Перегляд вихідного коду сторінки Стохастичне числення Іто

'''Числення Іто'''&nbsp;— математична теорія, що описує методи маніпулювання з [[Випадковий процес|випадковими процесами]], такими як [[броунівський рух]] (або [[вінерівський процес]]). Названа на честь творця, японського математика [[Іто Кійосі|Кійосі Іто]]. Часто застосовується в [[фінансова математика|фінансовій математиці]] і теорії [[стохастичне диференціальне рівняння|стохастичних диференціальних рівнянь]]. Центральним поняттям цієї теорії є [[інтеграл]] Іто
: <math>Y_t=\int\limits_0^t H_s\,dX_s,</math>
де <math>X</math>&nbsp;— броунівський рух або, в загальнішому формулюванні, [[напівмартингал]].
Можна показати, що шлях інтегрування для броунівського руху не можна описати стандартними техніками інтегрального числення. Зокрема, броунівський рух не є інтегрованою функцією в кожній точці шляху і має нескінченну [[Варіація функції|варіацію]] на будь-якому часовому інтервалі. Таким чином, інтеграл Іто не може бути визначений у сенсі [[Інтеграл Стілтьєса|інтеграла Рімана&nbsp;— Стілтьєса]]. Проте, інтеграл Іто можна визначити строго, якщо помітити, що підінтегральна функція <math>H</math> є адитивним процесом; це означає, що залежність від часу <math>t</math> його середнього значення визначається поведінкою тільки до моменту <math>t</math>.
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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]] (see [[Black-Scholes]]). Then, the Itō stochastic integral represents the payoff of a continuous-time trading strategy consisting of holding an amount ''H<sub>t</sub>'' 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 stops unlimited gains from being possible by 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]]s.

Important results of Itō calculus include the integration by parts formula and [[Itō's lemma]], which is a [[Integration by substitution|change of variables]] formula. These differ from the formulas of standard calculus, due to [[quadratic variation]] terms.
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== Позначення ==

: <math>\int\limits_0^t H\,dX\equiv\int\limits_0^t H_s\,dX_s</math>
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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'' · ''X''. Alternatively, the integral is often written in differential form ''dY = H dX'', which is equivalent to ''Y - Y<sub>0</sub> = H · 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 ≥ 0''.
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== Інтегрування броунівського руху ==

: <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>
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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 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>

It can be shown that this limit [[Convergence_of_random_variables|converges in probability]].

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 ∫<sub>0</sub><sup>t</sup> ''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<sub>n</sub>'' of left-continuous, adapted and locally bounded processes, in the sense that

: <math> \int\limits_0^t (H-H_n)^2\,ds\rightarrow 0</math>

in probability. Then, the Itō integral is

: <math>\int\limits_0^t H\,dB = \lim_{n\rightarrow\infty}\int\limits_0^t H_n\,dB</math>

where, again, the limit can be shown to converge in probability.
The stochastic integral satisfies the [[Itō isometry]]
: <math>\mathbb{E}\left( (H\cdot B_t)^2\right)=\mathbb{E} \left (\int\limits_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.
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== Процес Іто ==

: <math>X_t=X_0+\int\limits_0^t\sigma_s\,dB_s+\int\limits_0^t\mu_s\,ds.</math>
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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 σ is a predictable ''B''-integrable process, and 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;σ be ''B''-integrable and ''H''&nbsp;μ be Lebesgue integrable, so that ∫<sub>0</sub><sup>''t''</sup>(''H''²σ²&nbsp;+&nbsp;|''H''&nbsp;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
: <math>df(X_t)=f^\prime(X_t)\,dX_t + \frac{1}{2}f^{\prime\prime}(X_t)\sigma_t^2\,dt.</math>
This is the stochastic calculus version of the [[integration by substitution|change of variables]] formula and [[chain rule]]. 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]].
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== Семімартингали, як інтегратори ==

: <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>

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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'' · ''X'' exists, and can be calculated as a limit of Riemann sums. Let 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>

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_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>'' → ''H'' and |''H<sub>n</sub>''| ≤ ''J'' for a locally bounded process ''J'', then ∫<sub>''0''</sub><sup>''t''</sup> ''H<sub>n</sub> dX'' → ∫<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'' · ''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;·&nbsp;''X'' = ''Y'', if and only if ''Y<sub>0</sub>'' = ''0'' and ''K'' · ''Y'' = ''(KH)'' · ''X''. The set of ''X''-integrable processes is denoted by L(''X'').
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== Властивості ==

:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math>

:: <math>[H\cdot X]=H^2\cdot[X]</math>

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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 Δ''X<sub>t</sub>''. 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]]'''. 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
:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math>

* '''[[Dominated convergence theorem|Dominated convergence]]'''. Suppose that ''H<sub>n</sub>'' → ''H'' and ''|H<sub>n</sub>|'' ≤ ''J'', where ''J'' is an ''X''-integrable process. then ''H<sub>n</sub>'' · ''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,
:: <math>[H\cdot X]=H^2\cdot[X]</math>
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== Інтегрування частинами ==

: <math>X_tY_t = X_0Y_0+\int\limits_0^t X_{s-}\,dY_s + \int\limits_0^t Y_{s-}\,dX_s + [X,Y]_t</math>
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As with ordinary calculus, [[integration by parts]] 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 variation|quadratic covariation]] 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
:<math>X_tY_t = X_0Y_0+\int\limits_0^t X_{s-}\,dY_s + \int\limits_0^t Y_{s-}\,dX_s + [X,Y]_t</math>
where [X,Y] is the quadratic covariation process.
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== Лема Іто ==

: <math>df(X_t)= \sum_{i=1}^d f_{,i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^d f_{,ij}(X_{t})\,d[X^i,X^j]_t.</math>
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{{main|Itō's lemma}}

