Itō calculus



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
 * $$Y_t=\int_0^t H_s\,dX_s$$

where X 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. 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 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 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 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.

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 integral of a process H with respect to another process X up until a time t is written as
 * $$\int_0^t H\,dX\equiv\int_0^t H_s\,dX_s$$

This is itself a stochastic process with time parameter t, which is also written as H &middot; X. Alternatively, the integral is often written in differential form dY = H dX, which is equivalent to Y - Y0 = H &middot; X. As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that there is an underlying filtered probability space.
 * $$(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$$

The sigma algebra Ft represents the information available up until time t, and a process X is adapted if Xt is Ft-measurable. A Brownian motion B is understood to be an Ft-Brownian motion, which is just a standard Brownian motion with the property that Bt+s - Bt is independent of Ft for all s,t &ge; 0.

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 left-continuous, adapted and locally bounded process. If &pi;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


 * $$\int_{0}^{t} H \,d B =\lim_{n\rightarrow\infty} \sum_{t_{i-1},t_i\in\pi_n}H_{t_{i-1}}(B_{t_i}-B_{t_{i-1}}).$$

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 which is closed under taking limits of sequences and contains all adapted left continuous processes. If H is any predictable process such that &int;0t H2 ds < &infin; for every t &ge; 0 then the integral of H with respect to B can be defined, and H is said to be B-integrable. Any such process can be approximated by a sequence Hn of left-continuous, adapted and locally bounded processes, in the sense that


 * $$ \int_0^t (H-H_n)^2\,ds\rightarrow 0$$

in probability. Then, the Itō integral is


 * $$\int_0^t H\,dB = \lim_{n\rightarrow\infty}\int_0^t H_n\,dB$$

where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itō isometry
 * $$\mathbb{E}\left( (H\cdot B_t)^2\right)=\mathbb{E} \left (\int_0^t H_s^2\,ds\right )$$

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,
 * $$X_t=X_0+\int_0^t\sigma_s\,dB_s+\int_0^t\mu_s\,ds.$$

Here, B is a Brownian motion and it is required that &sigma; is a predictable B-integrable process, and &mu; is predictable and (Lebesgue) integrable. That is,
 * $$\int_0^t(\sigma_s^2+|\mu_s|)\,ds<\infty$$

for each t. The stochastic integral can be extended to such Itō processes,
 * $$\int_0^t H\,dX =\int_0^t H_s\sigma_s\,dB_s + \int_0^t H_s\mu_s\,ds.$$

This is defined for all locally bounded and predictable integrands. More generally, it is required that H &sigma; be B-integrable and H &mu; be Lebesgue integrable, so that &int;0t(H2&sigma;2 + |H &mu;|)ds is finite. Such predictable processes H are called X-integrable.

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
 * $$df(X_t)=f^\prime(X_t)\,dX_t + \frac{1}{2}f^{\prime\prime}(X_t)\sigma_t^2\,dt.$$

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 non-zero 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 &middot; X exists, and can be calculated as a limit of Riemann sums. Let &pi;n be a sequence of partitions of [0,t] with mesh going to zero,


 * $$\int_0^t H\,dX = \lim_{n\rightarrow\infty} \sum_{t_{i-1},t_i\in\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}}).$$

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 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 Hn &rarr; H and |Hn| &le; J for a locally bounded process J, then &int;0t Hn dX &rarr; &int;0t H dX 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 H &middot; 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 X-integrable, with integral H &middot; X = Y, if and only if Y0 = 0 and K &middot; Y = (KH) &middot; X. The set of X-integrable processes is denoted by L(X).

Properties

 * 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 Xt - Xt-, and is often denoted by &Delta;Xt. With this notation, &Delta;(H &middot; X)=H &Delta;X. 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 X-integrable. Then, J is K &middot; X integrable if and only if JK is X integrable, in which case
 * $$ J\cdot (K\cdot X) = (JK)\cdot X$$


 * Dominated convergence. Suppose that Hn &rarr; H and |Hn| &le; J, where J is an X-integrable process. then Hn &middot; X &rarr; H &middot; 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 X-integrable process will also be [X,Y]-integrable, and [H &middot; X,Y] = H &middot; [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,
 * $$[H\cdot X]=H^2\cdot[X]$$

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 non-zero quadratic variation, which only occurs for infinite variation processes (such as Brownian motion). If X and Y are semimartingales then
 * $$X_tY_t = X_0Y_0+\int_0^t X_{s-}\,dY_s + \int_0^t Y_{s-}\,dX_s + [X,Y]_t$$

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
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 d-dimensional semimartingale X = (X1,&hellip;,Xd) and twice continuously differentiable function f from Rd to R, it states that f(X) is a semimartingale and,
 * $$df(X_t)= \sum_{i=1}^d f_{,i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^d f_{,ij}(X_{t})\,d[X^i,X^j]_t.$$

This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xi,Xj ]. The formula can be generalized to non-continuous semimartingales by adding a pure jump term to ensure that the jumps of the left and right hand sides agree (see Itō's lemma).

