1 Introduction
The present paper is devoted to the approximate solutions of the Fredholm integral equations of the second kind on the interval , with the kernel of the form where the numerator is bounded and continuous a.e. with respect to the Lebesgue measure but can have the points of discontinuity on , due to the need to approximately solve such equations when considering some optimization problems associated with mixed Brownianfractional Brownian motion. To the best of our knowledge, in the numerous papers and books devoted to this topic, the kernel is continuous. Such kernels are called weakly singular. In no way claiming completeness of the bibliographic references, we only mention in this connection the classic textbook [10] which we find very useful when considering integral equations with singular kernels. As for approximate methods for solving integral equations, we mention the monographs [1, 2, 4, 7, 11] and papers [3, 5, 15], which show various approximation methods, but both in these and in other works, the numerator is assumed to be at least continuous, and often differentiable.
However, we are faced with real problems whose process of solving led to Fredholm equations with a weakly singular kernel, the numerator of which is not a continuous function. We called such a kernel as having an additional singularity. We present the problems which lead to the integral equations involving the kernels with additional singularity. These problems were discussed in detail in the papers [12, 14, 13]. More precisely, the paper [12] is devoted to the problem of optimization of small deviation for mixed fractional Brownian motion with trend for the case of the Hurst index ; whereas in the paper [14], the same problem was considered for the case . The paper [13] is devoted to the minimization of entropy in the system described by the mixed fractional Brownian motion with trend. Both cases, and were considered and as the result, the problem was reduced to the couple of the same integral equations as in the case where the minimization of small deviations was studied. In the case the integral equation contains the kernel with additional singularity whereas in the case the kernel is simply weakly singular. In his connection, in order to deal with the kernel with additional singularity applying wellknown methods for weakly singular kernels, we prove the theorem on the approximation of solution of integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are weakly singular but the numerator is continuous. We demonstrate numerically how our methods work being applied to our specific integral equations.
The paper is organized as follows. In Section 2, we present two problems which lead to the integral equations involving the kernels with additional singularity. Roughly speaking, they are the problem of minimization of small deviation and the entropy minimization problem. Both of them appear when considering dynamical system involving mixed fractional Brownian motion. In Subsection 2.1 we describe the problems themselves and then, in Subsection 2.2, we explain how to reduce these optimization problems to the integral equation and describe the structure of the involved integral kernels. The representations of the kernels are new, in comparison with the papers [12, 14, 13], and they are much more convenient for the numerical solution. In Section 3, we provide approximation of the involved kernels by kernels with continuous numerators. The main result of this section is the Theorem 3.6 on the approximation of solution of integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are weakly singular. Section 4 is devoted to a numerical solution of the considered Fredholm integral equations. We describe the modification of productintegration method of the numerical solution in Subsection 4.1. We illustrate it by numerical experiments, the graphs of corresponding kernels and solutions, provide a short sensitivity study of errors in Subsection 4.2.
2 Integral equation appearing in the problem of optimization of small deviation and entropy functionals
Here, we present the problems which lead to the integral equations involving the kernels with additional singularity.
2.1 Description of the problems of the optimization of small deviation and the entropy minimization problem
Let be a filtered probability space that supports all the stochastic processes presented below, and it is assumed that they are all adapted to this filtration. Now, introduce two independent stochastic processes, namely, the Wiener process and the fractional Brownian motion (fBm) with Hurst index , that is the Gaussian process with zero mean and the covariance function
Consider a mixed Gaussian process composed of and involving a nonrandom drift. More precisely, we consider the the mixed fractional Brownian motion with the drift, i.e., the process of the form
(1) 
where is a nonrandom function and space will be specified below. Consider the following problem: to annihilate the drift by the change of the probability measure. More precisely, to choose the other probability measure such that
where the Wiener process and the fBm are two independent processes under the measure . The main idea of the solution is to apply Girsanov theorem to fractional Brownian motion and Wiener process with drifts. To do so we need to distribute the trend among and in some optimal way as follows
In order to write down the RadonNikodym derivative let us recall here the (weighted) RiemannLiouville fractional integrals, see papers [9] and [12, 13].
Definition 1.
The RiemannLiouville left and rightsided fractional integral of order on is defined as
Define the weighted Riemann – Liouville integrals
(2) 
for and
(3) 
in case The constant is equal to .
Since and are independent, we can write where
(4) 
according to standard Girsanov theorem, and
(5) 
according to Girsanov theorem for a fractional Brownian motion, see e.g. [14, Lemma 3.1]. In the above representation is a Brownian motion, related to as follows:
The optimal drift distribution problem arose when solving two problems of different types, but all of them ultimately came down to solving a certain Fredholm integral equation of the second kind. Namely, the paper [12] was devoted to the problem of optimization of small deviation for mixed fractional Brownian motion with trend for the case whereas in the paper [14], the same problem was considered for the case The paper [13] studied the problem of minimization of the entropy functional appearing under the distribution of the drift, and this problem was studied for . Now our goal is twofold: first, to present the existing results from [12, 14, 13] and second, to demonstrate how to reduce the problem of minimization of entropy functional in the case
2.2 How to reduce the problem of the optimization to the integral equation
Let us start with the small deviations of a mixed fractional Brownian motion with trend. We are interested in the asymptotic as After passing to the measure we have from [13, Lemma 3.3] and [12, Lemma 3.3] the lower bound for this probability
Therefore, the maximization of its righthand side leads to the following optimization problem
(6) 
where if If then consists of all functions for which there exist such that and Furthermore, if is a minimizator in (6), then
Now consider the minimization of entropy functional, which can be formulated as follows: define the functions and in (4) and (5), which minimize the entropytype functional
(7) 
The next result was proved in [13] for the case but the proof remains the same for all
Lemma 2.1.
Entropy functional could be represented as
(8) 
It was shown in [12] for and in [13] for , that the minimization in (6) is a solution of the following fractional integral/differential equation
(9) 
The existence and uniqueness of solution for equation (9) was proved in Theorem 3.9 [12] in the case of Hurst index When it was proved in [14], that (9) is equivalent to
(10) 
for which has a unique solution After applying definition of weighted Riemann – Liouville integral (1), the operators and have the form of integral operator with the kernel where
(11) 
and Thus, the both optimization problems reduces the solution of the Fredholm integral equation of the second kind, which can be represented as
(12) 
In this paper, we study further equation (12) and prove in the next lemma that kernel (11) can be significantly simplified.
Let be the Beta function and be an incomplete beta function defined for given by .
Lemma 2.2.
The kernel (11) equals

