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Encyclopedia > 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. It is used to model systems that behave randomly. Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... In the mathematics of probability, a stochastic process is a random function. ...

The best-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modelling Brownian motion as described by Albert Einstein and other physical diffusion processes in space of particles subject to random forces. More recently, the Wiener process has been widely applied in financial mathematics to model the evolution in time of stock and bond prices. In mathematics, the Wiener process, so named in honor of Norbert Wiener, is a continuous-time Gaussian stochastic process with independent increments used in modelling Brownian motion and some random phenomena observed in finance. ... Norbert Wiener Norbert Wiener (November 26, 1894 - March 18, 1964) was an US mathematician, known as the founder of cybernetics. ... An example of 1000 simulated steps of Brownian motion in two dimensions. ... Albert Einstein photographed by Oren J. Turner in 1947. ... Diffusion, being the spontaneous spreading of matter (particles), heat, or momentum, is one type of transport phenomena. ... Mathematical finance is the branch of applied mathematics concerned with the financial markets. ...

The main flavours of stochastic calculus are the Itô calculus and its variational relative the Malliavin calculus. For technical reasons the Itô integral is the most useful for general classes of processes but the related Stratonovich integral is frequently useful in problem formulation (particularly in engineering disciplines) and the integrals can readily be expressed in terms of the Itô integral. The Dominated Convergence theorem does not hold for the Stratonovich integral, consequently it is very difficult to prove results without re-expressing the integrals in Itô form. ItÅ calculus, named after Kiyoshi ItÅ, treats mathematical operations on stochastic processes. ... The Malliavin calculus, named after Paul Malliavin, is a theory of variational stochastic calculus. ... The Stratonovich Integral is a stochastic integral, the commonest alternative to the ItÅ integral. ...

The key to the construction of a stochastic integral is the definition of a quadratic-variation process; the quadratic variation of a general L2 bounded martingale Xt may be defined as the increasing process [X]t such that In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i. ...

(i)[X]0 = 0
(ii)$Delta [X]_t = (Delta X_t )^2 quad forall t$
(iii)$X_t^2 - [X]_t$ is a UI martingale.

The proof that such a process may be constructed and is unique is a major hurdle in the development of stochastic calculus. However for an process Xt with continuous sample paths it may be shown to be equivalent to the following definition for a partition

$pi_t = { 0 = t_0 < t_1 < cdots < t_m=t}$

whose mesh is defined by

$delta(pi_t) = max_{k in [1,m]} | t_{k}-t_{k-1} |$

in terms of which the quadratic-variation process may be defined by

$V_t = lim_{delta(pi_t) to 0} sum_{pi} | X_{t_k} - X_{t_{k-1}} | ^2.$

A related process $langle X rangle_t$ is historically sometimes used as the basis of the integral; this process is defined as a previsible process satisfying the first and thirds conditions above. It can be shown that this process is the previsible projection of [X]t. While much of the theory can be developed from this starting point, to approach the theory stochastic integration of discontinuous processes it proves the wrong starting point.

This definition is extended to semimartingales by defining

$[X]_t = [X^{mathrm{cm}}]_t + sum_{0 le s le t} Delta X_s^2$

where Xcm is the canonical continuous martingale in the decomposition of X i.e.

$X_t = X^{mathrm{cm}}_t+ X^{mathrm{dm}}_t + A_t$

where A is of finite variation.

The definition of the quadratic variation process gives rise immediately to the definition of the covariation process can be defined by polarization

$[X,Y]_t := frac{1}{4} left ( [X+Y]_t - [X-Y]_t right )$

## Stochastic integral of simple process

For a sequence of stopping times satisfying $0 le T_1 le T_2 le cdots$, and for each k, Hk an $mathcal{F}_{T_k}$ measurable random variable, then a process H of the form

$H_t = 1_{ {0}}(t) H_0 + sum_k H_k 1_{(T_k, T_{k+1}]}(t)$

is said to be a simple process.

For X an L2 bounded local martingale define the Itô integral $(H cdot X)$ as In mathematical analysis, a real- or complex-valued function of a real variable is square-integrable on an interval if the integral over that interval of the square of its absolute value is finite. ... If we have a map where B is a poset, then f is a bounded function if the image of f, as a subset of B has an upper and lower bound. ... In probability theory, a (discrete-time) martingale is a discrete-time stochastic process (i. ...

