1. Introduction
We will consider stochastic differential equations of the following form:
where
is a continuous function,
are measurable functions, and
,
, denotes a fractional Brownian motion (fBm). The stochastic integral in Equation (
1) is a pathwise generalized Lebesgue–Stieltjes integral. Thus, we can use the pathwise approach to consider this FSDE. We call such an equation FSDE with a stochastic forcing term
.
Many authors have considered the problem of the existence and uniqueness of solutions to FSDEs without a stochastic forcing term [
1,
2,
3,
4,
5,
6,
7,
8,
9,
10,
11,
12,
13,
14]. The first attempt to consider the FSDE with a stochastic forcing term was made in an article by Kubilius and Medžiūnas [
15]. In this article, equations with constant and strictly positive diffusion coefficients with a “soft wall” are considered. That is, the value of the function
depends on the position of
x to a fixed point,
w, called the wall boundary. For example, we can take exponential forces with a wall (
w) defined by a function
characterized by the amplitude
and decay constant
. The term “soft wall” was introduced in reference [
16]. It should be noted that the “soft wall” model has a permeable wall. The process may cross the wall, but it is affected by the force of the selected quantity in the opposite direction. The force acts weakly when the process is far from the wall. As it approaches or crosses the wall, the force acts stronger. An illustration of the behavior of trajectories for such a model was considered in [
15,
17]. There, we considered the fractional Vasicek process with the soft wall. FSDE (
1) includes the “soft wall” model.
In reference [
17] we consider the conditions for the existence and uniqueness of solutions to Equation (
1) by using the implicit Picard iteration. In the final part of the proof, an error occurred due to an incorrect simplified notation. The theorem statement remains true, but we must use the implicit Euler approximation instead of the implicit Picard iteration. The new proof repeats the proof of Theorem 3 in [
17] with estimates of other norms.
The proof of the existence and uniqueness of the solution of Equation (
1) is based on estimates obtained in the article by Nualart and Rǎşcanu [
10] and the conditions for the coefficients
f and
g are almost the same as in their article.
This paper is organized in the following way. In
Section 2, we present the paper’s main results. In
Section 3, we define a deterministic differential equation corresponding to FSDE (
1) and consider its implicit Euler approximation properties. Moreover, it contains definitions of considered spaces of functions and a priori estimates for the Lebesgue–Stieltjes integral. In
Section 4, we prove the existence and uniqueness of a solution for a deterministic differential equation. In
Section 5, we consider the fractional Pearson diffusion process as an example.
2. Main Result
We will assume that the coefficients satisfy the following conditions:
(A1) is differentiable in x, and there exist some constants , and for every , there exists such that the following properties hold:
(i) Lipschitz continuity in
x(ii) Local uniform Hölder continuity of the derivative in
x(iii) Hölder continuity in
t(A2) There exists a bounded function , and for every , there exists such that the following properties hold:
(i) Local uniform Lipschitz continuity in
x(A3) Assume the following:
(i) Function , where = is strictly monotonic and surjective;
(ii) There is a constant
, such that we have the following:
(A4) Assume that is a differentiable function, and there exist some constants , , and for every , there exists such that we have the following:
(i) , for all ,
(ii) Local uniform Hölder continuity of the derivative
Remark 1 (see Remark 8 in [
15]).
Under assumptions , function D satisfies assumptions . We can now formulate our main result. We set the following:
Theorem 1. Suppose that the functions and satisfy the assumptions and with , . Let , where . If assumptions are satisfied, then there exists a stochastic process satisfying FSDE (1), where is the space of γ-Hölder continuous functions. If assumptions are satisfied and , where , then there exists a unique stochastic process satisfying FSDE (1). 3. Deterministic Differential Equations
3.1. Preliminaries
3.1.1. Spaces of Functions and Norms
Let us introduce some function spaces that will be used to analyze solutions of (
1).
We denote by
, where
, the space of real-valued measurable functions
such that we have the following:
The space
is a Banach space with respect to the norm
, and for
, the equivalent norm is defined by the following:
For any
, denote by
the space of
-Hölder continuous functions
equipped with a norm
, where we have the following:
Clearly, we have
for
and
We denote by
,
, the space of measurable functions
such that we have the following:
Note that
(see [
10]).
We also denote by
,
, the space of measurable functions
f on
such that we have the following:
Fix
. Let
denote a set of all possible partitions of
. For any
, we define the following:
Recall that
is called the
p-variation of
f on
. We denote by
(resp.
) the class of (resp. continuous) functions on
with bounded
p-variation,
.
Define , which is a seminorm on , and is 0 if and only if f is constant. For each f, is a non-increasing function of , i.e., if , then . Thus, if . If , then f is bounded.
Let and . Then is a norm, and is equipped with the p-variation norm is a Banach space.
