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Itō calculus, named after Kiyoshi Itō, extends the methods of calculus to stochastic processes such as Brownian motion (Wiener process). It has important applications in mathematical finance and stochastic differential equations. The central concept is the Itō stochastic integral. This is a generalization of the ordinary concept of a Riemann–Stieltjes integral. The generalization is in two respects. Firstly, we are now dealing with random variables (more precisely, stochastic processes). Secondly, we are integrating with respect to a nondifferentiable function (technically, stochastic process).
The Itō integral allows one to integrate one stochastic process (the integrand) with respect to another stochastic process (the integrator). It is common for the integrator to be the Brownian motion (also see Wiener process). The result of the integration is another stochastic process. In particular, the integral from 0 to any particular t is a random variable. This random variable is defined as a limit of a certain sequence of random variables. (There are several equivalent ways to construct a definition). Roughly speaking, we are choosing a sequence of partitions of the interval from 0 to t. Then we are constructing Riemann sums. However, it is important which point in each of the small intervals is used to compute the value of the function. Typically, the left end of the interval is used. (It is conceptualized in mathematical finance 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). Every time we are computing a Riemann sum, we are using a particular instantiation of the integrator. 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).
The usual notation for the Itō stochastic integral is:
 $Y\_t=\backslash int\_0^t\; H\_s\backslash ,dX\_s$
where X is a Brownian motion or, more generally, a semimartingale and H is a locally squareintegrable process adapted to the filtration generated by X (Revuz & Yor 1999, Chapter IV). The paths of Brownian motion fail to satisfy the requirements to be able to apply the standard techniques of calculus. In particular, it is not differentiable at any point and has infinite variation over every time interval. As a result, the integral cannot be defined in the usual way (see Riemann–Stieltjes integral). The main insight is that the integral can be defined as long as the integrand H is adapted, which loosely speaking means that its value at time t can only depend on information available up until this time.
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 Itō stochastic integral represents the payoff of a continuoustime trading strategy consisting of holding an amount H_{t} 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 high frequency trading: 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).
Important results of Itō calculus include the integration by parts formula and Itō's lemma, which is a change of variables formula. These differ from the formulas of standard calculus, due to quadratic variation terms.
Notation
The process Y defined as before as
 $Y\_t\; =\; \backslash int\_0^t\; H\backslash ,dX\backslash equiv\backslash int\_0^t\; H\_s\backslash ,dX\_s\; ,$
is itself a stochastic process with time parameter t, which is also sometimes written as Y = H · X (Rogers & Williams 2000). Alternatively, the integral is often written in differential form dY = H dX, which is equivalent to Y − Y_{0} = H · X. As Itō calculus is concerned with continuoustime stochastic processes, it is assumed that an underlying filtered probability space is given
 $(\backslash Omega,\backslash mathcal\{F\},(\backslash mathcal\{F\}\_t)\_\{t\backslash ge\; 0\},\backslash mathbb\{P\})\; .$
The sigma algebra F_{t} represents the information available up until time t, and a process X is adapted if X_{t} is F_{t}measurable. A Brownian motion B is understood to be an F_{t}Brownian motion, which is just a standard Brownian motion with the properties that B_{t} is F_{t}measurable and that B_{t+s} − B_{t} is independent of F_{t} for all s,t ≥ 0 (Revuz & Yor 1999).
