Itô Calculus
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Itô calculus, named after Kiyosi Itô, extends the methods of
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
to
stochastic process In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
es such as
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
(see
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
). It has important applications in
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
and
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s. The central concept is the Itô stochastic integral, a stochastic generalization of the
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
in analysis. The integrands and the integrators are now stochastic processes: :Y_t=\int_0^t H_s\,dX_s, where ''H'' is a locally square-integrable process adapted to the
filtration Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
generated by ''X'' , which is a
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
or, more generally, a
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
. The result of the integration is then another stochastic process. Concretely, the integral from 0 to any particular ''t'' is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
, 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 ''H'' is
adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
, which loosely speaking means that its value at time ''t'' can only depend on information available up until this time. Roughly speaking, one chooses a sequence of partitions of the interval from 0 to ''t'' and constructs
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s. 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 A mesh is a barrier made of connected strands of metal, fiber, or other flexible or ductile materials. A mesh is similar to a web or a net in that it has many attached or woven strands. Types * A plastic mesh may be extruded, oriented, ex ...
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 In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves a ...
, which is a
change of variables Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
formula. These differ from the formulas of standard calculus, due to
quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued sto ...
terms. In
mathematical finance Mathematical finance, also known as quantitative finance and financial mathematics, is a field of applied mathematics, concerned with mathematical modeling of financial markets. In general, there exist two separate branches of finance that require ...
, 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 A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift. It i ...
(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 ''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 Clairvoyance (; ) is the magical ability to gain information about an object, person, location, or physical event through extrasensory perception. Any person who is claimed to have such ability is said to be a clairvoyant () ("one who sees cl ...
: 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 sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s .


Notation

The process ''Y'' defined before as : Y_t = \int_0^t H\,dX\equiv\int_0^t H_s\,dX_s , is itself a stochastic process with time parameter ''t'', which is also sometimes written as ''Y'' = ''H'' · ''X'' . Alternatively, the integral is often written in differential form ''dY = H dX'', which is equivalent to ''Y'' − ''Y''0 = ''H'' · ''X''. As Itô calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying
filtered probability space Filtration is a physical separation process that separates solid matter and fluid from a mixture using a ''filter medium'' that has a complex structure through which only the fluid can pass. Solid particles that cannot pass through the filter ...
is given :(\Omega,\mathcal,(\mathcal_t)_,\mathbb) . The
σ-algebra In mathematical analysis and in probability theory, a σ-algebra (also σ-field) on a set ''X'' is a collection Σ of subsets of ''X'' that includes the empty subset, is closed under complement, and is closed under countable unions and countabl ...
''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 properties that ''B''''t'' is ''Ft''-measurable and that ''B''''t''+''s'' − ''B''''t'' is independent of ''Ft'' for all ''s'',''t'' ≥ 0 .


