Ito calculus - AbsoluteAstronomy.com

Itō calculus, named after Kiyoshi Itō

Kiyoshi Ito

was a Japanese mathematician whose work is now called Itō calculus. The basic concept of this calculus is the Itō integral, and among the most important results is Itō's lemma. The Itō calculus facilitates mathematical understanding of random events...

, extends the methods of calculus to stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

es such as Brownian motion

Brownian motion

Brownian motion or pedesis is the presumably random drifting of particles suspended in a fluid or the mathematical model used to describe such random movements, which is often called a particle theory.The mathematical model of Brownian motion has several real-world applications...

(Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

). It has important applications in mathematical finance

Mathematical finance

Mathematical finance is a field of applied mathematics, concerned with financial markets. The subject has a close relationship with the discipline of financial economics, which is concerned with much of the underlying theory. Generally, mathematical finance will derive and extend the mathematical...

and stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

s.
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 non-differentiable function (technically, stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic 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

Brownian motion

(also see Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

). The result of the integration is another stochastic process. In particular, the integral from

to any particular

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

. Then we are constructing Riemann sum

Riemann sum

In mathematics, a Riemann sum is a method for approximating the total area underneath a curve on a graph, otherwise known as an integral. It mayalso be used to define the integration operation. The method was named after German mathematician Bernhard Riemann....

s. 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

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:

where X is a Brownian motion

Brownian motion

or, more generally, a semimartingale

Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

and H is a locally square-integrable process adapted to the filtration generated by X .
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

Bounded variation

In mathematical analysis, a function of bounded variation, also known as a BV function, is a real-valued function whose total variation is bounded : the graph of a function having this property is well behaved in a precise sense...

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

Adapted process

In the study of stochastic processes, an adapted process is one that cannot "see into the future". An informal interpretation is that X is adapted if and only if, for every realisation and every n, Xn is known at time n...

, 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

Geometric Brownian motion

A geometric 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...

(see Black–Scholes). Then, the Itō 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 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 sum

Riemann sum

s .

Important results of Itō calculus include the integration by parts formula and Itō's lemma

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

, which is a change of variables

Integration by substitution

In calculus, integration by substitution is a method for finding antiderivatives and integrals. Using the fundamental theorem of calculus often requires finding an antiderivative. For this and other reasons, integration by substitution is an important tool for mathematicians...

formula. These differ from the formulas of standard calculus, due to quadratic variation

Quadratic variation

In mathematics, quadratic variation is used in the analysis of stochastic processes such as Brownian motion and martingales. Quadratic variation is just one kind of variation of a process.- Definition :...

terms.

Notation

The process Y defined as before as

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 − Y₀ = H · X. As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that an underlying filtered probability space is given

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 .

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

Convergence of random variables

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 statistics and stochastic processes...

of Riemann sum

Riemann sum

s; such a limit does not necessarily exist pathwise. Suppose that B is a Wiener process

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

(Brownian motion) and that H is a left-continuous, adapted

Adapted process

and locally bounded process.
If π_n is a sequence of partition

Partition of an interval

In mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the formIn mathematics, a partition, P of an interval [a, b] on the real line is a finite sequence of the form...

s 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

Random variable

In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...

It can be shown that this limit converges in probability

Convergence of random variables

.

For some applications, such as martingale representation theorem

Martingale representation theorem

In probability theory, the martingale representation theorem states that a random variable which 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 motion....

s and local times

Local time (mathematics)

In the mathematical theory of stochastic processes, local time is a stochastic process associated with diffusion processes such as Brownian motion, that characterizes the amount of time a particle has spent at a given level...

, the integral is needed for processes that are not continuous. The predictable process

Predictable process

In stochastic analysis, a part of the mathematical theory of probability, a predictable process is a stochastic process which the value is knowable at a prior time...

