Malliavin calculus

Overview

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, significant contributors such as Seiichiro Kusuoka, Daniel W. Stroock, Jean-Michel Bismut, Shinzo Watanabe, Ichirō Shigekawa, and so on completed the foundations for the field. 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

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.

Gaussian probability space

Consider a Wiener functional

F

{\displaystyle F}

(a functional from the classical Wiener space) and consider the task of finding a derivative for it. The natural idea would be to use the Gateaux derivative

D

g

F [ f ] :=

lim

τ → 0

F [ f + τ g ] − F [ f ]

τ

{\displaystyle D_{g}F[f]:=\lim _{\tau \to 0}{\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 irreducible Gaussian probability space

X = ( Ω ,

F

, P ,

H

)

{\displaystyle X=(\Omega ,{\mathcal {F}},P,{\mathcal {H}})}

. This is a (complete) probability space

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

∫

− ∞

∞

f ( x )

d λ ( x ) =

∫

− ∞

∞

f ( x + ε )

d λ ( x )

{\displaystyle \int _{-\infty }^{\infty }f(x)\,d\lambda (x)=\int _{-\infty }^{\infty }f(x+\varepsilon )\,d\lambda (x)}

and hence

∫

− ∞

∞

f ′

( x )

d λ ( x ) = 0.

{\displaystyle \int _{-\infty }^{\infty }f'(x)\,d\lambda (x)=0.}

Clark–Ocone formula

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: Consider the standard Wiener measure on the canonical space

C [ 0 , 1 ]

{\displaystyle C[0,1]}

, equipped with its canonical filtration. For

F : C [ 0 , 1 ] →

R

{\displaystyle F:C[0,1]\to \mathbb {R} }

satisfying

E ( F ( X

)

2

) < ∞

{\displaystyle E(F(X)^{2})<\infty }

which is Lipschitz and such that F has a strong derivative kernel, in the sense that for

φ

{\displaystyle \varphi }

in C[0,1]

lim

ε → 0

1 ε

Skorokhod integral

The Skorokhod integral operator, which is conventionally denoted δ, is defined as the adjoint of the Malliavin derivative in the white noise case when the Hilbert space is an

L

2

{\displaystyle L^{2}}

space; thus for u in the domain of the operator which is a subset of

L

2

( [ 0 , ∞ ) × Ω )

{\displaystyle L^{2}([0,\infty )\times \Omega )}

, for F in the domain of the Malliavin derivative, we require

E ( ⟨ D F , u ⟩ ) = E ( F δ ( u ) ) ,

{\displaystyle E(\langle DF,u\rangle )=E(F\delta (u)),}

where the inner product is that on

L

2

[ 0 , ∞ )

{\displaystyle L^{2}[0,\infty )}

, viz.

⟨ f , g ⟩ =

∫

Applications

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

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

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