Integration with respect to Brownian motion

The Itô integral can be defined in a manner similar to the
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
, that is as a limit in probability of
Riemann sum In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
s; such a limit does not necessarily exist pathwise. Suppose that ''B'' is a
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
(Brownian motion) and that ''H'' is a
right-continuous In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in valu ...
(
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
),
adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
and locally bounded process. If \ is a sequence of
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
s of , ''t''with mesh going to zero, then the Itô integral of ''H'' with respect to ''B'' up to time ''t'' is a
random variable A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
:\int_0^t H \,d B =\lim_ \sum_H_(B_-B_). It can be shown that this limit
converges in probability In probability theory, there exist several different notions of convergence of random variables. The convergence of sequences of random variables to some limit random variable is an important concept in probability theory, and its applications to ...
. For some applications, such as
martingale representation theorem In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian m ...
s and local times, the integral is needed for processes that are not continuous. The
predictable process In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process whose value is knowable at a prior time. The predictable processes form the smallest class that is closed under taking limits o ...
es form the smallest class that is closed under taking limits of sequences and contains all adapted left-continuous 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 ''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\to 0 in probability. Then, the Itô integral is :\int_0^t H\,dB = \lim_\int_0^t H_n\,dB where, again, the limit can be shown to converge in probability. The stochastic integral satisfies the
Itô isometry In mathematics, the Itô isometry, named after Kiyoshi Itô, is a crucial fact about Itô stochastic integrals. One of its main applications is to enable the computation of variances for random variables that are given as Itô integrals. Let W : ...
:\mathbb\left \left(\int_0^t H_s \, dB_s\right)^2\right\mathbb \left \int_0^t H_s^2\,ds\right /math> which holds when ''H'' 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 In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
stochastic process that 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 σ is a predictable ''B''-integrable process, and μ is predictable and (
Lebesgue Henri Léon Lebesgue (; June 28, 1875 – July 26, 1941) was a French mathematician known for his theory of integration, which was a generalization of the 17th-century concept of integration—summing the area between an axis and the curve of ...
) integrable. That is, :\int_0^t(\sigma_s^2+, \mu_s, )\,ds<\infty for each ''t''. The stochastic integral can be extended to such Itô 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''σ be ''B''-integrable and ''H''μ be Lebesgue integrable, so that :\int_0^t (H^2 \sigma^2 + , H\mu, )ds < \infty. Such predictable processes ''H'' are called ''X''-integrable. An important result for the study of Itô processes is
Itô's lemma In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves a ...
. In its simplest form, for any twice continuously differentiable function ''f'' on the reals and Itô process ''X'' as described above, it states that ''f''(''X'') is itself an Itô process satisfying :df(X_t)=f^\prime(X_t)\,dX_t + \fracf^ (X_t) \sigma_t^2 \, dt. This is the stochastic calculus version of the
change of variables Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
formula and
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
. 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 In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued sto ...
.


Semimartingales as integrators

The Itô integral is defined with respect to a
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
''X''. These are processes which can be decomposed as ''X'' = ''M'' + ''A'' for a
local martingale In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local m ...
''M'' and finite variation process ''A''. Important examples of such processes include
Brownian motion Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas). This pattern of motion typically consists of random fluctuations in a particle's position insi ...
, which is a martingale, and
Lévy process In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which disp ...
es. 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
partition Partition may refer to: Computing Hardware * Disk partitioning, the division of a hard disk drive * Memory partition, a subdivision of a computer's memory, usually for use by a single job Software * Partition (database), the division of a ...
s of , ''t''with mesh going to zero, :\int_0^t H\,dX = \lim_ \sum_H_(X_-X_). 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 In probability theory, the Girsanov theorem tells how stochastic processes change under changes in measure. The theorem is especially important in the theory of financial mathematics as it tells how to convert from the physical measure which des ...
, and for the study of
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s. However, it is inadequate for other important topics such as
martingale representation theorem In probability theory, the martingale representation theorem states that a random variable that is measurable with respect to the filtration generated by a Brownian motion can be written in terms of an Itô integral with respect to this Brownian m ...
s and local times. The integral extends to all predictable and locally bounded integrands, in a unique way, such that the
dominated convergence theorem In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
holds. That is, if ''Hn'' → ;''H'' and , ''Hn'',  ≤ ''J'' for a locally bounded process ''J'', then :\int_0^t H_n \,dX \to \int_0^t 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'' · ''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'' · ''X'' = ''Y'', if and only if ''Y''0 = 0 and ''K'' · ''Y'' = (''KH'') · ''X''. The set of ''X''-integrable processes is denoted by ''L''(''X'').


Properties

The following properties can be found in works such as and : * The stochastic integral is a
càdlàg In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
process. Furthermore, it is a
semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
. * 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'' − ''X''t−, and is often denoted by Δ''Xt''. With this notation, Δ(''H'' · ''X'') = ''H'' Δ''X''. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous. *
Associativity In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
. Let ''J'', ''K'' be predictable processes, and ''K'' be ''X''-integrable. Then, ''J'' is ''K'' · ''X'' integrable if and only if ''JK'' is ''X'' integrable, in which case *: J\cdot (K\cdot X) = (JK)\cdot X *
Dominated convergence In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which almost everywhere convergence of a sequence of functions implies convergence in the ''L''1 norm. Its power and utility are two of the primary ...
. Suppose that ''Hn'' → ''H'' and '', Hn, '' ≤ ''J'', where ''J'' is an ''X''-integrable process. then ''Hn'' · ''X'' → ''H'' · ''X''. Convergence is in probability at each time ''t''. In fact, it converges uniformly on compact sets 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'' · ''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, *: \cdot XH^2\cdot /math>


