \(dt\).