A Class of Fractional Stochastic Differential Equations with a Soft Wall

Abstract

In this paper we are interested in fractional stochactic differential equations (SDEs) with a soft wall. What do we mean by such a type of equation? It has been established that SDE with reflection can be imagined as equations having a hard wall. Now, by introducing repulsion instead of reflection, one obtains an SDE with a soft wall. In contrast to the SDE with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. We find conditions under which SDE with a soft wall has a unique solution and construct an implicit Euler approximation with a rate of convergence for this equation. Using the example of the fractional Vasicek process with soft walls, we illustrate the dependence of the behaviour of the solution on the repulsion force.

1. Introduction

Stochastic processes-based modelling has a long and rich history in a variety of fields. For the last decade, there was an effort to expand classical stochastic differential equations (SDEs) driven by Brownian motion into SDEs driven by fractional Brownian motion (fBm) (see, e.g., [1,2,3], etc.). A majority of these efforts were driven by problems, which arose in financial applications of SDEs (e.g., option pricing, stochastic volatility, interest rate modelling).

However, mathematicians are yet to fully explore potential application possibilities of stochastic processes in material sciences, especially biology/medicine. According to Fulinski [4], certain findings in these fields “seem to open the whole new chapter in the Brownian motions story”. Upon closer inspection, natural sciences offer a unique set of requirements and challenges in the application of SDEs. For example, cellular growth can be accelerated/decelerated/changed or even stopped by proteins, hormones, oxygen, temperature, spacial geometries, etc. For instance, when modelling retinal angiogenesis, Capasso and Wieczorek [5] emphasised seven main features/behaviours, which had to be accounted for in their stochastic model. This plethora of influencing factors is one of the defining traits of SDE application in biology. For mathematical purposes, these effects can be seen as an influence of a medium on a behaviour of a process. Vojta et al. [6] presented us with a list of such behaviours, which they call walls, and classify them depending on process discrete expressions. Many of these types are variations of well-known stochastic objects (e.g., Skorohod problem). Yet, one of the proposed types —soft wall— described by recursion relation

$$x_{n+1}=x_n+{\xi }_n+G\left(x_n\right),$$

where $x_n$—process value, ${\xi }_n$—discrete fractional Gaussian noise, G—repulsive force, is far less investigated and demonstrates a more subtle behaviour. The subtlety is produced by introduction of repulsion instead of reflection. In contrast to the standard SDEs with reflection, where the process cannot pass the hard wall, the soft wall is repulsive but not impenetrable. As the process crosses the soft wall boundary, it experiences the force of a chosen magnitude in the opposite direction. When the process is far from the wall, the force acts weakly. Since this muffled deflection of process trajectories avoids elasticity or inelasticity assumptions, it is a much more accurate mathematical model for many actual natural processes.

Therefore, we consider a one-dimensional SDE

$$Y_t=Y_0+G\left(Y_t\right)-G\left(Y_0\right)+{\int }_0^{t}f(s,Y_s)ds+Z_t,t\in [0,T],$$

(1)

driven by an arbitrary stochastic process $Z={\left(Z_t\right)}_{t⩾0}$, $Z_0=0$, with continuous paths, where functions $G:\mathbb{R}\to [0,\infty )$, $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ are continuous. As a special case, we can take $Z=B^H$, where $B^H$ is a fBm.

We will call such equation SDE with a soft wall, where the function G is interpreted as a repulsive force, which adjusts as the process position with respect to the wall changes. For instance, we can take the exponential repulsive force profile with a wall boundary at the point w, defined as

$$G\left(x\right)=G_{0}exp\{-\lambda (x-w)\},$$

characterised by amplitude $G_0$ and decay constant $\lambda$ (see [6]).

It has to be noted that the task of finding conditions under which SDE in the form

$$Y_t=Y_0+{\int }_0^{t}f(s,Y_s)ds+{\int }_0^{t}g(s,Y_s)d{B}_s^H,t\in [0,T]$$

(2)

or mixed SDE has a unique solution was considered by many authors [1,2,3,7,8,9,10,11,12,13] under a variety of condition sets and methods of proof. However, the introduction of force term G presents new challenges, where the previous approaches are not applicable.

Hence, the aim of this paper is to examine SDE (1) and to apply the obtained results for a more general class of SDEs. First, we explore conditions under which Equation (1) admits a unique solution. Then, we find a pathwise convergence rate for the implicit Euler scheme of the considered SDE solution. In addition, for SDE (1) with $Z=B^H$ and known function G, we construct a strongly consistent and asymptotically normal estimator of the Hurst parameter $0<H<1$, based on discrete observations of the underlying process. Moreover, we specify the rate of convergence of this estimator to the true value of H, when the diameter of the partition of the observation interval tends to zero.

It should be noted that in contrast to previous research (e.g., [6,14]), the results demonstrated in this work are much more mathematically rigid and synthetic in their nature. Since all approximation schemes presented are iterative, they are easy to use, control and analyse (as demonstrated in Section 6). The pathwise approach does not “hide” the process trajectories behind complicated mathematical structures; hence, it enables easy first-hand use of all statistical analysis tools available. Moreover, approximation convergence rates produced in our research assure the validity of the generated data, which guarantees genuine basis for any other research, without any further error analysis needed.

The paper is organised in the following way. In Section 2, we present the main results of the paper. In Section 3, we discuss condition (E), which at first glance appears to be unusual and restrictive. Section 4 contains proofs of these main theorems. Section 4.1 contains existence and uniqueness proofs of the Equation (1) solution and the convergence rate of its approximation. In Section 4.2, we find the existence and uniqueness conditions for SDEs that have a strictly positive diffusion coefficient. Moreover, we obtain the convergence rate of its approximation. In Section 5, a fractional Pearson diffusion process with a soft wall was considered an example of a stochastic process, satisfying conditions (A)–(E). In Section 6, the fractional Vasicek with the soft wall is considered as a modeling example. Finally, in the Appendix A, we investigate the asymptotic behaviour of the Hurst index estimator and in Appendix B we recall CLT for fBm, a discrete version of Gronwall’s lemma, and the chain rule formula for Hölder-continuous functions.

2. Main Results

$C\left(\right[a,b\left]\right):a<b$ denotes the space of continuous $\mathbb{R}$-valued functions defined on $[a,b]$. For $\lambda \in (0,1]$, $C^{\lambda }\left([a,b]\right)$ denotes the space of Hölder continuous functions equipped with a norm

$${\parallel g\parallel }_{\lambda }:={\left|f\right|}_{\infty }+{\left|g\right|}_{\lambda }{,\left|g\right|}_{\lambda }=\underset{\begin{array}{c}s,t\in [a,b]\\ s\ne t\end{array}}{sup}\frac{\left|g\right(t)-g(s\left)\right|}{{\left|s-t\right|}^{\lambda }}{,\left|g\right|}_{\infty }=\underset{t\in [a,b]}{sup}\left|g\left(t\right)\right|.$$

We consider SDE of the form

$$Y_t=Y_0+G\left(Y_t\right)-G\left(y_0\right)+{\int }_0^{t}f(s,Y_s)ds+Z_t,t\in [0,T],$$

(3)

driven by an arbitrary stochastic process $Z={\left(Z_t\right)}_{t⩾0},Z_0=0$, with continuous paths.

Using function $D\left(x\right):=x-G\left(x\right)$, Equation (3) can be simplified to

$$D\left(Y_t\right)=D\left(Y_0\right)+{\int }_0^{t}f(s,Y_s)ds+Z_t.$$

(4)

We proceed by introducing a set of conditions:

(A)

Linear growth of f for any $t\in [0,T]$ and any $x\in \mathbb{R}$

$$\left|f\right(t,x\left)\right|⩽K(1+|x\left|\right).$$

(B)

Lipschitz continuity of fin space: for any $t\in [0,T]$ and $x,y\in \mathbb{R}$

$$\left|f\right(t,x)-f(t,y\left)\right|⩽K|x-y|.$$

(C)

Hölder continuity in time: there exists $\beta \in (0,1]$ such that for any $s,t\in [0,T]$ and any $x\in \mathbb{R}$

$$|f(s,x)-f(t,x)|⩽{K|s-t|}^{\beta }.$$

(D)

Function D is strictly monotonic and surjective.

(E)

Constant $d>0$ exists, such that

$$\left|D\left(x\right)-D\left(y\right)\right|⩾d\left|x-y\right|.$$

(5)

Now, we formulate the solution existence and uniqueness theorem for the SDE (3).

Theorem 1.

Assume that conditions (A), (B), (D) and (E) are satisfied. Then, the Equation (3) has a unique solution $Y\in C\left(\right[0,T\left]\right)$. If stochastic process $Z={\left(Z_t\right)}_{t⩾0}$, $Z_0=0$, has a Hölder continuous paths of order $0<\gamma <1$, then $Y\in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$.

Continuing, we introduce the implicit Euler scheme associated with (3). Let $\pi =\{t_k^n=(k/n)T,1\le k\le n\}$ be a sequence of uniform partitions of the interval $[0,T]$ and $h_n=t_k^n-t_{k-1}^n$, $1\le k\le n$. We define the following implicit Euler approximation scheme:

$$D\left(Y_{n,k+1}\right)=D\left(Y_{n,k}\right)+f(t_k^n,Y_{n,k})h_n+(Z_{t_{k+1}^n}-Z_{t_k^n}),Y_{n,0}=Y_0.$$

(6)

For the simplicity of notation, we introduce the symbol $O_{\omega }$. Let $\left({\xi }_n\right)$ be a sequence of r.v.s, let $\varsigma$ be an a.s. non-negative r.v. and $\left(a_n\right)\subset (0,\infty )$ be a vanishing sequence. Then ${\xi }_n=O_{\omega }\left(a_n\right)$ means that $|{\xi }_n|\le \varsigma ·a_n$ for all n. In particular, ${\xi }_n=O_{\omega }\left(1\right)$ signifies that the sequence $\left({\xi }_n\right)$ is a.s. bounded.

Theorem 2.

Assume assumptions $\left(\mathbf{A}\right)--\left(\mathbf{E}\right)$ are satisfied, and Y is a solution of Equation (3) such that $Y\in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$. Then,

$$\underset{1⩽k⩽n}{max}\left|Y_{t_k^n}-Y_{n,k}\right|=O_{\omega }\left(h_n^{\theta }\right),$$

where $\theta =\beta \wedge \gamma$.

Corollary 3.

Under conditions of Theorem 2, it follows that

$$\begin{array}{c}\hfill \underset{0\le t\le T}{sup}|Y_t-Y_t^n|=O_{\omega }\left(h_n^{\theta }\right),\gamma \in (0,1),\end{array}$$

(7)

where

$$\begin{array}{cc}\hfill Y_t^n& =Y_{n,k}+\frac{t-t_k^n}{h_n}(Y_{n,k+1}-Y_{n,k}),t\in \left(t_k^n,t_{k+1}^n\right],k=0,\dots ,n-1.\hfill \end{array}$$

We now proceed by application of the results obtained for Equation (1) to equations with additional restrictions on the diffusion coefficient. Let us consider SDE

$$X_t=X_0+\Phi \left(X_t\right)-\Phi \left(X_0\right)+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s,t\in [0,T],$$

(8)

where $\Phi ,\alpha ,\sigma$ are continuous functions, Z is a process with Hölder continuous paths of order $1/2<\gamma <1$ and strictly positive diffusion coefficient, i.e., ${inf}_{x\in \mathbb{R}}\sigma \left(x\right)>0$. The stochastic integral in Equation (8) is a pathwise Riemann–Stieltjes integral, and thus, the whole equation is understood as a pathwise Riemann–Stieltjes integral equation. A pathwise Riemann–Stieltjes integral is well defined if $\sigma \left(X\right)$ is Hölder continuous with order $\lambda$ and such that $\lambda +\gamma >1$.

For the following theorem, we need condition:

$\left(\mathbf{H}\right)$$\Phi :\mathbb{R}\to \mathbb{R}$ is a continuously differentiable function, and there exists a constant $0⩽c<1$ such that ${\Phi }^{\prime }\left(x\right)⩽c$ for all $x\in \mathbb{R}$.

Theorem 4.

Assume that $\alpha \left(x\right)$ and $\sigma \left(x\right)$ are continuously differentiable functions and

${inf}_{x\in \mathbb{R}}\sigma \left(x\right)>0$. If the function $\Phi \left(x\right)$ satisfies condition (H) and for some constant $C>0$ and any $x\in \mathbb{R}$,

$$|{\alpha }^{\prime }\left(x\right)|⩽C,|{\sigma }^{\prime }\left(x\right)|⩽C,\left|\frac{\alpha \left(x\right)}{\sigma \left(x\right)}\right|⩽C$$

(9)

then SDE (8) has a unique solution $X\in C^{\gamma }\left([0,T]\right)$, $1/2<\gamma <1$.

Function

$$F\left(x\right)={\int }_0^x\frac{1}{\sigma \left(y\right)}dy,$$

is called the Lamperti transform, and $F^{-1}$ is the inverse Lamperti transform.

Theorem 5.

Assume that conditions of Theorem 4 hold. Then,

$$\underset{1⩽k⩽n}{max}\left|X_{t_k^n}-X_{n,k}\right|=O_{\omega }\left(h_n^{\gamma }\right),\gamma \in (1/2,1),$$

where $X_{n,k}=F^{-1}\left(Y_{n,k}\right)$.

3. Remarks on the Condition (E)

At first glance, condition (E) can seem unusual and restrictive (especially in conjunction with condition (D)). However, a wide class of functions satisfies it. We will propose only several more interesting examples of such functions.

Remark 6.

Assume that the function $D:\mathbb{R}\to \mathbb{R}$ is bi-Lipschitz (i.e., D is Lipschitz, and its inverse function $D^{-1}$ is also Lipschitz). Then, inequality (5) is satisfied.

For any $\widehat{x},\widehat{y}\in \mathbb{R}$ there exsist $x,y\in \mathbb{R}$ such that $D^{-1}\left(\widehat{x}\right)=x$ and $D^{-1}\left(\widehat{y}\right)=y$. Since $D^{-1}$ is a Lipschitz function, then for some L,

$$\left|x-y\right|=\left|D^{-1}\left(\widehat{x}\right)-D^{-1}\left(\widehat{y}\right)\right|⩽L\left|\widehat{x}-\widehat{y}\right|=L\left|D\left(x\right)-D\left(y\right)\right|.$$

(10)

Remark 7.

Assume that the function $D:\mathbb{R}\to \mathbb{R}$ is differentiable, invertable and such that $|D^{\prime }{|}_{\infty }⩾c>0$, then inequality (5) is satisfied.

Since $|D^{\prime }{|}_{\infty }⩾c>0$, and by using the Inverse Function Theorem, we obtain

$$\left|{\left(D^{-1}\right)}^{\prime }\left(x\right)\right|=\left|\frac{1}{D^{\prime }\left(D^{-1}\left(x\right)\right)}\right|⩽\frac{1}{c}.$$

Thus, the function $D^{-1}\left(x\right)$ is Lipschitz, and inequality (5) is satisfied.

Remark 8.

If $G:\mathbb{R}\to \mathbb{R}$ is a continuously differentiable function and there exists a constant $0⩽c<1$ such that $G^{\prime }\left(x\right)⩽c$, for all $x\in \mathbb{R}$, then $D\left(x\right)$ is strictly increasing and $\left|D\left(x\right)-D\left(y\right)\right|⩾(1-c)\left|x-y\right|$. Indeed, since $D^{\prime }\left(x\right)⩾1-c$ for all $x\in \mathbb{R}$, then $D\left(x\right)$ is strictly increasing, and from Lagrange’s mean value theorem, it follows that

$$(1-c)\left|x-y\right|⩽\left|\left(D\right(x)-D(y)\right|.$$

Remark 9.

If $G\left(x\right)$ is Lipschitz continuous with Lipschitz constant $0⩽c<1$, then $\left|D\left(x\right)-D\left(y\right)\right|⩾(1-c)\left|x-y\right|$. Since $\left|G\left(x\right)-G\left(y\right)\right|⩽c\left|x-y\right|$, we see at once that

$$\left|D\left(x\right)-D\left(y\right)\right|⩾\left|x-y\right|-\left|G\left(x\right)-G\left(y\right)\right|⩾(1-c)\left|x-y\right|.$$

4. Proofs of Theorems

To start with, we will construct and prove one auxiliary result. Consider the deterministic differential equation on $\mathbb{R}$

$$x_t=x_0+G\left(x_t\right)-G\left(x_0\right)+{\int }_0^{t}f(s,x_s)ds+{\phi }_t,t\in [0,T],$$

(11)

where $x_0\in \mathbb{R}$, $\phi \in C\left(\right[0,T\left]\right)$, $T>0$, $\phi \left(0\right)=0$, functions $f:[0,T]\times \mathbb{R}\to \mathbb{R}$ and $G:\mathbb{R}\to \mathbb{R}$ are continuous.

Theorem 10.

Assume that conditions (A), (B), (D) and (E) are satisfied. Then, the Equation (11) has a unique solution $x\in C\left(\right[0,T\left]\right)$. If $\phi \in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$, then $x\in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$.

Proof. 

Recall that the function $D\left(x\right)=x-G\left(x\right)$ is strictly monotonic and continuous. Then, it has the inverse $D^{-1}\left(x\right)$, which is strictly monotonic and continuous too.

Existence. We will apply the Picard iteration method. Define the Picard iterations sequence as follows. Let

$$D\left(x_t^{m+1}\right)=D\left(x_0\right)+{\int }_0^{t}f(s,x_s^m)ds+{\phi }_t,x_t^0=x_0,x_0^m=x_0,m⩾0.$$

Let $y_t^{m+1}:=D\left(x_t^{m+1}\right)$. Then

$$y_t^{m+1}=D\left(x_0\right)+{\int }_0^{t}f(s,x_s^m)ds+{\phi }_t$$

is a continuous function. Since $y_t^1=D\left(x_0\right)+{\int }_0^{t}f(s,x_0)ds+{\phi }_t$, it is evident that $y^1$ is a continuous function. By this and continuity of the inverse $D^{-1}$ one obtains that $x_t^1=D^{-1}\left(y_t^1\right)$ is continuous too. In the general case, if $x_t^m$ is a continuous function, then $y_t^{m+1}$ is a continuous function too. Thus, $x_t^{m+1}=D^{-1}\left(y_t^{m+1}\right)$ is continuous.

We now turn to the proof that the sequence ${\left(x_t^m\right)}_{0⩽t⩽T}$ converges uniformly on $[0,T]$ to a continuous limiting function $\tilde{x}\in C\left([0,T]\right)$.

Note that

$$D\left(x_t^{m+1}\right)-D\left(x_t^m\right)={\int }_0^t[f(s,x_s^m)-f(s,x_s^{m-1})]ds.$$

Thus, from the conditions (B) and $\left(\mathbf{D}\right)$, we obtain

$$d|x_t^{m+1}-x_t^m|⩽\left|(D\left(x_t^{m+1}\right)-D\left(x_t^m\right)\right|$$

and

$$\left|x_t^{m+1}-x_t^m\right|⩽\frac{K}{d}{\int }_0^t\left|x_s^m-x_s^{m-1}\right|ds.$$

By this inequality, we have

$$\begin{array}{cc}\hfill \left|x_t^{m+1}-x_t^m\right|⩽& \frac{K^2}{d^2}{\int }_0^t{\int }_0^{s_1}\left|x_{s_2}^{m-1}-x_{s_2}^{m-2}\right|d{s}_{2}d{s}_1\hfill \\ \hfill ⩽& \frac{K^m}{d^m}{\int }_0^t{\int }_0^{s_1}\cdots {\int }_0^{s_{m-1}}\left|x_{s_m}^1-x_{s_m}^0\right|d{s}_m\dots d{s}_{2}d{s}_1.\hfill \end{array}$$

From inequality

$$|x_t^1-x_0|⩽\frac{1}{d}\left({\int }_0^t|f(s,x_0)|ds+|{\phi }_t|\right),$$

it follows that

$$\underset{0⩽t⩽T}{sup}|x_t^1-x_0|⩽C_T,C_T:=d^{-1}\left(T\underset{0⩽t⩽T}{sup}|f(t,x_0)|+\underset{0⩽t⩽T}{sup}|{\phi }_t|\right).$$

Consequently,

$$\left|x_t^{m+1}-x_t^m\right|⩽\frac{{\left(Kt\right)}^m}{d^{m}m!}{C}_T.$$

Thus,

$$\sum _{m=0}^{\infty }\underset{0⩽t⩽T}{sup}\left|x_t^{m+1}-x_t^m\right|<\infty$$

and the sequence $x^m={\left(x_t^m\right)}_{0⩽t⩽T}$ converge uniformly on $[0,T]$ to a continuous limiting function $\tilde{x}\in C\left([0,T]\right)$.

D is a continuous function, and for all $t\in [0,T]$,

$$\begin{array}{cc}\hfill & \left|D\left({\tilde{x}}_t\right)-D\left(x_0\right)-{\int }_0^{t}f(s,{\tilde{x}}_s)ds-{\phi }_t\right|\hfill \\ \hfill & ⩽\left|D\left(x_t^m\right)-D\left({\tilde{x}}_t\right)\right|+{\int }_0^t\left|f(s,x_s^{m-1})-f(s,{\tilde{x}}_s)\right|ds\hfill \\ \hfill & ⩽\underset{0⩽t⩽T}{sup}\left|D\left(x_t^m\right)-D\left({\tilde{x}}_t\right)\right|+TK\underset{0⩽t⩽T}{sup}\left|{\tilde{x}}_t-x_t^{m-1}\right|\underset{m\to \infty }{\to }0;\hfill \end{array}$$

hence, $\tilde{x}$ is a solution of Equation (11).

Next, we prove that $\tilde{x}\in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$, if $\phi \in C^{\gamma }\left([0,T]\right)$, $\gamma \in (0,1)$. By the similar method as above, it follows that

$$|{\tilde{x}}_t-{\tilde{x}}_s|⩽\frac{1}{d}{\int }_s^t|f(u,{\tilde{x}}_u)|du+d^{-1}\left|{\phi }_t-{\phi }_s\right|.$$

Finally, from condition (A), we obtain that

$$\begin{array}{cc}\hfill |{\tilde{x}}_t-{\tilde{x}}_s|⩽& K{d}^{-1}{\int }_s^t(1+|{\tilde{x}}_u\left|\right)du+d^{-1}{\left|\phi \right|}_{\gamma }{(t-s)}^{\gamma }\hfill \\ \hfill ⩽& K{d}^{-1}\left(1+|\tilde{x}{|}_{\infty }\right)(t-s)+d^{-1}{\left|\phi \right|}_{\gamma }{(t-s)}^{\gamma }\hfill \\ \hfill ⩽& d^{-1}\left(K\left(1+|\tilde{x}{|}_{\infty }\right)(T\vee 1)+{\left|\phi \right|}_{\gamma }\right){(t-s)}^{\gamma }\hfill \end{array}$$

(12)

and the proof is complete.

Uniqueness. Assume that y is another solution of (11). The uniqueness of the solution $\tilde{x}$ of (11) follows from the inequality

$$\underset{0⩽t⩽T}{sup}|{\tilde{x}}_t-y_t|⩽K{d}^{-1}{\int }_0^T\underset{0⩽s⩽t}{sup}\left|{\tilde{x}}_s-y_s\right|ds$$

and Gronwall’s lemma. □

4.1. Proof of Theorems 1 and 2

Proof of Theorem 1.

The existence and uniqueness of a solution of Equation (3) is derived directly from the deterministic version of Theorem 10.

Proof of Theorem 2.

We rewrite the approximation (6) in the following way

$$\begin{array}{cc}\hfill D\left(Y_{n,k+1}\right)=& D\left(Y_{n,k}\right)+f(t_k^n,Y_{n,k})h_n+(Z_{t_{k+1}^n}-Z_{t_k^n})\hfill \\ \hfill =& D\left(Y_0\right)+h_n\sum _{j=0}^{k}f(t_j^n,Y_{n,j})+Z_{t_{k+1}^n}\hfill \end{array}$$

(13)

and discretise the process Y by

$$\begin{array}{cc}\hfill D(Y_{t_{k+1}^n})=& D(Y_{t_k^n})+f(t_k^n,Y_{t_k^n})h_n+(Z_{t_{k+1}^n}-Z_{t_k^n})+{\int }_{t_k^n}^{t_{k+1}^n}(f(s,Y_s)-f(t_k^n,Y_{t_k^n}))ds\hfill \\ \hfill =& D(Y_0)+h_n\sum _{j=0}^{k}f(t_j^n,Y_{t_j^n})+Z_{t_{k+1}^n}+R_{n,k+1},\hfill \end{array}$$

(14)

where

$$R_{n,k+1}:=\sum _{j=0}^k{\int }_{t_k^n}^{t_{k+1}^n}(f(s,Y_s)-f(t_j^n,Y_{t_j^n}))ds.$$

Then,

$$D(Y_{t_{k+1}^n})-D(Y_{n,k+1})=h_n\sum _{j=1}^k[f(t_j^n,Y_{n,j})-f(t_j^n,Y_{t_j^n})]+R_{n,k+1}.$$

Similarly, as in proof of Theorem 10, we obtain

$$\left|Y_{t_{k+1}^n}-Y_{n,k+1}\right|⩽\frac{K{h}_n}{d}\sum _{j=1}^k\left|Y_{t_j^n}-Y_{n,j}\right|+d^{-1}{R}_{n,k+1}.$$

(15)

It remains to estimate $|R_{n,k+1}|$. Assume that $t\in [t_k^n,t_{k+1}^n)$. Then (see (12)),

$$\left|Y_t-Y_{t_k^n}\right|⩽d^{-1}\left[K\left(1+{\left|Y\right|}_{\infty }\right)+{\left|Z\right|}_{\gamma }\right]h_n^{\gamma }.$$

Thus,

$$\begin{array}{cc}\hfill |R_{n,k+1}|⩽& n\underset{0⩽k⩽n-1}{max}{\int }_{t_k^n}^{t_{k+1}^n}|f(s,Y_s)-f(t_k^n,Y_{t_k^n})|ds\hfill \\ \hfill ⩽& Kn\underset{0⩽k⩽n-1}{max}{\int }_{t_k^n}^{t_{k+1}^n}{(s-t_k^n)}^{\beta }ds+Kn\underset{0⩽k⩽n-1}{max}{\int }_{t_k^n}^{t_{k+1}^n}|Y_s-Y_{t_k^n}|ds\hfill \\ \hfill ⩽& \frac{KT}{1+\beta }{h}_n^{\beta }+\frac{KT}{d}[K(1+{|Y|}_{\infty }^2)+{|Z|}_{\gamma }]h_n^{\gamma }⩽C_{\omega }{h}_n^{\theta },\hfill \end{array}$$

(16)

where $\theta =\beta \wedge \gamma$,

$$C_{\omega }:=\frac{KT}{1+\beta }{h}_n^{\beta -\theta }+\frac{KT}{d}[K(1+{|Y|}_{\infty }^2)+{|Z|}_{\gamma }]h_n^{\gamma -\theta }.$$

Combining inequalities (15), (16) and the discrete version of Gronwall’s Lemma A1 (see Appendix B), we obtain

$$\underset{0⩽k⩽n-1}{max}|Y_{t_{k+1}^n}-Y_{n,k+1}|⩽C_{\omega }{e}^{KT{(1-c)}^{-1}}{h}_n^{\theta }.$$

□

Proof of Corollary 3.

By definition of $Y^n$, for any $t\in (t_k^n,t_{k+1}^n]$,

$$\begin{array}{cc}\hfill Y_t-Y_t^n& =Y_t-\frac{t-t_k^n}{h_n}{Y}_{n,k+1}-\frac{t_{k+1}^n-t}{h_n}{Y}_{n,k}\hfill \\ \hfill & =\frac{t_k^n-t}{h_n}[{\int }_t^{t_{k+1}^n}f(s,Y_s)ds+Z_{t_{k+1}^n}-Z_t]\hfill \\ \hfill & +\frac{t_{k+1}^n-t}{h_n}[{\int }_{t_k^n}^{t}f(s,X_s)ds+Z_t-Z_{t_k^n}]\hfill \\ \hfill & +\frac{t-t_k^n}{h_n}(Y_{t_{k+1}^n}-Y_{n,k+1})+\frac{t_{k+1}^n-t}{h_n}(Y_{t_k^n}-Y_{n,k}).\hfill \end{array}$$

We see at once that the asymptotic behaviour of the first two terms is $O_{\omega }\left(n^{-\gamma }\right)$, which is clear from inequality (12), and the estimation of the last two terms follows from Theorem 2. This completes the proof. □

4.2. Proof of Theorems 4 and 5

If coefficients $\alpha \left(x\right)$ and $\sigma \left(x\right)$ of Equation (8) are Lipschitz functions and $X\in C^{\gamma }(0,T)$, $1/2<\gamma <1$, then Equation (8) is well defined. We need to find a process $X\in C^{\gamma }(0,T)$, $1/2<\gamma <1$ satisfying Equation (8).

