A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

1 Introduction

As we all know, fractional differential equations (FDEs) can be viewed as a generalization of ordinary differential equations, which can be used to model complex physical phenomena and processes with nonlocal properties. In recent twenty years, with the continuous development of fractional differential equations, they have very important applications in the fields of mechanics, electrical engineering, electromagnetic wave, population system

[31, 34, 14, 30, 37] and so on. Among them, multi-term FDEs are known as an important tool in describing viscoelastic damping materials, modelling nonlinear wave phenomenon in plasma and simulating anomalous diffusive process, such as anomalous relaxation in magnetic resonance imaging signal magnitude, mechanical models of oxygen delivery through capillaries [21, 29, 32, 1], etc. Thus, there are lots of excellent researches focusing on the analytical methods and numerical methods for multi-term FDEs, see [22, 8, 10, 5, 13] and the references therein.

Meanwhile, with scientific research constantly deepening, researchers find that there are always some noise disturbances that could not be ignored in real life, and in order to better describe the phenomena and process influenced by these noisy factors, researchers pay more attention to stochastic differential equations (SDEs). As we all know, many classical SDEs play an extremely important role in describing physical phenomena. For instance, Langevin equations are used to describe Brownian motion and explain Einstein relations [19] and the stochastic Navier-Stokes equation are often used to simulate various problems in fluid motion[35]. Nowadays, in addition to the field of physics, SDEs are widely used in option pricing, population growth and many other fields [11, 20, 28]. Especially, stochastic fractional differential equations (SFDEs) appeal more scholars’ attention and many studies have been carried out, such as the random motion of harmonically trapped charged particles in a constant external magnetic field, the relationship between fluctuation-dissipation theorem and physical behavior, the noise driving in some financial models, epidemiological research [25, 23, 27, 36] and so on. Obviously, it is quite difficult or even rarely impossible to obtain the exact solutions of SFDEs, thus there has been a growing interest to construct numerical methods for these equations. Until now, many numerical methods have been developed for SFDEs. For example, Kamrani in [18] investigated a numerical solution of SFDEs driven by additive noise and proved the convergence of the proposed method. Jin et al. in [17] studied the stochastic time-fractional diffusion problem driven by fractionally integrated Gaussian noise and developed a numerical scheme by employing the Galerkin finite element method in space, Gr $¨ u$ nwald-Letnikov formula in time and $L^{2}$ -projection for the noise. Doan et al. in [12]

constructed an Euler-Maruyama (EM) method for a kind of SFDEs driven by a multiplicative white noise with the Caputo fractional order

α \in (0.5, 1)

. Zhou et al. in [40] used the finite difference method to solve the stochastic fractional nonlinear wave equation and presented the performance of numerical solution and property of energy under effect of two different types of noise, i.e., additive noise and multiplicative noise. Additionally, there also exist many other good works of SFDEs, see [4, 39, 2, 3], for examples.

Motivated by the above numerical methods for SFDEs, we in this paper will construct and analyze a modified EM method for the following multi-term Riemann-Liouville SFDEs,

y^{'} (t) + m \sum i = 1_{0}^{R L} D_{t}^{α_{i}} y (t) = f (t, y (t)) + g (t, y (t)) \frac{d W_{t}}{d t}, t \in [0, T],

(1.1)

where

$_{0}^{R L} D_{t}^{α_{i}} y (t) (i = 1, 2, \dots, m)$ are the Riemann-Liouville fractional derivatives with $0 < α_{1} < α_{2} < \dots < α_{m} < 1$ ;
$W_{t}$ is a one dimensional ${F_{t}}_{{0 \leq t \leq T}}$
-adapted Brownian motion defined on the complete filtered probability space
${Ω, F, F = {F_{t}}_{t \in [0, \infty)}, P}$ ;
the initial value $y (0) = y_{0} \in R^{d}$ is an $F_{0}$
-measurable random variable such that
$E [| y_{0} |^{2}] < + \infty$ and $f, g : [0, T] \times R^{d} \to R^{d}$ are measurable functions.

