Itô calculus

where H is a locally square-integrable process adapted to the filtration generated by X (Revuz & Yor 1999, Chapter IV), which is a Brownian motion or, more generally, a semimartingale.

The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus.

Roughly speaking, one chooses a sequence of partitions of the interval from 0 to t and constructs Riemann sums.

Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions.

In mathematical finance, the described evaluation strategy of the integral is conceptualized as that we are first deciding what to do, then observing the change in the prices.

Then, the Itô stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount Ht of the stock at time t. In this situation, the condition that H is adapted corresponds to the necessary restriction that the trading strategy can only make use of the available information at any time.

This prevents the possibility of unlimited gains through clairvoyance: buying the stock just before each uptick in the market and selling before each downtick.

Similarly, the condition that H is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums (Revuz & Yor 1999, Chapter IV).

As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given

is a sequence of partitions of [0, t] with mesh width going to zero, then the Itô integral of H with respect to B up to time t is a random variable

For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous.

In its simplest form, for any twice continuously differentiable function f on the reals and Itô process X as described above, it states that

It differs from the standard result due to the additional term involving the second derivative of f, which comes from the property that Brownian motion has non-zero quadratic variation.

For a left continuous, locally bounded and adapted process H the integral H · X exists, and can be calculated as a limit of Riemann sums.

For example, it is sufficient for applications of Itô's Lemma, changes of measure via Girsanov's theorem, and for the study of stochastic differential equations.

The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem holds.

The uniqueness of the extension from left-continuous to predictable integrands is a result of the monotone class lemma.

In general, the stochastic integral H · X can be defined even in cases where the predictable process H is not locally bounded.

This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xi,Xj ].

For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales.

Such simple predictable processes are linear combinations of terms of the form Ht = A1{t > T} for stopping times T and FT-measurable random variables A, for which the integral is

This is extended to all simple predictable processes by the linearity of H · X in H. For a Brownian motion B, the property that it has independent increments with zero mean and variance Var(Bt) = t can be used to prove the Itô isometry for simple predictable integrands,

By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying

This uniquely defines ⟨M⟩, which is referred to as the predictable quadratic variation of M. The Itô isometry for square integrable martingales is then

As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying E[H2 · ⟨M⟩t] < ∞.

A continuous linear extension can be used to construct the integral for all left-continuous and adapted integrands with right limits everywhere (caglad or L-processes).

Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands (Bichteler 2002).

However, there are also different notions of "derivative" with respect to Brownian motion: Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula (Nualart 2006).

The following result allows to express martingales as Itô integrals: if M is a square-integrable martingale on a time interval [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted square integrable process

For a recent treatment of different interpretations of stochastic differential equations see for example (Lau & Lubensky 2007).