1 Introduction

We study the following class of scalar stochastic differential equations (SDE):

where a , b : [ 0 , T ] × ℝ → ℝ are measurable functions such that (1.1) has a unique solution and x 0 is independent of all { W t } t ≥ 0 . We assume that the coefficients a ⁢ ( t , x ) , b ⁢ ( t , x ) in (1.1) depend explicitly on t. SDEs of the type (1.1) rarely have explicit solutions, and therefore the need for numerical approximations for simulations of the solution process x t is apparent. In the case of non-linear drift and diffusion coefficients, classical methods may fail to strongly approximate (in the mean-square sense) the solution of (1.1); cf. [8], where the Euler method may explode in finite time.

The semi-discrete (SD) method, originally proposed in [1], is a strongly converging numerical method with the qualitative property of domain preservation; if for instance the solution process x t is nonnegative, then the approximation process y t is also nonnegative (see [6, 2, 4, 3, 5, 12, 13]). The ℒ 2 -convergence of the truncated SD method, see [14], was recently shown to be arbitrarily close to 1 2 .

The main idea behind the semi-discrete method is freezing on each subinterval appropriate parts of the drift and diffusion coefficients of the solution at the boundaries of the subinterval, ending up with an explicitly solvable SDE.

Our main goal is to further examine qualitative properties of the SD method relevant to the stability of the method and answer questions of the following type: Is the SD method able to preserve the asymptotic stability of the underlying SDE?

The answer of the question above is in the positive, and is given in our main result, Theorem 3.3. In Section 2, we give all necessary information about the truncated version of the semi-discrete method; the way of construction of the numerical scheme and some useful properties. Section 3 contains the main result with the proof. Section 4 provides a numerical example. Motivated by the SDE appearing in the example, we also propose a different SD scheme, using the Lamperti transformation to the original SDE. Numerical simulations support our theoretical findings. Finally, Section 5 contains concluding remarks.

2 Setting and assumptions

We will use some standard notation and setting that has already appeared in previous works of the SD method in a condensed way; we refer the interested reader mainly to [14]. Let T > 0 and ( Ω , ℱ , { ℱ t } 0 ≤ t ≤ T , ℙ ) be a complete probability space. Let also W t , ω : [ 0 , T ] × Ω → ℝ be a one-dimensional Wiener process adapted to the filtration { ℱ t } 0 ≤ t ≤ T . We rewrite SDE (1.1) in its integral form

and assume that it admits a unique strong solution.

We recall the SD scheme. We use the auxiliary real functions f ⁢ ( s , r , x , y ) and g ⁢ ( s , r , x , y ) with s , r ∈ [ 0 , T ] and x , y ∈ ℝ with the property f ⁢ ( s , s , x , x ) = a ⁢ ( s , x ) and g ⁢ ( s , s , x , x ) = b ⁢ ( s , x ) . Suppose the equidistant partition 0 = t 0 < t 1 < ⋯ < t N = T with step-size Δ = T N and assume that for every n ≤ N - 1 the SDE

with y 0 = x 0 a.s., has a unique strong solution. The first and third variable in f , g denote the discretized part of the original SDE. Now, we proceed to the construction of the truncated SD method. We define the truncated functions f Δ and g Δ by

for x , y ∈ ℝ , where we set x / | x | = 0 when x = 0 , the strictly increasing function μ : ℝ + → ℝ + is such that

for every s , r ≤ T , and the strictly decreasing function h : ( 0 , 1 ] → [ μ ⁢ ( 1 ) , ∞ ) satisfies

for a constant h ^ ≥ 1 ∨ μ ⁢ ( 1 ) . Let

with y 0 = x 0 a.s., admit a unique strong solution for every n ≤ N - 1 .

In the statement of the main result in [14], which we rewrite below, we use the compact form of the approximation process:

where s ^ = t n when s ∈ [ t n , t n + 1 ) .

3 Asymptotic stability

Now we are ready to study the ability of the truncated SD method to preserve the asymptotic stability of (2.1). For that reason, we also assume that a ⁢ ( 0 , 0 ) = 0 and b ⁢ ( 0 , 0 ) = 0 . Moreover, to guarantee the asymptotic stability of (2.1), we use an assumption similar to [7, Assumption 5.1].

Now, we state a result without proof concerning the asymptotic stability of (2.1); see also [7, Theorem 5.2], where autonomous coefficients are assumed.

Recall equation (2.5), which defines the truncated SD numerical scheme. We rewrite our proposed scheme, that is, the solution of (2.5) at the discrete points 0 , t 1 , … , t n + 1 , in the following way:

where Δ ⁢ W n are the Wiener increments, Δ = t n + 1 - t n is the step-size and y n stands for y t n . We assume the following decomposition of Φ Δ for the above representation (3.2):

where

The following theorem shows that the truncated SD method is able to preserve the asymptotic stability property of the underlying SDE.