In addition to the purpose of numerically solving the multi-term Riemann-Liouville SFDEs (1.1), we hope to construct more efficient numerical methods to avoid the huge computational cost caused by the nonlocal property of Riemann-Liouville fractional integral operators. Recently, lots of efficient methods based on the sum-of-exponentials (SOEs) technique proposed in [16] by Jiang et al. are presented to overcome this difficulty for FDEs, see [41, 6, 15, 33], for examples. There are also some articles about applying the SOEs technique to efficiently solve the SFDEs. For example, Dai et al in [9] proposed an EM method and its fast implementation based on the SOEs approximation for Lévy-driven stochastic Volterra integral equations with doubly singular kernels. Ma et al. [24] used the SOEs approximation to simulate the rough volatility and developed a fast two-step iteration algorithm. We in [38] also presented a fast EM method for a class of nonlinear SFDEs by the SOEs technique. Herein, based on the SOEs approximation and the proposed EM method, a fast EM method for (1.1) is also introduced and analyzed in this paper.

The rest of this paper is organized as follows. In Section 2, some notations and preliminaries are given. In Section 3, the modified EM method is derived and the strong convergence of this method is proved. Section 4 aims to construct a fast EM method based on SOEs technique and present the corresponding numerical theoretical results. Two numerical examples are shown to examine the theoretical results and illustrate the effectiveness of the proposed two methods. Finally, a brief conclusion is given.

2 Notations and preliminaries

In this section, we first introduce some useful notations that will be used throughout this paper. Let $E$ denote the expectation corresponding to $R$ . Define $L^{2} (Ω, F_{t}, R)$ as the space of all $F_{t}$ -measurable, mean square integral functions $f = (f_{1}, f_{2}, \dots, f_{d})^{⊤} : Ω \to R^{d}$ with standard Euclidean norm $∥ f ∥ = \sqrt{\sum_{i = 1}^{d} E [| f_{i} |^{2}]}$ . For two real numbers $A$ and $B$ , we denote $max {A, B} = A \lor B$ and $min {A, B} = A \land B$ . Moreover, the capital letter $C$ will be used to represent a positive constant whose value may change when it appears in different places. The following two definitions are taken from [26].

Definition 2.1

The Riemann-Liouville fractional integral of order $α (α \geq 0)$ of function $f : [0, + \infty) \to R^{d}$ is defined by

_{0} J_{t}^{α} f (t) = \frac{1}{Γ (α)} \int_{0}^{t} (t - τ)^{α - 1} f (τ) d τ,

with $J^{0} f (t) = f (t)$ , where $Γ (α) = \int_{0}^{+ \infty} e^{- t} t^{α - 1} d t$ is the Euler gamma function.

Definition 2.2

The Riemann-Liouville derivative of order $α (0 \leq α < 1)$ of function $f \in C ([0, + \infty))$ can be written as

_{0}^{R L} D_{t}^{α} f (t) = \frac{d}{d t} (\frac{1}{Γ (1 - α)} \int_{0}^{t} \frac{f (τ)}{(t - τ)^{α}} d τ) .

In the rest of this section, to guarantee the existence and uniqueness of the solution of equation (1.1), we impose three important hypotheses.

Assumption 2.1

(H $¨ o$ lder continuity) There exists a positive constant $L_{1}$ such that for all $t_{1}, t_{2} \in [0, T]$ and $y \in R^{d}$ , $f$ and $g$ satisfy the condition:

∥ f (t_{1}, y) - f (t_{2}, y) ∥ \lor ∥ g (t_{1}, y) - g (t_{2}, y) ∥ \leq L_{1} | t_{1} - t_{2} | .

Assumption 2.2

(Lipschitz continuity) There exists a positive constant $L_{2}$ such that for all $t \in [0, T]$ and $x_{1}, x_{2} \in R^{d}$ , $f$ and $g$ satisfy the inequality:

∥ f (t, x_{1}) - f (t, x_{2}) ∥ \lor ∥ g (t, x_{1}) - g (t, x_{2}) ∥ \leq L_{2} ∥ x_{1} - x_{2} ∥ .