Itō's lemma is the version of the [[chain rule]] or [[integration by substitution|change of variables]] 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'' = (''X''<sup>1</sup>,…,''X''<sup>''d''</sup>) and twice continuously differentiable function ''f'' from '''R'''<sup>''d''</sup> to '''R''', it states that ''f''(''X'') is a semimartingale and,
: <math>df(X_t)= \sum_{i=1}^d f_{,i}(X_t)\,dX^i_t + \frac{1}{2}\sum_{i,j=1}^d f_{,ij}(X_{t})\,d[X^i,X^j]_t.</math>
This differs from the chain rule used in standard calculus due to the term involving the quadratic covariation [''X''<sup>''i''</sup>,''X''<sup>''j''</sup>&nbsp;]. 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]]).
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== Мартингали-інтегратори ==

=== 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'' · ''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 ∫<sub>''0''</sub><sup>''t''</sup>''H² 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''])<sup>''1/2''</sup> is [[Stopping time#Localization|locally integrable]] then ''H'' · ''M'' exists and is a local martingale.
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=== Квадратично інтегровні мартингали ===

: <math>\mathbb{E}\left((H\cdot M_t)^2\right)=\mathbb{E}\left(\int\limits_0^t H^2\,d[M]\right).</math>
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For bounded integrands, the Itō stochastic integral preserves the space of ''square integrable'' martingales, which is the set of [[càdlàg]] martingales ''M'' such that E(''M<sub>t</sub>''²) 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
: <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;·&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;·&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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=== ''p''-інтегральні мартингали ===
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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<sub>t</sub>''|<sup>''p''</sup>) 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 cadlag process ''M'' is written as ''M''<sub>''t''</sub><sup>*</sup> = sup<sub>''s''&nbsp;≤''t''</sub>&nbsp;|''M<sub>s</sub>''|. For any ''p''&nbsp;≥&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'' ≥ 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;·&nbsp;''M'' is a ''p''-integrable martingale. More generally, this statement is true whenever (''H''²&nbsp;·&nbsp;[''M''])<sup>''p''/2</sup> is integrable.
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== Стохастична похідна ==

: <math>\mathbb{D}_{B_{t}} S_{t}= \frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} \langle B, B \rangle_{t}}  =\frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} t},</math>

: <math>\mathbb{D}_{B_{t}} \int\limits_{0}^{t} X_{s} \mathrm{d} B_{s} = X_{t},</math> &nbsp; and &nbsp; <math>\int\limits_{0}^{t} \mathbb{D}_{B_{s}} S_{s} \mathrm{d} B_{s}= S_{t} - S_{0} - V_{t}.</math>
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Itō calculus, as ground-breaking and remarkable as it is, for over 60 years has only been an integral calculus: there was no explicit pathwise differentiation theory behind it. However, in 2004 (published in 2006) Hassan Allouba defined the derivative of a given [[semimartingale]] ''S'' with respect to a Brownian motion ''B'' using the derivative of the covariation of ''S'' and ''B'' (also known as the cross-variation of ''S'' and ''B'') with respect to the quadratic variation of ''B''.  Given a continuous semimartingale <math>S_{t} = S_{0} + V_{t} + M_{t},</math> where ''V'' is a process of [[bounded variation]] on compacts and ''M'' is a local martingale, the (strong) derivative of ''S'' with respect to a Brownian motion ''B'' is defined as the stochastic process <math>\mathbb{D}_{B} S</math> given by

:<math>\mathbb{D}_{B_{t}} S_{t}= \frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} \langle B, B \rangle_{t}}  =\frac{\mathrm{d} \langle S, B \rangle_{t}}{\mathrm{d} t},</math>

where we used the fact that the covariation of Brownian motion ''B'' with itself is just its quadratic variation, which is ''t''.

This stochastic derivative turns out to have many of the properties of the usual derivative of elementary calculus.  It leads to a fundamental theorem of stochastic calculus 
for this stochastic derivative/integral pair:

:<math>\mathbb{D}_{B_{t}} \int\limits_{0}^{t} X_{s} \mathrm{d} B_{s} = X_{t},</math>  &nbsp;  and &nbsp;   <math>\int\limits_{0}^{t} \mathbb{D}_{B_{s}} S_{s} \mathrm{d} B_{s}= S_{t} - S_{0} - V_{t}.</math>

It also leads to a stochastic [[mean value theorem]], stochastic chain rules,  as well as other differentiation rules that are similar to those in elementary calculus.  A key difference is that where an [[indefinite integral]] (anti-derivative) in the usual elementary calculus sense is determined only up to an additive constant of integration, an indefinite integral in this stochastic calculus is determined only up to a process of bounded variation on compacts. These processes are the "constants" in this stochastic differentiation theory.   Also, if ''M'' and ''B'' are orthogonal (zero covariation) then  <math>\mathbb{D}_{B} M=0</math>; in particular, if ''M'' and ''B'' are independent then <math>\mathbb{D}_{B} M=0.</math>
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== Див. також ==

* [[Вінерівський процес]]
* [[інтеграл Стратоновича]]

== Посилання ==

* [http://synset.com/ru/Стохастический_мир Стохастический мир]&nbsp;— просте введення в стохастичне диференційне рівняння

== Література ==

* {{cite journal | author=Allouba, Hassan | title=A Differentiation Theory for Itô's Calculus | journal=Stochastic Analysis and Applications | year=2006 | volume=24 | pages=367-380 | id=DOI 10.1080/07362990500522411}}
* 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:Стохастичні диференціальні рівняння]]
[[Категорія:Теорія випадкових процесів]]
[[Категорія:Теорія ймовірностей]]