In
mathematics, the Skorokhod integral (also named Hitsuda-Skorokhod integral), often denoted
, is an
operator of great importance in the theory of
stochastic processes. It is named after the
Ukrainian mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
Anatoliy Skorokhod
Anatoliy Volodymyrovych Skorokhod ( uk, Анато́лій Володи́мирович Скорохо́д; September 10, 1930January 3, 2011) was a Soviet and Ukrainian mathematician.
Skorokhod is well-known for a comprehensive treatise on the ...
and
japanese
Japanese may refer to:
* Something from or related to Japan, an island country in East Asia
* Japanese language, spoken mainly in Japan
* Japanese people, the ethnic group that identifies with Japan through ancestry or culture
** Japanese diaspor ...
mathematician
Masuyuki Hitsuda. Part of its importance is that it unifies several concepts:
*
is an extension of the
Itô integral to non-
adapted process In the study of stochastic processes, an adapted process (also referred to as a non-anticipating or non-anticipative process) is one that cannot "see into the future". An informal interpretation is that ''X'' is adapted if and only if, for every re ...
es;
*
is the
adjoint
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
of the
Malliavin derivative, which is fundamental to the stochastic
calculus of variations (
Malliavin calculus
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 ...
);
*
is an infinite-dimensional generalization of the
divergence
In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of t ...
operator from classical
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subjec ...
.
The integral was introduced by Hitsuda in 1972 and by Skorokhod in 1975.
Definition
Preliminaries: the Malliavin derivative
Consider a fixed
probability space
In probability theory, a probability space or a probability triple (\Omega, \mathcal, P) is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models t ...
and a
Hilbert space ;
denotes
expectation with respect to
Intuitively speaking, the Malliavin derivative of a random variable
in
is defined by expanding it in terms of Gaussian random variables that are parametrized by the elements of
and differentiating the expansion formally; the Skorokhod integral is the adjoint operation to the Malliavin derivative.
Consider a family of
-valued
random variables , indexed by the elements
of the Hilbert space
. Assume further that each
is a Gaussian (
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
) random variable, that the map taking
to
is a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
, and that the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value (magnitude and sign) of a given data set.
For a data set, the '' ari ...
and
covariance
In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the ...
structure is given by
for all
and
in
. It can be shown that, given
, there always exists a probability space
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
to be
, and then extending this definition to "
smooth enough" random variables. For a random variable
of the form
where
is smooth, the Malliavin derivative is defined using the earlier "formal definition" and the chain rule:
In other words, whereas
was a real-valued random variable, its derivative
is an
-valued random variable, an element of the space
. Of course, this procedure only defines
for "smooth" random variables, but an approximation procedure can be employed to define
for
in a large subspace of
; the
domain of
is the
closure of the smooth random variables in the
seminorm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk ...
:
This_space_is_denoted_by_
_and_is_called_the_
Watanabe–Sobolev_space.
_The_Skorokhod_integral
For_simplicity,_consider_now_just_the_case_
.__The_Skorokhod_integral_
_is_defined_to_be_the_
-adjoint_of_the_Malliavin_derivative_
.__Just_as_
_was_not_defined_on_the_whole_of_
,_
_is_not_defined_on_the_whole_of_
:__the_domain_of_
_consists_of_those_processes_
_in_
_for_which_there_exists_a_constant_
_such_that,_for_all_
_in_
,
\big, _\mathbf_ \langle_\mathrm_F,_u_\rangle__\big, _\leq_C(u)_\, _F_\, _.
The_Skorokhod_integral_of_a_process__in__is_a_real-valued_random_variable__in_;_if__lies_in_the_domain_of_,_then__is_defined_by_the_relation_that,_for_all_,
\mathbf_ _\,_\delta_u=_\mathbf_ \langle_\mathrmF,_u_\rangle__
Just_as_the_Malliavin_derivative__was_first_defined_on_simple,_smooth_random_variables,_the_Skorokhod_integral_has_a_simple_expression_for_"simple_processes":__if__is_given_by
u_=_\sum_^_F__h_
with__smooth_and__in_,_then
\delta_u_=_\sum_^_\left(_F__W(h_)_-_\langle_\mathrm_F_,_h__\rangle__\right).
_Properties
*_The_isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
_property:__for_any_process__in__that_lies_in_the_domain_of_,_\mathbf_\big (\delta_u)^_\big=_\mathbf_\int_, _u_t_, ^_dt_+_\mathbf_\int_D_s_u_t\,_D_t_u_s\,ds\,_dt._If__is_an_adapted_process,_then__for_,_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_\mathrm__(\delta_u)_=_\langle_u,_h_\rangle__+_\delta_(\mathrm__u),_where__stands_for_,_the_random_variable_that_is_the_value_of_the_process__at_"time"__in_.
*_The_Skorokhod_integral_of_the_product_of_a_random_variable__in__and_a_process__in__is_given_by_the_formula_\delta_(F_u)_=_F_\,_\delta_u_-_\langle_\mathrm_F,_u_\rangle_.
__Alternatives_
An_alternative_to_the_Skorokhod_integral_is_the_ Ogawa_integral.
_References
*_
*__
*_
{{Stochastic_processes
Definitions_of_mathematical_integration
Stochastic_calculushtml" ;"title="F, ^] + \mathbf , \mathrmF \, _^\big)^.
This space is denoted by and is called the Watanabe–Sobolev space.
The Skorokhod integral
For simplicity, consider now just the case . The Skorokhod integral is defined to be the -adjoint of the Malliavin derivative . Just as was not defined on the whole of , is not defined on the whole of : the domain of consists of those processes in for which there exists a constant such that, for all in ,
The Skorokhod integral of a process in is a real-valued random variable in ; if lies in the domain of , then is defined by the relation that, for all ,
Just as the Malliavin derivative was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if is given by
with smooth and in , then
Properties
* The isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
property: for any process in that lies in the domain of , If is an adapted process, then for , 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 where stands for , the random variable that is the value of the process at "time" in .
* The Skorokhod integral of the product of a random variable in and a process in is given by the formula
Alternatives
An alternative to the Skorokhod integral is the Ogawa integral.
References
*
*
*
{{Stochastic processes
Definitions of mathematical integration
Stochastic calculus>F, ^+ \mathbf , \mathrmF \, _^\big)^.
This space is denoted by and is called the Watanabe–Sobolev space.
The Skorokhod integral
For simplicity, consider now just the case . The Skorokhod integral is defined to be the -adjoint of the Malliavin derivative . Just as was not defined on the whole of , is not defined on the whole of : the domain of consists of those processes in for which there exists a constant such that, for all in ,
The Skorokhod integral of a process in is a real-valued random variable in ; if lies in the domain of , then is defined by the relation that, for all ,
Just as the Malliavin derivative was first defined on simple, smooth random variables, the Skorokhod integral has a simple expression for "simple processes": if is given by
with smooth and in , then
Properties
* The isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' me ...
property: for any process in that lies in the domain of , If is an adapted process, then for , 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 where stands for , the random variable that is the value of the process at "time" in .
* The Skorokhod integral of the product of a random variable in and a process in is given by the formula
Alternatives
An alternative to the Skorokhod integral is the Ogawa integral.
References
*
*
*
{{Stochastic processes
Definitions of mathematical integration
Stochastic calculus