Itô calculus

File:ItoIntegralWienerProcess.svg

Itô integral Y_t(B) (Template:Ifsubst style="color:blue">blue) of a Brownian motion Template:Math (Template:Ifsubst style="color:red">red) with respect to itself, i.e., both the integrand and the integrator are Brownian. It turns out Template:Math.

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 and stochastic differential equations.

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: <math display="block">Y_t = \int_0^t H_s\,dX_s,</math> where Template:Math is a locally square-integrable process adapted to the filtration generated by Template:Math Template:Harv, 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 Template:Mvar 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 Template:Math is adapted, which loosely speaking means that its value at time Template:Mvar can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to Template:Mvar 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 H_t of the stock at time t. In this situation, the condition that Template:Math 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 Template:Math is adapted implies that the stochastic integral will not diverge when calculated as a limit of Riemann sums Template:Harv.

Notation

The process Template:Math defined before as <math display="block">Y_t = \int_0^t H\,dX\equiv\int_0^t H_s\,dX_s ,</math> is itself a stochastic process with time parameter t, which is also sometimes written as Template:Math Template:Harv. Alternatively, the integral is often written in differential form Template:Math, which is equivalent to Template:Math. As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given <math display="block">(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P}) .</math> The σ-algebra <math>\mathcal{F}_t</math> represents the information available up until time Template:Mvar, and a process Template:Math is adapted if Template:Math is <math>\mathcal{F}_t</math>-measurable. A Brownian motion Template:Math is understood to be an <math>\mathcal{F}_t</math>-Brownian motion, which is just a standard Brownian motion with the properties that Template:Math is <math>\mathcal{F}_t</math>-measurable and that Template:Math is independent of <math>\mathcal{F}_t</math> for all Template:Math Template:Harv.

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 Template:Math is a Wiener process (Brownian motion) and that Template:Math is a right-continuous (càdlàg), adapted and locally bounded process. If <math>\{\pi_n\}</math> is a sequence of partitions of Template:Closed-closed with mesh width going to zero, then the Itô integral of Template:Math with respect to Template:Math up to time Template:Mvar is a random variable <math display="block">\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}}).</math>

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 Template:Math is any predictable process such that Template:Math for every Template:Math then the integral of Template:Math with respect to Template:Math can be defined, and Template:Math is said to be Template:Math-integrable. Any such process can be approximated by a sequence H_n of left-continuous, adapted and locally bounded processes, in the sense that <math display="block"> \int_0^t (H-H_n)^2\,ds\to 0</math> in probability. Then, the Itô integral is <math display="block">\int_0^t H\,dB = \lim_{n\to\infty}\int_0^t H_n\,dB</math> where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the Itô isometry <math display="block">\mathbb{E}\left[ \left(\int_0^t H_s \, dB_s\right)^2\right] = \mathbb{E} \left[ \int_0^t H_s^2\,ds\right ]</math> which holds when Template:Math is bounded or, more generally, when the integral on the right hand side is finite.

Itô processes

File:ItoProcess1D.svg

A single realization of Itô process with Template:Math and Template:Math, where Template:Math is the Ricker wavelet. Off the tide of wavelet, the motion of Itô process is stable.

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, <math display="block">X_t=X_0+\int_0^t\sigma_s\,dB_s + \int_0^t\mu_s\,ds.</math>

Here, Template:Math is a Brownian motion and it is required that σ is a predictable Template:Math-integrable process, and μ is predictable and (Lebesgue) integrable. That is, <math display="block">\int_0^t(\sigma_s^2+|\mu_s|)\,ds<\infty</math> for each Template:Mvar. The stochastic integral can be extended to such Itô processes, <math display="block">\int_0^t H\,dX =\int_0^t H_s\sigma_s\,dB_s + \int_0^t H_s\mu_s\,ds.</math>

This is defined for all locally bounded and predictable integrands. More generally, it is required that Template:Math be Template:Math-integrable and Template:Math be Lebesgue integrable, so that <math display="block">\int_0^t \left(H^2 \sigma^2 + |H\mu| \right) ds < \infty.</math> Such predictable processes Template:Math are called Template:Math-integrable.

An important result for the study of Itô processes is Itô's lemma. In its simplest form, for any twice continuously differentiable function Template:Math on the reals and Itô process Template:Math as described above, it states that <math> Y_t=f(X_t) </math> is itself an Itô process satisfying <math display="block">d Y_t = f^\prime(X_t) \mu_t\,d t + \tfrac{1}{2} f^{\prime\prime} (X_t) \sigma_t^2 \, d t + f^\prime(X_t) \sigma_t \,dB_t .</math>

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 Template:Math, 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 Template:Math. These are processes which can be decomposed as Template:Math for a local martingale Template:Math and finite variation process Template:Math. Important examples of such processes include Brownian motion, which is a martingale, and Lévy processes. For a left continuous, locally bounded and adapted process Template:Math the integral Template:Math exists, and can be calculated as a limit of Riemann sums. Let Template:Math be a sequence of partitions of Template:Closed-closed with mesh going to zero, <math display="block">\int_0^t H\,dX = \lim_{n\to\infty} \sum_{t_{i-1},t_i\in\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}}).</math>