Integration by parts

As with ordinary calculus,
integration by parts In calculus, and more generally in mathematical analysis, integration by parts or partial integration is a process that finds the integral of a product of functions in terms of the integral of the product of their derivative and antiderivative. ...
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 ''X'' and ''Y'' are semimartingales then X_t Y_t = X_0Y_0+ \int_0^t X_ \,dY_s + \int_0^t Y_\,dX_s + ,Yt where 'X'', ''Y''is the quadratic covariation process. The result is similar to the integration by parts theorem for the
Riemann–Stieltjes integral In mathematics, the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition of this integral was first published in 1894 by Stieltjes. It serves as an inst ...
but has an additional
quadratic variation In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and other martingales. Quadratic variation is just one kind of variation of a process. Definition Suppose that X_t is a real-valued sto ...
term.


Itô's lemma

Itô's lemma is the version of the
chain rule In calculus, the chain rule is a formula that expresses the derivative of the composition of two differentiable functions and in terms of the derivatives of and . More precisely, if h=f\circ g is the function such that h(x)=f(g(x)) for every , ...
or
change of variables Change or Changing may refer to: Alteration * Impermanence, a difference in a state of affairs at different points in time * Menopause, also referred to as "the change", the permanent cessation of the menstrual period * Metamorphosis, or change, ...
formula which applies to the Itô integral. It is one of the most powerful and frequently used theorems in stochastic calculus. For a continuous ''n''-dimensional semimartingale ''X'' = (''X''1,...,''X''''n'') and twice continuously differentiable function ''f'' from R''n'' to R, it states that ''f''(''X'') is a semimartingale and, df(X_t)= \sum_^n f_(X_t)\,dX^i_t + \frac \sum_^n f_(X_) \, d ^i,X^jt. 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 include an explicit time-dependence in f, and in other ways (see
Itô's lemma In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves a ...
).


Martingale integrators


Local martingales

An important property of the Itô integral is that it preserves the
local martingale In mathematics, a local martingale is a type of stochastic process, satisfying the localized version of the martingale property. Every martingale is a local martingale; every bounded local martingale is a martingale; in particular, every local m ...
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 ''M''-integrable if and only if :\int_0^t H^2 \, d <\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 In mathematics, a locally integrable function (sometimes also called locally summable function) is a function which is integrable (so its integral is finite) on every compact subset of its domain of definition. The importance of such functions li ...
then ''H'' · ''M'' 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 In mathematics, a càdlàg (French: "''continue à droite, limite à gauche''"), RCLL ("right continuous with left limits"), or corlol ("continuous on (the) right, limit on (the) left") function is a function defined on the real numbers (or a subset ...
martingales ''M'' such that E 'Mt''2is finite for all ''t''. For any such square integrable martingale ''M'', the quadratic variation process 'M''is integrable, and the Itô isometry states that :\mathbb\left H\cdot M_t)^2\right \mathbb\left int_0^t_H^2\,d[Mright_.html" ;"title=".html" ;"title="int_0^t H^2\,d[M">int_0^t H^2\,d[Mright ">.html" ;"title="int_0^t H^2\,d[M">int_0^t H^2\,d[Mright This equality holds more generally for any martingale ''M'' such that ''H''2 · [''M'']''t'' is integrable. The Itô 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.


