Відмінності між версіями «Стохастичне числення Іто»

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The integral of a process ''H'' with respect to another process ''X'' up until a time ''t'' is written as
:<math>\int\limits_0^t H\,dX\equiv\int\limits_0^t H_s\,dX_s</math>
This is itself a stochastic process with time parameter ''t'', which is also written as ''H'' &middot;· ''X''. Alternatively, the integral is often written in differential form ''dY = H dX'', which is equivalent to ''Y - Y<sub>0</sub> = H &middot;· X''.
As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that there is an underlying [[filtration (mathematics)|filtered probability space]].
: <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})</math>
The sigma algebra ''F<sub>t</sub>'' represents the information available up until time ''t'', and a process ''X'' is adapted if ''X<sub>t</sub>'' is ''F<sub>t</sub>''-measurable. A Brownian motion ''B'' is understood to be an ''F<sub>t</sub>''-Brownian motion, which is just a standard Brownian motion with the property that ''B<sub>t+s</sub>&nbsp;— B<sub>t</sub>'' is independent of ''F<sub>t</sub>'' for all ''s, t &ge; 0''.
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The Itō integral can be defined in a manner similar to the [[Riemann-Stieltjes integral]], that is as a [[Convergence_of_random_variables|limit in probability]] of [[Riemann sum]]s; such a limit does not necessarily exist pathwise. Suppose that ''B'' is a [[Wiener process]] (Brownian motion) and that ''H'' is a left-continuous, [[adapted process|adapted]] and locally bounded process.
If &pi;π<sub>''n''</sub> is a sequence of [[Partition_of_an_interval|partition]]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]]
 
: <math>\int\limits_{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>
For some applications, such as [[martingale representation theorem]]s and [[local time (mathematics)|local times]], the integral is needed for processes that are not continuous.
The '''predictable''' processes form the smallest class which is closed under taking limits of sequences and contains all adapted left continuous processes.
If ''H'' is any predictable process such that &int;<sub>0</sub><sup>t</sup> ''H² ds'' < &infin; for every ''t'' &ge; ''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<sub>n</sub>'' of left-continuous, adapted and locally bounded processes, in the sense that
 
An '''Itō process''' is defined to be an [[adapted process|adapted]] stochastic process which can be expressed as the sum of an integral with respect to Brownian motion and an integral with respect to time,
:<math>X_t=X_0+\int\limits_0^t\sigma_s\,dB_s+\int\limits_0^t\mu_s\,ds.</math>
Here, ''B'' is a Brownian motion and it is required that &sigma;σ is a predictable ''B''-integrable process, and &mu;μ is predictable and ([[Lebesgue integration|Lebesgue]]) integrable. That is,
:<math>\int\limits_0^t(\sigma_s^2+|\mu_s|)\,ds<\infty</math>
for each ''t''. The stochastic integral can be extended to such Itō processes,
:<math>\int\limits_0^t H\,dX =\int\limits_0^t H_s\sigma_s\,dB_s + \int\limits_0^t H_s\mu_s\,ds.</math>
This is defined for all locally bounded and predictable integrands. More generally, it is required that ''H''&nbsp;&sigma;σ be ''B''-integrable and ''H''&nbsp;&mu;μ be Lebesgue integrable, so that &int;<sub>0</sub><sup>''t''</sup>(''H''²&sigma;σ²&nbsp;+&nbsp;|''H''&nbsp;&mu;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]]. In its simplest form, for any twice continuously differentiable function ''f'' on the reals and Itō process ''X'' as described above, it states that ''f''(''X'') is itself an Itō process satisfying
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The Itō integral is defined with respect to a [[semimartingale]] ''X''. These are processes which can be decomposed as ''X'' = ''M'' + ''A'' for a local martingale ''M'' and [[bounded variation|finite variation]] process ''A''. Important examples of such processes include [[Wiener process|Brownian motion]], which is a martingale, and [[Lévy process]]es.
For a left continuous, locally bounded and adapted process ''H'' the integral ''H'' &middot;· ''X'' exists, and can be calculated as a limit of Riemann sums. Let &pi;π<sub>''n''</sub> be a sequence of [[Partition_of_an_interval|partition]]s of [0,''t''] with mesh going to zero,
 
: <math>\int\limits_0^t H\,dX = \lim_{n\rightarrow\infty} \sum_{t_{i-1},t_i\in\pi_n}H_{t_{i-1}}(X_{t_i}-X_{t_{i-1}}).</math>
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_theorem|Girsanov's theorem]], and for the study of [[stochastic differential equation]]s. However, it is inadequate for other important topics such as [[martingale representation theorem]]s and [[local time (mathematics)|local times]].
 
The integral extends to all predictable and locally bounded integrands, in a unique way, such that the [[Dominated convergence theorem|dominated convergence]] theorem holds. That is, if ''H<sub>n</sub>'' &rarr; ''H'' and |''H<sub>n</sub>''| &le; ''J'' for a locally bounded process ''J'', then &int;<sub>''0''</sub><sup>''t''</sup> ''H<sub>n</sub> dX'' &rarr; &int;<sub>''0''</sub><sup>''t''</sup> ''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'' &middot;· ''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''&nbsp;&middot;·&nbsp;''X'' = ''Y'', if and only if ''Y<sub>0</sub>'' = ''0'' and ''K'' &middot;· ''Y'' = ''(KH)'' &middot;· ''X''. The set of ''X''-integrable processes is denoted by L(''X'').
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* 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 ''t'' is ''X<sub>t</sub>&nbsp;— X<sub>t-</sub>'', and is often denoted by &Delta;Δ''X<sub>t</sub>''. With this notation, &Delta;Δ''(H'' &middot;· ''X)=H'' &Delta;Δ''X''. A particular consequence of this is that integrals with respect to a continuous process are always themselves continuous.
 