Integration with respect to Brownian motion
The Itō 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 B is a Wiener process (Brownian motion) and that H is a leftcontinuous, adapted and locally bounded process. If {π_{n}} is a sequence of partitions of [0, t] with mesh going to zero, then the Itō integral of H with respect to B up to time t is a random variable
 $\backslash int\_0^t\; H\; \backslash ,d\; B\; =\backslash lim\_\{n\backslash rightarrow\backslash infty\}\; \backslash sum\_(B\_\{t\_i\}B\_\{t\_\{i1\}\}).$
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 leftcontinuous processes. If H is any predictable process such that ∫_{0}^{t} H^{2} ds < ∞ for every t ≥ 0 then the integral of H with respect to B can be defined, and H is said to be Bintegrable. Any such process can be approximated by a sequence H_{n} of leftcontinuous, adapted and locally bounded processes, in the sense that
 $\backslash int\_0^t\; (HH\_n)^2\backslash ,ds\backslash to\; 0$
in probability. Then, the Itō integral is
 $\backslash int\_0^t\; H\backslash ,dB\; =\; \backslash lim\_\{n\backslash to\backslash infty\}\backslash int\_0^t\; H\_n\backslash ,dB$
where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itō isometry
 $\backslash mathbb\{E\}\backslash left[\; \backslash left(\backslash int\_0^t\; H\_s\; \backslash ,\; dB\_s\backslash right)^2\backslash right]=\backslash mathbb\{E\}\; \backslash left[\; \backslash int\_0^t\; H\_s^2\backslash ,ds\backslash right\; ]$
which holds when H is bounded or, more generally, when the integral on the right hand side is finite.
Itō processes
An Itō process is defined to be an adapted stochastic process which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,
 $X\_t=X\_0+\backslash int\_0^t\backslash sigma\_s\backslash ,dB\_s\; +\; \backslash int\_0^t\backslash mu\_s\backslash ,ds.$
Here, B is a Brownian motion and it is required that σ is a predictable Bintegrable process, and μ is predictable and (Lebesgue) integrable. That is,
 $\backslash int\_0^t(\backslash sigma\_s^2+\backslash mu\_s)\backslash ,ds<\backslash infty$
for each t. The stochastic integral can be extended to such Itō processes,
 $\backslash int\_0^t\; H\backslash ,dX\; =\backslash int\_0^t\; H\_s\backslash sigma\_s\backslash ,dB\_s\; +\; \backslash int\_0^t\; H\_s\backslash mu\_s\backslash ,ds.$
This is defined for all locally bounded and predictable integrands. More generally, it is required that Hσ be Bintegrable and Hμ be Lebesgue integrable, so that
 $\backslash int\_0^t\; (H^2\; \backslash sigma^2\; +\; H\backslash mu\; )ds\; <\; \backslash infty.$
Such predictable processes H are called Xintegrable.
An important result for the study of Itō processes is Itō's lemma. In its simplest form, for any twice continuously differentiable function f on the reals and Itō process X as described above, it states that f(X) is itself an Itō process satisfying
 $df(X\_t)=f^\backslash prime(X\_t)\backslash ,dX\_t\; +\; \backslash frac\{1\}\{2\}f^\{\backslash prime\backslash prime\}\; (X\_t)\; \backslash sigma\_t^2\; \backslash ,\; dt.$
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 f, which comes from the property that Brownian motion has nonzero quadratic variation.
Semimartingales as integrators
The Itō integral is defined with respect to a semimartingale X. These are processes which can be decomposed as X = M + A for a local martingale M and finite variation process A. Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. 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. Let π_{n} be a sequence of partitions of [0, t] with mesh going to zero,
 $\backslash int\_0^t\; H\backslash ,dX\; =\; \backslash lim\_\{n\backslash rightarrow\backslash infty\}\; \backslash sum\_\{t\_\{i1\},t\_i\backslash in\backslash pi\_n\}H\_\{t\_\{i1\}\}(X\_\{t\_i\}X\_\{t\_\{i1\}\}).$
This limit converges in probability. The stochastic integral of leftcontinuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itō'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 H_{n} → ;H and H_{n} ≤ J for a locally bounded process J, then
 $\backslash int\_0^t\; H\_n\; dX\; \backslash to\; \backslash int\_0^t\; H\; dX,$
in probability. The uniqueness of the extension from leftcontinuous 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. If K = 1 / (1 + H) then K and KH are bounded. Associativity of stochastic integration implies that H is Xintegrable, with integral H · X = Y, if and only if Y_{0} = 0 and K · Y = (KH) · X. The set of Xintegrable processes is denoted by L(X).