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 Template:Math and Template:Math for a locally bounded process Template:Math, then <math display="block">\int_0^t H_n \,dX \to \int_0^t H \,dX, </math> 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 Template:Math can be defined even in cases where the predictable process Template:Math is not locally bounded. If Template:Math then Template:Math and Template:Math are bounded. Associativity of stochastic integration implies that Template:Math is Template:Math-integrable, with integral Template:Math, if and only if Template:Math and Template:Math. The set of Template:Math-integrable processes is denoted by Template:Math.

Properties

The following properties can be found in works such as Template:Harv and Template:Harv:

• 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 Template:Mvar is Template:Math, and is often denoted by Template:Math. With this notation, Template:Math. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous.
• Associativity. Let Template:Math, Template:Math be predictable processes, and Template:Math be Template:Math-integrable. Then, Template:Math is Template:Math integrable if and only if Template:Math is Template:Math-integrable, in which case <math display="block"> J\cdot (K\cdot X) = (JK)\cdot X</math>
• Dominated convergence. Suppose that Template:Math and Template:Math, where Template:Math is an Template:Math-integrable process. then Template:Math. Convergence is in probability at each time Template:Mvar. In fact, it converges uniformly on compact sets in probability.
• The stochastic integral commutes with the operation of taking quadratic covariations. If Template:Math and Template:Math are semimartingales then any Template:Math-integrable process will also be Template:Math-integrable, and Template:Math. A consequence of this is that the quadratic variation process of a stochastic integral is equal to an integral of a quadratic variation process, <math display="block">[H\cdot X] = H^2\cdot[X]</math>

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 Template:Math and Template:Math are semimartingales then <math display="block">X_t Y_t = X_0 Y_0 + \int_0^t X_{s-} \, dY_s + \int_0^t Y_{s-} \, dX_s + [X,Y]_t </math> where Template:Math 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

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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 Template:Mvar-dimensional semimartingale Template:Math and twice continuously differentiable function Template:Math from Template:Math to Template:Math, it states that Template:Math is a semimartingale and, <math display="block">df(X_t)= \sum_{i=1}^n f_{i}(X_t)\,dX^i_t + \frac{1}{2} \sum_{i,j=1}^n f_{i,j}(X_{t}) \, d[X^i,X^j]_t.</math> This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation Template:Math. The formula can be generalized to include an explicit time-dependence in <math>f,</math> 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 Template:Math is a local martingale and Template:Math is a locally bounded predictable process then Template:Math is also a local martingale. For integrands which are not locally bounded, there are examples where Template:Math is not a local martingale. However, this can only occur when Template:Math is not continuous. If Template:Math is a continuous local martingale then a predictable process Template:Math is Template:Math-integrable if and only if <math display="block">\int_0^t H^2 \, d[M] <\infty,</math> for each Template:Mvar, and Template:Math is always a local martingale.

The most general statement for a discontinuous local martingale Template:Math is that if Template:Math is locally integrable then Template:Math 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 Template:Math such that Template:Math is finite for all Template:Mvar. For any such square integrable martingale Template:Math, the quadratic variation process Template:Math is integrable, and the Itô isometry states that <math display="block">\mathbb{E}\left [(H\cdot M_t)^2\right ]=\mathbb{E}\left [\int_0^t H^2\,d[M]\right ].</math> This equality holds more generally for any martingale Template:Math such that Template:Math is integrable. The Itô isometry is often used as an important step in the construction of the stochastic integral, by defining Template:Math 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 Template:Math, and bounded predictable integrand, the stochastic integral preserves the space of Template:Math-integrable martingales. These are càdlàg martingales such that Template:Math is finite for all Template:Mvar. However, this is not always true in the case where Template:Math. 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 Template:Math is written as Template:Math. For any Template:Math and bounded predictable integrand, the stochastic integral preserves the space of càdlàg martingales Template:Math such that Template:Math is finite for all Template:Mvar. If Template:Math then this is the same as the space of Template:Math-integrable martingales, by Doob's inequalities.

The Burkholder–Davis–Gundy inequalities state that, for any given Template:Math, there exist positive constants Template:Math, Template:Math that depend on Template:Math, but not Template:Math or on Template:Mvar such that <math display="block">c\mathbb{E} \left [ [M]_t^{\frac{p}{2}} \right ] \le \mathbb{E}\left [(M^*_t)^p \right ]\le C\mathbb{E}\left [ [M]_t^{\frac{p}{2}} \right ]</math> for all càdlàg local martingales Template:Math. These are used to show that if Template:Math is integrable and Template:Math is a bounded predictable process then <math display="block">\mathbb{E}\left [ ((H\cdot M)_t^*)^p \right ] \le C\mathbb{E}\left [(H^2\cdot[M]_t)^{\frac{p}{2}} \right ] < \infty</math> and, consequently, Template:Math is a Template:Math-integrable martingale. More generally, this statement is true whenever Template:Math is integrable.