Assumption 2.3

(Linear growth condition) There exists a positive constant $L_{3}$ such that for all $t \in [0, T]$ and all $y \in R^{d}$ , $f$ and $g$ satisfy the condition:

∥ f (t, y) ∥ \lor ∥ g (t, y) ∥ \leq L_{3} (1 + ∥ y ∥) .

3 The modified EM method for multi-term Riemann-Liouville SFDEs

3.1 The modified EM method

In order to get the numerical method of the Riemann-Liouville SFDEs (1.1), we first transform equation (1.1) into the following stochastic Volterra integral equation (SVIE) by taking the Riemann-Liouville fractional integral operator $_{0} J_{t}$ on both sides of equation (1.1), that is

y (t)

=

y_{0} - m \sum i = 1 \frac{1}{Γ (1 - α_{i})} \int_{0}^{t} (t - s)^{- α_{i}} y (s) d s + \int_{0}^{t} f (s, y (s)) d s + \int_{0}^{t} g (s, y (s)) d W_{s},

(3.1)

where the property of Riemann-Liouville fractional integral and Riemann-Liouville fractional derivative is used, i.e.

_{0} J_{t} (_{0}^{R L} D_{t}^{α} y (t)) = \frac{1}{Γ (1 - α_{i})} \int_{0}^{t} \frac{y (s)}{(t - s)^{α_{i}}} d s .

In another way, for each $y_{0} \in L^{2} (Ω, F_{t}, R)$ , when the above equality holds for $t \in [0, T]$ , a $F$ -adapted process $y (t)$ is called a solution of equation (1.1) on the interval $[0, T]$ with the initial condition $y (0) = y_{0}$ .

For every integer $N \geq 1$ , the modified EM method for equation (3.1) can be presented as

	$Y^{(N)} (t_{n})$	$=$	$y_{0} - m \sum i = 1 \frac{1}{Γ (1 - α_{i})} n - 1 \sum j = 0 (t_{n} - t_{j})^{- α_{i}} Y^{(N)} (t_{j}) h$		(3.2)
			$+ n - 1 \sum j = 0 f (t_{j}, Y^{(N)} (t_{j})) h + n - 1 \sum j = 0 g (t_{j}, Y^{(N)} (t_{j})) △ W_{j},$		(3.2)

where $t_{n} = n h (n = 0, 1, 2, \dots, N)$ denote the grid points with step size $h = \frac{T}{N}$ , $△ W_{j} = W (t_{j + 1}) - W (t_{j}) (j = 0, 1, 2, \dots, N - 1)$ denote the increments of Brownian motion. To facilitate its convergence analysis, we introduce a continuous-time version

	$Y^{(N)} (t)$	$=$	$y_{0} - m \sum i = 1 \frac{1}{Γ (1 - α_{i})} \int_{0}^{t} (t - τ_{N} (s))^{- α_{i}} Y^{(N)} (τ_{N} (s)) d s$		(3.3)
			$+ \int_{0}^{t} f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d s + \int_{0}^{t} g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d W (s),$		(3.3)

where $τ_{N} (s) = \frac{n T}{N}, n = 0, 1, \dots, N - 1$ for $s \in (\frac{n T}{N}, \frac{(n + 1) T}{N}]$ .

3.2 Strong convergence of the modified EM method

To prove the strong convergence of the modified EM method (3.3), we first list some necessary lemmas.

Lemma 3.1

If $α \in (0, 1]$ , then for any $s^{'} \leq s < ~ t < t$ , the following inequality holds

\frac{1}{(~ t - s^{'})^{α}} - \frac{1}{(t - s^{'})^{α}} \leq \frac{1}{(~ t - s)^{α}} - \frac{1}{(t - s)^{α}} .