Consider SDE

$$Y_t=Y_0+V\left(Y_t\right)-V\left(Y_0\right)+{\int }_0^{t}f\left(Y_s\right)ds+Z_t,$$

(17)

where

$$f\left(x\right)=\frac{\alpha \left(F^{-1}\left(x\right)\right)}{\sigma \left(F^{-1}\left(x\right)\right)},V\left(x\right)={\int }_0^{x}g\left(u\right)du,$$

$$g\left(u\right)={\Phi }^{\prime }\left(F^{-1}\left(u\right)\right),F\left(x\right)={\int }_0^x\frac{1}{\sigma \left(y\right)}dy.$$

Recall that the Lamperti transform $F\left(x\right)$ has the inverse function $F^{-1}:\mathbb{R}\to \mathbb{R}$, which is strictly increasing and differentiable.

The proof of Theorem 4 consists of two steps. First, we find the conditions under which the Equation (17) has a unique solution in $C^{\gamma }(0,T):1/2<\gamma <1$. Secondly, we prove that $X_t=F^{-1}\left(Y_t\right):Y_0=F\left(X_0\right)$ satisfies Equation (8).

Proof of Corollary 4.

We find requirements for coefficients $\alpha \left(x\right)$ and $\sigma \left(x\right)$ in (17) to satisfy condition (B).

Denote

$$f\left(x\right)=\widehat{f}\left(F^{-1}\left(x\right)\right),\widehat{f}\left(x\right)=\frac{\alpha \left(x\right)}{\sigma \left(x\right)}.$$

Since the inverse function $F^{-1}:\mathbb{R}\to \mathbb{R}$ is differentiable, then

$${\left(F^{-1}\right)}^{\prime }\left(x\right)=\sigma \left(F^{-1}\left(x\right)\right),x\in \mathbb{R}$$

and

$$\begin{array}{cc}\hfill f^{\prime }\left(x\right)=& {\left(\widehat{f}\left(F^{-1}\left(x\right)\right)\right)}^{\prime }={\widehat{f}}^{\prime }\left(F^{-1}\left(x\right)\right){\left(F^{-1}\right)}^{\prime }\left(x\right)\hfill \\ \hfill =& {\widehat{f}}^{\prime }\left(F^{-1}\left(x\right)\right)\sigma \left(F^{-1}\left(x\right)\right)=\left({\alpha }^{\prime }-\frac{\alpha {\sigma }^{\prime }}{\sigma }\right)\left(F^{-1}\left(x\right)\right).\hfill \end{array}$$

Conditions of the theorem imply that

$$\left|f^{\prime }\left(x\right)\right|=\left|\left({\alpha }^{\prime }-\frac{\alpha {\sigma }^{\prime }}{\sigma }\right)\left(F^{-1}\left(x\right)\right)\right|⩽C+C^2.$$

Thus, conditions (A) and (B) are satisfied at the same time. Finally, condition (H) is satisfied for the function $V\left(x\right)$. Indeed, since $V^{\prime }\left(x\right)={\Phi }^{\prime }\left(F^{-1}\left(x\right)\right)$ and ${\Phi }^{\prime }\left(x\right)⩽c$, then $V^{\prime }\left(x\right)⩽c$. From Remark 8, we obtain that conditions (D)–(E) are satisfied.

Thus, $Y\in C^{\gamma }(0,T):1/2<\gamma <1$ and is a unique solution of Equation (17).

Now, we return to the consideration of the process $X_t=F^{-1}\left(Y_t\right)$. Function $\sigma \left(F^{-1}\left(x\right)\right)$ is continuously differentiable. Indeed,

$${\left(\sigma \left(F^{-1}\left(x\right)\right)\right)}^{\prime }={\sigma }^{\prime }\left(F^{-1}\left(x\right)\right)\sigma \left(F^{-1}\left(x\right)\right)$$

and the right side of equality is a composition of two continuous functions ${\sigma }^{\prime }\left(x\right)\sigma \left(x\right)$ and $F^{-1}\left(x\right)$. Thus, function ${\left(F^{-1}\right)}^{\prime }\left(x\right)=\sigma \left(F^{-1}\left(x\right)\right)$ is locally Lipschitz, and the process $\sigma \left(F^{-1}\left(Y_t\right)\right)$ is bounded on $[0,T]$, i.e., there exists a random variable $M\left(\omega \right)$ such that

$$\underset{0⩽t⩽T}{sup}\left|\sigma \left(F^{-1}\left(Y_t\right)\right)\right|⩽M\left(\omega \right).$$

Consequently,

$$|X_t-X_s|=\left|F^{-1}\left(Y_t\right)-F^{-1}\left(Y_s\right)\right|⩽M|Y_t-Y_s|⩽M{G}_{\gamma ,T}{\left|t-s\right|}^{\gamma },$$

where $G_{\gamma ,T}$ is a random variable (see Appendix B.2). Therefore, we obtain $X\in C^{\gamma }\left([0,T]\right)$, $1/2<\gamma <1$.

Now, we prove that the process $X_t=F^{-1}\left(Y_t\right)$, $Y_0=F\left(X_0\right)$ satisfies Equation (8). Note that the function $F^{\prime }\left(x\right)$ is locally Lipschitz. Indeed,

$$F^{\prime \prime }\left(x\right)=-\frac{{\sigma }^{\prime }\left(x\right)}{{\sigma }^2\left(x\right)},\underset{x\in \mathbb{R}}{inf}\sigma \left(x\right)>0,$$

and $\sigma \left(x\right)$ is a continuous function, then $F^{\prime \prime }\left(x\right)$ is a continuous function, and therefore, $F^{\prime }\left(x\right)$ is locally Lipschitz. In addition, we will note that functions $V\left(x\right)$ and $\Phi \left(x\right)$ are Lipschitz.