* '''[[Associativity]]'''. Let ''J'', ''K'' be predictable processes, and ''K'' be ''X''-integrable. Then, ''J'' is ''K'' &middot;· ''X'' integrable if and only if ''JK'' is ''X'' integrable, in which case
:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math>
 
* '''[[Dominated convergence theorem|Dominated convergence]]'''. Suppose that ''H<sub>n</sub>'' &rarr; ''H'' and ''|H<sub>n</sub>|'' &le; ''J'', where ''J'' is an ''X''-integrable process. then ''H<sub>n</sub>'' &middot;· ''X'' &rarr; ''H'' &middot;· ''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'' &middot;· ''X,Y''] = ''H'' &middot;· [''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,
:: <math>[H\cdot X]=H^2\cdot[X]</math>
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=== Локальні мартингали ===
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An important property of the Itō integral is that it preserves the [[local martingale]] property. If ''M'' is a local martingale and ''H'' is a locally bounded predictable process then ''H'' &middot;· ''M'' is also a local martingale.
For integrands which are not locally bounded, there are examples where ''H'' &middot;· ''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;<sub>''0''</sub><sup>''t''</sup>''H² d''[''M''] is finite for each ''t'', and ''H'' &middot;· ''M'' is always a local martingale.
 
The most general statement for a discontinuous local martingale ''M'' is that if (''H²'' &middot;· [''M''])<sup>''1/2''</sup> is [[Stopping time#Localization|locally integrable]] then ''H'' &middot;· ''M'' exists and is a local martingale.
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For any such square integrable martingale ''M'', the quadratic variation process [''M''] is integrable, and the '''Itō isometry''' states that
: <math>\mathbb{E}\left((H\cdot M_t)^2\right)=\mathbb{E}\left(\int\limits_0^t H^2\,d[M]\right).</math>
This equality holds more generally for any martingale ''M'' such that ''H''²&nbsp;&middot;·&nbsp;[''M'']<sub>''t''</sub> is integrable. The Itō isometry is often used as an important step in the construction of the stochastic integral, by defining ''H''&nbsp;&middot;·&nbsp;''M'' to be the unique extension of this isometry from a certain class of simple integrands to all bounded and predictable processes.
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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 cadlag process ''M'' is written as ''M''<sub>''t''</sub><sup>*</sup> = sup<sub>''s''&nbsp;&le;''t''</sub>&nbsp;|''M<sub>s</sub>''|. For any ''p''&nbsp;&ge;&nbsp;1 and bounded predictable integrand, the stochastic integral preserves the space of cadlag martingales ''M'' such that E((''M''<sub>''t''</sub><sup>*</sup>)<sup>''p''</sup>) is finite for all ''t''.
If ''p''&nbsp;>&nbsp;1 then this is the same as the space of ''p''-integrable martingales, by [[Doob's martingale inequality|Doob's inequalities]].
 
The '''Burkholder-Davis-Gundy inequalities''' state that, for any given ''p'' &ge; 1, there exists positive constants ''c,C'' such that
: <math>c\mathbb{E}([M]_t^{p/2})\le \mathbb{E}((M^*_t)^p)\le C\mathbb{E}([M]_t^{p/2})</math>
for all cadlag local martingales ''M''.
These are used to show that if (''M''<sub>''t''</sub><sup>*</sup>)<sup>p</sup> is integrable and ''H'' is a bounded predictable process then
: <math>\mathbb{E}(((H\cdot M)_t^*)^p) \le C\mathbb{E}((H^2\cdot[M]_t)^{p/2})<\infty</math>
and, consequently, ''H''&nbsp;&middot;·&nbsp;''M'' is a ''p''-integrable martingale. More generally, this statement is true whenever (''H''²&nbsp;&middot;·&nbsp;[''M''])<sup>''p''/2</sup> is integrable.
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* Hagen Kleinert, ''Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets'', 4th edition, World Scientific (Singapore), [[2004]], (ISBN 981-238-107-4). Пятое издание доступно в виде [http://www.physik.fu-berlin.de/~kleinert/b5 pdf].
* He Sheng-Wu, Wang Jia-Gang, Yan Jia-An, ''Semimartingale Theory and Stochastic Calculus'', Science Press, CRC Press Inc., [[1992]] (ISBN 7-03-003066-4, 0-8493-7715-3)
* Ioannis Karatzas and Steven E. Shreve, ''Brownian Motion and Stochastic Calculus'', Springer, [[1991 ]] г. (ISBN 0-387-97655-8)
* Philip E. Protter, ''Stochastic Integration and Differential Equations'', Springer, 2001 (ISBN 3-540-00313-4)
* Bernt K. Øksendal, ''Stochastic Differential Equations: An Introduction with Applications'', Springer, [[2003]] (ISBN 3-540-04758-1)
* Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators.
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[[Категорія:Теорія випадкових процесів]]
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[[en:Itō calculus]]
[[ru:Стохастическое исчисление Ито]]
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[[zh:伊藤积分]]
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