Existence of the integral

Proofs that the Itô 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 Template:Math for stopping times Template:Math and Template:Math-measurable random variables Template:Math, for which the integral is <math display="block">H\cdot X_t\equiv \mathbf{1}_{\{t>T\}}A(X_t-X_T).</math> This is extended to all simple predictable processes by the linearity of Template:Math in Template:Math.

For a Brownian motion Template:Math, the property that it has independent increments with zero mean and variance Template:Math can be used to prove the Itô isometry for simple predictable integrands, <math display="block"> \mathbb{E} \left [ (H\cdot B_t)^2\right ] = \mathbb{E} \left [\int_0^tH_s^2\,ds\right ].</math> By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying <math display="block"> \mathbb{E} \left[ \int_0^t H^2 \, ds \right ] < \infty,</math> in such way that the Itô isometry still holds. It can then be extended to all Template:Math-integrable processes by localization. This method allows the integral to be defined with respect to any Itô process.

For a general semimartingale Template:Math, the decomposition Template:Math into a local martingale Template:Math plus a finite variation process Template:Math can be used. Then, the integral can be shown to exist separately with respect to Template:Math and Template:Math and combined using linearity, Template:Math, 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 Itô integral for semimartingales will follow from any construction for local martingales.

For a càdlàg square integrable martingale Template:Math, a generalized form of the Itô isometry can be used. First, the Doob–Meyer decomposition theorem is used to show that a decomposition Template:Math exists, where Template:Math is a martingale and Template:Math is a right-continuous, increasing and predictable process starting at zero. This uniquely defines Template:Math, which is referred to as the predictable quadratic variation of Template:Math. The Itô isometry for square integrable martingales is then <math display="block">\mathbb{E} \left [(H\cdot M_t)^2\right ]= \mathbb{E} \left [\int_0^tH^2_s\,d\langle M\rangle_s\right],</math> 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 Template:Math. 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 Itô 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 Itô isometry, the use of the Doléans measure for submartingales, or the use of the Burkholder–Davis–Gundy inequalities instead of the Itô 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 Template:Math is càdlàg, adapted, and the set {H · X_t: |H| ≤ 1 is simple previsible} is bounded in probability for each time Template:Math, which is an alternative definition for Template:Math 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 Itô's lemma Template:Harv. Also, a Khintchine inequality can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands Template:Harv.

Differentiation in Itô calculus

The Itô 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 Template:Harv.

Martingale representation

The following result allows to express martingales as Itô integrals: if Template:Math is a square-integrable martingale on a time interval Template:Closed-closed with respect to the filtration generated by a Brownian motion Template:Math, then there is a unique adapted square integrable process <math>\alpha</math> on Template:Closed-closed such that <math display="block">M_{t} = M_{0} + \int_{0}^{t} \alpha_{s} \, \mathrm{d} B_{s}</math> almost surely, and for all Template:Math Template:Harv. This representation theorem can be interpreted formally as saying that α is the "time derivative" of Template:Math with respect to Brownian motion Template:Math, since α is precisely the process that must be integrated up to time Template:Mvar to obtain Template:Math, as in deterministic calculus.

Itô calculus for physicists

In physics, usually stochastic differential equations (SDEs), such as Langevin equations, are used, rather than stochastic integrals. Here an Itô stochastic differential equation (SDE) is often formulated via <math display="block"> \dot{x}_k = h_k + g_{kl} \xi_l,</math> where <math>\xi_j</math> is Gaussian white noise with <math display="block">\langle\xi_k(t_1)\,\xi_l(t_2)\rangle = \delta_{kl}\delta(t_1-t_2)</math> and Einstein's summation convention is used.

If <math>y = y(x_k)</math> is a function of the Template:Math, then Itô's lemma has to be used: <math display="block"> \dot{y}=\frac{\partial y}{\partial x_j}\dot{x}_j+\frac{1}{2}\frac{\partial^2 y}{\partial x_k \, \partial x_l} g_{km}g_{ml}. </math>

An Itô SDE as above also corresponds to a Stratonovich SDE which reads <math display="block"> \dot{x}_k = h_k + g_{kl} \xi_l - \frac{1}{2} \frac{\partial g_{kl}}{\partial {x_m}} g_{ml}.</math>

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 Template:Harv.

See also

• Stochastic calculus
• Itô's lemma
• Otto calculus
• Stratonovich integral
• Semimartingale
• Wiener process

References

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• Hagen Kleinert (2004). Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, 4th edition, World Scientific (Singapore); Paperback Template:ISBN. Fifth edition available online: PDF-files, with generalizations of Itô's lemma for non-Gaussian processes.
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• Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators.