Inprobability theory and related fields,Malliavin calculus is a set of mathematical techniques and ideas that extend the mathematical field ofcalculus of variations from deterministic functions tostochastic processes. In particular, it allows the computation ofderivatives ofrandom variables. Malliavin calculus is also called thestochastic calculus of variations. P. Malliavin first initiated the calculus on infinite dimensional space. Then, significant contributors such as S. Kusuoka, D. Stroock,J-M. Bismut,Shinzo Watanabe, I. Shigekawa, and so on completed the foundations for the field.

Malliavin calculus is named afterPaul Malliavin whose ideas led to aproof thatHörmander's condition implies the existence and smoothness of adensity for the solution of astochastic differential equation;Hörmander's original proof was based on the theory ofpartial differential equations. The calculus has been applied tostochastic partial differential equations as well.

The calculus allowsintegration by parts with random variables; this operation is used inmathematical finance to compute the sensitivities offinancial derivatives. The calculus has applications in, for example,stochastic filtering.

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

Consider aWiener functional${\displaystyle F}$ (a functional from theclassical Wiener space) and consider the task of finding a derivative for it. The natural idea would be to use theGateaux derivative

${\displaystyle D_{g}F[f]:=\underset{\tau \to 0}{lim}\frac{F[f+\tau g]-F[f]}{\tau }}$,

however this does not always exist. Therefore it does make sense to find a new differential calculus for such spaces by limiting the directions.

The toy model of Malliavin calculus is an irreducibleGaussian probability space${\displaystyle X=(\mathrm{\Omega },\mathcal{F},P,\mathcal{H})}$. This is a (complete) probability space${\displaystyle (\mathrm{\Omega },\mathcal{F},P)}$ together with a closedsubspace${\displaystyle \mathcal{H}\subset L^2(\mathrm{\Omega },\mathcal{F},P)}$ such that all${\displaystyle H\in \mathcal{H}}$ are mean zero Gaussian variables and${\displaystyle \mathcal{F}=\sigma (H:H\in \mathcal{H})}$. If one chooses a basis for${\displaystyle \mathcal{H}}$ then one calls${\displaystyle X}$ anumerical model. On the other hand, for any separable Hilbert space${\displaystyle \mathcal{G}}$ exists a canonical irreducible Gaussian probability space${\displaystyle \mathrm{Seg}(\mathcal{G})}$ named theSegal model (named afterIrving Segal) having${\displaystyle \mathcal{G}}$ as its Gaussian subspace. In this case for a${\displaystyle g\in \mathcal{G}}$ one notates the associated random variable in${\displaystyle \mathrm{Seg}(\mathcal{G})}$ as${\displaystyle W(g)}$.

Properties of a Gaussian probability space that do not depend on the particular choice of basis are calledintrinsic and such that do depend on the choiceextrensic.^[1] We denote the countably infinite product of real spaces as${\displaystyle {\mathbb{R}}^{\mathbb{N}}=\prod _{i=1}^{\mathrm{\infty }}\mathbb{R}}$.

Recall the modern version of theCameron-Martin theorem

Consider alocally convex vector space${\displaystyle E}$ with a cylindrical Gaussian measure${\displaystyle \gamma }$ on it. For an element in the topological dual${\displaystyle f\in E^{\prime }}$ define the distance to the mean

${\displaystyle t_{\gamma }(f):=f-{\int }_{E}f(x)\gamma (\mathrm{d}x),}$

which is a map${\displaystyle t_{\gamma }:E^{\prime }\to L^2(E,\gamma )}$, and denote the closure in${\displaystyle L^2(E,\gamma )}$ as

${\displaystyle E_{\gamma }^{\prime }:=\mathrm{clos}\left\{t_{\gamma }(f):f\in E^{\prime }\right\}}$

Let${\displaystyle {\gamma }_m:=\gamma (\cdot -m)}$ denote the translation by${\displaystyle m\in E}$. Then${\displaystyle E_{\gamma }^{\prime }}$ respectively thecovariance operator${\displaystyle R_{\gamma }:E_{\gamma }^{\prime }\to (E_{\gamma }^{\prime }{)}^{\ast }}$ on it induces areproducing kernel Hilbert space${\displaystyle R}$ called the Cameron-Martin space such that for any${\displaystyle m\in R}$ there isequivalence${\displaystyle {\gamma }_m\sim \gamma }$.^[2]

In fact one can use here theFeldman–Hájek theorem to find that for any other${\displaystyle h\notin R}$ such measure would be singular.

