What is the branch of calculus developed by  Paul Malliavin 
https://en.wikipedia.org/wiki/Malliavin_calculus
Malliavin calculus - Wikipedia
2004-07-17
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	<p>From Wikipedia, the free encyclopedia</p>
	</div><div>
<p>In probability theory and related fields, <b>Malliavin calculus</b> is a set of mathematical techniques and ideas that extend the mathematical field of <a href="/wiki/Calculus_of_variations">calculus of variations</a> from deterministic functions to <a href="/wiki/Stochastic_processes">stochastic processes</a>. In particular, it allows the computation of <a href="/wiki/Derivative">derivatives</a> of <a href="/wiki/Random_variable">random variables</a>. Malliavin calculus is also called the <b><a href="/wiki/Stochastic_calculus">stochastic calculus</a> of variations</b>. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations.
</p><p>Malliavin calculus is named after <a href="/wiki/Paul_Malliavin">Paul Malliavin</a> whose ideas led to a proof that <a href="/wiki/H%C3%B6rmander%27s_condition">Hörmander's condition</a> implies the existence and smoothness of a <a href="/wiki/Probability_density_function">density</a> for the solution of a <a href="/wiki/Stochastic_differential_equation">stochastic differential equation</a>; <a href="/wiki/Lars_H%C3%B6rmander">Hörmander</a>'s original proof was based on the theory of <a href="/wiki/Partial_differential_equation">partial differential equations</a>. The calculus has been applied to <a href="/wiki/Stochastic_partial_differential_equation">stochastic partial differential equations</a> as well.
</p><p>The calculus allows <a href="/wiki/Integration_by_parts">integration by parts</a> with <a href="/wiki/Random_variable">random variables</a>; this operation is used in <a href="/wiki/Mathematical_finance">mathematical finance</a> to compute the sensitivities of <a href="/wiki/Derivative_(finance)">financial derivatives</a>. The calculus has applications in, for example, <a href="/wiki/Stochastic_filtering">stochastic filtering</a>.
</p>
<h2>Overview and history[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=1">edit</a>]</h2>
<p>Malliavin introduced Malliavin calculus to provide a stochastic proof that <a href="/wiki/H%C3%B6rmander%27s_condition">Hörmander's condition</a> implies the existence of a <a href="/wiki/Probability_density_function">density</a> for the solution of a <a href="/wiki/Stochastic_differential_equation">stochastic differential equation</a>; <a href="/wiki/Lars_H%C3%B6rmander">Hörmander</a>'s original proof was based on the theory of <a href="/wiki/Partial_differential_equation">partial differential equations</a>. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to <a href="/wiki/Stochastic_partial_differential_equation">stochastic partial differential equations</a>.
</p>
<h2>Invariance principle[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=2">edit</a>]</h2>
<p>The usual invariance principle for <a href="/wiki/Lebesgue_integration">Lebesgue integration</a> over the whole real line is that, for any real number ε and integrable function <i>f</i>, the
following holds
</p>
<dl><dd> and hence </dd></dl>
<p>This can be used to derive the <a href="/wiki/Integration_by_parts">integration by parts</a> formula since, setting <i>f</i> = <i>gh</i>, it implies
</p>
<dl><dd></dd></dl>
<p>A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let be a square-integrable <a href="/wiki/Predictable_process">predictable process</a> and set
</p>
<dl><dd></dd></dl>
<p>If is a <a href="/wiki/Wiener_process">Wiener process</a>, the <a href="/wiki/Girsanov_theorem">Girsanov theorem</a> then yields the following analogue of the invariance principle:
</p>
<dl><dd></dd></dl>
<p>Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:
</p>
<dl><dd></dd></dl>
<p>Here, the left-hand side is the <a href="/wiki/Malliavin_derivative">Malliavin derivative</a> of the random variable in the direction and the integral appearing on the right hand side should be interpreted as an <a href="/wiki/It%C3%B4_integral">Itô integral</a>. This expression also remains true (by definition) if is not adapted, provided that the right hand side is interpreted as a <a href="/wiki/Skorokhod_integral">Skorokhod integral</a>.<sup>[<i><a href="/wiki/Wikipedia:Citation_needed">citation needed</a></i>]</sup>
</p>
<h2>Clark–Ocone formula[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=3">edit</a>]</h2>
<p>One of the most useful results from Malliavin calculus is the <a href="/wiki/Clark%E2%80%93Ocone_theorem">Clark–Ocone theorem</a>, which allows the process in the <a href="/wiki/Martingale_representation_theorem">martingale representation theorem</a> to be identified explicitly. A simplified version of this theorem is as follows:
</p><p>For satisfying which is Lipschitz and such that <i>F</i> has a strong derivative kernel, in the sense that
for in <i>C</i>[0,1]
</p>
<dl><dd></dd></dl>
<p>then
</p>
<dl><dd></dd></dl>
<p>where <i>H</i> is the previsible projection of <i>F</i>'(<i>x</i>, (<i>t</i>,1]) which may be viewed as the derivative of the function <i>F</i> with respect to a suitable parallel shift of the process <i>X</i> over the portion (<i>t</i>,1] of its domain.
</p><p>This may be more concisely expressed by
</p>
<dl><dd></dd></dl>
<p>Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionals <i>F</i> by replacing the derivative kernel used above by the "<a href="/wiki/Malliavin_derivative">Malliavin derivative</a>" denoted in the above statement of the result.<sup>[<i><a href="/wiki/Wikipedia:Citation_needed">citation needed</a></i>]</sup>
</p>
<h2>Skorokhod integral[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=4">edit</a>]</h2>
<p>The <a href="/wiki/Skorokhod_integral">Skorokhod integral</a> 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 ,
for <b>F</b> in the domain of the Malliavin derivative, we require
</p>
<dl><dd></dd></dl>
<p>where the inner product is that on viz
</p>
<dl><dd></dd></dl>
<p>The existence of this adjoint follows from the <a href="/wiki/Riesz_representation_theorem">Riesz representation theorem</a> for linear operators on <a href="/wiki/Hilbert_spaces">Hilbert spaces</a>.
</p><p>It can be shown that if <i>u</i> is adapted then
</p>
<dl><dd></dd></dl>
<p>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.
</p>
<h2>Applications[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=5">edit</a>]</h2>
<p>The calculus allows <a href="/wiki/Integration_by_parts">integration by parts</a> with <a href="/wiki/Random_variable">random variables</a>; this operation is used in <a href="/wiki/Mathematical_finance">mathematical finance</a> to compute the sensitivities of <a href="/wiki/Derivative_(finance)">financial derivatives</a>. The calculus has applications for example in <a href="/wiki/Stochastic_control">stochastic filtering</a>.
</p>
<h2>References[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=6">edit</a>]</h2>
<ul><li>Kusuoka, S. and Stroock, D. (1981) "Applications of Malliavin Calculus I", <i>Stochastic Analysis, Proceedings Taniguchi International Symposium Katata and Kyoto</i> 1982, pp 271–306</li>
<li>Kusuoka, S. and Stroock, D. (1985) "Applications of Malliavin Calculus II", <i>J. Faculty Sci. Uni. Tokyo Sect. 1A Math.</i>, 32 pp 1–76</li>
<li>Kusuoka, S. and Stroock, D. (1987) "Applications of Malliavin Calculus III", <i>J. Faculty Sci. Univ. Tokyo Sect. 1A Math.</i>, 34 pp 391–442</li>
<li>Malliavin, Paul and Thalmaier, Anton. <i>Stochastic Calculus of Variations in Mathematical Finance</i>, Springer 2005, <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/3-540-43431-3">3-540-43431-3</a></li>
<li><cite>Nualart, David (2006). <a href="https://archive.org/details/malliavincalculu0000nual"><i>The Malliavin calculus and related topics</i></a> (Second ed.). Springer-Verlag. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-28328-7"><bdi>978-3-540-28328-7</bdi></a>.</cite></li>
<li>Bell, Denis. (2007) <i>The Malliavin Calculus</i>, Dover. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-486-44994-7">0-486-44994-7</a>; <a href="https://books.google.com/books?id=IVm8AQAAQBAJ">ebook</a></li>
<li>Sanz-Solé, Marta (2005) <i>Malliavin Calculus, with applications to stochastic partial differential equations</i>. EPFL Press, distributed by CRC Press, Taylor &amp; Francis Group.</li>
<li>Schiller, Alex (2009) <a href="https://web.archive.org/web/20110903171132/http://www.alexschiller.com/media/Thesis.pdf"><i>Malliavin Calculus for Monte Carlo Simulation with Financial Applications</i></a>. Thesis, Department of Mathematics, Princeton University</li>
<li><a href="/wiki/Bernt_%C3%98ksendal">Øksendal, Bernt K.</a>(1997) <a href="https://web.archive.org/web/20190527120605/http://www.quantcode.com/modules/wflinks/visit.php%3Fcid%3D11%26lid%3D4"><i>An Introduction To Malliavin Calculus With Applications To Economics</i></a>. Lecture Notes, Dept. of Mathematics, University of Oslo (Zip file containing Thesis and addendum)</li>
<li><a href="/wiki/Giulia_Di_Nunno">Di Nunno, Giulia</a>, Øksendal, Bernt, Proske, Frank (2009) "Malliavin Calculus for Lévy Processes with Applications to Finance", Universitext, Springer. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-78571-2">978-3-540-78571-2</a></li></ul>
<h2>External links[<a href="/w/index.php?title=Malliavin_calculus&amp;action=edit&amp;section=7">edit</a>]</h2>
<ul><li><a href="/wiki/File:Wikiquote-logo.svg"></a> Quotations related to <a href="https://en.wikiquote.org/wiki/Special:Search/Malliavin_calculus">Malliavin calculus</a> at Wikiquote</li>
<li><cite><a href="/wiki/Peter_Friz">Friz, Peter K.</a> (2005-04-10). <a href="https://web.archive.org/web/20070417205303/http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf">"An Introduction to Malliavin Calculus"</a> (PDF). Archived from <a href="http://www.statslab.cam.ac.uk/~peter/malliavin/Malliavin2005/mall.pdf">the original</a> (PDF) on 2007-04-17. Retrieved 2007-07-23.</cite> Lecture Notes, 43 pages</li></ul>
<ul><li><cite>Zhang, H. (2004-11-11). <a href="http://frank-oertel-math.de/PhD_thesis_on_Malliavin_Calculus_incl_copy_of_FO.pdf">"The Malliavin Calculus"</a> (PDF). Retrieved 2004-11-11.</cite> Thesis, 100 pages</li></ul>
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 From Wikipedia, the free encyclopedia

