Itô calculus, named after Kiyosi Itô, extends the methods of calculus to stochastic processes such as Brownian motion (see Wiener process). It has important applications in mathematical finance, in stochastic differential equations, and more recently even in machine learning.
The central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators are now stochastic processes:
where is a locally square-integrable process adapted to the filtration generated by , which is a Brownian motion or, more generally, a semimartingale. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular is a random variable, defined as a limit of a certain sequence of random variables. The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. So with the integrand a stochastic process, the Itô stochastic integral amounts to an integral with respect to a function which is not differentiable at any point and has infinite variation over every time interval. The main insight is that the integral can be defined as long as the integrand is adapted, which loosely speaking means that its value at time can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to and constructs Riemann sums. Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. It is crucial which point in each of the small intervals is used to compute the value of the function. The limit then is taken in probability as the mesh of the partition is going to zero. Numerous technical details have to be taken care of to show that this limit exists and is independent of the particular sequence of partitions. Typically, the left end of the interval is used.
Important results of Itô calculus include the integration by parts formula and Itô's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms. This can be contrasted to the Stratonovich integral as an alternative formulation; it does follow the chain rule, and does not require Itô's lemma. The two integral forms can be converted to one-another. The Stratonovich integral is obtained as the limiting form of a Riemann sum that employs the average of stochastic variable over each small timestep, whereas the Itô integral considers it only at the beginning.
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. The integrand is how much stock we hold, the integrator represents the movement of the prices, and the integral is how much money we have in total including what our stock is worth, at any given moment. The prices of stocks and other traded financial assets can be modeled by stochastic processes such as Brownian motion or, more often, geometric Brownian motion (see Black–Scholes). 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 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 is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums.
Notation The process defined before as
is itself a stochastic process with time parameter t, which is also sometimes written as . Alternatively, the integral is often written in differential form , which is equivalent to . As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given
The σ-algebra represents the information available up until time , and a process is adapted if is -measurable. A Brownian motion is understood to be an -Brownian motion, which is just a standard Brownian motion with the properties that is -measurable and that is independent of for all .
Integration with respect to Brownian motion The Itô integral can be defined in a manner similar to the Riemann–Stieltjes integral, that is as a limit in probability of Riemann sums; such a limit does not necessarily exist pathwise. Suppose that is a Wiener process (Brownian motion) and that is a right-continuous (càdlàg), adapted and locally bounded process. If is a sequence of partitions of with mesh width going to zero, then the Itô integral of with respect to up to time is a random variable
It can be shown that this limit converges in probability.
For some applications, such as martingale representation theorems and local times, the integral is needed for processes that are not continuous. The predictable processes form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous processes. If is any predictable process such that for every then the integral of with respect to can be defined, and is said to be -integrable. Any such process can be approximated by a sequence Hn of left-continuous, adapted and locally bounded processes, in the sense that
in probability. Then, the Itô integral is
where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itô isometry
which holds when is bounded or, more generally, when the integral on the right hand side is finite.
Itô processes
An Itô process is defined to be an adapted stochastic process that can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,
Here, is a Brownian motion and it is required that σ is a predictable -integrable process, and μ is predictable and (Lebesgue) integrable. That is,
for each . The stochastic integral can be extended to such Itô processes,
This is defined for all locally bounded and predictable integrands. More generally, it is required that be -integrable and be Lebesgue integrable, so that
Such predictable processes are called -integrable.
An important result for the study of Itô processes is Itô's lemma. In its simplest form, for any twice continuously differentiable function on the reals and Itô process as described above, it states that is itself an Itô process satisfying
This is the stochastic calculus version of the change of variables formula and chain rule. It differs from the standard result due to the additional term involving the second derivative of , which comes from the property that Brownian motion has non-zero quadratic variation.
Semimartingales as integrators
The Itô integral is defined with respect to a semimartingale . These are processes which can be decomposed as for a local martingale and finite variation process . Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process the integral exists, and can be calculated as a limit of Riemann sums. Let be a sequence of partitions of with mesh going to zero,
This limit converges in probability. The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. 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. However, it is inadequate for other important topics such as martingale representation theorems and local times.
The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem holds. That is, if and for a locally bounded process , then
in probability. The uniqueness of the extension from left-continuous to predictable integrands is a result of the monotone class lemma.
In general, the stochastic integral can be defined even in cases where the predictable process is not locally bounded. If then and are bounded. Associativity of stochastic integration implies that is -integrable, with integral , if and only if and . The set of -integrable processes is denoted by .
Properties The following properties can be found in works such as and :
The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. The discontinuities of the stochastic integral are given by the jumps of the integrator multiplied by the integrand. The jump of a càdlàg process at a time is , and is often denoted by . With this notation, . A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous. Associativity. Let , be predictable processes, and be -integrable. Then, is integrable if and only if is -integrable, in which case Dominated convergence. Suppose that and , where is an -integrable process. then . Convergence is in probability at each time . In fact, it converges uniformly on compact sets in probability. The stochastic integral commutes with the operation of taking quadratic covariations. If and are semimartingales then any -integrable process will also be -integrable, and . A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process,
Integration by parts As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itô integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itô calculus deals with processes with non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If and are semimartingales then
where is the quadratic covariation process.
The result is similar to the integration by parts theorem for the Riemann–Stieltjes integral but has an additional quadratic variation term.
Itô's lemma
Itô's lemma is the version of the chain rule or change of variables formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous -dimensional semimartingale and twice continuously differentiable function from to , it states that is a semimartingale and,
This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation . The formula can be generalized to include an explicit time-dependence in and in other ways (see Itô's lemma).
Martingale integrators
Local martingales An important property of the Itô integral is that it preserves the local martingale property. If is a local martingale and is a locally bounded predictable process then is also a local martingale. For integrands which are not locally bounded, there are examples where is not a local martingale. However, this can only occur when is not continuous. If is a continuous local martingale then a predictable process is -integrable if and only if
for each , and is always a local martingale.
The most general statement for a discontinuous local martingale is that if is locally integrable then exists and is a local martingale.
Square integrable martingales For bounded integrands, the Itô stochastic integral preserves the space of square integrable martingales, which is the set of càdlàg martingales such that is finite for all . For any such square integrable martingale , the quadratic variation process is integrable, and the Itô isometry states that