Skorokhod integral

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In mathematics, the Skorokhod integral, often denoted δ, is an operator of great importance in the theory of stochastic processes. It is named after the Ukrainian mathematician Anatoliy Skorokhod. Part of its importance is that it unifies several concepts:

Definition[edit]

Preliminaries: the Malliavin derivative[edit]

Consider a fixed probability space (Ω, Σ, P) and a Hilbert space H; E denotes expectation with respect to P

\mathbf{E} [X] := \int_{\Omega} X(\omega) \, \mathrm{d} \mathbf{P}(\omega).

Intuitively speaking, the Malliavin derivative of a random variable F in Lp(Ω) 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

\mathbf{E} [W(h)] = 0,
\mathbf{E} [W(g)W(h)] = \langle g, h \rangle_{H},

for all g and h in H. It can be shown that, given H, there always exists a probability space (Ω, Σ, 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}), \ldots, W(h_{n})),

where f : Rn → R is smooth, the Malliavin derivative is defined using the earlier “formal definition” and the chain rule:

\mathrm{D} F := \sum_{i = 1}^{n} \frac{\partial f}{\partial x_{i}} (W(h_{1}), \ldots, W(h_{n})) h_{i}.

In other words, whereas F was a real-valued random variable, its derivative DF is an H-valued random variable, an element of the space Lp(Ω;H). Of course, this procedure only defines DF for “smooth” random variables, but an approximation procedure can be employed to define DF for F in a large subspace of Lp(Ω); the domain of D is the closure of the smooth random variables in the seminorm :

\| F \|_{1, p} := \big( \mathbf{E}[|F|^{p}] + \mathbf{E}[\| \mathrm{D}F \|_{H}^{p}] \big)^{1/p}.

This space is denoted by D1,p and is called the Watanabe-Sobolev space.

The Skorokhod integral[edit]

For simplicity, consider now just the case p = 2. The Skorokhod integral δ is defined to be the L2-adjoint of the Malliavin derivative D. Just as D was not defined on the whole of L2(Ω), δ is not defined on the whole of L2(Ω; H): the domain of δ consists of those processes u in L2(Ω; H) for which there exists a constant C(u) such that, for all F in D1,2,

\big| \mathbf{E} [ \langle \mathrm{D} F, u \rangle_{H} ] \big| \leq C(u) \| F \|_{L^{2} (\Omega)}.

The Skorokhod integral of a process u in L2(Ω; H) is a real-valued random variable δu in L2(Ω); if u lies in the domain of δ, then δu is defined by the relation that, for all F ∈ D1,2,

\mathbf{E} [F \, \delta u] = \mathbf{E} [ \langle \mathrm{D}F, u \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 = \sum_{j = 1}^{n} F_{j} h_{j}

with Fj smooth and hj in H, then

\delta u = \sum_{j = 1}^{n} \left( F_{j} W(h_{j}) - \langle \mathrm{D} F_{j}, h_{j} \rangle_{H} \right).

Properties[edit]

  • The isometry property: for any process u in L2(Ω; H) that lies in the domain of δ,
\mathbf{E} \big[ (\delta u)^{2} \big] = \mathbf{E} \big[ \| u \|_{H}^{2} \big] + \mathbf{E} \big[ \| \mathrm{D} u \|_{H \otimes H}^{2} \big].
If u is an adapted process, then the second term on the right-hand side is zero, the Skorokhod and Itō integrals coincide, and the above equation becomes the Itō isometry.
  • The derivative of a Skorokhod integral is given by the formula
\mathrm{D}_{h} (\delta u) = \langle u, h \rangle_{H} + \delta (\mathrm{D}_{h} u),
where DhX stands for (DX)(h), the random variable that is the value of the process DX at “time” h in H.
  • The Skorokhod integral of the product of a random variable F in D1,2 and a process u in dom(δ) is given by the formula
\delta (F u) = F \, \delta u - \langle \mathrm{D} F, u \rangle_{H}.

References[edit]