|
|
| Line 1: |
Line 1: |
| The '''Malliavin calculus''', named after [[Paul Malliavin]], extends the [[calculus of variations]] from functions to [[stochastic processes]]. The Malliavin calculus is also called the '''[[stochastic calculus]] of variations'''. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.
| | BSOD or the Blue Screen of Death, (additionally known as blue screen bodily memory dump), is an error which happens on a Windows program - whenever the computer merely shuts down or automatically reboots. This error may happen just as a computer is booting up or some Windows application is running. When the Windows OS discovers an unrecoverable error it hangs the system or leads to memory dumps.<br><br>Firstly, you need to utilize a Antivirus or safety tool and run a scan on a computer. It can be done that a computer is infected with virus or malware that slows down your computer. If there is nothing found in the scanning report, it might be a RAM which cause the issue.<br><br>H/w connected error handling - when hardware causes BSOD installing newest fixes for the hardware and/ or motherboard could enable. You can moreover add modern hardware that is compatible with all the program.<br><br>Paid registry products found on the other hand, I have found, are usually cheap. They provide regular, free updates or at least inexpensive updates. This follows because the software maker demands to ensure their product is best in staying before its competitors.<br><br>The [http://bestregistrycleanerfix.com/tune-up-utilities tuneup utilities] could come because standard with a back up and restore center. This ought to be an simple to implement process.That signifies that should you encounter a issue with your PC following utilizing a registry cleaning you can just restore a settings.<br><br>The system is made plus built for the purpose of helping we accomplish jobs plus not be pestered by windows XP error messages. When there are mistakes, what do you do? Some individuals pull their hair and cry, while those sane ones have their PC repaired, whilst those actually wise ones analysis to have the errors fixed themselves. No, these errors were not moreover tailored to rob you off your money and time. There are details to do to really prevent this from happening.<br><br>By restoring the state of your system to an earlier date, error 1721 will not appear inside Windows 7, Vista and XP. There is a tool called System Restore that we have to use in this procedure.<br><br>There are numerous businesses that provide the service of troubleshooting a PC each time we call them, all we have to do is signal up with them plus for a little fee, you have a machine always functioning well plus serve you greater. |
| | |
| Malliavin ideas led to a proof that [[Hörmander's condition]] implies the existence and smoothness of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. The calculus has been applied to [[stochastic partial differential equation]]s as well.
| |
| | |
| The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic filtering]].
| |
| | |
| ==Overview and history==
| |
| Paul Malliavin's [[stochastic calculus]] of variations extends the [[calculus of variations]] from functions to [[stochastic processes]]. In particular, it allows the computation of [[derivative]]s of [[random variable]]s.
| |
| | |
| Malliavin invented his calculus to provide a stochastic proof that [[Hörmander's condition]] implies the existence of a [[probability density function|density]] for the solution of a [[stochastic differential equation]]; [[Lars Hörmander|Hörmander]]'s original proof was based on the theory of [[partial differential equation]]s. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to [[stochastic partial differential equation]]s.
| |
| | |
| == Invariance principle ==
| |
| The usual invariance principle for [[Lebesgue integration]] over the whole real line is that, for any real number ε and integrable function ''f'', the
| |
| following holds
| |
| | |
| :<math> \int_{-\infty}^\infty f(x)\, d \lambda(x) = \int_{-\infty}^\infty f(x+\varepsilon)\, d \lambda(x) .</math>
| |
| | |
| This can be used to derive the [[integration by parts]] formula since, setting ''f'' = ''gh'' and differentiating with respect to ε on both sides, it implies
| |
| | |
| :<math> \int_{-\infty}^\infty f' \,d \lambda = \int_{-\infty}^\infty (gh)' \,d \lambda = \int_{-\infty}^\infty g h'\, d \lambda +
| |
| \int_{-\infty}^\infty g' h\, d \lambda.</math>
| |
| | |
| A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let <math>h_s</math> be a square-integrable [[predictable process]] and set
| |
| | |
| :<math> \varphi(t) = \int_0^t h_s\, d s .</math>
| |
| | |
| If <math>X</math> is a [[Wiener process]], the [[Girsanov theorem]] then yields the following analogue of the invariance principle:
| |
| | |
| :<math> E(F(X + \varepsilon\varphi))= E \left [F(X) \exp \left ( \varepsilon\int_0^1 h_s\, d X_s -
| |
| \frac{1}{2}\varepsilon^2 \int_0^1 h_s^2\, ds \right ) \right ].</math>
| |
| | |
| Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:
| |
| | |
| :<math>E(\langle DF(X), \varphi\rangle) = E\Bigl[ F(X) \int_0^1 h_s\, dX_s\Bigr].