3.1.2. Riemann–Stieltjes Integral
Assume that
and
, where
. The generalized Lebesgue–Stieltjes integral (see [
10])
exists for all
and for any
where
and
is a Gamma function. Furthermore, the integral
exists if
.
If
and
with
, then the generalized Lebesgue–Stieltjes integral exists and coincides with the Riemann–Stieltjes integral (see [
18]).
From Young’s Stieltjes integrability theorem [
19] (see p. 264) the Riemann–Stieltjes integral
can be defined for functions having bounded
p-variation on
(see [
20]).
Let
and
with
,
,
. If
f and
h have no common discontinuities, then the extended Riemann–Stieltjes integral
exists, and the Love–Young inequality.
holds for any
, where
,
denotes the Riemann zeta function, i.e.,
.
3.1.3. Estimation of the Generalized Lebesgue–Stieltjes Integrals
From now on, we fix
. For any function
define
where
f satisfies the assumptions
.
Proposition 1 (see [
10]).
Assume that f satisfies the assumptions . If then and we have the following: for all , where , , are positive constants depending only on α, T, .If are such that and , then we have the following:for all , where depends only on α, T, and from . Given two functions,
and
, we denote the following:
where
g satisfies the assumptions
with constant
.
Proposition 2 (see Proposition 4.1 [
10]).
If , then we have the following:where , is the Beta function. Proposition 3 (see [
10]).
If then andfor all , where the constants and are independent of λ, u, h (they depend on T and the constants , , α, β from .If are such that and , then we have the following:for all , whereand the constant is independent of λ, u, v, h depends on T and the constants from . Remark 2. If and then 3.1.4. Integration with Respect to fBm
The trajectories of
,
, are almost surely locally
-Hölder continuous functions for all
. To be more precise, for all
and
, there exists a nonnegative random variable
such that
for all
, and
for all
.
The pathwise generalized Lebesgue–Stieltjes integral for one-dimensional fBm
can be defined as follows:
if
(see [
6,
21] p. 225), where
and
are fractional derivatives.
For
, we can choose
such that
. An easy computation shows that almost all trajectories of
belong to the space
. Indeed, since
, then for any
, we have the following:
Thus
for almost all
.
If
is a stochastic process whose trajectories belong to the space
, then the pathwise generalized Lebesgue–Stieltjes integral
exists and we can express it according to (
13). Moreover, if the trajectories of the process
u belong to the space
, then the indefinite integral
is a Hölder continuous function of order
(see [
10]).
3.2. The Implicit Euler Approximation and Auxiliary Results
Let
be fixed. Let
,
. Consider the deterministic differential equation on
, as follows:
where
, the coefficients
satisfy assumptions
and
, and the function
satisfies assumption
.
Let
be a sequence of uniform partitions of the interval
, and let
,
,
. We define the implicit Euler approximations for the Equation (
14) as follows:
and their continuous interpolations are as follows:
where
and
if
,
We rewrite implicit Euler approximations (
15) and (
16) in a more compact way, as follows:
with
and
where
The implicit Euler approximations scheme (
17) is correctly defined. From the recursive expression (
17), we calculate
. The properties of the function
give us a single value of
. Since
is a continuous function, then
is a continuous function. Indeed, since
and
are continuous functions, then
is a continuous function.
Now, we consider the properties of the implicit Euler approximation.
Lemma 1. Let Assumption be satisfied. Then , , for any fixed .
Proof. We first note that the functions and have bounded variations on for a fixed n. Thus, for the fixed n, they are bounded and have p-bounded variation, where . From now on, we assume that .
First, observe that for
Thus,
for fixed
.
Now consider
. Since
then
and
Assume that
for some
and
. Then
If
, then from (
20) and the Love–Young inequality, we obtain the following:
From Assumption
, we have the following:
Since
is bounded for any fixed
, then
for any fixed
. □
The next lemma enables us to apply the estimate (
6) to the integral
.
Lemma 2. Let Assumptions and be satisfied. If for any fixed , then for any fixed and .
Proof. From Lemma 1, it follows that
for any fixed
. We prove this. The following is evident:
Now, we estimate the second term of the norm (
5). From Lemma 1, it follows that there exists a constant
, depending on
n, such that we have the following:
Note that
for
. Therefore, we have
Since
and (see [
22] p. 494)
then
By a similar argument, we obtain the following:
Consequently,
has the claimed property. □
The following result is crucial to prove our main results.
Proposition 4. Let and the functions and satisfy assumptions , and , respectively. Moreover, let assumption hold. Then there exists a constant C such that we have the following: Proof. It is easy to check that
Indeed, we note that
From Lemma 1, we have that for any fixed , the norms , , are finite.