 In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations.

 Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well.

 The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.

 Overview and history [ edit ]

 Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations.

 Invariance principle [ edit ]

 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

 and hence

 This can be used to derive the integration by parts formula since, setting f = gh, it implies

 A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let be a square-integrable predictable process and set

 If is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle:

 Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:

 Here, the left-hand side is the Malliavin derivative of the random variable in the direction and the integral appearing on the right hand side should be interpreted as an Itô integral. This expression also remains true (by definition) if is not adapted, provided that the right hand side is interpreted as a Skorokhod integral.[ citation needed ]

 Clark–Ocone formula [ edit ]

 Main article: 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 satisfying which is Lipschitz and such that F has a strong derivative kernel, in the sense that for in C[0,1]

 then

 where H is the previsible projection of F'(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

 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 in the above statement of the result.[ citation needed ]

 Skorokhod integral [ edit ]

 Main article: 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 , for F in the domain of the Malliavin derivative, we require

 where the inner product is that on viz

 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

 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 [ edit ]

 The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications for example in stochastic filtering.

 This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. Please help to improve this article by introducing more precise citations. ( June 2011 ) (Learn how and when to remove this template message)
 

 References [ edit ]

 * 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
 * Nualart, David (2006). The Malliavin calculus and related topics (Second ed.). Springer-Verlag. ISBN 978-3-540-28328-7.
 * Bell, Denis. (2007) The Malliavin Calculus, Dover. ISBN 0-486-44994-7; ebook
 * Sanz-Solé, Marta (2005) Malliavin Calculus, with applications to stochastic partial differential equations. EPFL Press, distributed by CRC Press, Taylor & Francis Group.
 * Schiller, Alex (2009) Malliavin Calculus for Monte Carlo Simulation with Financial Applications. Thesis, Department of Mathematics, Princeton University
 * Øksendal, Bernt K.(1997) 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 [ edit ]

 * Quotations related to Malliavin calculus at Wikiquote
 * Friz, Peter K. (2005-04-10). "An Introduction to Malliavin Calculus" (PDF) . Archived from the original (PDF) on 2007-04-17 . Retrieved 2007-07-23 . Lecture Notes, 43 pages
 * Zhang, H. (2004-11-11). "The Malliavin Calculus" (PDF) . Retrieved 2004-11-11 . Thesis, 100 pages
 Retrieved from "https://en.wikipedia.org/w/index.php?title=Malliavin_calculus&oldid=1115107254"
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From Wikipedia, the free encyclopedia
	 
In probability theory and related fields, Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field of calculus of variations from deterministic functions to stochastic processes. In particular, it allows the computation of derivatives of random variables. Malliavin calculus is also called the  stochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, the significant contributors such as S. Kusuoka, D. Stroock, Bismut, S. Watanabe, I. Shigekawa, and so on finally completed the foundations.
 Malliavin calculus is named after Paul Malliavin whose ideas led to a proof that Hörmander's condition implies the existence and smoothness of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. The calculus has been applied to stochastic partial differential equations as well.
 The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications in, for example, stochastic filtering.
Overview and history[edit]
Malliavin introduced Malliavin calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations.
Invariance principle[edit]
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
  and hence  
This can be used to derive the integration by parts formula since, setting f = gh, it implies
   
A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let be a square-integrable predictable process and set
   
If is a Wiener process, the Girsanov theorem then yields the following analogue of the invariance principle:
   
Differentiating with respect to ε on both sides and evaluating at ε=0, one obtains the following integration by parts formula:
   
Here, the left-hand side is the Malliavin derivative of the random variable in the direction and the integral appearing on the right hand side should be interpreted as an Itô integral. This expression also remains true (by definition) if is not adapted, provided that the right hand side is interpreted as a Skorokhod integral.[ citation needed ]
Clark–Ocone formula[edit]
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 satisfying which is Lipschitz and such that F has a strong derivative kernel, in the sense that
for in C[0,1]
   
then
   
where H is the previsible projection of F'(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
   