for
where numerator is bounded on , meanwhile, has no limit at points and .

for
Proof.
Item (i): it is easy to see that kernel is symmetric, so that it is enough to consider only the case Then
(13) 
In turn, transform the last integral in (13) with the change of variables Then
and we get
Item (ii): it was proved in [14] that kernel is symmetric and nonnegative, consequently, we consider only the case Then
(14) 
Introduce the similar change of variables for the second integral in (14) as . Then
and we obtain
The lemma is proved. ∎
3 Approximation theorem for integral operator
We start with very simple auxiliary approximation result for the sequence of operators. Let be a real Hilbert space.
Lemma 3.1.
Let be a compact positive operator, and let be a sequence of compact operators with spectrum such that as .
Then the spectrum is asymptotically included into , in the sense that
Proof.
Suppose the contrary: let there exist some and subsequent such that as and Then there exists such that However, let
be any sequence of eigenvalues of operators
, , and letLet us write the following obvious equalities:
whence
Furthermore, as , and , and it immediately follows that The resulting contradiction proves the theorem. ∎
Now we are apply Lemma 3.1 to the sequence of the integral operators in the space for some . Namely, consider the integral operator defined by its kernel function via formula
(15) 
where is taken from some space of functions defined on , and this space will be specified later. We assume that has a singularity, more precisely, it has the form
(16) 
where is a bounded function on and . Let us recall the following general statement from [10, p.397, item 6.4].
Proposition 3.2.
Let kernel of operator from (15) satisfy the following conditions: there exist , and such that and for which
Then operator is a compact operator from into with the norm
Corollary 3.3.
Consider the integral operator (15) with the kernel (16), where is a bounded function on and . Then we can put for any , and additionally we can choose in such a way that . Then we can put and all conditions of Proposition 3.2 will be fulfilled. Therefore is a compact operator from into , and so is a space that was claimed to be specified later.
Now, consider the Fredholm integral equation of the second kind
(17) 
where is a given function, and the integral operator has the form (15) and the kernel is taken from (16). The standard situation is when the function is continuous. However, in the applications which we will consider later, function will be bounded however, it may have a finite number of fatal discontinuities, i.e., points in which the limit of function does not exist. We call such kernels as the kernels with additional singularity. In this connection, let us prove an auxiliary result concerning the possibility of approximation of the kernel by the respective kernels with continuous numerators.
Lemma 3.4.
Let the function be bounded, , and continuous a.s. except finite number of points. Let for Let be fixed. Then there exists a sequence of totally bounded continuous functions (we can take the same constant for them), such that
where depend on and as .
Proof.
Let us describe the construction of continuous function . Let be one of the points of discontinuity, let be the total number of such points, and let is sufficiently large. Each point of discontinuity, in particular, , can be surrounded by sufficiently small closed box where and is the maximum norm in Euclidean space. Assume that outside the union of these small boxes, so, it is necessary to determine only inside each box. Let us put for and
(18) 
The range of the values of does not exceed the range of values of . The point is situated on the square
therefore every is a continuous function, Moreover,
and
Thus, are totally bounded.
Now, denote the projection of the union of the small boxes surrounding the points of discontinuity of , on . Evidently, the total Lebesgue measure of does not exceed as . Therefore,
as , and can be introduced and treated similarly, whence the proof follows. ∎
Remark 1.
Of course, there can be different ways of construction of the functions . In what follows, for us will be important to construct them in such a way that the approximating operators be selfadjoint.
Remark 2.
Now, let us return to equation (17) and specify the assumption regarding the integral operator .
Lemma 3.5.
Let the integral operator is compact from into and positive, in particular, selfadjoint, and let . Then there exists a unique function which satisfies (17).
Proof.
It is just sufficient to mention that all eigenvalues of operator are real and nonnegative, therefore the respective homogeneous equation has only trivial solution, and the proof immediately follows from the Fredholm alternative.∎
Now, let us establish the main result of this section, namely, the theorem on the approximation of solution of integral equation with the kernel containing additional singularity by the solutions of the integral equations whose kernels are of type (16), but the numerator is continuous.
Theorem 3.6.
Let be the kernel defined in (16), where the numerator has the following properties

is bounded and symmetric.

is continuous, except finite number of points.

is a positively definite kernel.
Let be a unique solution of equation (17). Then the sequence of functions satisfying conditions of Lemma 3.4 can be chosen in such a way that the respective integral operators are selfadjoint, for sufficiently large the equation
has a unique solution , and
Proof.
Let us choose according to Lemma 3.4. Then for the respective integral operators, according to Lemma 3.4 and Proposition 3.2 we have that as . Show that is symmetric which yields that is selfadjoint. Let be the union of the points of discontinuity of Denoting by for we have from symmetry of that if then as well. Then from (18) we have for all points outside Assume that is large enough that all are disjoint. Let for some then and it follows from the construction of that
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