For $X,Y\in C^{\gamma }\left([0,T]\right)$, $1/2<\gamma <1$, the processes ${\left(F^{-1}\right)}^{\prime }\left(Y_t\right)$, $V^{\prime }\left(Y_t\right)$, $F^{\prime }\left(X_t\right)$, and ${\Phi }^{\prime }\left(X_t\right)$ are Riemann–Stieltjes integrable with respect to X and Y (see Theorem A3 in Appendix B). Now, we can apply the chain rule formula for $F^{-1}\left(X_t\right)$, $V\left(Y_t\right)$ and $\Phi \left(X_t\right)$ (see Theorem A3 in Appendix B). By the chain rule and substitution rule (see Proposition A1 in Appendix B), we obtain

$$V\left(Y_t\right)-V\left(Y_0\right)={\int }_0^t{V}^{\prime }\left(Y_s\right)d{Y}_s={\int }_0^{t}g\left(Y_s\right)d{Y}_s,\Phi \left(X_t\right)-\Phi \left(X_0\right)={\int }_0^t{\Phi }^{\prime }\left(X_s\right)d{X}_s,$$

and

$$\begin{array}{cc}\hfill X_t=& X_0+{\int }_0^t{\left(F^{-1}\right)}^{\prime }\left(Y_s\right)d{Y}_s=\left(F^{-1}\right)\left(Y_0\right)+{\int }_0^t\sigma \left(F^{-1}\left(Y_s\right)\right)d{Y}_s\hfill \\ \hfill =& X_0+{\int }_0^t\sigma \left(X_s\right)dV\left(Y_s\right)+{\int }_0^t\sigma \left(X_s\right)f\left(F\left(X_s\right)\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s\hfill \\ \hfill =& X_0+{\int }_0^t\sigma \left(X_s\right)g\left(F\left(X_s\right)\right)dF\left(X_s\right)+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s\hfill \\ \hfill =& X_0+{\int }_0^t\sigma \left(X_s\right)g\left(F\left(X_s\right)\right){\sigma }^{-1}\left(X_s\right)d{X}_s+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s\hfill \\ \hfill =& X_0+{\int }_0^{t}g\left(F\left(X_s\right)\right)d{X}_s+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s\hfill \\ \hfill =& X_0+{\int }_0^t{\Phi }^{\prime }\left(X_s\right)d{X}_s+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s\hfill \\ \hfill =& X_0+\Phi \left(X_t\right)-\Phi \left(x_0\right)+{\int }_0^t\alpha \left(X_s\right)ds+{\int }_0^t\sigma \left(X_s\right)d{Z}_s.\hfill \end{array}$$

Thus, X is a solution of the Equation (8), and this solution is unique (it follows from the uniqueness of the solution Y and properties of the Lamperti transform F). □

Proof of Corollary 5.

First, observe that

$$\begin{array}{cc}\hfill \left|X_{t_k^n}-F^{-1}(Y_{n,k})\right|=& \left|F^{-1}(Y_{t_k^n})-F^{-1}(Y_{n,k})\right|\hfill \\ \hfill =& \left|\sigma (F^{-1}(Y_{t_k^n}+{\theta }_{n,k}(Y_{t_k^n}-Y_{n,k})))\right|·\left|Y_{t_k^n}-Y_{n,k}\right|,\hfill \end{array}$$

where ${\theta }_{n,k}\in (0,1)$. Theorem 2 implies that

$$\underset{1⩽k⩽n}{max}|Y_{t_k^n}+{\theta }_{n,k}(Y_{t_k^n}-Y_{n,k})|⩽\underset{0⩽t⩽T}{sup}|Y_t|+C_{\omega }{h}^{\gamma },$$

where $C_{\omega }$ is a random variable. Since the function $\sigma \left(F^{-1}\left(x\right)\right)$ is locally Lipschitz, we have

$$\sigma \left(F^{-1}\left(Y_{t_k^n}+{\theta }_{n,k}(Y_{t_k^n}-Y_{n,k})\right)\right)$$

is bounded for almost all $\omega$. Applying Theorem 2, we conclude that

$$\underset{1⩽k⩽n}{max}\left|X_{t_k^n}-F^{-1}\left(Y_{n,k}\right)\right|=O_{\omega }\left(h_n^{\gamma }\right),\gamma \in (1/2,1).$$

□

5. Example: Fractional Pearson Diffusion Process with Soft Wall

First, as an example of a stochastic process, which satisfies conditions (A)–(E), we will consider the fractional version of the Pearson diffusion process. This process is widely used in physical and chemical sciences, engineering, rheology, environmental sciences and financial mathematics [15].

Let

$$X_t=x_0+\Phi \left(X_t\right)-\Phi \left(x_0\right)+{\int }_0^t\alpha \left(X_t\right)dt+{\int }_0^t\sigma \left(X_t\right)d{B}_t^H,t\ge 0,$$

(18)

with

$$\alpha \left(x\right)=b-ax,\sigma \left(x\right)=\sqrt{{\sigma }_0+{\sigma }_{1}x+{\sigma }_2{x}^2},{\sigma }_2>0.$$

Assume that ${\sigma }_i$, $i=0,1,2$ are such that the square root is well defined and ${inf}_{x\in \mathbb{R}}\sigma \left(x\right)>0$, i.e., ${\sigma }_0+{\sigma }_{1}x+{\sigma }_2{x}^2>0$.

Note that

$$|{\alpha }^{\prime }\left(x\right)|⩽|a|,{\sigma }^{\prime }\left(x\right)=\frac{{\sigma }_1+2{\sigma }_{2}x}{2\sqrt{{\sigma }_0+{\sigma }_{1}x+{\sigma }_2{x}^2}}.$$

Since

$${\sigma }_0+{\sigma }_{1}x+{\sigma }_2{x}^2⩾{\sigma }_2{\left(x+\frac{{\sigma }_1}{2{\sigma }_2}\right)}^2,$$

then

$$\left|{\sigma }^{\prime }\left(x\right)\right|⩽\frac{|{\sigma }_1+2{\sigma }_{2}x|}{2\sqrt{{\sigma }_2}\left|x+\frac{{\sigma }_1}{2{\sigma }_2}\right|}=\sqrt{{\sigma }_2}.$$

It remains to prove the existence of a constant $K>0:\frac{\left|\alpha \left(x\right)\right|}{\left|\sigma \left(x\right)\right|}⩽K$. By function

$$\frac{b-ax}{\sqrt{{\sigma }_0+{\sigma }_{1}x+{\sigma }_2{x}^2}}$$

being continuous and therefore bounded on a compact set, we obtain

$$\left|\frac{\alpha \left(x\right)}{\sigma \left(x\right)}\right|⩽K$$

for some constant $K>0$.