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 in the above statement of the result.[ citation needed ]
Skorokhod integral[edit]
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 ,
for F in the domain of the Malliavin derivative, we require
   
where the inner product is that on viz
   
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
   
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[edit]
The calculus allows integration by parts with random variables; this operation is used in mathematical finance to compute the sensitivities of financial derivatives. The calculus has applications for example in stochastic filtering.
References[edit]
 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 
 Nualart, David (2006).  The Malliavin calculus and related topics  (Second ed.). Springer-Verlag. ISBN  978-3-540-28328-7 . 
Bell, Denis. (2007) The Malliavin Calculus, Dover. ISBN 0-486-44994-7; ebook 
Sanz-Solé, Marta (2005) Malliavin Calculus, with applications to stochastic partial differential equations. EPFL Press, distributed by CRC Press, Taylor &amp; Francis Group.
Schiller, Alex (2009)  Malliavin Calculus for Monte Carlo Simulation with Financial Applications . Thesis, Department of Mathematics, Princeton University
 Øksendal, Bernt K.(1997)  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[edit]
    Quotations related to Malliavin calculus at Wikiquote
  Friz, Peter K. (2005-04-10). "An Introduction to Malliavin Calculus" (PDF). Archived from the original (PDF) on 2007-04-17. Retrieved 2007-07-23. Lecture Notes, 43 pages 
  Zhang, H. (2004-11-11). "The Malliavin Calculus" (PDF). Retrieved 2004-11-11. Thesis, 100 pages
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<table><tbody><tr><th><p>Paul Malliavin</p></th></tr><tr><td><a href="/wiki/File:Paul_Malliavin,_Marie-Paule_Malliavin.jpeg"></a><p>Paul Malliavin at the <a href="/wiki/International_Congress_of_Mathematicians">ICM</a> 2006 in Madrid, with Marie-Paule Malliavin</p></td></tr><tr><th>Born</th><td>September 10, 1925<br /><div><p><a href="/wiki/Neuilly-sur-Seine">Neuilly-sur-Seine</a></p></div></td></tr><tr><th>Died</th><td>June 3, 2010 (aged 84)<br /><div><p><a href="/wiki/Paris">Paris</a></p></div></td></tr><tr><th>Nationality</th><td>French</td></tr><tr><th>Alma mater</th><td><a href="/wiki/University_of_Paris">University of Paris</a></td></tr><tr><th>Known for</th><td><a href="/wiki/Malliavin_derivative">Malliavin derivative</a><br /><a href="/wiki/Malliavin%27s_absolute_continuity_lemma">Malliavin's absolute continuity lemma</a><br /><a href="/wiki/Malliavin_calculus">Malliavin calculus</a><br />Probabilistic proof of <a href="/wiki/H%C3%B6rmander%27s_condition">Hörmander's theorem</a></td></tr><tr><th>Awards</th><td>Servant Prize (1972) Prix Gaston Julia Prize (1974)</td></tr><tr><td><b>Scientific career</b></td></tr><tr><th>Fields</th><td><a href="/wiki/Mathematics">Mathematics</a></td></tr><tr><th>Institutions</th><td><a href="/wiki/Pierre_and_Marie_Curie_University">Pierre and Marie Curie University</a></td></tr><tr><th><a href="/wiki/Doctoral_advisor">Doctoral advisor</a></th><td><a href="/wiki/Szolem_Mandelbrojt">Szolem Mandelbrojt</a></td></tr><tr><th>Doctoral students</th><td><div>
<ul><li><a href="/wiki/Mar%C3%ADa_Emilia_Caballero">María Emilia Caballero</a></li>
<li><a href="/wiki/Ana_Bela_Cruzeiro">Ana Bela Cruzeiro</a></li></ul>
</div></td></tr><tr><th>Influenced</th><td><div>
<ul><li><a href="/wiki/Shinzo_Watanabe">Shinzo Watanabe</a></li>
<li><a href="/wiki/Daniel_Stroock">Daniel Stroock</a></li></ul>
</div></td></tr></tbody></table>
<p><b>Paul Malliavin</b> (<small>French: </small><a href="/wiki/Help:IPA/French">[maljavɛ̃]</a>; September 10, 1925 – June 3, 2010) was a French <a href="/wiki/Mathematician">mathematician</a> who made important contributions to <a href="/wiki/Harmonic_analysis">harmonic analysis</a> and <a href="/wiki/Stochastic_analysis">stochastic analysis</a>. 
He is known for the <a href="/wiki/Malliavin_calculus">Malliavin calculus</a>, an infinite dimensional calculus for functionals on the <a href="/wiki/Wiener_space">Wiener space</a> and his probabilistic proof of <a href="/wiki/H%C3%B6rmander%27s_condition">Hörmander's theorem</a>.
He was <a href="/wiki/Professor">Professor</a> at the <a href="/wiki/Pierre_and_Marie_Curie_University">Pierre and Marie Curie University</a> and a member of the <a href="/wiki/French_Academy_of_Sciences">French Academy of Sciences</a> from 1979 to 2010.<sup><a href="#cite_note-academie-1">[1]</a></sup>
</p>
<h2>Scientific contributions[<a href="/w/index.php?title=Paul_Malliavin&amp;action=edit&amp;section=1">edit</a>]</h2>
<p>Malliavin's early work was in <a href="/wiki/Harmonic_analysis">harmonic analysis</a>, where he derived important results on the spectral synthesis problem, providing definitive answers to fundamental questions in this field, including a complete characterization of 'band-limited' functions whose Fourier transform has compact support, known as the <a href="/w/index.php?title=Beurling-Malliavin_theorem&amp;action=edit&amp;redlink=1">Beurling-Malliavin theorem</a>.<sup><a href="#cite_note-2">[2]</a></sup>
</p><p>In <a href="/wiki/Stochastic_analysis">stochastic analysis</a>, Malliavin is known for his work on the stochastic calculus of variation, now known as the <a href="/wiki/Malliavin_calculus">Malliavin calculus</a>, a mathematical theory which has found many applications in <a href="/wiki/Monte_Carlo_simulation">Monte Carlo simulation</a> and <a href="/wiki/Mathematical_finance">mathematical finance</a>.
</p><p>As stated by <a href="/wiki/Daniel_W._Stroock">Stroock</a> and <a href="/wiki/Marc_Yor">Yor</a>: "Like Norbert Wiener, Paul Malliavin came to probability theory from harmonic analysis, and, like Wiener, his analytic origins were apparent in everything he did there."<sup><a href="#cite_note-3">[3]</a></sup>
Malliavin introduced a differential operator on <a href="/wiki/Wiener_space">Wiener space</a>, now called the <a href="/wiki/Malliavin_derivative">Malliavin derivative</a>, and derived an <a href="/wiki/Integration_by_parts">integration by parts</a> formula for Wiener functionals. Using this integration by parts formula, Malliavin initiated a probabilistic approach to <a href="/wiki/H%C3%B6rmander%27s_condition">Hörmander's theorem</a> for hypo-elliptic operators and gave a condition for the existence of smooth densities for Wiener functionals in terms of their <a href="/w/index.php?title=Malliavin_covariance_matrix&amp;action=edit&amp;redlink=1">Malliavin covariance matrix</a>.
</p>
<h2>Selected publications[<a href="/w/index.php?title=Paul_Malliavin&amp;action=edit&amp;section=2">edit</a>]</h2>
<ul><li><i><a href="http://www.numdam.org/item?id=ASENS_1955_3_72_1_93_0">La quasi-analyticité généralisée sur un intervalle borné</a></i>, Annales scientifiques de l’École Normale Supérieure 3<sup>e</sup> série 72, 1955, pp. 93–110</li>
<li><i><a href="http://www.numdam.org/item?id=PMIHES_1959__2__61_0">Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts</a></i>, Publications Mathématiques de l’IHÉS 2, 1959, pp. 61–68</li>
<li><i><a href="http://www.numdam.org/item?id=BSMF_1959__87__181_0">Calcul symbolique et sous-algèbres de L<sub>1</sub>(G)</a></i>, Bulletin de la Société Mathématique de France 87, 1959, pp. 181–186, <i><a href="http://www.numdam.org/item?id=BSMF_1959__87__187_0">suite</a></i>, pp. 187–190</li>
<li>with <a href="/wiki/Lee_Albert_Rubel">Lee A. Rubel</a>: <i><a href="http://www.numdam.org/item?id=BSMF_1961__89__175_0">On small entire functions of exponential type with given zeros</a></i>, Bulletin de la Société Mathématique de France 89, 1961, pp. 175–206</li>
<li><i><a href="http://www.numdam.org/item?id=SL_1961__3__A8_0">Spectre des fonctions moyenne-périodiques. Totalité d’une suite d’exponentielles sur un segment</a></i>, Séminaire Lelong. Analyse 3 Exposé No. 11, 1961</li>
<li><i><a href="http://www.numdam.org/item?id=SJL_1962-1963____224_0">Un théorème taubérien relié aux estimations de valeurs propres</a></i>, Séminaire Jean Leray, 1962–1963, pp. 224–231</li>
<li><cite>Malliavin, Paul (1972). <i>Géométrie différentielle intrinsèque</i>. Paris: Hermann. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-2-7056-5696-6"><bdi>978-2-7056-5696-6</bdi></a>.</cite></li>
<li><i><a href="http://www.numdam.org/item?id=SJL_1973-1974___2_A1_0">Géométrie riemannienne stochastique</a></i>, Séminaire Jean Leray 2 Exposé No. 1, 1973–1974</li>
<li><cite>Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators". <i>Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976)</i>. New York: Wiley. pp. 195–263.</cite> <a href="/wiki/MR_(identifier)">MR</a><a href="https://mathscinet.ams.org/mathscinet-getitem?mr=536013">536013</a></li>
<li><i>Geometrie differentielle stochastique</i>, Presses de l’Universite de Montreal, 1978</li>
<li>with Hélène Airault, Leslie Kay, Gérard Letac: <i>Integration and Probability</i>, Springer, 1995</li>
<li>with H. Airault: <i><a href="http://www.numdam.org/item?id=AMBP_1996__3_1_1_0">Some heat operators on P(R<sup>d</sup>)</a></i>, Annales mathématiques Blaise Pascal 3 no. 1, 1996, pp. 1–11</li>
<li><i>Stochastic Analysis</i>, Springer, 1997<sup><a href="#cite_note-4">[4]</a></sup></li>
<li><cite>Malliavin, Paul; Thalmaier, Anton (2005). <a href="https://books.google.com/books?id=VsmdN58iWAYC"><i>Stochastic Calculus of Variations in Mathematical Finance</i></a>. Springer. <a href="/wiki/ISBN_(identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-3-540-43431-3"><bdi>978-3-540-43431-3</bdi></a>.</cite><sup><a href="#cite_note-5">[5]</a></sup></li></ul>
<h2>References[<a href="/w/index.php?title=Paul_Malliavin&amp;action=edit&amp;section=3">edit</a>]</h2>
<div><ol>
<li><b><a href="#cite_ref-academie_1-0">^</a></b> <cite><a href="https://web.archive.org/web/20120315164912/http://www.academie-sciences.fr/academie/membre/Malliavin_Paul.htm">"In memoriam"</a>. <i>Academy of Sciences</i> (in French). June 3, 2010. Archived from <a href="http://www.academie-sciences.fr/academie/membre/Malliavin_Paul.htm">the original</a> on March 15, 2012. Retrieved June 4, 2010.</cite>
</li>
<li><b><a href="#cite_ref-2">^</a></b> 
<cite>Beurling, Arne; Malliavin, Paul (1962). <a href="https://doi.org/10.1007%2FBF02545792">"On Fourier transforms of measures with compact support"</a>. <i>Acta Mathematica</i>. <b>107</b> (3–4): 291–309. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.1007%2FBF02545792">10.1007/BF02545792</a>.</cite>
</li>
<li><b><a href="#cite_ref-3">^</a></b> <cite>Stroock, Daniel; Yor, Marc (2011). <a href="https://www.ams.org/notices/201104/rtx110400568p.pdf">"Remembering Paul Malliavin"</a> (PDF). <i>Notices of the American Mathematical Society</i>. <b>58</b>: 568.</cite>
</li>
<li><b><a href="#cite_ref-4">^</a></b> <cite>Driver, Bruce K. (1998). <a href="https://www.ams.org/journals/bull/1998-35-01/S0273-0979-98-00739-3/S0273-0979-98-00739-3.pdf">"Review: <i>Stochastic analysis</i>, by Paul Malliavin"</a> (PDF). <i>Bull. Amer. Math. Soc. (N.S.)</i>. <b>35</b> (1): 99–104. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.1090%2Fs0273-0979-98-00739-3">10.1090/s0273-0979-98-00739-3</a>.</cite>
</li>
<li><b><a href="#cite_ref-5">^</a></b> <cite><a href="/wiki/David_Nualart">Nualart, David</a> (2007). <a href="https://www.ams.org/journals/bull/2007-44-03/S0273-0979-07-01146-9/S0273-0979-07-01146-9.pdf">"Review: <i>Stochastic calculus of variations in mathematical finance</i>, by Paul Malliavin and Anton Thalmaier"</a> (PDF). <i>Bull. Amer. Math. Soc. (N.S.)</i>. <b>44</b> (3): 487–492. <a href="/wiki/Doi_(identifier)">doi</a>:<a href="https://doi.org/10.1090%2Fs0273-0979-07-01146-9">10.1090/s0273-0979-07-01146-9</a>.</cite>
</li>
</ol></div>
<h2>External links[<a href="/w/index.php?title=Paul_Malliavin&amp;action=edit&amp;section=4">edit</a>]</h2>
<ul><li><a href="https://mathgenealogy.org/id.php?id=33938">Paul Malliavin</a> at the <a href="/wiki/Mathematics_Genealogy_Project">Mathematics Genealogy Project</a></li>
<li><cite><a href="https://www.ams.org/notices/201104/rtx110400568p.pdf">"Remembering Paul Malliavin"</a> (PDF), <i><a href="/wiki/Notices_of_the_American_Mathematical_Society">Notices of the American Mathematical Society</a></i>, <b>58</b> (4): 568–579, 2011</cite></li></ul>
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 Paul Malliavin