Proof. Define $p (x) = \frac{1}{(~ t - x)^{α}} - \frac{1}{(t - x)^{α}}$ with $x < ~ t < t$ . Then we have $p^{'} (x) = α [\frac{1}{(~ t - s)^{α + 1}} - \frac{1}{(t - s)^{α + 1}}]$ . According to the property of power function, it holds that $p^{'} (x) > 0$ . Thus it implies that $p (s^{'}) \leq p (s)$ together with $s^{'} \leq s$ . $□$

Lemma 3.2

Suppose $α \in (0, 1)$ , then there exists a positive constant $C$ independent of $h$ such that for any $t \in (t_{n}, t_{n + 1}], n = 0, 1, 2, \dots, N - 1$ ,

\int_{0}^{t} | (t - τ_{N} (s))^{- α} - (t - s)^{- α} | d s \leq C h^{1 - α},

where $τ_{N} (s)$ have been defined in the above subsection.

Proof. For $t \in (t_{n}, t_{n + 1}]$ , it is easily derived that

			$\int_{0}^{t} \| (t - τ_{N} (s))^{- α} - (t - s)^{- α} \| d s$
		$=$	$\int_{0}^{t} (t - s)^{- α} - (t - τ_{N} (s))^{- α} d s$
		$\leq$	$n - 1 \sum j = 0 \int_{t_{j}}^{t_{j + 1}} (t_{n} - s)^{- α} - (t_{n + 1} - t_{j})^{- α} d s + \int_{t_{n}}^{t} (t - s)^{- α} - (t - t_{n})^{- α} d s$
		$\leq$	$\frac{t_{n}}{1 - α} - n - 1 \sum j = 0 (t_{n + 1} - t_{j})^{- α} h + \frac{α}{1 - α} (t - t_{n})^{1 - α}$
		$\leq$	$\frac{(n h)^{1 - α}}{1 - α} - n ((n + 1) h)^{- α} h + \frac{α}{1 - α} h^{1 - α}$
		$\leq$	$C h^{1 - α} .$

$□$

The following inequality can easily be obtained by the property of norm.

Lemma 3.3

For all $y_{i} \in R^{d}$ , $i = 1, 2, \dots, n$ , the following inequality holds

∥ n \sum i = 1 y_{i} ∥^{2} \leq n n \sum i = 1 ∥ y_{i} ∥^{2} .

To carry out the proof of the strong convergence of the modified EM method (3.3

), the bounded estimate for the numerical solutions is given.

Theorem 3.1

Suppose Assumption 2.3 holds, then for all $N \in N^{*}$ , the numerical solution $Y^{(N)} (t)$ of the modified EM method (3.3) is bounded, that is

E [∥ Y^{(N)} (t) ∥^{2}] \leq C a n d E [∥ Y^{(N)} (t_{n}) ∥^{2}] \leq C, \forall t, t_{n} \in [0, T],

where $C$ is a positive constant independent of $N$ .

Proof. From equation (3.3) and Lemma 3.3, it is deduced that

			$\frac{1}{4} E [∥ Y^{(N)} (t) ∥^{2}]$
		$\leq$	$E [∥ y_{0} ∥^{2}] + E [∥ m \sum i = 1 \frac{1}{Γ (1 - α_{i})} \int_{0}^{t} (t - τ_{N} (s))^{- α_{i}} Y^{(N)} (τ_{N} (s)) d s ∥^{2}]$
			$+ E [∥ \int_{0}^{t} f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d s ∥^{2}] + E [∥ \int_{0}^{t} g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d W (s) ∥^{2}] .$

Then using the H $¨ o$ lder inequality and It $^o$ isometry as well as Lemma 3.3 implies that

			$\frac{1}{4} E [∥ Y^{(N)} (t) ∥^{2}]$
		$\leq$	$E [∥ y_{0} ∥^{2}] + m \sum i = 1 \frac{m}{Γ^{2} (1 - α_{i})} \int_{0}^{t} \frac{1}{(t - τ_{N} (s))^{α_{i}}} d s \int_{0}^{t} \frac{E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}]}{(t - τ_{N} (s))^{α_{i}}} d s$

