1. Introduction
For the analysis of dynamical systems whose mathematical models include stochastic differential equations, numerical methods for their solution are often used. To obtain high accuracy of the approximation of output processes, it is necessary to apply numerical methods with high order of strong or mean-square convergence based on Taylor–Itô or Taylor–Stratonovich expansions [
1,
2,
3,
4]. These methods involve modeling iterated Itô or Stratonovich stochastic integrals, respectively, which can be represented as multiple stochastic integrals of functions
where
,
and
,
,
is the unit step function:
If , then the notation is used instead of for simplicity.
In the general theory of multiple and iterated stochastic integrals, a wider class of functions is of interest:
where
. The function
is called the factorized Volterra-type function [
5]. Here and further,
denotes the space of square-integrable functions
,
. Also, the space of continuous functions
is denoted by
.
The multiple Itô stochastic integral introduced in [
6] is defined for any function
, and such an integral can be represented as a multiple random series [
7]. The multiple Stratonovich stochastic integral [
8,
9] is more complicated, and when it is represented as a multiple random series, the trace convergence problem arises [
10]. The representation of multiple stochastic integrals by multiple random series is based on the expansion of
f in the orthogonal series using a basis of
.
For clarity, we can use an analogy with the theory of linear operators. In fact, the differences between functions, for which we can define multiple Itô and Stratonovich stochastic integrals, are partly similar to the differences between Hilbert–Schmidt operators and trace class operators [
11], respectively.
If we consider multiple Itô and Stratonovich stochastic integrals for numerical methods, which are used for solving stochastic differential equations, then it is enough to restrict ourselves to functions (
2). In fact, we can only study functions (
1) for this purpose [
3]. However, multiple (or iterated) stochastic integrals should be specified with respect to all possible combinations of components of the multidimensional Wiener process [
1,
3,
12]. In [
3,
13,
14], the trace convergence problem in this context is studied in detail with additional smoothness conditions on the weights
and with a restriction on both a parameter
k and a basis of
.
Note that some multiple Stratonovich stochastic integrals coincide with corresponding multiple Itô stochastic integrals, but for other Stratonovich integrals, the trace convergence problem remains. One variant of the briefly described problem is solved in this paper for the weights from and without additional restrictions on both a parameter k and a basis of .
The motivation for this study is that the solution to the trace convergence problem provides a theoretical basis for the representation of iterated Stratonovich stochastic integrals as multiple random series and for their mean-square approximation based on partial sums of these series. This is an important component for the implementation of numerical methods for solving stochastic differential equations with high order of strong or mean-square convergence. Such methods can be used to simulate stochastic processes in different fields [
15,
16,
17].
2. Preliminary Discussion and Problem Statement
Let
and
be independent Wiener processes defined on a probability space
. Denote by
and
two linear operators that establish a correspondence between a function and multiple stochastic integrals for that function. The operator
corresponds to the multiple Itô stochastic integral, and the operator
corresponds to the multiple Stratonovich stochastic integral,
:
where
k is the integral multiplicity, which is the same as the number of arguments of
f, and the symbol ∘ is to distinguish Itô and Stratonovich stochastic integrals.
Further, we use the following notations:
is an orthonormal basis of
, and
are independent random variables having standard normal distribution for
and
. Then, according to properties of multiple stochastic integrals [
18,
19], we have
where ∗ means the Wick product defined for this case in terms of Hermite polynomials [
20,
21],
If
, then this function can be represented as the orthogonal series [
22], i.e.,
where
Formally (without considering the convergence issues), we can write that
and multiple Itô and Stratonovich stochastic integrals do not generally coincide.
Consider a simple example for stochastic integrals of multiplicity
under condition
. Let
and
where
It is known [
6,
7] that
where the multiple random series on the right-hand side converges in the mean-square sense (the equivalent relation is given in [
3] using different notations). But the multiple random series on the right-hand side of the equality
can diverge, and the equality itself may not make sense, since
where
is the Kronecker delta, i.e.,
and the convergence of series
does not follow from condition
. This series can diverge or it can converge conditionally, then its sum depends on a basis
.