| |
| </math> | |
| | |
| Here, the left-hand side is the [[Malliavin derivative]] of the random variable <math>F</math> in the direction <math>\varphi</math> and the integral appearing on the right hand side should be interpreted as an [[Itô integral]]. This expression also remains true (by definition) if <math>h</math> is not adapted, provided that the right hand side is interpreted as a [[Skorokhod integral]].{{Citation needed|date=August 2011}}
| |
| | |
| == Clark-Ocone formula ==
| |
| {{Main|Clark–Ocone theorem}}
| |
| | |
| One of the most useful results from Malliavin calculus is the [[Clark-Ocone theorem]], which allows the process in the [[martingale representation theorem]] to be identified explicitly. A simplified version of this theorem is as follows:
| |
| | |
| For <math>F: C[0,1] \to \R</math> satisfying <math> E(F(X)^2) < \infty</math> which is Lipschitz and such that ''F'' has a strong derivative kernel, in the sense that
| |
| for <math>\varphi</math> in ''C''[0,1]
| |
| | |
| :<math> \lim_{\varepsilon \to 0} (1/\varepsilon)(F(X+\varepsilon \varphi) - F(X) ) = \int_0^1 F'(X,dt) \varphi(t)\ \mathrm{a.e.}\ X</math> | |
| | |
| then
| |
| | |
| :<math>F(X) = E(F(X)) + \int_0^1 H_t \,d X_t ,</math>
| |
| | |
| where ''H'' is the previsible projection of ''F''<nowiki>'</nowiki>(''x'', (''t'',1]) which may be viewed as the derivative of the function ''F'' with respect to a suitable parallel shift of the process ''X'' over the portion (''t'',1] of its domain.
| |
| | |
| This may be more concisely expressed by | |
| | |
| :<math>F(X) = E(F(X))+\int_0^1 E (D_t F | \mathcal{F}_t ) \, d X_t .</math>
| |
| | |
| Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals ''F'' by replacing the derivative kernel used above by the "[[Malliavin derivative]]" denoted <math>D_t</math> in the above statement of the result. {{Citation needed|date=August 2011}}
| |
| | |
| == Skorokhod integral ==
| |
| {{Main|Skorokhod integral}}
| |
| The [[Skorokhod integral]] operator which is conventionally denoted δ is defined as the adjoint of the Malliavin derivative thus for u in the domain of the operator which is a subset of <math>L^2([0,\infty) \times \Omega)</math>, | |
| for '''F''' in the domain of the Malliavin derivative, we require
| |
| | |
| : <math> E (\langle DF, u \rangle ) = E (F \delta (u) ),</math>
| |
| | |
| where the inner product is that on <math>L^2[0,\infty)</math> viz
| |
| | |
| : <math> \langle f, g \rangle = \int_0^\infty f(s) g(s) \, ds.</math>
| |
| | |
| The existence of this adjoint follows from the [[Riesz representation theorem]] for linear operators on [[Hilbert spaces]].
| |
| | |
| It can be shown that if ''u'' is adapted then
| |
| | |
| : <math> \delta(u) = \int_0^\infty u_t\, d W_t ,</math>
| |
| | |
| where the integral is to be understood in the Itô sense. Thus this provides a method of extending the Itô integral to non adapted integrands.
| |
| | |
| ==Applications==
| |
| The calculus allows [[integration by parts]] with [[random variable]]s; this operation is used in [[mathematical finance]] to compute the sensitivities of [[derivative (finance)|financial derivative]]s. The calculus has applications for example in [[stochastic control|stochastic filtering]].