''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 càdlàg process ''M'' is written as ''M*t'' = sup''s'' ≤''t'' , ''Ms'', . 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 ''p''-integrable 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\mathbb \left [Mt^_\right_.html"_;"title=".html"_;"title="[M">[Mt^_\right_">.html"_;"title="[M">[Mt^_\right_\le_\mathbb\left_[(M^*_t)^p_\right_.html" ;"title="">[Mt^_\right_.html" ;"title=".html" ;"title="[M">[Mt^ \right ">.html" ;"title="[M">[Mt^ \right \le \mathbb\left [(M^*_t)^p \right ">">[Mt^_\right_.html" ;"title=".html" ;"title="[M">[Mt^ \right ">.html" ;"title="[M">[Mt^ \right \le \mathbb\left [(M^*_t)^p \right le C\mathbb\left [Mt^ \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 :\mathbb\left [ ((H\cdot M)_t^*)^p \right ] \le C\mathbb\left H^2\cdot[Mt)^_\right_.html" ;"title=".html" ;"title="H^2\cdot[M">H^2\cdot[Mt)^ \right ">.html" ;"title="H^2\cdot[M">H^2\cdot[Mt)^ \right \infty and, consequently, ''H'' · ''M'' is a ''p''-integrable martingale. More generally, this statement is true whenever (''H''2 ·  'M''''p''/2 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 ''Ht'' = ''A''1 for stopping times ''T'' and ''FT''-measurable random variables ''A'', for which the integral is :H\cdot X_t\equiv \mathbf_A(X_t-X_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(''Bt'') = ''t'' can be used to prove the Itô isometry for simple predictable integrands, : \mathbb \left [ (H\cdot B_t)^2\right ] = \mathbb \left int_0^tH_s^2\,ds\right By a
continuous linear extension In functional analysis, it is often convenient to define a linear transformation on a complete, normed vector space X by first defining a linear transformation \mathsf on a dense subset of X and then extending \mathsf to the whole space via the the ...
, the integral extends uniquely to all predictable integrands satisfying : \mathbb \left \int_0^t H^2 \, ds \right < \infty, in such way that the Itô isometry still holds. It can then be extended to all ''B''-integrable processes by
localization Localization or localisation may refer to: Biology * Localization of function, locating psychological functions in the brain or nervous system; see Linguistic intelligence * Localization of sensation, ability to tell what part of the body is a ...
. This method allows the integral to be defined with respect to any Itô process. For a general semimartingale ''X'', the decomposition ''X'' = ''M'' + ''A'' into a local martingale ''M'' plus a 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 Itô integral for semimartingales will follow from any construction for local martingales. For a càdlàg square integrable martingale ''M'', a generalized form of the Itô isometry can be used. First, the
Doob–Meyer decomposition theorem The Doob–Meyer decomposition theorem is a theorem in stochastic calculus stating the conditions under which a submartingale may be decomposed in a unique way as the sum of a martingale and an increasing predictable process. It is named for ...
is used to show that a decomposition exists, where ''N'' is a martingale and is a right-continuous, increasing and predictable process starting at zero. This uniquely defines , which is referred to as the ''predictable quadratic variation'' of ''M''. The Itô isometry for square integrable martingales is then :\mathbb \left H\cdot M_t)^2\right \mathbb \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 . 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
submartingale In probability theory, a martingale is a sequence of random variables (i.e., a stochastic process) for which, at a particular time, the conditional expectation of the next value in the sequence is equal to the present value, regardless of all ...
s, 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 ''X'' is càdlàg, adapted, and the set 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 Itô's lemma . Also, a
Khintchine inequality In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Latin alphabet, is a theorem from probability, and is also frequently used in analysis. Heuristically, it says that if we pick N complex ...
can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands .


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 In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows ...
provides a theory of differentiation for random variables defined over
Wiener space In mathematics, classical Wiener space is the collection of all Continuous_function#Continuous_functions_between_metric_spaces, continuous functions on a given domain of a function, domain (usually a subinterval of the real line), taking values i ...
, including an integration by parts formula .


Martingale representation

The following result allows to express martingales as Itô integrals: if ''M'' is a square-integrable martingale on a time interval , ''T''with respect to the filtration generated by a Brownian motion ''B'', then there is a unique
adapted In biology, adaptation has three related meanings. Firstly, it is the dynamic evolutionary process of natural selection that fits organisms to their environment, enhancing their evolutionary fitness. Secondly, it is a state reached by the po ...
square integrable process α on , ''T''such that :M_ = M_ + \int_^ \alpha_ \, \mathrm B_ almost surely, and for all ''t'' ∈  , ''T''. 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.