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 H² 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 H_n of left-continuous, adapted and locally bounded processes, in the sense that

in probability. Then, the Itō integral is

where, again, the limit can be shown to converge in probability.
The stochastic integral satisfies the Itō isometry

Ito isometry

In mathematics, the Itō isometry, named after Kiyoshi Itō, is a crucial fact about Itō stochastic integrals. One of its main applications is to enable the computation of variances for stochastic processes....

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

Adapted process

stochastic process which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,

Here, B is a Brownian motion and it is required that σ is a predictable B-integrable process, and μ is predictable and (Lebesgue

Lebesgue integration

In mathematics, Lebesgue integration, named after French mathematician Henri Lebesgue , refers to both the general theory of integration of a function with respect to a general measure, and to the specific case of integration of a function defined on a subset of the real line or a higher...

) integrable. That is,

for each t. The stochastic integral can be extended to such Itō processes,

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 ∫₀^t(H²σ² + |H μ|) ds is finite. Such predictable processes H are called X-integrable.

An important result for the study of Itō processes is Itō's lemma

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

. In its simplest form, for any twice continuously differentiable function ƒ on the reals and Itō process X as described above, it states that ƒ(X) is itself an Itō process satisfying

This is the stochastic calculus version of the change of variables

Integration by substitution

formula and chain rule

Chain rule

In calculus, the chain rule is a formula for computing the derivative of the composition of two or more functions. That is, if f is a function and g is a function, then the chain rule expresses the derivative of the composite function in terms of the derivatives of f and g.In integration, the...

. It differs from the standard result due to the additional term involving the second derivative of ƒ, which comes from the property that Brownian motion has non-zero quadratic variation

Quadratic variation

Semimartingales as integrators

The Itō integral is defined with respect to a semimartingale

Semimartingale

In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

X. These are processes which can be decomposed as X = M + A for a local martingale

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; however, in general a local martingale is not a martingale, because its...

M and finite variation

Bounded variation

process A. Important examples of such processes include Brownian motion

Wiener process

In mathematics, the Wiener process is a continuous-time stochastic process named in honor of Norbert Wiener. It is often called standard Brownian motion, after Robert Brown...

, which is a martingale

Martingale (probability theory)

In probability theory, a martingale is a model of a fair game where no knowledge of past events can help to predict future winnings. In particular, a martingale is a sequence of random variables for which, at a particular time in the realized sequence, the expectation of the next value in the...

, and Lévy process

Lévy process

In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is any continuous-time stochastic process that starts at 0, admits càdlàg modification and has "stationary independent increments" — this phrase will be explained below...

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 of an interval

s of [0, t] with mesh going to zero,

This limit converges in probability.
The stochastic integral of left-continuous processes is general enough for studying much of stochastic calculus. For example, it is sufficient for applications of Itō's Lemma, changes of measure via Girsanov's theorem

Girsanov theorem

In probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure...

, and for the study of stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

s. However, it is inadequate for other important topics such as martingale representation theorem

Martingale representation theorem

s and local times

Local time (mathematics)

.

The integral extends to all predictable and locally bounded integrands, in a unique way, such that the dominated convergence theorem

Dominated convergence theorem

In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...

holds. That is, if H_n → ;H and |H_n| ≤ J for a locally bounded process J, then ∫₀^t H_n dX → ∫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 Y₀ = 0 and K · Y = (KH) · X. The set of X-integrable processes is denoted by L(X).

Properties

The following properties can be found for example in and :

The stochastic integral is a càdlàg
Càdlàg
In mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...

process. Furthermore, it is a semimartingale
Semimartingale
In probability theory, a real valued process X is called a semimartingale if it can be decomposed as the sum of a local martingale and an adapted finite-variation process....

.

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
Associativity
In mathematics, associativity is a property of some binary operations. It means that, within an expression containing two or more occurrences in a row of the same associative operator, the order in which the operations are performed does not matter as long as the sequence of the operands is not...

. 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

Dominated convergence
Dominated convergence theorem
In measure theory, Lebesgue's dominated convergence theorem provides sufficient conditions under which two limit processes commute, namely Lebesgue integration and almost everywhere convergence of a sequence of functions...