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 &middot; M is also a local martingale. For integrands which are not locally bounded, there are examples where H &middot; 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 M-integrable if and only if &int;0tH2 d[M] is finite for each t, and H &middot; M is always a local martingale.

The most general statement for a discontinuous local martingale M is that if (H2 &middot; [M])1/2 is locally integrable then H &middot; 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(Mt2) 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
 * $$\mathbb{E}\left((H\cdot M_t)^2\right)=\mathbb{E}\left(\int_0^t H^2\,d[M]\right).$$

This equality holds more generally for any martingale M such that H2 &middot; [M]t is integrable. The Itō isometry is often used as an important step in the construction of the stochastic integral, by defining H &middot; M to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes.

p-Integrable martingales
For any p > 1, and bounded predictable integrand, the stochastic integral preserves the space of p-integrable martingales. These are càdlàg martingales such that E(|Mt|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 cadlag process M is written as Mt* = sups &le;t |Ms|. For any p &ge; 1 and bounded predictable integrand, the stochastic integral preserves the space of cadlag martingales M such that E((Mt*)p) is finite for all t. If p > 1 then this is the same as the space of p-integrable martingales, by Doob's inequalities.

The Burkholder-Davis-Gundy inequalities state that, for any given p &ge; 1, there exists positive constants c,C such that
 * $$c\mathbb{E}([M]_t^{p/2})\le \mathbb{E}((M^*_t)^p)\le C\mathbb{E}([M]_t^{p/2})$$

for all cadlag local martingales M. These are used to show that if (Mt*)p is integrable and H is a bounded predictable process then
 * $$\mathbb{E}(((H\cdot M)_t^*)^p) \le C\mathbb{E}((H^2\cdot[M]_t)^{p/2})<\infty$$

and, consequently, H &middot; M is a p-integrable martingale. More generally, this statement is true whenever (H2 &middot; [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 Ht = A1{t > T} for stopping times T and FT -measurable random variables A, for which the integral is
 * $$H\cdot X_t\equiv 1_{\{t>T\}}A(X_t-X_T).$$

This is extended to all simple predictable processes by the linearity of H &middot; 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 Itō isometry for simple predictable integrands,
 * $$ E\left( (H\cdot B_t)^2\right) = E\left(\int_0^tH_s^2\,ds\right).$$

By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying E(&int;0tH 2ds) < &infin; in such way that the Itō isometry still holds. It can then be extended to all B-integrable 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&middot;X = H&middot;M + H&middot;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 cadlag 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 M2 = N+  exists, where N is a martingale and  is a right-continuous, increasing and predictable process starting at zero. This uniquely defines , which is refered to as the predictable quadratic variation of M. The Itō isometry for square integrable martingales is then
 * $$E\left((H\cdot M_t)^2\right)=E\left(\int_0^tH^2_s\,d\langle M\rangle_s\right),$$

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(H2&middot;t) < &infin;. 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 Dol&eacute;ans 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 cadlag, adapted, and the set {H&middot;Xt: |H |&le;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 left-continuous and adapted integrands with right limits everywhere (caglad or L-processes). This is general enough to be able to apply techniques such as Itō's lemma. Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands.

Further extensions of Itō calculus: stochastic derivative
Itō calculus, as ground-breaking and remarkable as it is, for over 60 years has only been an integral calculus: there was no explicit pathwise differentiation theory behind it. However, in 2004 (published in 2006) Hassan Allouba defined the derivative of a given semimartingale S with respect to a Brownian motion B using the derivative of the covariation of S and B (also known as the cross-variation of S and B) with respect to the quadratic variation of B. Given a continuous semimartingale $$S_{t} = S_{0} + V_{t} + M_{t},$$ where V is a process of bounded variation on compacts and M is a local martingale, the (strong) derivative of S with respect to a Brownian motion B is defined as the stochastic process $$\mathbb{D}_{B} S$$ given by


 * $$\mathbb{D}_{B_{t}} S_{t}= \frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} \langle B, B \rangle_{t}} =\frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} t},$$

where we used the fact that the covariation of Brownian motion B with itself is just its quadratic variation, which is t.

This stochastic derivative turns out to have many of the properties of the usual derivative of elementary calculus. It leads to a fundamental theorem of stochastic calculus for this stochastic derivative/integral pair:


 * $$\mathbb{D}_{B_{t}} \int_{0}^{t} X_{s} \mathrm{d} B_{s} = X_{t},$$    and     $$\int_{0}^{t} \mathbb{D}_{B_{s}} S_{s} \mathrm{d} B_{s}= S_{t} - S_{0} - V_{t}.$$

It also leads to a stochastic mean value theorem, stochastic chain rules, as well as other differentiation rules that are similar to those in elementary calculus. A key difference is that where an indefinite integral (anti-derivative) in the usual elementary calculus sense is determined only up to an additive constant of integration, an indefinite integral in this stochastic calculus is determined only up to a process of bounded variation on compacts. These processes are the "constants" in this stochastic differentiation theory. Also, if M and B are orthogonal (zero covariation) then $$\mathbb{D}_{B} M=0$$; in particular, if M and B are independent then $$\mathbb{D}_{B} M=0.$$