Properties
The following properties can be found in works such as (Revuz & Yor 1999) and (Rogers & Williams 2000):
 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 t is X_{t} − X_{t−}, and is often denoted by ΔX_{t}. With this notation, Δ(H · X) = H ΔX. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous.
 Associativity. Let J, K be predictable processes, and K be Xintegrable. Then, J is K · X integrable if and only if JK is X integrable, in which case
 $J\backslash cdot\; (K\backslash cdot\; X)\; =\; (JK)\backslash cdot\; X$
 Dominated convergence. Suppose that H_{n} → H and H_{n} ≤ J, where J is an Xintegrable process. then H_{n} · X → H · X. Convergence is in probability at each time t. In fact, it converges uniformly on compacts in probability.
 The stochastic integral commutes with the operation of taking quadratic covariations. If X and Y are semimartingales then any Xintegrable process will also be [X, Y]integrable, and [H · X, Y] = H · [X, Y]. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process,
 $[H\backslash cdot\; X]=H^2\backslash cdot[X]$
Integration by parts
As with ordinary calculus, integration by parts is an important result in stochastic calculus. The integration by parts formula for the Itō integral differs from the standard result due to the inclusion of a quadratic covariation term. This term comes from the fact that Itō calculus deals with processes with nonzero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If X and Y are semimartingales then
 $X\_tY\_t\; =\; X\_0Y\_0+\backslash int\_0^t\; X\_\{s\}\backslash ,dY\_s\; +\; \backslash int\_0^t\; Y\_\{s\}\backslash ,dX\_s\; +\; [X,Y]\_t$
where [X, Y] 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.
Itō's lemma
Main article:
Itō's lemma
Itō's lemma is the version of the chain rule or change of variables formula which applies to the Itō integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous ddimensional semimartingale X = (X^{1},...,X^{d}) and twice continuously differentiable function f from R^{d} to R, it states that f(X) is a semimartingale and,
 $df(X\_t)=\; \backslash sum\_\{i=1\}^d\; f\_\{i\}(X\_t)\backslash ,dX^i\_t\; +\; \backslash frac\{1\}\{2\}\backslash sum\_\{i,j=1\}^d\; f\_\{i,j\}(X\_\{t\})\backslash ,d[X^i,X^j]\_t.$
This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [X^{i},X^{j} ]. The formula can be generalized to noncontinuous semimartingales by adding a pure jump term to ensure that the jumps of the left and right hand sides agree (see Itō's lemma).
Martingale integrators
Local martingales
An important property of the Itō integral is that it preserves the local martingale property. If M is a local martingale and H is a locally bounded predictable process then H · M is also a local martingale. For integrands which are not locally bounded, there are examples where H · M is not a local martingale. However, this can only occur when M is not continuous. If M is a continuous local martingale then a predictable process H is Mintegrable if and only if
 $\backslash int\_0^t\; H^2\; d[M]\; <\backslash infty,$
for each t, and H · M is always a local martingale.
The most general statement for a discontinuous local martingale M is that if (H^{2} · [M])^{1/2} is locally integrable then H · M exists and is a local martingale.
Square integrable martingales
For bounded integrands, the Itō stochastic integral preserves the space of square integrable martingales, which is the set of càdlàg martingales M such that E[M_{t}^{2}] is finite for all t. For any such square integrable martingale M, the quadratic variation process [M] is integrable, and the Itō isometry states that
 $\backslash mathbb\{E\}\backslash left\; [(H\backslash cdot\; M\_t)^2\backslash right\; ]=\backslash mathbb\{E\}\backslash left\; [\backslash int\_0^t\; H^2\backslash ,d[M]\backslash right\; ].$
This equality holds more generally for any martingale M such that H^{2} · [M]_{t} is integrable. The Itō isometry is often used as an important step in the construction of the stochastic integral, by defining H · M to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes.
pIntegrable martingales
For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of pintegrable martingales. These are càdlàg martingales such that E(M_{t}^{p}) is finite for all t. However, this is not always true in the case where p = 1. There are examples of integrals of bounded predictable processes with respect to martingales which are not themselves martingales.
