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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 modeling 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. 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. QUADRATIC-VARIATION PROCESS The key to the construction of a stochastic integral is the definition of a Quadratic-variation Process ; the quadratic variation of a general $L^2$ (see L2 Space ) bounded Martingale $X\_t$ may be defined as the increasing process $${Link without Title} _t such that :(i)$${Link without Title} _0 = 0 :(ii)$\backslash Delta${Link without Title} _t = (\Delta X_t )^2 \quad orall t :(iii)$X\_t^2\; -${Link without Title} _t is a uniformly integrable 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 $X\_t$ with continuous sample paths it may be shown to be equivalent to the following definition for a partition : whose mesh is defined by

where $X^\{\backslash mathrm\{cm\}\}$ is the canonical continuous martingale in the decomposition of $X$ i.e.

:$X\_t\; =\; X^\{\backslash mathrm\{cm\}\}\_t+\; X^\{\backslash mathrm\{dm\}\}\_t\; +\; A\_t$

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

:


STOCHASTIC INTEGRAL OF SIMPLE PROCESS


For a sequence of stopping times satisfying
$0\; \backslash le\; T\_1\; \backslash le\; T\_2\; \backslash le\; \backslash cdots$, and for each $k$, $H\_k$ an
$\backslash mathcal\{F\}\_\{T\_k\}$ measurable random variable, then
a process $H$ of the form

:$H\_t\; =\; 1\_\{\; \backslash \{0\backslash \}\}(t)\; H\_0\; +\; \backslash sum\_k\; H\_k\; 1\_\{(T\_k,\; T\_\{k+1\}]\}(t)$

is said to be a simple process.

For $X$ an ''L''² Bounded local Martingale define the Itô
integral $(H\; \backslash cdot\; X)$ as

:$(H\backslash cdot\; X)\_t\; =\backslash sum\_k\; H\_k\; (X\_\{T\_\{k+1\}\backslash wedge\; t\}\; -\; X\_\{T\_k\backslash wedge\; t\}\; )$

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


ITô ISOMETRY


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

  Not A Norm, Since <math>\H\ X 0</math> does not imply that <math>H</math>
  :<math> L^2(X) \{ H \mathrm{\ previsible\ such\ that\ } \H\_X < \infty \}</math>
  :<math> \ (H \cdot X) \^2 2 \mathbb{E}(H \cdot X )^2 = \ H \^2_X</math>
  <math>+ \sum {0 \le S \le T} \Delta F(X S) - \sum {i 1}^m rac{\partial f}{\partial x_i}(X_{s-})\Delta X^{(i)}_s </math>




\Delta Y_s.


DISCONTINUOUS PROCESS

It might appear that the Itô integral defined here could be extended to discontinuous martingales by decomposing a square-integrable martingale ''X'' in the form

:$X\_t\; =\; X^\{\backslash mathrm\{cm\}\}\_t\; +\; V\_t$

where $X^\{\backslash mathrm\{cm\}\}$ is a square-integrable continuous martingale and ''V'' is a process
of integrable variation. Unfortunately this is decomposition is not always possible, but the space of square-integrable processes of integrable variation is dense in the orthogonal complement of the space of continuous square-integrable martingales.


EXTERNAL LINKS

• Notes on Stochastic Calculus A short elementary description of the basic Itô integral.