Thus, the operator
is defined only on some linear subspace of
, although the domain of the operator
coincides with
. In fact, the equality
makes sense if the series (
5) converges absolutely, and its sum does not depend on a basis
. It this case, we can write that
but the integral on the right-hand side should be understood in a special way, since any function
is defined up to a set of measure zero, while the set
has measure zero (on the plane). This integral is equal to the expectation of the multiple Stratonovich stochastic integral
.
If
, then there is no such a problem, and the equality
holds (“a.s.” means “almost surely” or “with probability 1”).
For stochastic integrals of multiplicity
, the series
can appear depending on values
, where
are expansion coefficients (
3) of
, and the indices, over which the summation is not carried out, are parameters. Under certain conditions, the following equalities
hold.
The number of possible variants of such series increases with the multiplicity
k. For example, if
, then it is required to consider the series
where
are expansion coefficients (
3) of
, and the set of series is determined by values
. Here the indices, over which the summation is not carried out, are also parameters.
Such series appear only if some values among
are equal. In particular, for
under condition
, we need to consider only one series
or for
under condition
, we need to consider only three series
Now we return to stochastic integrals of multiplicity
. To establish convergence conditions for the series (
5), it is useful to apply the theory of trace class operators, but the simplest example shows that we have to be careful. Indeed, consider the Volterra integral operator
defined as
where
is the kernel function.
It is known [
22,
23] that the operator
is not traceable, but it can be shown that the property
is satisfied for an arbitrary basis
, where
Note that the specified sum of that series corresponds to the well-known relation for Itô and Stratonovich stochastic integrals [
3,
24]:
where
and
where
means the expectation operator that associates a random variable with its expected value.
Nevertheless, it is possible to apply the theory of trace class operators by symmetrization of
. Linear operators
and
have one useful property: they can be considered on the set of equivalence classes constructed by symmetrization. In particular,
where
and expansion coefficients (
4) do not change under condition
for such functions,
is the symmetrization operator.
In fact, if
, then
hence
where
, and the linear operator with trivial kernel
c is traceable.
Iterated stochastic integrals can be represented by solutions to corresponding systems of stochastic differential equations. For the integral
we have
, where
is the component of the solution to the system of Itô stochastic differential equations
and
.
Similarly, for the integral
we have
, where
is the component of the solution to the system of Stratonovich stochastic differential equations
and
.
Note that for the system (
6) we can derive the equivalent system of Stratonovich stochastic differential equations
and the system (
7) can be transformed into the equivalent system of Itô stochastic differential equations
Here, in addition to above notations, we use the following: , , is the Kronecker delta.
The structure of obtained equations shows that the differences between Itô and Stratonovich stochastic differential equations exist only when some values
with neighboring indices are equal. And these differences form traces, which are sums of expansion coefficients
of the function
given by the formula (
2):
Traces are formed only by summing over neighboring pairs of indices. If we assume that possible parameters are fixed (all indices, over which the summation is not carried out, are parameters), then it suffices to consider the following series:
The problem statement is to prove the absolute convergence of these series and to express their sums as a functional depending on the weights .
Next sections present proofs of the absolute convergence of these series regardless of a basis . The main results are formulated separately for cases and , .
3. Main Result for the Case
Consider the Hilbert–Schmidt operator
with the kernel
given by the relation (“a.e.” means “almost everywhere”)
This operator is the trace class operator [
23,
25] if there exist functions
such that
Conversely, if
is the trace class operator, then there exists a (nonunique) representation (
9) for its kernel.
One of the simplest examples of trace class operators is the operator with the kernel
Here, it suffices to show that
f can be represented by the equality (
9). For this, we assume that
and
. Then
Let
be the averaging operator [
23]:
where
, i.e.,
is a linear operator, which associates a function
f with a continuous function that has well-defined value at each point
as the average value of
f on the square
centered at this point (
f should be defined by zero outside the square
). Then
a.e. on
, where
Theorem 1 ([
23,
25])
. Let be the trace class operator with the kernel and let be a basis of . Thenwhere are expansion coefficients (
4)
of f relative to the basis , and Remark 1. The series in the relation (
11)
is called the trace of the operator . It converges absolutely, and its sum does not depend on a basis . Next, we prove two technical lemmas, after that we can formulate and prove one of the main results.