The maximum process of a càdlàg process M is written as M*_{t} = sup_{s ≤t} M_{s}. For any p ≥ 1 and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales M such that E[(M*_{t})^{p}] is finite for all t. If p > 1 then this is the same as the space of pintegrable martingales, by Doob's inequalities.
The Burkholder–Davis–Gundy inequalities state that, for any given p ≥ 1, there exist positive constants c, C that depend on p, but not M or on t such that
 $c\backslash mathbb\{E\}\; \backslash left\; [\; [M]\_t^\{\backslash frac\{p\}\{2\}\}\; \backslash right\; ]\; \backslash le\; \backslash mathbb\{E\}\backslash left\; [(M^*\_t)^p\; \backslash right\; ]\backslash le\; C\backslash mathbb\{E\}\backslash left\; [\; [M]\_t^\{\backslash frac\{p\}\{2\}\}\; \backslash right\; ]$
for all càdlàg local martingales M. These are used to show that if (M*_{t})^{p} is integrable and H is a bounded predictable process then
 $\backslash mathbb\{E\}\backslash left\; [\; ((H\backslash cdot\; M)\_t^*)^p\; \backslash right\; ]\; \backslash le\; C\backslash mathbb\{E\}\backslash left\; [(H^2\backslash cdot[M]\_t)^\{\backslash frac\{p\}\{2\}\}\; \backslash right\; ]<\backslash infty$
and, consequently, H · M is a pintegrable martingale. More generally, this statement is true whenever (H^{2} · [M])^{p/2} is integrable.
Existence of the integral
Proofs that the Itō integral is well defined typically proceed by first looking at very simple integrands, such as piecewise constant, left continuous and adapted processes where the integral can be written explicitly. Such simple predictable processes are linear combinations of terms of the form H_{t} = A1_{{t > T}} for stopping times T and F_{T}measurable random variables A, for which the integral is
 $H\backslash cdot\; X\_t\backslash equiv\; \backslash mathbf\{1\}\_\{\backslash \{t>T\backslash \}\}A(X\_tX\_T).$
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(B_{t}) = t can be used to prove the Itō isometry for simple predictable integrands,
 $\backslash mathbb\{E\}\; \backslash left\; [\; (H\backslash cdot\; B\_t)^2\backslash right\; ]\; =\; \backslash mathbb\{E\}\; \backslash left\; [\backslash int\_0^tH\_s^2\backslash ,ds\backslash right\; ].$
By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying
 $\backslash mathbb\{E\}\; \backslash left[\; \backslash int\_0^t\; H^2\; ds\; \backslash right\; ]\; <\; \backslash infty,$
in such way that the Itō isometry still holds. It can then be extended to all Bintegrable processes by localization. This method allows the integral to be defined with respect to any Itō process.
For a general semimartingale X, the decomposition X = M + A for a local martingale M and finite variation process A can be used. Then, the integral can be shown to exist separately with respect to M and A and combined using linearity, H · X = H · M + H · A, to get the integral with respect to X. The standard Lebesgue–Stieltjes integral allows integration to be defined with respect to finite variation processes, so the existence of the Itō integral for semimartingales will follow from any construction for local martingales.
For a càdlàg square integrable martingale M, a generalized form of the Itō isometry can be used. First, the Doob–Meyer decomposition theorem is used to show that a decomposition M^{2} = N + <M> exists, where N is a martingale and <M> is a rightcontinuous, increasing and predictable process starting at zero. This uniquely defines <M>, which is referred to as the predictable quadratic variation of M. The Itō isometry for square integrable martingales is then
 $\backslash mathbb\{E\}\; \backslash left\; [(H\backslash cdot\; M\_t)^2\backslash right\; ]=\; \backslash mathbb\{E\}\; \backslash left\; [\backslash int\_0^tH^2\_s\backslash ,d\backslash langle\; M\backslash rangle\_s\backslash right],$
which can be proved directly for simple predictable integrands. As with the case above for Brownian motion, a continuous linear extension can be used to uniquely extend to all predictable integrands satisfying E[H^{2} · <M>_{t}] < ∞. This method can be extended to all local square integrable martingales by localization. Finally, the Doob–Meyer decomposition can be used to decompose any local martingale into the sum of a local square integrable martingale and a finite variation process, allowing the Itō integral to be constructed with respect to any semimartingale.