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 From Wikipedia, the free encyclopedia
 (Redirected from Malliavin )
 Paul Malliavin 
 Paul Malliavin at the ICM 2006 in Madrid, with Marie-Paule Malliavin
 Born  September 10, 1925 
  Neuilly-sur-Seine 
 Died  June 3, 2010 (aged 84) 
  Paris 
 Nationality French 
 Alma mater University of Paris 
  Malliavin derivative 
 Known for Malliavin's absolute continuity lemma 
  Malliavin calculus 
  Probabilistic proof of Hörmander's theorem 
 Awards  Servant Prize (1972) Prix Gaston Julia Prize (1974)
 Scientific career 
 Fields  Mathematics 
 Institutions Pierre and Marie Curie University 
 Doctoral advisor Szolem Mandelbrojt 
 Doctoral students * María Emilia Caballero 
  * Ana Bela Cruzeiro 
 Influenced  * Shinzo Watanabe 
  * Daniel Stroock 
  
 

 Paul Malliavin (French: [maljavɛ̃]; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. He is known for the Malliavin calculus, an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem. He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences from 1979 to 2010.[1]

 Scientific contributions [ edit ]

 Malliavin's early work was in harmonic analysis, where he derived important results on the spectral synthesis problem, providing definitive answers to fundamental questions in this field, including a complete characterization of 'band-limited' functions whose Fourier transform has compact support, known as the Beurling-Malliavin theorem.[2]

 In stochastic analysis, Malliavin is known for his work on the stochastic calculus of variation, now known as the Malliavin calculus, a mathematical theory which has found many applications in Monte Carlo simulation and mathematical finance.

 As stated by Stroock and Yor: "Like Norbert Wiener, Paul Malliavin came to probability theory from harmonic analysis, and, like Wiener, his analytic origins were apparent in everything he did there."[3] Malliavin introduced a differential operator on Wiener space, now called the Malliavin derivative, and derived an integration by parts formula for Wiener functionals. Using this integration by parts formula, Malliavin initiated a probabilistic approach to Hörmander's theorem for hypo-elliptic operators and gave a condition for the existence of smooth densities for Wiener functionals in terms of their Malliavin covariance matrix.