Lemma 1. Operators with symmetric kernelswhere and , are trace class operators. Proof of Lemma 1. Let
and
with
and
. According to the representation (
9), we have
The term
defines the trace class operator with the kernel (
10). In addition, we can restrict ourselves to conditions
and
, so the function (
12) defines the trace class operator.
Let
and
,
and
. Using the representation (
9), we obtain
Similarly, the function (
13) also defines the trace class operator, since the term
defines the trace class operator with the kernel (
10). □
Lemma 2. Let be polynomials. Then the operator with the symmetric kernelis the trace class operator. Proof of Lemma 2. The operator
with the symmetric kernel
is the trace class operator.
Indeed, for
we have the kernel
, which satisfies the condition (
10). For
, the required result follows from Lemma 1. For functions (
12) and (
13), conditions
and
, respectively, should be satisfied.
The function
f is represented as a linear combination of functions
g, and it defines the trace class operator
, since the space of trace class operators is linear [
26]. □
Theorem 2. Let and let be a basis of . Then Proof of Theorem 2. Define functions
as follows:
Then their expansion coefficients
and
relative to the basis
are defined by the formula (
4):
and for them the condition
holds due to the symmetry,
Let
i.e.,
, where
is the symmetrization operator. Then expansion coefficients
of
f are determined as
Moreover, we can write that
and this means that the equality (
14) is equivalent to the relation (
11), which holds for the trace class operator
with some kernel
f according to Theorem 1.
If are polynomials, then the operator with the kernel f is the trace class operator according to Lemma 2, and an arbitrary function from can be approximated using polynomials (polynomials are dense in ).
Further, let
and
where
and
are expansion coefficients of
relative to orthonormal Legendre polynomials
, i.e.,
Then we can establish the following equality for arbitrary
n and
m:
where the series converges absolutely, and its sum does not depend on a basis
.
If we fix
n in the equality (
15), then it defines a bounded linear functional in
, which is given by the function
(the trace of operator is also a bounded linear functional but in the space of trace class operators [
26]). Letting
, we obtain a bounded linear functional given by the function
:
Similarly, letting
, we obtain the relation
which proves the theorem. □
4. Main Result for the Case for
Define the Hilbert–Schmidt operator
with the kernel
,
for
:
It is known [
23,
25] that if the function
f is represented as
where
, then
is the trace class operator.
Let
be the averaging operator [
23]:
where
, i.e.,
is a linear operator, which associates a function
f with a continuous function that has well-defined value at each point
as the average value of
f on the hypercube
centered at this point (
f should be defined by zero outside the hypercube
). In this case, we have
a.e. on
, where
Theorem 3 ([
23,
25])
. Let be the trace class operator with the kernel and let be a basis of . Thenwhere are expansion coefficients (
3)
of f relative to the basis , and Remark 2. - 1.
The series in the relation (
17)
is the trace of the operator . It converges absolutely, and its sum does not depend on a basis . - 2.
Obviously, Theorem 1 is the particular case of Theorem 3.
- 3.
There is the trace-oriented definition of the averaging operator [25]. However, the above definition naturally agrees with the definition of the multiple Stratonovich stochastic integral from [9,19], and the main results presented in this paper are directly related to such integrals.
Now we give the technical lemma and then formulate and prove a more general result compared to Theorem 2.
Lemma 3. If the function defines the trace class operator , , then the operator with the kernelis the trace class operator. Proof of Lemma 3. Since
is the trace class operator, for its kernel
there exists a representation (
9), i.e.,
but then the representation (
16) holds for the function
f if
where
. Hence,
is the trace class operator. □
Theorem 4. Let , , and let be a basis of . Thenwhere . Proof of Theorem 4. Define functions
as follows:
and let
If
are polynomials, then the operator
with the kernel
is the trace class operator according to Lemma 2. Therefore, the operator
with the kernel
is the trace class operator according to Lemma 3, where
and for
the function
is obtained from
by permutation of variables in pairs
if the binary representation of
p is
and
. So, values of
f are not change by permutation of variables
and
,
, i.e.,
f is the symmetrized function relatively pairs
:
where
is the corresponding symmetrization operator.