Applying the Lamperti transform F to Equation (18), we obtain

$$Y_t=y_0+V\left(Y_t\right)-V\left(y_0\right)+{\int }_0^t\frac{b-a{Y}_t}{\sqrt{{\sigma }_0+{\sigma }_1{Y}_s+{\sigma }_2{Y}_s^2}}ds+B_t^H,$$

where $y_0=F\left(x_0\right)$ and

$$V\left(x\right)={\int }_0^{x}g\left(u\right)du,g\left(u\right)={\Phi }^{\prime }\left(F^{-1}\left(u\right)\right).$$

The approximation scheme for the process (18) has a form

$$X_{n,k}=F^{-1}\left(Y_{n,k}\right),$$

where

$$Y_{n,k+1}-V\left(Y_{n,k+1}\right)=Y_{n,k}-V\left(Y_{n,k}\right)+\frac{b-a{Y}_{n,k}}{\sqrt{{\sigma }_0+{\sigma }_1{Y}_{n,k}+{\sigma }_2{Y}_{n,k}^2}}{h}_n+(B_{t_{k+1}^n}^H-B_{t_k^n}^H).$$

6. Modelling: Fractional Vasicek Process

The Vasicek model is one of the earliest classical stochastic models. Usually, it used to describe the fluctuation of interest rates, but it also can be utilised in many other fields such as biology, medical and environmental sciences [16]. Since the fractional Vasicek model admits to conditions (A)–(B) and has a far less bulky expression with fewer parameters than the Pearson diffusion process, we have chosen it as our modeling example.

Let

$$X_t=x_0+{\int }_0^t(\beta -\alpha X\left(s\right))ds+\sigma B^H\left(t\right),t\ge 0,\alpha ,\beta \in \mathbb{R},\sigma \ge 0.$$

(19)

6.1. Profile of Soft-Wall Resistant Force

We introduce a soft-wall resistant force G into the process (19) and obtain

$$X_t=x_0+G\left(X_t\right)-G\left(x_0\right)+{\int }_0^t(\beta -\alpha X\left(s\right))ds+\sigma B^H\left(t\right),t\ge 0,\alpha ,\beta \in \mathbb{R},\sigma \ge 0.$$

(20)

Force G has the following conditions (D)–(E), satisfying profile

$$G\left(x\right)=\left\{\begin{array}{cc}{G}_1\left(x\right)=k_1{e}^{-\lambda (x-a_1)},\hfill & ifx>a_1\hfill \\ G_2\left(x\right)=k_2(a_1-x)+G_1\left(a_1\right),\hfill & ifx\le a_1,\hfill \end{array}\right.$$

(21)

where $\lambda \in (0,1)$; $a_1,k_1,k_2>0$.

This specific construction of force G is chosen in order to simulate the rapid change of properties in a process surrounding medium by the simplest controllable means—linear function.

As we can see from Figure 1a, profile (21) produces a rapid increase of resistance after crossing a soft-wall boundary at $a_1$. However, conditions for function G and its derivative $G^{\prime }$ do not limit the number of these changes. For example, we can have a two-step change in the force profile. Naturally, these two steps can both increase the force change rate (Figure 1b), or the first step can increase the rate and the second one decrease it (Figure 1c).

Adding another inflection point $a_2$ to profile (21), we obtain the following two-step profile

$$G\left(x\right)=\left\{\begin{array}{cc}{G}_1\left(x\right)=k_1{e}^{-\lambda (x-a_1)},\hfill & ifx>a_1\hfill \\ G_2\left(x\right)=k_2(a_1-x)+G_1\left(a_1\right),\hfill & if{a}_2\le x\le a_1\hfill \\ k_3(a_2-x)+G_2\left(a_2\right),\hfill & ifx<a_2,\hfill \end{array}\right.$$

(22)

where $\lambda \in (0,1)$; $a_1,a_2,k_1,k_2,k_3>0$.

6.2. Process Trajectories Simulation under Soft-Wall conditions

Using process approximation scheme (6) for the fractional Vasicek process (20) and soft-wall resistant force profiles (21) and (22) illustrated bellow in Figure 2 we obtain the following trajectories of the fractional Vasicek process.

As we can see from Figure 3 by comparing soft-wall process trajectories with trajectories without soft-wall resistance force, force G “pushes” the process trajectories towards the boundary. The strength of this “push” is dependent on the magnitude of derivative $G^{\prime }$ (large enough derivatives have almost a straightening effect on the trajectories).

Even more fascinating soft-wall behaviour can be observed for a double “increase–decrease” soft-wall force profile.

In Figure 4 we clearly see that the volatility of trajectories is not dependent on the value of G, but on the magnitude of its derivative $G^{\prime }$. Hence, the double soft-wall force profile (Figure 5) simulates some kind of resistant membrane between boundaries $a_1$ and $a_2$ being punctured by the fractional Vasicek process (Figure 4). Notice how, after leaving the area $[a_2,a_1]$, the increments of the the soft-wall affected trajectories become almost identical to the increments of the standard process.

7. Conclusions

In this work, we formed a mathematically rigid framework from a stochastic point of view to describe processes with a soft-wall condition in the form of fractional SDE. Subsequently, we proposed and proved the validity of conditions set for FSDE solution existence and uniqueness. This condition set covers a wide class of functions and processes. Furthermore, we presented and investigated a process approximation scheme, which has a good convergence rate and is easy to use for process trajectories simulations. This fact was proven and explored by simulating the Vasicek process with soft-wall conditions of varying levels of complexity. Therefore, our results are a new and original addition in the field of fractional SDE and have broad application perspectives in a variety of fields.