| |
| | |
| {{No footnotes|date=June 2011}}
| |
| | |
| == References ==
| |
| * Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", ''Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto'' 1982, pp 271–306
| |
| * Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", ''J. Faculty Sci. Uni. Tokyo Sect. 1A Math.'', 32 pp 1–76
| |
| * Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", ''J. Faculty Sci. Univ. Tokyo Sect. 1A Math.'', 34 pp 391–442
| |
| * Malliavin, Paul and Thalmaier, Anton. ''Stochastic Calculus of Variations in Mathematical Finance'', Springer 2005, ISBN 3-540-43431-3
| |
| * {{cite book
| |
| | last = Nualart
| |
| | first = David
| |
| | title = The Malliavin calculus and related topics
| |
| | edition = Second edition
| |
| |publisher = Springer-Verlag
| |
| | year = 2006
| |
| | isbn = 978-3-540-28328-7
| |
| }}
| |
| * Bell, Denis. (2007) ''The Malliavin Calculus'', Dover. ISBN 0-486-44994-7
| |
| * Schiller, Alex (2009) [http://www.alexschiller.com/media/Thesis.pdf ''Malliavin Calculus for Monte Carlo Simulation with Financial Applications'']. Thesis, Department of Mathematics, Princeton University
| |
| * [[Bernt Øksendal|Øksendal, Bernt K.]].(1997) [http://www.quantcode.com/modules/wflinks/visit.php?cid=11&lid=4 ''An Introduction To Malliavin Calculus With Applications To Economics'']. Lecture Notes, Dept. of Mathematics, University of Oslo (Zip file containing Thesis and addendum)
| |
| *Di Nunno, Giulia, Øksendal, Bernt, Proske, Frank (2009) "Malliavin Calculus for Lévy Processes with Applications to Finance", Universitext, Springer. ISBN 978-3-540-78571-2
| |
| | |
| == External links ==
| |
| | |
| * {{cite web
| |
| |url = http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf
| |
| |title = An Introduction to Malliavin Calculus
| |
| |accessdate = 2007-07-23
| |
| |last = Friz
| |
| |first = Peter K.
| |
| |date = 2005-04-10
| |
| |format = PDF
| |
| |archiveurl = http://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf |archivedate = 2007-04-17}} Lecture Notes, 43 pages
| |
| | |
| [[Category:Stochastic calculus]]
| |
| [[Category:Integral calculus]]
| |
| [[Category:Mathematical finance]]
| |
| [[Category:Calculus of variations]]
| |
BSOD or the Blue Screen of Death, (additionally known as blue screen bodily memory dump), is an error which happens on a Windows program - whenever the computer merely shuts down or automatically reboots. This error may happen just as a computer is booting up or some Windows application is running. When the Windows OS discovers an unrecoverable error it hangs the system or leads to memory dumps.
Firstly, you need to utilize a Antivirus or safety tool and run a scan on a computer. It can be done that a computer is infected with virus or malware that slows down your computer. If there is nothing found in the scanning report, it might be a RAM which cause the issue.
H/w connected error handling - when hardware causes BSOD installing newest fixes for the hardware and/ or motherboard could enable. You can moreover add modern hardware that is compatible with all the program.
Paid registry products found on the other hand, I have found, are usually cheap. They provide regular, free updates or at least inexpensive updates. This follows because the software maker demands to ensure their product is best in staying before its competitors.
The tuneup utilities could come because standard with a back up and restore center. This ought to be an simple to implement process.That signifies that should you encounter a issue with your PC following utilizing a registry cleaning you can just restore a settings.
The system is made plus built for the purpose of helping we accomplish jobs plus not be pestered by windows XP error messages. When there are mistakes, what do you do? Some individuals pull their hair and cry, while those sane ones have their PC repaired, whilst those actually wise ones analysis to have the errors fixed themselves. No, these errors were not moreover tailored to rob you off your money and time. There are details to do to really prevent this from happening.
By restoring the state of your system to an earlier date, error 1721 will not appear inside Windows 7, Vista and XP. There is a tool called System Restore that we have to use in this procedure.
There are numerous businesses that provide the service of troubleshooting a PC each time we call them, all we have to do is signal up with them plus for a little fee, you have a machine always functioning well plus serve you greater.