 Selected publications [ edit ]

 * La quasi-analyticité généralisée sur un intervalle borné, Annales scientifiques de l’École Normale Supérieure 3e série 72, 1955, pp. 93–110
 * Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts, Publications Mathématiques de l’IHÉS 2, 1959, pp. 61–68
 * Calcul symbolique et sous-algèbres de L1(G), Bulletin de la Société Mathématique de France 87, 1959, pp. 181–186, suite, pp. 187–190
 * with Lee A. Rubel: On small entire functions of exponential type with given zeros, Bulletin de la Société Mathématique de France 89, 1961, pp. 175–206
 * Spectre des fonctions moyenne-périodiques. Totalité d’une suite d’exponentielles sur un segment, Séminaire Lelong. Analyse 3 Exposé No. 11, 1961
 * Un théorème taubérien relié aux estimations de valeurs propres, Séminaire Jean Leray, 1962–1963, pp. 224–231
 * Malliavin, Paul (1972). Géométrie différentielle intrinsèque. Paris: Hermann. ISBN 978-2-7056-5696-6.
 * Géométrie riemannienne stochastique, Séminaire Jean Leray 2 Exposé No. 1, 1973–1974
 * Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators". Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). New York: Wiley. pp. 195–263. MR536013
 * Geometrie differentielle stochastique, Presses de l’Universite de Montreal, 1978
 * with Hélène Airault, Leslie Kay, Gérard Letac: Integration and Probability, Springer, 1995
 * with H. Airault: Some heat operators on P(Rd), Annales mathématiques Blaise Pascal 3 no. 1, 1996, pp. 1–11
 * Stochastic Analysis, Springer, 1997[4]
 * Malliavin, Paul; Thalmaier, Anton (2005). Stochastic Calculus of Variations in Mathematical Finance. Springer. ISBN 978-3-540-43431-3.[5]

 References [ edit ]

 1. ^ "In memoriam". Academy of Sciences (in French). June 3, 2010. Archived from the original on March 15, 2012 . Retrieved June 4, 2010 .
 2. ^ Beurling, Arne; Malliavin, Paul (1962). "On Fourier transforms of measures with compact support". Acta Mathematica. 107 (3–4): 291–309. doi:10.1007/BF02545792.
 3. ^ Stroock, Daniel; Yor, Marc (2011). "Remembering Paul Malliavin" (PDF) . Notices of the American Mathematical Society. 58: 568.
 4. ^ Driver, Bruce K. (1998). "Review: Stochastic analysis, by Paul Malliavin" (PDF) . Bull. Amer. Math. Soc. (N.S.). 35 (1): 99–104. doi:10.1090/s0273-0979-98-00739-3.
 5. ^ Nualart, David (2007). "Review: Stochastic calculus of variations in mathematical finance, by Paul Malliavin and Anton Thalmaier" (PDF) . Bull. Amer. Math. Soc. (N.S.). 44 (3): 487–492. doi:10.1090/s0273-0979-07-01146-9.

 External links [ edit ]

 * Paul Malliavin at the Mathematics Genealogy Project
 * "Remembering Paul Malliavin" (PDF) , Notices of the American Mathematical Society, 58 (4): 568–579, 2011
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From Wikipedia, the free encyclopedia
	  (Redirected from Malliavin)  
    Paul Malliavin       Paul Malliavin at the ICM 2006 in Madrid, with Marie-Paule Malliavin    Born September 10, 1925   Neuilly-sur-Seine      Died June 3, 2010 (aged 84)   Paris      Nationality French   Alma mater  University of Paris    Known for  Malliavin derivative  Malliavin's absolute continuity lemma  Malliavin calculus Probabilistic proof of Hörmander's theorem    Awards Servant Prize (1972) Prix Gaston Julia Prize (1974)    Scientific career    Fields  Mathematics    Institutions  Pierre and Marie Curie University     Doctoral advisor   Szolem Mandelbrojt    Doctoral students  
  María Emilia Caballero 
 Ana Bela Cruzeiro  
    Influenced  
  Shinzo Watanabe 
 Daniel Stroock  
    
 Paul Malliavin (French:  [maljavɛ̃]; September 10, 1925 – June 3, 2010) was a French mathematician who made important contributions to harmonic analysis and stochastic analysis. 
He is known for the Malliavin calculus, an infinite dimensional calculus for functionals on the Wiener space and his probabilistic proof of Hörmander's theorem.
He was Professor at the Pierre and Marie Curie University and a member of the French Academy of Sciences from 1979 to 2010. [1] 
Scientific contributions[edit]
Malliavin's early work was in harmonic analysis, where he derived important results on the spectral synthesis problem, providing definitive answers to fundamental questions in this field, including a complete characterization of 'band-limited' functions whose Fourier transform has compact support, known as the Beurling-Malliavin theorem. [2] 
 In stochastic analysis, Malliavin is known for his work on the stochastic calculus of variation, now known as the Malliavin calculus, a mathematical theory which has found many applications in Monte Carlo simulation and mathematical finance.
 As stated by Stroock and Yor: "Like Norbert Wiener, Paul Malliavin came to probability theory from harmonic analysis, and, like Wiener, his analytic origins were apparent in everything he did there." [3] 
Malliavin introduced a differential operator on Wiener space, now called the Malliavin derivative, and derived an integration by parts formula for Wiener functionals. Using this integration by parts formula, Malliavin initiated a probabilistic approach to Hörmander's theorem for hypo-elliptic operators and gave a condition for the existence of smooth densities for Wiener functionals in terms of their Malliavin covariance matrix.
Selected publications[edit]
   La quasi-analyticité généralisée sur un intervalle borné , Annales scientifiques de l’École Normale Supérieure 3e série 72, 1955, pp. 93–110
  Impossibilité de la synthèse spectrale sur les groupes abéliens non compacts , Publications Mathématiques de l’IHÉS 2, 1959, pp. 61–68
  Calcul symbolique et sous-algèbres de L1(G) , Bulletin de la Société Mathématique de France 87, 1959, pp. 181–186,  suite , pp. 187–190
with Lee A. Rubel:  On small entire functions of exponential type with given zeros , Bulletin de la Société Mathématique de France 89, 1961, pp. 175–206
  Spectre des fonctions moyenne-périodiques. Totalité d’une suite d’exponentielles sur un segment , Séminaire Lelong. Analyse 3 Exposé No. 11, 1961
  Un théorème taubérien relié aux estimations de valeurs propres , Séminaire Jean Leray, 1962–1963, pp. 224–231
 Malliavin, Paul (1972). Géométrie différentielle intrinsèque. Paris: Hermann. ISBN  978-2-7056-5696-6 . 
  Géométrie riemannienne stochastique , Séminaire Jean Leray 2 Exposé No. 1, 1973–1974
 Malliavin, Paul (1978). "Stochastic calculus of variations and hypoelliptic operators". Proceedings of the International Symposium on Stochastic Differential Equations (Res. Inst. Math. Sci., Kyoto Univ., Kyoto, 1976). New York: Wiley. pp. 195–263. MR 536013 
 Geometrie differentielle stochastique, Presses de l’Universite de Montreal, 1978
with Hélène Airault, Leslie Kay, Gérard Letac: Integration and Probability, Springer, 1995
with H. Airault:  Some heat operators on P(Rd) , Annales mathématiques Blaise Pascal 3 no. 1, 1996, pp. 1–11
 Stochastic Analysis, Springer, 1997 [4]  
 Malliavin, Paul; Thalmaier, Anton (2005).  Stochastic Calculus of Variations in Mathematical Finance . Springer. ISBN  978-3-540-43431-3 .  [5]   
References[edit]
 
  ^   "In memoriam". Academy of Sciences (in French). June 3, 2010. Archived from the original on March 15, 2012. Retrieved June 4, 2010.
  ^  
Beurling, Arne; Malliavin, Paul (1962). "On Fourier transforms of measures with compact support". Acta Mathematica. 107 (3–4): 291–309. doi:10.1007/BF02545792.
  ^  Stroock, Daniel; Yor, Marc (2011). "Remembering Paul Malliavin" (PDF). Notices of the American Mathematical Society. 58: 568.
  ^  Driver, Bruce K. (1998). "Review: Stochastic analysis, by Paul Malliavin" (PDF). Bull. Amer. Math. Soc. (N.S.). 35 (1): 99–104. doi:10.1090/s0273-0979-98-00739-3.
  ^   Nualart, David (2007). "Review: Stochastic calculus of variations in mathematical finance, by Paul Malliavin and Anton Thalmaier" (PDF). Bull. Amer. Math. Soc. (N.S.). 44 (3): 487–492. doi:10.1090/s0273-0979-07-01146-9.
 