Many other proofs exist which apply similar methods but which avoid the need to use the Doob–Meyer decomposition theorem, such as the use of the quadratic variation [M] in the Itō isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itō isometry. The latter applies directly to local martingales without having to first deal with the square integrable martingale case.
Alternative proofs exist only making use of the fact that X is càdlàg, adapted, and the set {H · X_{t}: H ≤ 1 is simple previsible} is bounded in probability for each time t, which is an alternative definition for X to be a semimartingale. A continuous linear extension can be used to construct the integral for all leftcontinuous and adapted integrands with right limits everywhere (caglad or Lprocesses). This is general enough to be able to apply techniques such as Itō's lemma (Protter 2004). Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands (Bichteler 2002).
Differentiation in Itō calculus
The Itō calculus is first and foremost defined as an integral calculus as outlined above. However, there are also different notions of "derivative" with respect to Brownian motion:
Malliavin derivative
Malliavin calculus provides a theory of differentiation for random variables defined over Wiener space, including an integration by parts formula (Nualart 2006).
Martingale representation
The following result allows to express martingales as Itô integrals: if M is a squareintegrable 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 α on [0, T] such that
 $M\_\{t\}\; =\; M\_\{0\}\; +\; \backslash int\_\{0\}^\{t\}\; \backslash alpha\_\{s\}\; \backslash ,\; \backslash mathrm\{d\}\; B\_\{s\}$
almost surely, and for all t ∈ [0, T] (Rogers & Williams 2000, Theorem 36.5). This representation theorem can be interpreted formally as saying that α is the “time derivative” of M with respect to Brownian motion B, since α is precisely the process that must be integrated up to time t to obtain M_{t} − M_{0}, as in deterministic calculus.
Itō calculus for physicists
In physics, usually stochastic differential equations, also called Langevin equations, are used, rather than general stochastic integrals. A physicist would formulate an Itō stochastic differential equation (SDE) as
 $\backslash dot\{x\}\_k=h\_k+g\_\{kl\}\; \backslash xi\_l,$
where $\backslash xi\_j$ is Gaussian white noise with
 $\backslash langle\backslash xi\_k(t\_1)\backslash ,\backslash xi\_l(t\_2)\backslash rangle=\backslash delta\_\{kl\}\backslash delta(t\_1t\_2)$
and Einstein's summation convention is used.
If $y=y(x\_k)$ is a function of the x_{k}, then Itō's lemma has to be used:
 $\backslash dot\{y\}=\backslash frac\{\backslash partial\; y\}\{\backslash partial\; x\_j\}\backslash dot\{x\}\_j+\backslash tfrac\{1\}\{2\}\backslash frac\{\backslash partial^2\; y\}\{\backslash partial\; x\_k\; \backslash ,\; \backslash partial\; x\_l\}\; g\_\{km\}g\_\{ml\}.$
An Itō SDE as above also corresponds to a Stratonovich SDE which reads
 $\backslash dot\{x\}\_k\; =\; h\_k\; +\; g\_\{kl\}\; \backslash xi\_l\; \; \backslash frac\{1\}\{2\}\; \backslash frac\{\backslash partial\; g\_\{kl\}\}\{\backslash partial\; \{x\_m\}\}\; g\_\{ml\}.$
SDEs frequently occur in physics in Stratonovich form, as limits of stochastic differential equations driven by colored noise if the correlation time of the noise term approaches zero.
For a recent treatment of different interpretations of stochastic differential equations see for example (Lau & Lubensky 2007).
See also
References

 PDFfiles, with generalizations of Itō's lemma for nonGaussian processes.








 Mathematical Finance Programming in TIBasic, which implements Ito calculus for TIcalculators.


 Methods  

 Improper Integrals  

 Stochastic integrals  


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