*Ito’s lemma*, otherwise known as the *Ito formula*, expresses functions of stochastic processes in terms of stochastic integrals. In standard calculus, the differential of the composition of functions satisfies . This is just the chain rule for differentiation or, in integral form, it becomes the change of variables formula.

In stochastic calculus, Ito’s lemma should be used instead. For a twice differentiable function applied to a continuous semimartingale , it states the following,

This can be understood as a Taylor expansion up to second order in , where the quadratic term is the quadratic variation of the process .

A d-dimensional process is said to be a semimartingale if each of its components, , are semimartingales. The first and second order partial derivatives of a function are denoted by and , and I make use of the summation convention where indices which occur twice in a single term are summed over. Then, the statement of Ito’s lemma is as follows.

Theorem 1 (Ito’s Lemma)Letbe a continuous d-dimensional semimartingale taking values in an open subset. Then, for any twice continuously differentiable function,is a semimartingale and,

(1)

To be clear, written out in integral form and explicitly summing over all indices, this is equivalent to the following,

(2) |

The proof of the result is given below. Ito’s lemma has the following consequence for the covariation with a semimartingale . The quadratic covariation terms in (2) are FV processes and, by the properties of covariations, do not contribute, giving

#### Example: The Doléans exponential

The *Doléans exponential* of a semimartingale is defined as a process solving the integral equation

(3) |

so, in differential notation, . The symbol is sometimes used for the Doléans exponenial. If are the Doléans exponentials of processes then the quadratic covariation is and, applying integration by parts,

So, the following product formula is satisfied,

For continuous semimartingales, Ito’s lemma can be used to solve for the Doléans exponential. Assuming that the process remains positive, then this gives,

Therefore, , and the Doléans exponential of a continuous semimartingale is given by

#### Example: Geometric Brownian motion

A *geometric Brownian motion* with volatility and drift is defined as the solution to the stochastic differential equation

where is a standard Brownian motion. So, is the Doléans exponential of . As Brownian motion has quadratic variation , this gives and, from the solution above for the Doléans exponential,

#### Proof of Ito’s Lemma

Writing for some times , a second order Taylor expansion gives

(4) |

As is continuous and contained in the open set , the image of its sample path over any bounded interval is almost surely a compact subset of . It follows that the remainder term is almost surely of the order of uniformly over the interval . That is, whenever on .

Now, for a positive integer n, partition the interval into n equal segments. That is, set for . Using the notation , summing (4) over the intervals gives,

(5) |

Other than the final remainder term, which we will show converges to zero, this expression is a Riemann sum approximation to equation (2). I show separately that each of the terms does indeed converge to the expected limit as . As I will make use of these results again in the extension of Ito’s formula to noncontinuous processes, the more general context of cadlag processes is used where possible. However, for the application to this post, only continuous processes are required, so that .

Lemma 2Ifis a semimartingale andis a cadlag adapted process then

in probability as.

*Proof:* For any the following limit is clear,

Furthermore, the left hand side is dominated by the locally bounded process . So, the result follows from dominated convergence

⬜

Applying this to the first summation term of (5) gives

(6) |

in probability as .

Next, the second order terms of the Taylor expansion will converge to the required integral with respect to the quadratic covariation.

Lemma 3Ifare semimartingales andis a cadlag adapted process then

in probability as.

*Proof:* First, rearranging the expression a bit gives

Lemma 2 can then be applied to each of the terms on the right hand side to give convergence in probability,

The final equality here is the integration by parts formula. ⬜

Applying this result to the second summation term of (5) gives

(7) |

in probability as .

Finally, the remainder terms in the Taylor expansion will converge to zero, as required.

Lemma 4Ifis a continuous d-dimensional semimartingale andis almost surely of orderon the intervalthen,

(8)

in probability as.

*Proof:* As the sample paths of are continuous, it follows that goes to zero as . So, as is of order , the sequence of random variables

will also go to zero.

The sum over on the right hand side tends to in probability as and, in particular, is bounded in probability. As this shows that the right hand side goes to zero in probability as . ⬜

Taking the limit in equation (5) and applying (6,7,8) gives the required integral expression,

Finally, as is expressed as a sum of stochastic integrals, it is a semimartingale.

When proving Ito’s lemma, you use Taylor approximation and then show that the terms converge.

The convergence is in probability, but I believe that Ito’s lemma holds almost surely.

Is there no contradiction there?

Hi.

No, there’s no contradiction. If you have a sequence of random variables

Uwhich you can show converge in probability to two limits^{n}UandV, then it follows thatU=Valmost surely. That is the situation here, whereUis equal to^{n}f(X) for each_{T}nbut, also, by breaking it into terms corresponding to the Taylor expansion, we show that it converges in probability to the right hand side of Ito’s formula. Hence, Ito’s formula holds almost surely.Concerning Lemma 2: Why is the running supremum process of a cadlag adapted process locally bounded?

I think I understand that now (it is easy to find the localizting sequence), but why does the dominated convergence theorem hold for dominating functions that are locally bounded?

Locally bounded predictable processes are Y-integrable (Lemma 5 of post on properties of the stochastic integral), so dominated convergence holds.

Hi there. I’m just newbie to the world of semimartingales. I have a question regarding the second term of Ito’s formula applied to the Doleans exponential. It says there -1/2U^{-2}d[U] = -1/2d[X]. Are you saying that -1/2U^{-2}d[U] = -1/2U^{-2}U^{2}d[X] ?

Thank you for even reading this comment.

I’m not sure what you are asking, but it is true that .

What is U_ in your Doleans example?

Well, U is the Doleans exponential by definition. If you are asking about the ‘-‘ subscript, this refers to the left-limit (I use this notation throughout my notes).

George, what remainder form are you using for here to get that the remainder term is of order uniformly over [s,t]? I can get this from the Lagrange form if f is of C^3, but I don’t know of a remainder formula that gives this for just C^2 function even if it is compactly supported.

I just did a quick check online and didn’t see quite the statement that I would like, although I am pretty sure it is standard. Look at the integral form for the remainder.

For the multidimensional case, the same formula applies if the integral is along a line from x to y, and we use the chain rule to express as partial derivates.

By uniform continuity of , the integral is .