External links[edit]
  Paul Malliavin at the Mathematics Genealogy Project 
  "Remembering Paul Malliavin" (PDF),  Notices of the American Mathematical Society , 58 (4): 568–579, 2011
Paul Malliavin - Wikipedia
https://en.wikipedia.org/wiki/Malliavin
https://www.mi.fu-berlin.de/math/groups/stoch/teaching/_Stochastics/2020so_Stochastics4.html
Stochastic Partial Differential Equations: Classical and New
2019-10-08T00:00:00Z

<html><body><div><div><h2>Summer Term 2020</h2>
<p>lecture and exercise by <strong><a href="http://www.mi.fu-berlin.de/math/groups/stoch/members/Professors/perkowski.html">Prof. Dr. Nicolas Perkowski</a></strong></p>
<hr/>
<p><strong>Time and place</strong></p>
<ul>
<li>
<p><strong>Lecture:</strong> Video lectures are available online (see below).</p>
</li>
<li>
<p><strong>Exercise Session:</strong> Wednesdays, 10:15 - 11:45, online.</p>
</li>
<li><strong>Final Exam:</strong>  to be announced in due course</li>
</ul>
<hr/>
<p><strong>Prerequisits:</strong> Stochastics I-II and Analysis I — III. Recommended: Stochastic Analysis and Functional Analysis. Previous knowledge in PDE theory is not required.</p>
<h2>Assessment</h2>
<p>To receive credits fo the course you need to</p>
<ul>
<li>actively participate in the exercise session </li>
<li>work on and successfully solve the weekly exercises </li>
<li>pass the final exam (see above)</li>
</ul>
<p>If you are an FU student you only need to register for the course via CM (Campus Management).<br/>If you are not an FU student, you are required to register via KVV/Whiteboard.    </p>
<h2>Exercises </h2>
<p>Problem sets will be put online every Wednesday and can be found under <em>Assignements</em> in the KVV/Whiteboard portal. You do not have to submit your solutions. The solutions will be discussed in the online tutorial.</p>
<h2>Course Overview/ Content:</h2>
<ul>
<li>
<p>Ito calculus for Gaussian random measures</p>
</li>
<li>
<p>Semilinear stochastic PDEs in one dimension</p>
</li>
<li>
<p>Basic rough path theory</p>
</li>
<li>
<p>Schauder estimates</p>
</li>
<li>
<p>Gaussian hypercontractivity</p>
</li>
<li>
<p>Paraproducts and paracontrolled distributions</p>
</li>
<li>
<p>Local existence and uniqueness for semilinear SPDEs in higher dimensions</p>
</li>
<li>
<p>Properties of solutions</p>
</li>
</ul>
<h2>References</h2>
<p><a href="https://www.mi.fu-berlin.de/math/groups/stoch/members/publ_Perkowski/LN202007_SPDEs.pdf">Lecture notes.</a></p>
<p><a href="https://www.mi.fu-berlin.de/math/groups/stoch/members/publ_Perkowski/LN202007_SPDEs_Exercises.pdf">Exercises.</a></p>
<h2>Videos</h2>
<ul>
<li>Videos for Wednesday, April 22:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/41052c84-f91b-475d-a85f-5fc67dca26be#/videos/41052c84-f91b-475d-a85f-5fc67dca26be">Welcome</a></li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/a8b4c1bb-c41b-4d46-89e2-de5fb1ee6eae#/videos/a8b4c1bb-c41b-4d46-89e2-de5fb1ee6eae">Introduction</a> (Section 0 in the notes)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/91af9fec-8394-4440-94ce-eebabc211012#/videos/91af9fec-8394-4440-94ce-eebabc211012">Covariance measures</a> (1.1-1.2)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/2bd1e296-5818-4576-acd7-2f966a1b578d#/videos/2bd1e296-5818-4576-acd7-2f966a1b578d">Gaussian martingale measures</a> (1.3-1.4)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/5ac68b65-e8f4-4c26-9848-a0bf6ff65bc1#/videos/5ac68b65-e8f4-4c26-9848-a0bf6ff65bc1">Examples of Gaussian martingale measures</a> (1.5-1.8)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, April 29:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/e7758bf7-8cf2-4fb0-a9b1-3d49f447bc12#/videos/e7758bf7-8cf2-4fb0-a9b1-3d49f447bc12">Construction of the Ito integral </a>(1.9-1.16)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/c850b4c0-4f1a-48e1-b9ff-5b2d6c69cfe3#/videos/c850b4c0-4f1a-48e1-b9ff-5b2d6c69cfe3">Weak and mild solutions</a> (1.17-1.20)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/b3ecee01-658b-430b-9f94-8030aa080252#/videos/b3ecee01-658b-430b-9f94-8030aa080252">K alpha covariances</a> (1.21-1.23)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, May 6:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/b3e5b161-eadd-4c05-ac02-57f1c95ce7f4#/videos/b3e5b161-eadd-4c05-ac02-57f1c95ce7f4">Existence and uniqueness of mild solutions</a> (1.24-1.25)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/8272634c-42d1-44b2-91cc-a2732457285c#/videos/8272634c-42d1-44b2-91cc-a2732457285c">Examples of SPDEs</a> (1.26)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/ed972647-b4e4-4ed5-b199-41b322826ce6#/videos/ed972647-b4e4-4ed5-b199-41b322826ce6">Towards the regularity of solutions</a> (1.27-1.28)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, May 13:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/88ec1ec6-863f-4cc2-bea7-79511ee03e79#/videos/88ec1ec6-863f-4cc2-bea7-79511ee03e79">Regularity of solutions</a> (1.29-1.32)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/f6c70730-975b-4f4a-b0ff-ccf41c089cd5#/videos/f6c70730-975b-4f4a-b0ff-ccf41c089cd5">Quartic variation for space-time white noise in 1d</a> (1.33)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/6d93aca1-a6b3-4e3c-8257-474116809051#/videos/6d93aca1-a6b3-4e3c-8257-474116809051">An example we cannot solve</a> (1.34)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/ee7b3352-9358-4547-b669-ea120c42bea3#/videos/ee7b3352-9358-4547-b669-ea120c42bea3">The d-dimensional torus</a> (1.35)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/bc9221e7-e158-4b1f-9f23-e071a1c73176#/videos/bc9221e7-e158-4b1f-9f23-e071a1c73176">Energy estimates, a glimpse in the variational approach</a> (1.36)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/5172735e-5df6-4718-8f76-209226573b16#/videos/5172735e-5df6-4718-8f76-209226573b16">&#34;Stochastic parabolicity&#34;, Ito vs Stratonovich</a> (1.37)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, May 20:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/e7d48d00-e0c0-44a3-8320-d0cbb20f06d0#/videos/e7d48d00-e0c0-44a3-8320-d0cbb20f06d0">Some &#34;singular&#34; SPDEs</a> (1.38-1.41)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/86f44d4f-b400-4ae0-85e9-95700df45af0#/videos/86f44d4f-b400-4ae0-85e9-95700df45af0">The sewing lemma</a> (2.1-2.2)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/56e650fc-85c5-4217-9a64-b85ecd363633#/videos/56e650fc-85c5-4217-9a64-b85ecd363633">The Young integral</a> (2.3-2.5)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/7fde6baa-e163-454e-beca-f5ec9c1bc647#/videos/7fde6baa-e163-454e-beca-f5ec9c1bc647">Some technical results</a> (2.6-2.8)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/8257878a-ed43-4a41-ae5d-d0ca390d8e1f#/videos/8257878a-ed43-4a41-ae5d-d0ca390d8e1f">Young integral equations</a> (2.9)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, May 27:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/0219fa09-36b1-411a-a387-cd66af86c1d3#/videos/0219fa09-36b1-411a-a387-cd66af86c1d3">Application of the Young theory to fractional Brownian