Відмінності між версіями «Стохастичне числення Іто»

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'''Числення Іто'''  — математична теорія, що описує методи маніпулювання з випадковими процесами, такими як [[броунівський рух]] (або [[вінерівський процес]]). Названа на честь творця, японського математика [[Іто Кійоси|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 більш загальному 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 Рімана — 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.
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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
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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== 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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== 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,
 
== Семімартингали, как 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>
 
 
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’sGirsanov'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.
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== Властивості ==
 
:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math>
 
* 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;''X<sub>t</sub>''. With this notation, &Delta;''(H'' &middot; ''X)=H'' &Delta;''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
 
== 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
 
== 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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=== 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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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’sDoob's inequalities]].
 
The '''Burkholder-Davis-Gundy inequalities''' state that, for any given ''p'' &ge; 1, there exists positive constants ''c,C'' such that
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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}^S_{t}= X_\frac{s} \mathrm{d} B_{s}\langle = X_{t}S,</math> B &nbsp; and &nbsp; <math>\int\limits_{0}^rangle_{t} \mathbb{D}_{B_\mathrm{s}d} S_\langle B, B \rangle_{st}} =\frac{\mathrm{d} B_{s}=\langle S_S, B \rangle_{t} - S_}{0\mathrm{d} - V_{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
 
== Див. також ==
 
* [[Вінерівський процес]]
* [[інтеграл Стратоновича]]
 
== Посилання ==
 
* [http://synset.com/ru/Стохастический_мир Стохастический мир] &nbsp;— простое введение в стохастические дифференциальные уравнения
 
== Література ==