Brownian functional

A certain random variable defined on the Wiener space (cf. Wiener space, abstract). Let $C$ be the space of continuous functions from $\mathbf R_+$ to $\mathbf R^n$ vanishing at zero, let $\mathcal B$ be its Borel $\sigma$-field and let $m$ be the Wiener measure, which is the probability measure on $(C,\mathcal B)$ making the coordinate mappings a Brownian motion. Then a Brownian functional is a measurable mapping defined on the probability space $(C,\mathcal B,m)$ with (generally) real values.

Such random variables are distinguished for several reasons:

a) they are met in a great number of applications, including filtering and mathematical finance;

b) they arise in connection with classical potential theory, cf. [a1];

c) stochastic analysis naturally yields Brownian functionals defined by stochastic integrals or stochastic differential equations, cf. [a2], [a3].

In many recent works, the motivation to study Brownian functionals comes from their irregularity even when their definition is simple. A Wiener stochastic integral with respect to the one-dimensional Brownian motion is generally discontinuous, cf. [a4]. From dimension two onwards appears a particularly deep irregularity; for example, Lévy's area, [a3], [a5],

\[\frac12\int\limits_0^1B_s^1\,dB_s^2-\frac12\int\limits_0^1B_s^2\,dB_s^1\]

is not Riemann-integrable, even when truncated, and the same happens for solutions of stochastic differential equations not satisfying the commutativity condition, [a3]. This non-Riemann-integrability is essential and does not depend on the Borelian version. It causes difficulties in numerical computations and simulation, [a4].

Some regularity results exist nevertheless. Brownian functionals are often approximately continuous (cf. Approximate continuity) in a neighbourhood of a point of the Wiener space which is a regular function (e.g., belongs to $C^\infty$). This has been first proved by D. Stroock and S.R. Varadhan ([a6] and [a3]) for solutions of stochastic differential equations (cf. Stochastic differential equation) with regular coefficients in connection with the theorem on the support of a diffusion. Hence it makes sense to define their values at such points. Although the set of these regular Brownian paths is negligible with respect to the Wiener measure, certain Brownian functionals are completely determined by their restriction to this set. This fact is the keystone of several recent works on the concept of a skeleton of a Brownian functional, [a7], [a8].

Moreover, an important trend of research in the stochastic calculus of variation, [a10], [a11], [a12], allows one to obtain regularity results for the laws of Brownian functionals following ideas initiated by P. Malliavin. It is also possible to pullback on the Wiener space measures or distributions defined on $\mathbf R^d$. This gives distributions in the sense of Watanabe, [a9], [a10], which are generalized Brownian functionals (similarly to Schwartz distributions, which are generalized functions, cf. Generalized function).

An interesting tool for studying Brownian functionals and generalized Brownian functionals are Wiener chaos expansions (cf. also Wiener chaos decomposition; [a9], [a11], [a12], [a13]), which, in classical form, express a square-integrable Brownian functional as the sum of a series of multiple Wiener–Itô integrals. Using this approach, the existence of the skeleton of a Brownian functional is connected with transformation of multiple Wiener–Itô integrals into multiple Stratonovich integrals (cf. also Stratonovich integral), which involves questions about the existence of traces for certain operators or kernels, [a14], [a15].

References

[a1]	S.C. Port, Ch.J. Stones, "Brownian motion and classical potential theory" , Acad. Press (1978)
[a2]	Ph. Protter, "Stochastic integration and differential equations" , Springer (1990)
[a3]	N. Ikeda, Sh. Watanabe, "Stochastic differential equations and diffusion processes" , North-Holland and Kodansha (1981)
[a4]	N. Bouleau, D. Lépingle, "Numerical methods for stochastic processes" , Wiley (1994)
[a5]	H. Sugita, "Various topologies on the Wiener space and Lévy's stochastic area" Probab. Th. Rel. Fields , 91 (1992) pp. 283–296
[a6]	D. Stroock, S.R. Varadhan, "On the support of a diffusion process with application to the strong maximum principle" , Proc. Sixth Berkeley Symp. Math. Stat. and Probab. , III , Univ. California Press (1972) pp. 333–359
[a7]	H. Sugita, "Properties of holomorphic Wiener functions, skeletons, contractions and local Taylor expansions" Probab. Th. Rel. Fields , 100 (1994) pp. 117–130
[a8]	S. Fang, J. Ren, "Sur les squelettes et les dérivées de Malliavin des fonctions holomorphes sur l'espace de Wiener complexe" J. Math. Kyoto Univ. , 33 (1993) pp. 749–764
[a9]	Sh. Watanabe, "On stochastic differential equations and Malliavin calculus" Tata Inst. Fundam. Research , 73 , Springer (1979)
[a10]	Sh. Watanabe, "Malliavin calculus in term of generalized Wiener functionals" , Theory and Applications of Random Fields , Lecture Notes in Control and Information Sciences , 49 , Springer (1983)
[a11]	D. Nualart, "The Malliavin calculus and related topics" , Springer (1995)
[a12]	N. Bouleau, F. Hirsch, "Dirichlet forms and analysis on Wiener space" , W. de Gruyter (1991)
[a13]	J.A. Yan, "Développement des distributions suivant les chaos de Wiener et applications à l'analyse stochastique" , Sem. de Probab. XXI , Lecture Notes in Mathematics , 1247 , Springer (1987) pp. 27–32
[a14]	R. Léandre, P.A. Meyer, "Sur le développement d'une diffusion en chaos de Wiener" , Sem. de Probab. XXII , Lecture Notes in Mathematics , 1372 , Springer (1989) pp. 161–164
[a15]	D. Nualart, M. Zakai, "Multiple Wiener–Itô integrals possessing a continuous extension" Probab. Th. Rel. Fields , 85 (1990) pp. 134–145