Стохастичне числення Іто: відмінності між версіями
[неперевірена версія] | [неперевірена версія] |
м r2.7.1) (робот змінив: zh:伊藤微积分 |
м r2.7.2+) (робот змінив: zh:伊藤积分; косметичні зміни |
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The integral of a process ''H'' with respect to another process ''X'' up until a time ''t'' is written as |
The integral of a process ''H'' with respect to another process ''X'' up until a time ''t'' is written as |
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:<math>\int\limits_0^t H\,dX\equiv\int\limits_0^t H_s\,dX_s</math> |
:<math>\int\limits_0^t H\,dX\equiv\int\limits_0^t H_s\,dX_s</math> |
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This is itself a stochastic process with time parameter ''t'', which is also written as ''H'' |
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''. |
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As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that there is an underlying [[filtration (mathematics)|filtered probability space]]. |
As Itō calculus is concerned with continuous-time stochastic processes, it is assumed that there is an underlying [[filtration (mathematics)|filtered probability space]]. |
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: <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})</math> |
: <math>(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})</math> |
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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> — B<sub>t</sub>'' is independent of ''F<sub>t</sub>'' for all ''s, t |
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> — B<sub>t</sub>'' is independent of ''F<sub>t</sub>'' for all ''s, t ≥ 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. |
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. |
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If |
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]] |
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: <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> |
: <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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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. |
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. |
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The '''predictable''' processes form the smallest class which is closed under taking limits of sequences and contains all adapted left continuous processes. |
The '''predictable''' processes form the smallest class which is closed under taking limits of sequences and contains all adapted left continuous processes. |
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If ''H'' is any predictable process such that |
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. |
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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 |
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 |
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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, |
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, |
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:<math>X_t=X_0+\int\limits_0^t\sigma_s\,dB_s+\int\limits_0^t\mu_s\,ds.</math> |
:<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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Here, ''B'' is a Brownian motion and it is required that |
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, |
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:<math>\int\limits_0^t(\sigma_s^2+|\mu_s|)\,ds<\infty</math> |
:<math>\int\limits_0^t(\sigma_s^2+|\mu_s|)\,ds<\infty</math> |
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for each ''t''. The stochastic integral can be extended to such Itō processes, |
for each ''t''. The stochastic integral can be extended to such Itō processes, |
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:<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> |
:<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> |
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This is defined for all locally bounded and predictable integrands. More generally, it is required that ''H'' |
This is defined for all locally bounded and predictable integrands. More generally, it is required that ''H'' σ be ''B''-integrable and ''H'' μ be Lebesgue integrable, so that ∫<sub>0</sub><sup>''t''</sup>(''H''²σ² + |''H'' μ|)''ds'' is finite. Such predictable processes ''H'' are called ''X''-integrable. |
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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 |
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. |
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. |
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For a left continuous, locally bounded and adapted process ''H'' the integral ''H'' |
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, |
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: <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> |
: <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 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 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]]. |
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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>'' |
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. |
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The uniqueness of the extension from left-continuous to predictable integrands is a result of the [[monotone class lemma]]. |
The uniqueness of the extension from left-continuous to predictable integrands is a result of the [[monotone class lemma]]. |
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In general, the stochastic integral ''H'' |
In general, the stochastic integral ''H'' · ''X'' can be defined even in cases where the predictable process ''H'' is not locally bounded. |
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If ''K'' = 1 / (1 + ''|H|'') then ''K'' and ''KH'' are bounded. Associativity of stochastic integration implies that ''H'' is ''X''-integrable, with integral ''H'' |
If ''K'' = 1 / (1 + ''|H|'') then ''K'' and ''KH'' are bounded. Associativity of stochastic integration implies that ''H'' is ''X''-integrable, with integral ''H'' · ''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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* The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. |
* The stochastic integral is a càdlàg process. Furthermore, it is a semimartingale. |
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* 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> — X<sub>t-</sub>'', and is often denoted by |
* 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> — 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. |
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* '''[[Associativity]]'''. Let ''J'', ''K'' be predictable processes, and ''K'' be ''X''-integrable. Then, ''J'' is ''K'' |
* '''[[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 |
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:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math> |
:: <math> J\cdot (K\cdot X) = (JK)\cdot X</math> |
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* '''[[Dominated convergence theorem|Dominated convergence]]'''. Suppose that ''H<sub>n</sub>'' |
* '''[[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. |