Applying the linear growth condition of Assumption 2.3 and the property of power function with the fact $τ_{N} (s) \leq s$ , it can be deduced that

			$\frac{1}{4} E [∥ Y^{(N)} (t) ∥^{2}]$
		$\leq$
			$+ t \int_{0}^{t} 2 L_{3}^{2} (1 + E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}]) d s + \int_{0}^{t} 2 L_{3}^{2} (1 + E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}]) d s .$

Arranging the above inequality, it becomes

			$E [∥ Y^{(N)} (t) ∥^{2}]$
		$\leq$	$4 E [∥ y_{0} ∥^{2}] + m \sum i = 1 \frac{4 m}{Γ^{2} (1 - α_{i})} \frac{t^{1 - α_{i}}}{1 - α_{i}} \int_{0}^{t} (t - s)^{- α_{i}} E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
			$+ (8 t + 8) L_{3}^{2} \int_{0}^{t} 1 + E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
		$\leq$	$4 E [∥ y_{0} ∥^{2}] + (8 T^{2} + 8 T) L_{3}^{2} + (8 T + 8) L_{3}^{2} \int_{0}^{t} E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
			$+ m \sum i = 1 \frac{4 m T^{1 - α_{i}}}{Γ (1 - α_{i}) Γ (2 - α_{i})} \int_{0}^{t} (t - s)^{- α_{i}} E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s .$

There exists a positive constant $α \in {α_{1}, α_{2}, \dots, α_{m}}$ such that

E [∥ Y^{(N)} (t) ∥^{2}]

\leq

4 E [∥ y_{0} ∥^{2}] + (8 T^{2} + 8 T) L_{3}^{2} + C \int_{0}^{t} (t - s)^{- α} E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s

Taking the supremum on both sides of the above inequality and according to the Gronwall’s inequality, we can arrive at

E [∥ Y^{(N)} (t) ∥^{2}] \leq C .

Similarly, we can prove that $E [∥ Y^{(N)} (t_{n}) ∥^{2}] \leq C$ . This proof is completed. $□$

Lemma 3.4

Suppose Assumption 2.3 holds, then for all $N \in N^{*}$ , there exists a positive constant $C$ independent of $N$ such that for all $t \in (t_{n}, t_{n + 1}], t_{n} \in [0, T]$ ,

E [∥ Y^{(N)} (t) - Y^{(N)} (t_{n}) ∥^{2}] \leq C h^{min {2 (1 - α_{m}), 1}} .

Proof. For arbitrary $t \in (t_{n}, t_{n + 1}], t_{n} \in [0, T]$ , it follows from equation (3.3) that

			$Y^{(N)} (t) - Y^{(N)} (t_{n})$
		$=$
			$+ \int_{t_{n}}^{t} (t - τ_{N} (s))^{- α_{i}} Y^{(N)} (τ_{N} (s)) d s}$
			$+ \int_{t_{n}}^{t} f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d s + \int_{t_{n}}^{t} g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d W (s) .$

Applying Lemma 3.3, we obtain

			$\frac{1}{4} E [∥ Y^{(N)} (t) - Y^{(N)} (t_{n}) ∥^{2}]$
		$\leq$	$m m \sum i = 1 E [∥ \frac{1}{Γ (1 - α_{i})} \int_{0}^{t_{n}} [(t - τ_{N} (s))^{- α_{i}} - (t_{n} - τ_{N} (s))^{- α_{i}}] Y^{(N)} (τ_{N} (s)) d s ∥^{2}]$
			$+ m m \sum i = 1 E [∥ \frac{1}{Γ (1 - α_{i})} \int_{t_{n}}^{t} (t - τ_{N} (s))^{- α_{i}} Y^{(N)} (τ_{N} (s)) d s ∥^{2}]$
			$+ E [∥ \int_{t_{n}}^{t} f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d s ∥^{2}] + E [∥ g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) d W (s) ∥^{2}] .$