Itô calculus for physicists

In physics, usually
stochastic differential equation A stochastic differential equation (SDE) is a differential equation in which one or more of the terms is a stochastic process, resulting in a solution which is also a stochastic process. SDEs are used to model various phenomena such as stock p ...
s (SDEs), such as Langevin equations, are used, rather than stochastic integrals. Here an Itô stochastic differential equation (SDE) is often formulated via : \dot_k=h_k+g_ \xi_l, where \xi_j is Gaussian white noise with :\langle\xi_k(t_1)\,\xi_l(t_2)\rangle=\delta_\delta(t_1-t_2) and
Einstein's summation convention In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
is used. If y=y(x_k) is a function of the ''xk'', then
Itô's lemma In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves a ...
has to be used: : \dot=\frac\dot_j+\frac\frac g_g_. An Itô SDE as above also corresponds to a Stratonovich SDE which reads : \dot_k = h_k + g_ \xi_l - \frac \frac g_. SDEs frequently occur in physics in Stratonovich form, as limits of stochastic differential equations driven by
colored noise In audio engineering, electronics, physics, and many other fields, the color of noise or noise spectrum refers to the power spectrum of a noise signal (a signal produced by a stochastic process). Different colors of noise have significantl ...
if the correlation time of the noise term approaches zero. For a recent treatment of different interpretations of stochastic differential equations see for example .


Itô interpretation and supersymmetric theory of SDEs

In the supersymmetric theory of SDEs, stochastic evolution is defined via stochastic evolution operator (SEO) acting on
differential form In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s of the phase space. The Itô-Stratonovich dilemma takes the form of the ambiguity of the operator ordering that arises on the way from the path integral to the operator representation of stochastic evolution. The Itô interpretation corresponds to the operator ordering convention that all the momentum operators act after all the position operators. The SEO can be made unique by supplying it with its most natural mathematical definition of the
pullback In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward. Precomposition Precomposition with a function probably provides the most elementary notion of pullback: i ...
induced by the noise-configuration-dependent SDE-defined
diffeomorphism In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two m ...
s and averaged over the noise configurations. This disambiguation leads to the Stratonovich interpretation of SDEs that can be turned into the Itô interpretation by a specific shift of the flow vector field of the SDE.


See also

*
Stochastic calculus Stochastic calculus is a branch of mathematics that operates on stochastic processes. It allows a consistent theory of integration to be defined for integrals of stochastic processes with respect to stochastic processes. This field was created an ...
*
Itô's lemma In mathematics, Itô's lemma or Itô's formula (also called the Itô-Doeblin formula, especially in French literature) is an identity used in Itô calculus to find the differential of a time-dependent function of a stochastic process. It serves a ...
*
Stratonovich integral In stochastic processes, the Stratonovich integral (developed simultaneously by Ruslan Stratonovich and Donald Fisk) is a stochastic integral, the most common alternative to the Itô integral. Although the Itô integral is the usual choice in a ...
*
Semimartingale In probability theory, a real valued stochastic process ''X'' is called a semimartingale if it can be decomposed as the sum of a local martingale and a càdlàg adapted finite-variation process. Semimartingales are "good integrators", forming the ...
*
Wiener process In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...


References

* * *
Hagen Kleinert Hagen Kleinert (born 15 June 1941) is professor of theoretical physics at the Free University of Berlin, Germany (since 1968)Honorary Doctorat the West University of Timișoaraandat thin Bishkek. He is alsHonorary Memberof th For his contributio ...
(2004). ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore); Paperback . Fifth edition available online
PDF-files
with generalizations of Itô's lemma for non-Gaussian processes. * * * * * * * * * Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators. {{DEFAULTSORT:Ito Calculus Definitions of mathematical integration Stochastic calculus Mathematical finance Integral calculus