motions </a>(2.10-2.11)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/36ad76f8-43e0-4122-ae15-e6364504b6d5#/videos/36ad76f8-43e0-4122-ae15-e6364504b6d5">Motivating examples for rough paths</a> (2.12-2.14)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/4d4b03bd-5df0-480f-bf30-9d1d50184960#/videos/4d4b03bd-5df0-480f-bf30-9d1d50184960">Definition of a rough path</a> (2.15-2.18)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/5a3117cd-14a6-4b53-b02e-eb40776e9649#/videos/5a3117cd-14a6-4b53-b02e-eb40776e9649">First applications of rough paths</a> (2.19-2.21)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, June 3:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/07eaaffc-a3ac-42a3-90cf-5b324e2afd0d#/videos/07eaaffc-a3ac-42a3-90cf-5b324e2afd0d">Controlled paths</a> (2.22)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/ee798ce3-f475-47d7-9aa1-27840744353a#/videos/ee798ce3-f475-47d7-9aa1-27840744353a">Controlled rough path integral</a> (2.23-2.24)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/277d21cd-b24e-4c08-86e4-3ff227adc8f6#/videos/277d21cd-b24e-4c08-86e4-3ff227adc8f6">Rough differential equations </a>(2.25-2.29)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/29cdcc1f-fb33-4618-a7c4-71152297e7c1#/videos/29cdcc1f-fb33-4618-a7c4-71152297e7c1">Brownian motion as a rough path</a> (2.30-2.31)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, June 10:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/d4bcf33d-3d7d-41dc-974d-6bb1586ca71a#/videos/d4bcf33d-3d7d-41dc-974d-6bb1586ca71a">Linear operations on tempered distributions</a> (3.1-3.3)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/3dc115f4-eecf-4a3b-99e3-6342cf384b63#/videos/3dc115f4-eecf-4a3b-99e3-6342cf384b63">Fourier transforms and convolutions</a> (3.4-3.8)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/4be1694d-3723-46e0-919c-12c74a5de967#/videos/4be1694d-3723-46e0-919c-12c74a5de967">Smooth dyadic partition of unity</a> (3.9-3.11)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/c424e87c-d620-42e2-95b5-5ef4f92623c1#/videos/c424e87c-d620-42e2-95b5-5ef4f92623c1">Besov spaces and Bernstein-type inequality</a> (3.12-3.13)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, June 17:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/385739c9-f47a-4b13-9762-b717fcb8aba4#/videos/385739c9-f47a-4b13-9762-b717fcb8aba4">Applications of the Bernstein-type inequality</a> (3.14-3.15)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/6d9ab977-5e93-40a2-9cbe-83a3b3982bb2#/videos/6d9ab977-5e93-40a2-9cbe-83a3b3982bb2">Lemma about functions that are localized in Fourier space</a> (3.16)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/e28dc5ea-7c5f-4620-a10b-6984d1ee0e27#/videos/e28dc5ea-7c5f-4620-a10b-6984d1ee0e27">The paraproduct and the resonant product </a>(3.17-3.20)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/ed11d9ae-8a8d-48f5-b712-e295ccb2a266#/videos/ed11d9ae-8a8d-48f5-b712-e295ccb2a266">Examples for products of distributions</a> (3.21-3.23)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, June 24:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/f331b62a-d9e8-4be3-ba5e-151e06746f31#/videos/f331b62a-d9e8-4be3-ba5e-151e06746f31">Besov spaces and heat kernel on the torus</a> (3.24)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/9806e180-b361-4ddf-babe-0f8ada3ba686#/videos/9806e180-b361-4ddf-babe-0f8ada3ba686">A Kolmogorov type criterion for space-time Hölder-Besov regularity</a> (4.1)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/9c7a585f-ec67-4b40-ab27-967074a21a73#/videos/9c7a585f-ec67-4b40-ab27-967074a21a73">Regularity of the stochastic convolution</a> (4.2-4.4)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/3a81972c-3dfd-426b-a171-3ac42cb00a9e#/videos/3a81972c-3dfd-426b-a171-3ac42cb00a9e">Schauder estimates</a> (4.5-4.6)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/bd9f1064-76b4-4ec2-a054-5b15ee2f9825#/videos/bd9f1064-76b4-4ec2-a054-5b15ee2f9825">The Phi41 equation</a> (4.7)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, July 1:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/bb69b57d-577e-4643-a65f-71bb453b7ccf#/videos/bb69b57d-577e-4643-a65f-71bb453b7ccf">The Phi42 equation</a> (4.10-4.11)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/be7feb3b-c2c9-422d-bda8-4d85a769266d#/videos/be7feb3b-c2c9-422d-bda8-4d85a769266d">Hermite polynomials</a> (4.12)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/d9cb7acb-7a36-4356-bad5-a8525c06bf15#/videos/d9cb7acb-7a36-4356-bad5-a8525c06bf15">Wiener-Ito integrals</a> (4.14-4.18)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/3c2e210a-5c7b-496f-ad93-e3ff3d48e1f8#/videos/3c2e210a-5c7b-496f-ad93-e3ff3d48e1f8">Link between Hermite polynomials and Wiener-Ito integrals</a> (4.19)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/574dc7a3-9093-49fb-9a2c-70e8767b4c72#/videos/574dc7a3-9093-49fb-9a2c-70e8767b4c72">Gaussian hypercontractivity</a> (4.20-4.23)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, July 8:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/8d7721f4-e322-4b33-b292-a4c40cf20243#/videos/8d7721f4-e322-4b33-b292-a4c40cf20243">Application to the Phi42 equation</a> (4.23)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/5b7466d1-1e14-46c6-9286-9f37d43521bb#/videos/5b7466d1-1e14-46c6-9286-9f37d43521bb">Local subcriticality</a> (4.27-4.28)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/f6ee69dc-d476-4d5a-84f9-c188f24a9242#/videos/f6ee69dc-d476-4d5a-84f9-c188f24a9242">Tree notation</a></li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/40f8be77-67ae-415e-b12e-4ffe9cd5eba4#/videos/40f8be77-67ae-415e-b12e-4ffe9cd5eba4">Statement of the commutator estimate</a> (5.2-5.3)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/b96c60bc-221a-4363-83c4-4d05ed366e04#/videos/b96c60bc-221a-4363-83c4-4d05ed366e04">Proof of the commutator estimate</a> (5.4)</li>
</ul>
</ul>
</li>
<li>Videos for Wednesday, July 15:
<ul>
<ul>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/94abf782-1cde-41bb-9c1d-d3b64451107f#/videos/94abf782-1cde-41bb-9c1d-d3b64451107f">Definition of paracontrolled distribution</a> (5.5-5.8)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/a19c5b33-dc01-42a2-b287-5a12a7761801#/videos/a19c5b33-dc01-42a2-b287-5a12a7761801">Comparison of modified paraproduct and usual paraproduct</a> (5.9-5.10)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/936a598b-f9b6-422b-ae57-a38933db899e#/videos/936a598b-f9b6-422b-ae57-a38933db899e">Operations on paracontrolled distributions</a> (5.11-5.14)</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/f3b99045-cff6-477c-a022-8f5c98914a47#/videos/f3b99045-cff6-477c-a022-8f5c98914a47">Paracontrolled Picard iteration</a> (5.15)</li>
</ul>
</ul>
</li>
<li><a href="https://fu-berlin.eu.vbrickrev.com/#/videos/a5de5dbf-c141-4012-a358-37227ced303c#/videos/a5de5dbf-c141-4012-a358-37227ced303c">Suggestion of some possible projects for the exam</a></li>
</ul></div></div></body></html>
Summer Term 2020
lecture and exercise by  Prof. Dr. Nicolas Perkowski  