By H $¨ o$ lder inequality and It $^o$ isometry, the above inequality turns into

			$\frac{1}{4} E [∥ Y^{(N)} (t) - Y^{(N)} (t_{n}) ∥^{2}]$
		$\leq$	$m \sum i = 1 \frac{m}{Γ^{2} (1 - α_{i})} \int_{0}^{t_{n}} \| (t - τ_{N} (s))^{- α_{i}} - (t_{N} - τ_{N} (s))^{- α_{i}} \| d s$
			$\cdot \int_{0}^{t_{n}} \| (t - τ_{N} (s))^{- α_{i}} - (t_{n} - τ_{N} (s))^{- α_{i}} \| E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
			$+ m \sum i = 1 \frac{m}{Γ^{2} (1 - α_{i})} \int_{t_{n}}^{t} (t - τ_{N} (s))^{- α_{i}} d s \int_{t_{n}}^{t} (t - τ_{N} (s))^{- α_{i}} E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
			$+ (t - t_{n}) \int_{t_{n}}^{t} E [∥ f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) ∥^{2}] d s + \int_{t_{n}}^{t} E [∥ g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) ∥^{2}] d s .$

This together with Lemma 3.1, Assumption 2.3 and Theorem 3.1 as well as the fact $τ_{N} (s) \leq s$ implies that

			$\frac{1}{4} E [∥ Y^{(N)} (t) - Y^{(N)} (t_{n}) ∥^{2}]$
		$\leq$	$m \sum i = 1 \frac{C m}{Γ^{2} (1 - α_{i})} \int_{0}^{t_{n}} (t_{n} - s)^{- α_{i}} - (t - s)^{- α_{i}} d s \int_{0}^{t_{n}} (t_{n} - s)^{- α_{i}} - (t - s)^{- α_{i}} d s$
			$+ m \sum i = 1 \frac{C m}{Γ^{2} (1 - α_{i})} \int_{t_{n}}^{t} (t - s)^{- α_{i}} d s \int_{t_{n}}^{t} (t - s)^{- α_{i}} d s + 2 L_{3}^{2} (1 + C) [(t - t_{n})^{2} + (t - t_{n})]$
		$\leq$	$m \sum i = 1 \frac{C m}{Γ^{2} (2 - α_{i})} {[(t - t_{n})^{1 - α_{i}} + t_{n}^{1 - α_{i}} - t^{1 - α_{i}}]}_{n}^{2} + m \sum i = 1 \frac{m C}{Γ^{2} (2 - α_{i})} (t - t_{n})^{2 - 2 α_{i}}$
			$+ 2 L_{3}^{2} (1 + C) [(t - t_{n})^{2} + (t - t_{n})] .$

Noticing $t - t_{n} \leq h$ , thus we can get

E [∥ Y^{(N)} (t) - Y^{(N)} (t_{n}) ∥^{2}] \leq C h^{min {2 (1 - α_{m}), 1}} .

This proof is completed. $□$

Theorem 3.2

(Strong convergence) Under Assumptions 2.1-2.3, there exists a positive constant $C$ independent of $N$ such that for all $t \in [0, T]$ ,

E [∥ Y^{(N)} (t) - y (t) ∥^{2}] \leq C h^{min {2 (1 - α_{m}), 1}} .

Proof. From SVIE (3.1) and the modified EM method (3.3), by adding some intermediate items and using Lemma 3.3, H $¨ o$ lder inequality and It $^o$ isometry, we derive that