Let${\displaystyle \gamma }$ be the canonical Gaussian measure, by transferring the Cameron-Martin theorem from${\displaystyle ({\mathbb{R}}^{\mathbb{N}},\mathcal{B}({\mathbb{R}}^{\mathbb{N}}),{\gamma }^{\mathbb{N}}={\otimes }_{n\in \mathbb{N}}\gamma )}$ into a numerical model${\displaystyle X}$, the additive group of${\displaystyle \mathcal{H}}$ will define a quasi-automorphism group on${\displaystyle \mathrm{\Omega }}$. A construction can be done as follows: choose an orthonormal basis in${\displaystyle \mathcal{H}}$, let${\displaystyle {\tau }_{\alpha }(x)=x+\alpha }$ denote the translation on${\displaystyle {\mathbb{R}}^{\mathbb{N}}}$ by${\displaystyle \alpha }$, denote the map into the Cameron-Martin space by${\displaystyle j:\mathcal{H}\to {\ell }^2}$, denote

and${\displaystyle q:L^{\mathrm{\infty }-0}({\mathbb{R}}^{\mathbb{N}},\mathcal{B}({\mathbb{R}}^{\mathbb{N}}),{\gamma }^{\mathbb{N}})\to L^{\mathrm{\infty }-0}(\mathrm{\Omega },\mathcal{F},P),}$

we get a canonical representation of the additive group${\displaystyle \rho :\mathcal{H}\to \mathrm{End}(L^{\mathrm{\infty }-0}(\mathrm{\Omega },\mathcal{F},P))}$ acting on theendomorphisms by defining

${\displaystyle \rho (h)=q\circ {\tau }_{j(h)}\circ q^{-1}.}$

One can show that the action of${\displaystyle \rho }$ is extrinsic meaning it does not depend on the choice of basis for${\displaystyle \mathcal{H}}$, further${\displaystyle \rho (h+h^{\prime })=\rho (h)\rho (h^{\prime })}$for${\displaystyle h,h^{\prime }\in \mathcal{H}}$ and for theinfinitesimal generator of${\displaystyle (\rho (h){)}_h}$ that

${\displaystyle \underset{\epsilon \to 0}{lim}\frac{\rho (\epsilon h)-I}{\epsilon }=M_h}$

where${\displaystyle I}$ is the identity operator and${\displaystyle M_h}$ denotes the multiplication operator by the random variable${\displaystyle h\in \mathcal{H}}$ (acting on the endomorphisms). In the case of an arbitrary Hilbert space${\displaystyle \mathcal{G}}$ and the Segal model${\displaystyle \mathrm{Seg}(\mathcal{G})}$ one has${\displaystyle j:\mathcal{G}\to {\ell }^2}$ (and thus${\displaystyle \rho :\mathcal{G}\to \mathrm{End}(L^{\mathrm{\infty }-0}(\mathrm{\Omega },\mathcal{F},P))}$. Then the limit above becomes the multiplication operator by the random variable${\displaystyle W(g)}$ associated to${\displaystyle g\in \mathcal{G}}$.^[3]

For${\displaystyle F\in L^{\mathrm{\infty }-0}(\mathrm{\Omega },\mathcal{F},P)}$ and${\displaystyle h\in \mathcal{H}}$ one now defines the directional derivative

${\displaystyle ⟨DF,h⟩=D_{h}F=\underset{\epsilon \to 0}{lim}\frac{\left(\rho (\epsilon h)-I\right)F}{\epsilon }.}$

Given a Hilbert space${\displaystyle H}$ and a Segal model${\displaystyle \mathrm{Seg}(H)}$ with its Gaussian space${\displaystyle \mathcal{H}=\{W(h):h\in H\}}$. One can now deduce for${\displaystyle F\in L^{\mathrm{\infty }-0}(\mathrm{\Omega },\mathcal{F},P)}$ the integration by parts formula

${\displaystyle \mathbb{E}[D_{h}F]=\mathbb{E}[M_{W(h)}F]=\mathbb{E}[W(h)F]}$.^[4]

The usual invariance principle forLebesgue integration over the whole real line is that, for any real number ε and integrable functionf, thefollowing holds

${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)d\lambda (x)={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x+\epsilon )d\lambda (x)}$ and hence${\displaystyle {\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}^{\prime }(x)d\lambda (x)=0.}$

This can be used to derive theintegration by parts formula since, settingf =gh, it implies