 Time and place 
 Lecture: Video lectures are available online (see below).
 Exercise Session: Wednesdays, 10:15 - 11:45, online.
 Final Exam:  to be announced in due course
 Prerequisits: Stochastics I-II and Analysis I — III. Recommended: Stochastic Analysis and Functional Analysis. Previous knowledge in PDE theory is not required.
Assessment
To receive credits fo the course you need to
actively participate in the exercise session 
work on and successfully solve the weekly exercises 
pass the final exam (see above)
If you are an FU student you only need to register for the course via CM (Campus Management).If you are not an FU student, you are required to register via KVV/Whiteboard.    
Exercises 
Problem sets will be put online every Wednesday and can be found under Assignements in the KVV/Whiteboard portal. You do not have to submit your solutions. The solutions will be discussed in the online tutorial.
Course Overview/ Content:
Ito calculus for Gaussian random measures
Semilinear stochastic PDEs in one dimension
Basic rough path theory
Schauder estimates
Gaussian hypercontractivity
Paraproducts and paracontrolled distributions
Local existence and uniqueness for semilinear SPDEs in higher dimensions
Properties of solutions
References
 Lecture notes. 
 Exercises. 
Videos
Videos for Wednesday, April 22:
 Welcome 
 Introduction (Section 0 in the notes)
 Covariance measures (1.1-1.2)
 Gaussian martingale measures (1.3-1.4)
 Examples of Gaussian martingale measures (1.5-1.8)
Videos for Wednesday, April 29:
 Construction of the Ito integral (1.9-1.16)
 Weak and mild solutions (1.17-1.20)
 K alpha covariances (1.21-1.23)
Videos for Wednesday, May 6:
 Existence and uniqueness of mild solutions (1.24-1.25)
 Examples of SPDEs (1.26)
 Towards the regularity of solutions (1.27-1.28)
Videos for Wednesday, May 13:
 Regularity of solutions (1.29-1.32)
 Quartic variation for space-time white noise in 1d (1.33)
 An example we cannot solve (1.34)
 The d-dimensional torus (1.35)
 Energy estimates, a glimpse in the variational approach (1.36)
 &#34;Stochastic parabolicity&#34;, Ito vs Stratonovich (1.37)
Videos for Wednesday, May 20:
 Some &#34;singular&#34; SPDEs (1.38-1.41)
 The sewing lemma (2.1-2.2)
 The Young integral (2.3-2.5)
 Some technical results (2.6-2.8)
 Young integral equations (2.9)
Videos for Wednesday, May 27:
 Application of the Young theory to fractional Brownian motions (2.10-2.11)
 Motivating examples for rough paths (2.12-2.14)
 Definition of a rough path (2.15-2.18)
 First applications of rough paths (2.19-2.21)
Videos for Wednesday, June 3:
 Controlled paths (2.22)
 Controlled rough path integral (2.23-2.24)
 Rough differential equations (2.25-2.29)
 Brownian motion as a rough path (2.30-2.31)
Videos for Wednesday, June 10:
 Linear operations on tempered distributions (3.1-3.3)
 Fourier transforms and convolutions (3.4-3.8)
 Smooth dyadic partition of unity (3.9-3.11)
 Besov spaces and Bernstein-type inequality (3.12-3.13)
Videos for Wednesday, June 17:
 Applications of the Bernstein-type inequality (3.14-3.15)
 Lemma about functions that are localized in Fourier space (3.16)
 The paraproduct and the resonant product (3.17-3.20)
 Examples for products of distributions (3.21-3.23)
Videos for Wednesday, June 24:
 Besov spaces and heat kernel on the torus (3.24)
 A Kolmogorov type criterion for space-time Hölder-Besov regularity (4.1)
 Regularity of the stochastic convolution (4.2-4.4)
 Schauder estimates (4.5-4.6)
 The Phi41 equation (4.7)
Videos for Wednesday, July 1:
 The Phi42 equation (4.10-4.11)
 Hermite polynomials (4.12)
 Wiener-Ito integrals (4.14-4.18)
 Link between Hermite polynomials and Wiener-Ito integrals (4.19)
 Gaussian hypercontractivity (4.20-4.23)
Videos for Wednesday, July 8:
 Application to the Phi42 equation (4.23)
 Local subcriticality (4.27-4.28)
 Tree notation 
 Statement of the commutator estimate (5.2-5.3)
 Proof of the commutator estimate (5.4)
Videos for Wednesday, July 15:
 Definition of paracontrolled distribution (5.5-5.8)
 Comparison of modified paraproduct and usual paraproduct (5.9-5.10)
 Operations on paracontrolled distributions (5.11-5.14)
 Paracontrolled Picard iteration (5.15)
 Suggestion of some possible projects for the exam
Stochastic Partial Differential Equations: Classical and New
https://www.mi.fu-berlin.de/math/groups/stoch/teaching/_Stochastics/2020so_Stochastics4.html
http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos/watch/202102090900-Stadler.html
Video From 21w5167: Optimization under Uncertainty: Learning and Decision Making (Online)
2021-02-09
Georg Stadler; New York University
<div><div>
<p>Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints</p>
<p>Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints</p>
<p>Georg Stadler, Optimal control of PDEs under uncertainty with joint chance state constraints, Optimization under Uncertainty: Learning and Decision Making (Online), BIRS, BIRS talk, 21w5167, math, mathematics, video</p>
<p>Banff International Research Station</p>
<p>Banff International Research Station</p>
<p>presentation</p>
<p>mpeg4</p>
<p>80M</p>
<p>http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos</p>
<p>http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos/watch/202102090900-Stadler.html</p>
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 Video From 21w5167: Optimization under Uncertainty: Learning and Decision Making (Online)

 Georg Stadler , New York University
 Tuesday, February 9, 2021 09:00 - 09:53
 Optimal control of PDEs under uncertainty with joint chance state constraints
 Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints
 Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints
 Georg Stadler, Optimal control of PDEs under uncertainty with joint chance state constraints, Optimization under Uncertainty: Learning and Decision Making (Online), BIRS, BIRS talk, 21w5167, math, mathematics, video
 Banff International Research Station
 Banff International Research Station
 presentation
 mpeg4
 80M
 http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos
 http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos/watch/202102090900-Stadler.html

 Download this video (80M)

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Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints
Georg Stadler, Video: Optimal control of PDEs under uncertainty with joint chance state constraints
Georg Stadler, Optimal control of PDEs under uncertainty with joint chance state constraints, Optimization under Uncertainty: Learning and Decision Making (Online), BIRS, BIRS talk, 21w5167, math, mathematics, video
Banff International Research Station
Banff International Research Station
presentation
mpeg4
80M
http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos
http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos/watch/202102090900-Stadler.html
Video From 21w5167: Optimization under Uncertainty: Learning and Decision Making (Online)
http://www.birs.ca/events/2021/5-day-workshops/21w5167/videos/watch/202102090900-Stadler.html