Skorokhod integral explained

In mathematics, the Skorokhod integral, also named Hitsuda–Skorokhod integral, often denoted

\delta , is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod and Japanese mathematician Masuyuki Hitsuda. Part of its importance is that it unifies several concepts:\delta is an extension of the Itô integral to non-adapted processes;\delta is the adjoint of the Malliavin derivative, which is fundamental to the stochastic calculus of variations (Malliavin calculus);\delta is an infinite-dimensional generalization of the divergence operator from classical vector calculus.

The integral was introduced by Hitsuda in 1972^[1] and by Skorokhod in 1975.^[2]

Definition

Preliminaries: the Malliavin derivative

(\Omega, \Sigma, P ) and a Hilbert space H ; E denotes expectation with respect to P

$$\backslash mathbf\; [X]\; :=\; \backslash int\_\backslash Omega\; X(\backslash omega)\; \backslash ,\; \backslash mathrm\; \backslash mathbf(\backslash omega).$$

Intuitively speaking, the Malliavin derivative of a random variable

F in L ^{p(\Omega)} is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of H and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.

Consider a family of

R -valued random variables W(h) , indexed by the elements h of the Hilbert space H . Assume further that each W(h) is a Gaussian (normal) random variable, that the map taking h to W(h) is a linear map, and that the mean and covariance structure is given by

$$\backslash mathbf\; [W(h)]\; =\; 0,$$$$\backslash mathbf\; [W(g)\; W(h)]\; =\; \backslash langle\; g,\; h\; \backslash rangle\_H,$$

for all

g and h in H . It can be shown that, given H , there always exists a probability space (\Omega, \Sigma, P ) and a family of random variables with the above properties. The Malliavin derivative is essentially defined by formally setting the derivative of the random variable W(h) to be h , and then extending this definition to "smooth enough" random variables. For a random variable F of the form

$$F\; =\; f(W(h\_1),\; \backslash ldots,\; W(h\_n)),$$

where

f : R ^{n} \to R is smooth, the Malliavin derivative is defined using the earlier "formal definition" and the chain rule:

$$\backslash mathrm\; F\; :=\; \backslash sum\_^n\; \backslash frac\; (W(h\_1),\; \backslash ldots,\; W(h\_n))\; h\_i.$$

In other words, whereas

F was a real-valued random variable, its derivative D F is an H -valued random variable, an element of the space L ^{p(\Omega;} H) . Of course, this procedure only defines D F for "smooth" random variables, but an approximation procedure can be employed to define D F for F in a large subspace of L ^{p(\Omega)} ; the domain of D is the closure of the smooth random variables in the seminorm :

$$\backslash |\; F\; \backslash |\_\; :=\; \backslash big(\backslash mathbf[|F|^p]\; +\; \backslash mathbf[\backslash |\; \backslash mathrm\{D\}F\; \backslash |\_H^p]\; \backslash big)^.$$

This space is denoted by

D ^{1,} and is called the Watanabe–Sobolev space.

The Skorokhod integral

For simplicity, consider now just the case

p = 2 . The Skorokhod integral \delta is defined to be the L ^{2} -adjoint of the Malliavin derivative D . Just as D was not defined on the whole of L ^{2(\Omega)} , \delta is not defined on the whole of L ^{2(\Omega;} H) : the domain of \delta consists of those processes u in L ^{2(\Omega;} H) for which there exists a constant C(u) such that, for all F in D ^{1,} ,

$$\backslash big|\; \backslash mathbf\; [\backslash langle\; \backslash mathrm\{D\}\; F,\; u\; \backslash rangle\_\{H\}\; ]\; \backslash big|\; \backslash leq\; C(u)\; \backslash |\; F\; \backslash |\_.$$

The Skorokhod integral of a process

u in L ^{2(\Omega;} H) is a real-valued random variable \delta u in L ^{2(\Omega)} ; if u lies in the domain of \delta , then \delta u is defined by the relation that, for all F \in D ^{1,} ,

$$\backslash mathbf\; [F\; \backslash ,\; \backslash delta\; u]\; =\; \backslash mathbf\; [\backslash langle\; \backslash mathrm\{D\}F,\; u\; \backslash rangle\_\{H\}\; ].$$

Just as the Malliavin derivative

D was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if u is given by

$$u\; =\; \backslash sum\_^\; F\_\; h\_$$

with

F _{j} smooth and h _{j} in H , then

$$\backslash delta\; u\; =\; \backslash sum\_^\; \backslash left(F\_\; W(h\_)\; -\; \backslash langle\; \backslash mathrm\; F\_,\; h\_\; \backslash rangle\_\; \backslash right).$$

Properties

• The isometry property: for any process

u in D ^{1,} that lies in the domain of \delta , $$\backslash mathbf\; \backslash big[(\backslash delta\; u)^\{2\}\; \backslash big]\; =\; \backslash mathbf\; \backslash int\; |\; u\_t\; |^\; dt\; +\; \backslash mathbf\; \backslash int\; D\_s\; u\_t\backslash ,\; D\_t\; u\_s\backslash ,ds\backslash ,\; dt.$$ If u is an adapted process, then D _{s} u _{t} = 0 for s > t , so the second term on the right-hand side vanishes. The Skorokhod and Itô integrals coincide in that case, and the above equation becomes the Itô isometry.

• The derivative of a Skorokhod integral is given by the formula $$\backslash mathrm\_\; (\backslash delta\; u)\; =\; \backslash langle\; u,\; h\; \backslash rangle\_\; +\; \backslash delta\; (\backslash mathrm\_\; u),$$ where

D _{h} X stands for ( D X)(h) , the random variable that is the value of the process D X at "time" h in H .

• The Skorokhod integral of the product of a random variable

F in D ^{1,} and a process u in \operatorname{dom}(\delta) is given by the formula $$\backslash delta\; (F\; u)\; =\; F\; \backslash ,\; \backslash delta\; u\; -\; \backslash langle\; \backslash mathrm\; F,\; u\; \backslash rangle\_.$$

Alternatives

An alternative to the Skorokhod integral is the Ogawa integral.

References

• • Book: Ocone , Daniel L. . A guide to the stochastic calculus of variations. Stochastic analysis and related topics (Silivri, 1986). Lecture Notes in Math. 1316. 1 - 79. Springer. Berlin. 1988.
• Web site: Sanz-Solé . Marta . Marta Sanz-Solé. Applications of Malliavin Calculus to Stochastic Partial Differential Equations (Lectures given at Imperial College London, 7 - 11 July 2008). 2008. 2008-07-09.

Notes and References

1. Masuyuki. Hitsuda. Formula for Brownian partial derivatives. Second Japan-USSR Symp. Probab. Th.2.. 1972. 111–114.
2. Kuo. Hui-Hsiung. 2014. The Itô calculus and white noise theory: a brief survey toward general stochastic integration. Communications on Stochastic Analysis. 8. 1. 10.31390/cosa.8.1.07. free.