Functions of type (
20) allow to obtain an approximation to the function
where, for example,
and then
x coincides with the function
given by the formula (
2), and its expansion coefficients
satisfy the relation (
8).
We can represent
y as a product of two functions of
arguments:
where the first function depends on variables
for
and
, and the second one depends on variables
for
and
, where
under condition
. Thus, we separate variables so that the list of arguments of each function does not include both variables that form any pair
:
Further, represent function
as
where
and
are expansion coefficients of
relative to orthonormal Legendre polynomials
, i.e.,
Moreover, for functions
we have
since
is a linear bounded operator [
23].
The function
defines the trace class operator because
is the function of type (
20). Its expansion coefficients
relative to the basis
are given by the formula (
3):
Expansion coefficients
of
can be similarly determined. According to both the linearity and the symmetry, they are related to
by
Since the convergence in the norm implies the weak convergence, expansion coefficients for limit functions can be defined as follows:
Further, we can write that
and in accordance with Theorem 3 we have the following equality for arbitrary
n and
m:
where the series converge absolutely, and their sums do not depend on a basis
. Similar relations hold for functions
and
f.
We can fix
n in the equality (
21). Then it defines a bounded linear functional in
, which is given by the function
(the trace of operator is also a bounded linear functional but in the space of trace class operators [
26]). Letting
, we obtain a bounded linear functional given by the function
:
Letting
, we obtain the following result:
hence
i.e., the theorem is proved. □
Remark 3. - 1.
A transition to symmetrized Volterra-type kernels is made in proving Theorems 2 and 4. The series in relations (
14)
and (
19)
converge regardless of a basis , and as a consequence they converge absolutely. - 2.
Certainly, Theorem 2 is the particular case of Theorem 4. However, it is useful to formulate and prove Theorem 2 separately for two reasons. Firstly, for the case , the proof is more simple, and it gives an idea of the proof in the general case. Secondly, this theorem is sufficient for solving the trace convergence problem if we consider not multiple series but iterated ones, for example,instead of -
For solving the trace convergence problem with iterated series, it suffices to apply Theorem 2 iteratively. This approach seems appropriate for applications to iterated Stratonovich stochastic integrals.
- 3.
How the trace convergence problem is related to the definition of the multiple Stratonovich stochastic integral is shown in [27].
As an example, we can find the sum of series
where
are expansion coefficients (
8) of the function
given by the formula (
1) for
and
,
.
Applying Theorem 4, we obtain
and then we should integrate sequentially, i.e.,
For the particular case
when
, we conclude that
where expansion coefficients
corresponds to the function
, i.e.,
These values correspond to expectations of the simplest iterated Stratonovich stochastic integrals of even multiplicities:
where
(
),
means the expectation operator.
Next, we present some numerical results. In
Table 1, partial sums of series
are given under conditions that
and the basis
is chosen as follows:
i.e., we consider three cases: the Fourier basis, cosines (for expansion of even functions in Fourier series), and sines (for expansion of odd functions in Fourier series). For
and for both bases (F) and (C), partial sums coincide with the exact value, since in this case
is the constant function. Otherwise, partial sums approximate the corresponding exact values.
Table 2 gives the estimates of the expectation of iterated Stratonovich stochastic integrals
and
under the same conditions. We use the following partial sums of multiple random series for their simulation:
where
are independent random variables having standard normal distribution for
and
(we assume that
). These estimates are obtained from
realizations of iterated Stratonovich stochastic integrals. Obviously, they correspond to partial sums from
Table 1.
Moreover, these numerical results together with data from [
4,
21] show that the Fourier basis, which used for approximation of iterated Stratonovich stochastic integrals in the Milstein method and then in the strong 1.5 order method [
1,
2] does not provide high accuracy. This is noted in [
3] based on a comparison the Fourier basis with Legendre polynomials. However, the presented result indicates that trigonometric functions can be effectively used for the approximation of iterated Stratonovich stochastic integrals, but it is only necessary to restrict ourselves to cosines.