			$\frac{1}{3} E [∥ Y^{(N)} (t) - y (t) ∥^{2}]$
		$\leq$	$m \sum i = 1 \frac{3 m}{Γ^{2} (1 - α_{i})} {\int_{0}^{t} \| (t - τ_{N} (s))^{- α_{i}} - (t - s)^{- α_{i}} \| d s$
			$\cdot \int_{0}^{t} \| (t - τ_{N} (s))^{- α_{i}} - (t - s)^{- α_{i}} \| E [∥ Y^{(N)} (τ_{N} (s)) ∥^{2}] d s$
			$+ \int_{0}^{t} (t - s)^{- α_{i}} d s \int_{0}^{t} (t - s)^{- α_{i}} E [∥ Y^{(N)} (τ_{N} (s)) - Y^{(N)} (s) ∥^{2}] d s$
			$+ \int_{0}^{t} (t - s)^{- α_{i}} d s \int_{0}^{t} (t - s)^{- α_{i}} E [∥ Y^{(N)} (s) - y (s) ∥^{2}] d s}$
			$+ 3 t {\int_{0}^{t} E [∥ f (τ_{N} (s), Y^{(N)} (τ_{N} (s))) - f (s, Y^{(N)} (τ_{N} (s))) ∥^{2}] d s$

			$+ 3 {\int_{0}^{t} E [∥ g (τ_{N} (s), Y^{(N)} (τ_{N} (s))) - g (s, Y^{(N)} (τ_{N} (s))) ∥^{2}] d s$

Based on Assumptions 2.1 and 2.2, Lemmas 3.2 and 3.4 and Theorem 3.1 as well as the fact $s - τ_{N} (s) \leq h$ , it is easy to obtain that

			$\frac{1}{3} E [∥ Y^{(N)} (t) - y (t) ∥^{2}]$
		$\leq$	$m \sum i = 1 \frac{3 m C}{Γ^{2} (1 - α_{i})} {h^{2 (2 - α_{i})} + h^{min {2 (1 - α_{m}), 1}} + \int_{0}^{t} \frac{E [∥ Y^{(N)} (s) - y (s) ∥^{2}]}{(t - s)^{α_{i}}} d s}$
			$+ 3 t {L_{1}^{2} h^{2} t + L_{2}^{2} C h^{min {2 (1 - α_{m}), 1}} + L_{2}^{2} \int_{0}^{t} E [∥ Y^{(N)} (s) - y (s) ∥^{2}] d s}$
			$+ 3 {L_{1}^{2} h^{2} t + L_{2}^{2} C h^{min {2 (1 - α_{m}), 1}} + L_{2}^{2} \int_{0}^{t} E [∥ Y^{(N)} (s) - y (s) ∥^{2}] d s} .$

Arranging the above inequality yields

E [∥ Y^{(N)} (t) - y (t) ∥^{2}] \leq C h^{min {2 (1 - α_{m}, 1)}} + C \int_{0}^{t} (t - s)^{- α} E [∥ Y^{(N)} (s) - y (s) ∥^{2}] d s,

where $α \in {α_{1}, α_{2}, \dots, α_{m}}$ . Then using the Gronwall’s inequality and arbitrariness of $t \in [0, T]$ , we arrive at

E [∥ Y^{(N)} (t) - y (t) ∥^{2}] \leq C h^{min {2 (1 - α_{m}), 1}} .

The proof is completed. $□$

4 The fast EM method for multi-term Riemann-Liouville SFDE

4.1 The fast EM method

To construct the fast EM method, it is necessary to introduce the sum-of-exponentials (SOEs) approximation.

Lemma 4.1

[16] For a given $α \in (0, 1)$ , let $ϵ$ denote tolerance error, $δ$ denote cut-off time restriction and $T$ denote final time, there are a positive integer $N_{e x p}^{(α)}$ and positive constants $ω_{j}^{(α)}$ and $s_{j}^{(α)}$ , $j = 1, 2, \dots, N_{e x p}^{(α)}$ such that for any $t \in [δ, T]$

∣ ∣ ∣ ∣ ∣ t^{- α} - N_{e x p}^{(α)} \sum j = 1 ω_{j}^{(α)} e^{- s_{j}^{(α)} t} ∣ ∣ ∣ ∣ ∣ \leq ϵ,

where $N_{e x p}^{(α)} = O ((log ϵ^{- 1}) (log log ϵ^{- 1} + log (T δ^{- 1})) + (log δ^{- 1}) (log log ϵ^{- 1} + log δ^{- 1}))$ .