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* 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'' |
* 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, |
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:: <math>[H\cdot X]=H^2\cdot[X]</math> |
:: <math>[H\cdot X]=H^2\cdot[X]</math> |
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=== Локальні мартингали === |
=== 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'' |
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. |
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For integrands which are not locally bounded, there are examples where ''H'' |
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. |
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If ''M'' is a continuous local martingale then a predictable process ''H'' is ''M''-integrable if and only if |
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. |
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The most general statement for a discontinuous local martingale ''M'' is that if (''H²'' |
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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For any such square integrable martingale ''M'', the quadratic variation process [''M''] is integrable, and the '''Itō isometry''' states that |
For any such square integrable martingale ''M'', the quadratic variation process [''M''] is integrable, and the '''Itō isometry''' states that |
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: <math>\mathbb{E}\left((H\cdot M_t)^2\right)=\mathbb{E}\left(\int\limits_0^t H^2\,d[M]\right).</math> |
: <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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This equality holds more generally for any martingale ''M'' such that ''H''² |
This equality holds more generally for any martingale ''M'' such that ''H''² · [''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'' · ''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. |
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. |
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The maximum process of a cadlag process ''M'' is written as ''M''<sub>''t''</sub><sup>*</sup> = sup<sub>''s'' |
The maximum process of a cadlag process ''M'' is written as ''M''<sub>''t''</sub><sup>*</sup> = sup<sub>''s'' ≤''t''</sub> |''M<sub>s</sub>''|. For any ''p'' ≥ 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''. |
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If ''p'' > 1 then this is the same as the space of ''p''-integrable martingales, by [[Doob's martingale inequality|Doob's inequalities]]. |
If ''p'' > 1 then this is the same as the space of ''p''-integrable martingales, by [[Doob's martingale inequality|Doob's inequalities]]. |
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The '''Burkholder-Davis-Gundy inequalities''' state that, for any given ''p'' |
The '''Burkholder-Davis-Gundy inequalities''' state that, for any given ''p'' ≥ 1, there exists positive constants ''c,C'' such that |
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: <math>c\mathbb{E}([M]_t^{p/2})\le \mathbb{E}((M^*_t)^p)\le C\mathbb{E}([M]_t^{p/2})</math> |
: <math>c\mathbb{E}([M]_t^{p/2})\le \mathbb{E}((M^*_t)^p)\le C\mathbb{E}([M]_t^{p/2})</math> |
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for all cadlag local martingales ''M''. |
for all cadlag local martingales ''M''. |
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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 |
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 |
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: <math>\mathbb{E}(((H\cdot M)_t^*)^p) \le C\mathbb{E}((H^2\cdot[M]_t)^{p/2})<\infty</math> |
: <math>\mathbb{E}(((H\cdot M)_t^*)^p) \le C\mathbb{E}((H^2\cdot[M]_t)^{p/2})<\infty</math> |
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and, consequently, ''H'' |
and, consequently, ''H'' · ''M'' is a ''p''-integrable martingale. More generally, this statement is true whenever (''H''² · [''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]. |
* 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]. |
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* 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) |
* 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) |
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* Ioannis Karatzas and Steven E. Shreve, ''Brownian Motion and Stochastic Calculus'', Springer, [[1991 |
* Ioannis Karatzas and Steven E. Shreve, ''Brownian Motion and Stochastic Calculus'', Springer, [[1991]] г. (ISBN 0-387-97655-8) |
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* Philip E. Protter, ''Stochastic Integration and Differential Equations'', Springer, 2001 (ISBN 3-540-00313-4) |
* Philip E. Protter, ''Stochastic Integration and Differential Equations'', Springer, 2001 (ISBN 3-540-00313-4) |
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* Bernt K. Øksendal, ''Stochastic Differential Equations: An Introduction with Applications'', Springer, [[2003]] (ISBN 3-540-04758-1) |
* Bernt K. Øksendal, ''Stochastic Differential Equations: An Introduction with Applications'', Springer, [[2003]] (ISBN 3-540-04758-1) |
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* Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators. |
* Mathematical Finance Programming in TI-Basic, which implements Ito calculus for TI-calculators. |
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[[Категорія:Теорія випадкових процесів]] |
[[Категорія:Теорія випадкових процесів]] |
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[[Категорія:Теорія ймовірностей]] |
[[Категорія:Теорія ймовірностей]] |
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[[en:Itō calculus]] |
[[en:Itō calculus]] |
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[[ru:Стохастическое исчисление Ито]] |
[[ru:Стохастическое исчисление Ито]] |
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[[sv:Itōprocess]] |
[[sv:Itōprocess]] |
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[[zh:伊藤 |
[[zh:伊藤积分]] |
Версія за 01:11, 14 квітня 2012
Числення Іто — математична теорія, що описує методи маніпулювання з випадковими процесами, такими як броунівський рух (або вінерівський процес). Названа на честь творця, японського математика Кійосі Іто. Часто застосовується в фінансовій математиці і теорії стохастичних диференціальних рівнянь. Центральним поняттям цієї теорії є інтеграл Іто
де — 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 варіацію на будь-якому часовому інтервалі. Таким чином, інтеграл Іто не може бути визначений у сенсі інтеграла Рімана — Стілтьєса. Проте, інтеграл Іто можна визначити строго, якщо помітити, що підінтегральна функція є адитивним процесом; це означає, що залежність від часу його середнього значення визначається поведінкою тільки до моменту .
Позначення
Інтегрування броунівського руху
Процес Іто
Семімартингали, як інтегратори
Властивості
Інтегрування частинами
Лема Іто
Мартингали-інтегратори
Локальні мартингали
Квадратично інтегровні мартингали
p-інтегральні мартингали
Стохастична похідна
- and
Див. також
Посилання
- Стохастический мир — простое введение в стохастические дифференциальные уравнения
Література
- Allouba, Hassan (2006). A Differentiation Theory for Itô's Calculus. Stochastic Analysis and Applications. 24: 367—380. 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 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.
Ця стаття містить неперекладені фрагменти іноземною мовою. |