The proof of the estimate
is similar to the proof of the estimate
in [
10]
where
.
Since
, we have
. From Lemma 2, it follows that for the integral
, we can apply the estimate (
6). Applying Proposition 2, we obtain the following:
where
,
is the Beta function.
Note that
Since
it follows that
where
From (
28), (
29), and inequality
we have the following:
where
, and
First, observe that
where
Furthermore, from (
30), we have the following:
and
where
Applying (
31), we obtain the following:
and
where
From (
25), (
27), and (
33), we have the following:
where
Note that for
we have the following:
Thus,
and from Lemma 7.6 in [
10]
where
and
are positive constants depending only on
,
□
Now, we can strengthen the result of Lemma 1.
Proposition 5. Under the assumptions of Proposition 4, we obtain .
Proof. Recall that from Lemma 1, we have
,
,
for any fixed
. Thus, for any fixed
, we have the following:
Indeed, similar to how we proved (
26), we have the following:
First, observe that for
Thus,
and the boundedness of the norm
follows from Proposition 4.
From (
6), it follows that
Furthermore, from (
30), we obtain the following:
We obtain the proof of the estimate for the second term in (
34) in the same way as presented in Lemma 2. We first compute the following integral:
Assume that
,
, and
. Similar to proofs (
23) and (
24), we have the following:
and
Applying (
31), we obtain the following:
and
Finally,
Thus the norm
is bounded for all
n and the proof is complete. □
4. Existence and Uniqueness of the Solution
We find conditions when the deterministic differential Equation (
14) has a unique solution.
Theorem 2. Let and the functions and satisfy assumptions and , respectively, where is defined in (3). If assumptions are satisfied, then for any , such that , Equation (14) has a solution . If assumptions are satisfied and , then Equation (14) has a unique solution . Proof. Existence of the solution. From Proposition 5 and assumption , we have that the sequence of functions is relatively compact in .
Thus, we can choose a subsequence
, which converges in
to a limit
, i.e.,
We show that
x is a solution to Equation (
14). For simplicity of notation, we write
n instead of
. Recall the following:
Thus,
Since
and function
D is continuous, the first term converges to zero. It remains to be proven that the second and third terms also converge to zero.
First, observe that there exists a constant,
N, such that
and
. It follows from Proposition 5 and (
35).
Next, we estimate the second term in (
36). Since
, then from Proposition 1, it follows that
. Applying (
4) to any
, we obtain the following:
To estimate the third term in (
36), we note that
for
and
. Applying Proposition 3, Remark 2, and (
4), we obtain the following:
where
From the definition of
, it follows that
. The proof is completed. □
To prove the uniqueness of the solution, we need the following result:
Lemma 3 (see Lemma 7.1 in [
10]).
Let Φ be a function satisfying assumptions . Then for all and , , , , we have the following: Proof. By the mean value theorem, we can write the following:
From the conditions of the lemma, we obtain the statement of the lemma. □
Uniqueness of the solution. Let
x and
be two solutions belonging to
. Then there exists
N, such that
and
. Furthermore,
and
(see Propositions 1 and 3) and
. Thus, we have the following:
We will obtain estimates of the second and third terms from Propositions 1 and 3. It remains to evaluate the first term. From Lemma 3, we have the following:
for
. Thus,
For any
,
, we can choose a sufficiently large
, such that we have the following:
Thus,
and, consequently,
.
Proof of Theorem 1. Fix
. Let
be such that
. Denote
and
. Then
. Since
then
,
, and
. Note that from the inequality
it follows that
.
If then and . This completes the proof. □
5. Example of a
Fractional Pearson Diffusion with a Stochastic Force
Consider the Pearson diffusion process with a stochastic force, as follows:
where
Assume that the coefficients
,
, are such that
and
. Then
.
For the existence of a unique solution to problem (
37), it is necessary to check the conditions of Theorem 1. Note the following:
where
is a critical point of the function
.
An easy computation shows the following:
and
Thus, the Pearson diffusion process with a stochastic force has a unique solution under the above conditions.
6. Discussion
The mathematical literature has extensively analyzed stochastic differential equations (SDEs) driven by a fractional Brownian motion. Most of these efforts have been motivated by problems arising in the financial applications of SDEs, such as option pricing, stochastic volatility, and interest rate modeling. However, there are few results concerning SDEs with boundary conditions. Typically, SDEs involving reflection at the boundary are considered. Our focus is on introducing and solving stochastic differential equations that are subject to a force allowing a process to cross a boundary while preventing it from moving far from it. Examining such a model can be interpreted as studying the influence of the environment on the behavior of the process. These types of processes can be applied in the natural sciences. This work represents an initial attempt to consider such processes. Introducing two- or three-dimensional SDEs with such a force would be of great practical interest.