Considering the modified method (3.2) at $t_{n + 1}$ and applying Lemma 4.1 to approximate $(t - τ_{N} (s))^{- α_{i}}$ in the integral from $0$ to $t_{n}$ , then equation (3.2) can be reformed as follows:

	$X^{(N)} (t_{n + 1})$	(4.1)
$=$	$y_{0} - m \sum i = 1 \frac{1}{Γ (1 - α_{i})} [\int_{0}^{t_{n}} \frac{X^{(N)} (τ_{N} (s))}{(t_{n + 1} - τ_{N} (s))^{α_{i}}} d s + \int_{t_{n}}^{t_{n + 1}} \frac{X^{(N)} (τ_{N} (s))}{(t - τ_{N} (s))^{α_{i}}} d s]$
	$+ \int_{0}^{t} f (τ_{N} (s), X^{(N)} (τ_{N} (s))) d s + \int_{0}^{t} g (τ_{N} (s), X^{(N)} (τ_{N} (s))) d W (s)$
$\approx$	$y_{0} - m \sum i = 1 \frac{1}{Γ (1 - α_{i})} ⎡ ⎢ ⎢ ⎣ N_{e x p}^{(α_{i})} \sum j = 1 ω_{j}^{(α_{i})} U_{j}^{(α_{i})} (t_{n + 1}) + h^{1 - α_{i}} X^{(N)} (t_{n}) ⎤ ⎥ ⎥ ⎦$
	$+ k \sum$

A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

The semi-implicit Euler-Maruyama method for nonlinear non-autonomous stochastic differential equations driven by a class of Lévy processes

EM-WaveHoltz: A flexible frequency-domain method built from time-domain solvers

An explicit Euler method for McKean-Vlasov SDEs driven by fractional Brownian motion

The rate of Lp-convergence for the Euler-Maruyama method of the stochastic differential equations with Markovian switching

Fractional-order Backpropagation Neural Networks: Modified Fractional-order Steepest Descent Method for Family of Backpropagation Neural Networks

Trapezoidal methods for fractional differential equations: theoretical and computational aspects

Dual Approach as Empirical Reliability for Fractional Differential Equations

1 Introduction

2 Notations and preliminaries

Definition 2.1

Definition 2.2

Assumption 2.1

Assumption 2.2

Assumption 2.3

3 The modified EM method for multi-term Riemann-Liouville SFDEs

3.1 The modified EM method

3.2 Strong convergence of the modified EM method

Lemma 3.1

Lemma 3.2

Lemma 3.3

Theorem 3.1

Lemma 3.4

Theorem 3.2

4 The fast EM method for multi-term Riemann-Liouville SFDE

4.1 The fast EM method

Lemma 4.1

A modified EM method and its fast implementation for multi-term Riemann-Liouville stochastic fractional differential equations

Related Research

The semi-implicit Euler-Maruyama method for nonlinear non-autonomous stochastic differential equations driven by a class of Lévy processes

EM-WaveHoltz: A flexible frequency-domain method built from time-domain solvers

An explicit Euler method for McKean-Vlasov SDEs driven by fractional Brownian motion

The rate of Lp-convergence for the Euler-Maruyama method of the stochastic differential equations with Markovian switching

Fractional-order Backpropagation Neural Networks: Modified Fractional-order Steepest Descent Method for Family of Backpropagation Neural Networks

Trapezoidal methods for fractional differential equations: theoretical and computational aspects

Dual Approach as Empirical Reliability for Fractional Differential Equations

1 Introduction

2 Notations and preliminaries

Definition 2.1

Definition 2.2

Assumption 2.1

Assumption 2.2

Assumption 2.3

3 The modified EM method for multi-term Riemann-Liouville SFDEs

3.1 The modified EM method

3.2 Strong convergence of the modified EM method

Lemma 3.1

Lemma 3.2

Lemma 3.3

Theorem 3.1

Lemma 3.4

Theorem 3.2

4 The fast EM method for multi-term Riemann-Liouville SFDE

4.1 The fast EM method

Lemma 4.1