. Suppose that H_n → H and |H_n| ≤ J, where J is an X-integrable 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 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,

Integration by parts

As with ordinary calculus, integration by parts

Integration by parts

In calculus, and more generally in mathematical analysis, integration by parts is a rule that transforms the integral of products of functions into other integrals...

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

Quadratic variation

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

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

Quadratic variation

term.

Itō's lemma

Itō's lemma is the version of the chain rule

Chain rule

or change of variables

Integration by substitution

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,…,X^d) and twice continuously differentiable function f from R^d to R, it states that f(X) is a semimartingale and,

This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [Xⁱ,X^j ]. 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

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

Local martingales

An important property of the Itō integral is that it preserves the local martingale

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 M-integrable if and only if ∫₀^tH² d[M] is finite for each t, and H · M is always a local martingale.

The most general statement for a discontinuous local martingale M is that if (H² · [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

Càdlàg

In mathematics, a càdlàg , RCLL , or corlol function is a function defined on the real numbers that is everywhere right-continuous and has left limits everywhere...

martingales M such that E(M_t²) 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

This equality holds more generally for any martingale M such that H² · [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.

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(|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 p-integrable martingales, by Doob's inequalities

Doob's martingale inequality

In mathematics, Doob's martingale inequality is a result in the study of stochastic processes. It gives a bound on the probability that a stochastic process exceeds any given value over a given interval of time...

.

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

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

and, consequently, H · M is a p-integrable martingale. More generally, this statement is true whenever (H² · [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

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,

By a continuous linear extension, the integral extends uniquely to all predictable integrands satisfying E(∫₀^tH ²ds) < ∞ 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·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 M² = N + <M> exists, where N is a martingale and <M> is a right-continuous, 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

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 · <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 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

Khintchine inequality

In mathematics, the Khintchine inequality, named after Aleksandr Khinchin and spelled in multiple ways in the Roman alphabet, is a theorem from probability, and is also frequently used in analysis...

can be used to prove the dominated convergence theorem and extend the integral to general predictable integrands .

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

The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. In other words it provides the mechanics to compute derivatives of random variables....

provides a theory of differentiation for random variables defined over Wiener space, 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 [0, T] with respect to the filtration generated by a Brownian motion B, then there is a unique adapted

Adapted process

square integrable process α on [0, T] such that

almost surely, and for all t ∈ [0, 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 − M₀, as in deterministic calculus.

Itō calculus for physicists

In physics, usually stochastic differential equation

Stochastic differential equation

A stochastic differential equation is a differential equation in which one or more of the terms is a stochastic process, thus resulting in a solution which is itself a stochastic process....

s, also called Langevin equation

Langevin equation

In statistical physics, a Langevin equation is a stochastic differential equation describing the time evolution of a subset of the degrees of freedom. These degrees of freedom typically are collective variables changing only slowly in comparison to the other variables of the system...

s,
are used, rather than general stochastic integrals.
A physicist would formulate an Itō stochastic differential equation (SDE) as

where

is Gaussian white noise with

and Einstein's summation convention

Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate formulae...

is used.

If

is a function of the

, then Itō's lemma

Ito's lemma

In mathematics, Itō's lemma is used in Itō stochastic calculus to find the differential of a function of a particular type of stochastic process. It is named after its discoverer, Kiyoshi Itō...

has to be used:

An Itō SDE as above also corresponds to a Stratonovich SDE

Stratonovich integral

In stochastic processes, the Stratonovich integral is a stochastic integral, the most common alternative to the Itō integral...

which reads

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
.

Notation

Integration with respect to Brownian motion

Itō processes

Semimartingales as integrators

Properties

Integration by parts

Itō's lemma

Local martingales

Square integrable martingales

p-Integrable martingales

Existence of the integral

Differentiation in Itō calculus

Malliavin derivative

Martingale representation

Itō calculus for physicists

See also