${\displaystyle 0={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{f}^{\prime }d\lambda ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}(gh{)}^{\prime }d\lambda ={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}g{h}^{\prime }d\lambda +{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{g}^{\prime }hd\lambda .}$

A similar idea can be applied in stochastic analysis for the differentiation along a Cameron-Martin-Girsanov direction. Indeed, let${\displaystyle h_s}$ be a square-integrablepredictable process and set

${\displaystyle \phi (t)={\int }_0^t{h}_{s}ds.}$

If${\displaystyle W}$ is aWiener process, theGirsanov theorem then yields the following analogue of the invariance principle:

${\displaystyle E(F(W+\epsilon \phi ))=E\left[F(W)\exp \left(\epsilon {\int }_0^1{h}_{s}d{w}_s-\frac{1}{2}{\epsilon }^2{\int }_0^1{h}_s^{2}ds\right)\right].}$

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

${\displaystyle E(⟨DF(W),\phi ⟩)=E[F(W){\int }_0^1{h}_{s}d{w}_s].}$

Here, the left-hand side is theMalliavin derivative of the random variable${\displaystyle F}$ in the direction${\displaystyle \phi }$ and the integral appearing on the right hand side should be interpreted as anItô integral.

One of the most useful results from Malliavin calculus is theClark–Ocone theorem, which allows the process in themartingale representation theorem to be identified explicitly. A simplified version of this theorem is as follows:

Consider the standard Wiener measure on the canonical space${\displaystyle C[0,1]}$, equipped with its canonical filtration. For${\displaystyle F:C[0,1]\to \mathbb{R}}$ satisfying which is Lipschitz and such thatF has a strong derivative kernel, in the sense thatfor${\displaystyle \phi }$ inC[0,1]

${\displaystyle \underset{\epsilon \to 0}{lim}\frac{1}{\epsilon }(F(X+\epsilon \phi )-F(X))={\int }_0^1{F}^{\prime }(X,dt)\phi (t)\mathrm{a}.\mathrm{e}.X}$

then

${\displaystyle F(X)=E(F(X))+{\int }_0^1{H}_{t}d{X}_t,}$

whereH is the previsible projection ofF'(x, (t,1]) which may be viewed as the derivative of the functionF with respect to a suitable parallel shift of the processX over the portion (t,1] of its domain.

This may be more concisely expressed by

${\displaystyle F(X)=E(F(X))+{\int }_0^{1}E(D_{t}F\mid {\mathcal{F}}_t)d{X}_t.}$

Much of the work in the formal development of the Malliavin calculus involves extending this result to the largest possible class of functionalsF by replacing the derivative kernel used above by the "Malliavin derivative" denoted${\displaystyle D_t}$ in the above statement of the result.^{[citation needed]}

TheSkorokhod integral operator, which is conventionally denoted δ, is defined as the adjoint of the Malliavin derivative in thewhite noise case when the Hilbert space is an${\displaystyle L^2}$ space; thus foru in the domain of the operator which is a subset of${\displaystyle L^2([0,\mathrm{\infty })\times \mathrm{\Omega })}$,forF in the domain of the Malliavin derivative, we require

${\displaystyle E(⟨DF,u⟩)=E(F\delta (u)),}$

where the inner product is that on${\displaystyle L^2[0,\mathrm{\infty })}$, viz.

${\displaystyle ⟨f,g⟩={\int }_0^{\mathrm{\infty }}f(s)g(s)ds.}$

The existence of this adjoint follows from theRiesz representation theorem for linear operators onHilbert spaces.

It can be shown that ifu is adapted then

${\displaystyle \delta (u)={\int }_0^{\mathrm{\infty }}{u}_{t}d{W}_t,}$

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.

The calculus allowsintegration by parts withrandom variables; this operation is used inmathematical finance to compute thesensitivities offinancial derivatives. The calculus has applications for example instochastic filtering.

This article includes alist of references,related reading, orexternal links,but its sources remain unclear because it lacksinline citations. Please helpimprove this article byintroducing more precise citations.(June 2011) (Learn how and when to remove this message)

• 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

• Quotations related toMalliavin calculus at Wikiquote
• Friz, Peter K. (2005-04-10)."An Introduction to Malliavin Calculus"(PDF). Archived fromthe original(PDF) on 2007-04-17. Retrieved2007-07-23. Lecture Notes, 43 pages

• Zhang, H. (2004-11-11)."The Malliavin Calculus"(PDF). Retrieved2004-11-11. Thesis, 100 pages

• Stochastic calculus
• Integral calculus
• Mathematical finance
• Calculus of variations
• Malliavin calculus

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