$(Hcdot X)_t =sum_k H_k (X_{T_{k+1}wedge t} - X_{T_kwedge t} )$

This process can be proved to be itself an L2 bounded martingale and thus by the usual L2 martingale convergence theorem it is only necessary to consider the limiting process $(H cdot X)_infty$ which is consequently an element of $L^2 (mathcal{F}_infty)$.

## Itô isometry

Given the quadratic-variation process, a seminorm may be introduced on the space of previsible stochastic processes

$|H|^2_X = int H^2_s , d [X]_s$

where the integral is to be understood in the usual lebesgue sense. This is not a norm, since $|H|_X=0$ does not imply that H is the zero process. Let Henri Léon Lebesgue (June 28, 1875 - July 26, 1941) was a French mathematician, most famous for his theory of integration. ...

$L^2(X) = { H mathrm{ previsible such that } |H|_X < infty }$

The Itô isometry between L2(X) and $L^2(mathcal{F}_infty)$ is given by

$| (H cdot X) |^2_2 = mathbb{E}(H cdot X )^2 = | H |^2_X$

This can be shown to hold for simple processes following the definitions above and then via the usual Banach space arguments the isometry allows the definition of the Itô integral to be extended to the space of previsible processes $H in L^2(X)$.

## Semimartingales as integrators

The general Itô integration theory extends naturally to the semimartingales as integrators. For a semimartingale Y which has a Doob-Meyer decomposition

Yt = Mt + At

where M is a local martingale starting from zero, and A is a process of finite variation (this decomposition is unique for continuous process, but not in general). The Itô integral of a previsible process Xwith respect to Y is defined by

$(X cdot Y)_t = (X cdot M)_t + (X cdot A)_t$

where the first integral is defined by the natural extension of the Itô integral from martingale integrators to local martingale integrators, and the second integral is understood in the usual Lebesgue-Stieltjes sense. Because of the non-uniqueness of the semi-martingale decomposition it is necessary to prove that any result holds independently of the decomposition.

## Itô's formula

One of the most powerful and frequently used theorems in Stochastic calculus states that if f is a C2 function from $mathbb{R}^d to R$ and $X_t=(X^{(1)}_t, ldots, X^{(d)}_t)$ is a d-dimensional semimartingale then

 f(Xt) = $f(X_0) + sum_{i=1}^d int_0^t frac{partial f}{partial x_i} (X_{s-}) , d X^{(k)}_s$ $+ frac{1}{2} sum_{i=1}^d sum_{j=1}^d int_0^t frac{partial^2 f}{partial x_ipartial x_j} (X_{s-}) d [X^{(i)}, X^{(j)}]^{mathrm{cm}}_s$ $+ sum_{0 le s le t} Delta f(X_s) - sum_{i=1}^m frac{partial f}{partial x_i}(X_{s-})Delta X^{(i)}_s$

where the continuous martingale part of the quadratic covariation process of two semimartingales X and Y is defined by

$[X,Y]^{mathrm{cm}}_t = [ X , Y ]_t -sum_{s le t} Delta X_s Delta Y_s.$

Results from FactBites:

 Stochastic calculus - TheBestLinks.com - Albert Einstein, Brownian motion, Differential equation, Diffusion, ... (185 words) Stochastic calculus is a branch of mathematics that provides the formal framework and mathematical tools needed for modelling stochastic processes, which are specified through one or more integral and/or differential equations involving both deterministic and random (i.e. The most well-known stochastic process to which stochastic calculus is applied is the Wiener process (named in honor of Norbert Wiener), which is used for modelling Brownian motion as described by Albert Einstein and other physical diffusion processes in space of particles subject to random forces. The main flavours of stochastic calculus are the Ito calculus and the Malliavin calculus.
 INTRODUCTION TO STOCHASTIC CALCULUS WITH APPLICATIONS (393 words) It may be used as a textbook by advanced undergraduates and graduate students in stochastic calculus and financial mathematics. For mathematicians, this book could be a first text on stochastic calculus; it is good companion to more advanced texts by a way of examples and exercises. It shows all readers the applications of stochastic calculus methods and takes readers to the technical level required in research and sophisticated modelling.
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