4 Example

We will use the numerical example of [7, Example 5.4], that is, we consider an autonomous SDE of the form (2.1) with a ⁢ ( x ) = - 10 ⁢ x 3 and b ⁢ ( x ) = x 2 , with initial condition x 0 ∈ ℝ , that is,

Using standard arguments, one may show that the solution process of SDE (4.1) is positive; see Section B. Assumption 3.1 holds with κ ⁢ ( u ) = 19 ⁢ u 4 . Therefore, by Theorem 3.2, SDE (4.1) is almost surely asymptotically stable. The classical Euler–Maruyama method is not able to reproduce this asymptotic stability; see [7, Appendix]. In the following, we show that the truncated SD method can reproduce this asymptotic stability. Since in the construction of the semi-discrete method the way of discretizing is not unique (but rather indicated by the equation itself), we will try two versions of the SD method by freezing different parts of the diffusion coefficient. We first choose the auxiliary functions f , g 1 and g 2 in the following way:

Thus, (2.2) becomes

and

respectively, with y 0 = y ^ 0 = x 0 a.s. SDEs (4.2) and (4.3) are linear equations ((4.2) is linear in the narrow sense and is known as Langevin equation) with variable coefficients which admit a unique strong solution; cf. [9, Chapter 4.4] and Section A. In particular,

and

Note that (2.3) holds with μ ⁢ ( u ) = 10 ⁢ | u | 2 since

Therefore, in the notation of Theorem 2.2, γ = 1 and C ¯ = 10 . Finally,

for any Δ ∈ ( 0 , 1 ] . Clearly, h ⁢ ( 1 ) ≥ μ ⁢ ( 1 ) and

for any Δ ∈ ( 0 , 1 ] and 0 < ϵ 1 ≤ 1 3 . Therefore, we take h ^ = 11 . The truncated versions of the semi-discrete method (TSD) read

and

for n ∈ ℕ , where

and therefore

4.4 Simulation paths

We present simulations for the numerical approximation of (4.1) with x 0 = 10 and compare with the truncated Euler–Maruyama method (TEM), which reads

(4.17) y n + 1 T ⁢ E ⁢ M = y n - 10 ⁢ ( | y n | ∧ μ ¯ - 1 ⁢ ( h ¯ ⁢ ( Δ ) ) ⁢ y n | y n | ) 3 ⁢ Δ + ( | y n | ∧ μ ¯ - 1 ⁢ ( h ¯ ⁢ ( Δ ) ) ) 2 ⁢ Δ ⁢ W n

for n ∈ ℕ , where h ¯ ⁢ ( Δ ) = Δ - 1 / 4 and μ ¯ ⁢ ( u ) = 10 ⁢ u 3 . According to the results in [7], it is shown that method (4.17) is asymptotically stable for any Δ ≤ 0.095 . Therefore, for such small step sizes we compare all methods presented here and for bigger Δ only the SD schemes (4.4), (4.5), (4.6) and (4.7). We also present the Lamperti semi-discrete schemes (LSD) (4.12) and (4.16). Moreover, the TEM method does not preserve positivity. Figures 1, 2 and 3 show sample simulations paths of TSD and TEM, respectively, for various stepsizes. Note that the truncated TSD, exponential truncated expTSD and the Lamperti semi-discrete schemes LSD1 (4.12) and LSD2 (4.16) work for all Δ < 1 .

Figure 1

Trajectories of (4.4)–(4.7), (4.12), (4.16) and (4.17) for the approximation of (4.1) with Δ < 0.095 . In (b) we find a zoom of Figure (a).

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Figure 2

Trajectories of (4.4)–(4.7) and (4.12), (4.16) for the approximation of (4.1) with Δ = 0.25 . In (b) we find a zoom of Figure (a).

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In the numerical simulation of the stochastic integral of the (truncated) TSD methods (4.4) and (4.6), we used the approximation

∫ t n t n + 1 e 10 ⁢ π Δ 2 ⁢ ( y n Δ ) ⁢ s ⁢ 𝑑 W s ≈ e 10 ⁢ π Δ 2 ⁢ ( y n Δ ) ⁢ t n ⁢ Δ ⁢ W n ,

that is, we calculated the integrand in the lower limit of integration. The above inequality is of the order Δ r , with 0 < r < 1 2 ; see Section D.

Figure 3

Trajectories of (4.4)–(4.7) and (4.12), (4.16) for the approximation of (4.1) with Δ = 0.5 . In (b) we find a zoom of Figure (a).

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We also present in Figure 4 the difference between the Lamperti schemes and the truncated Euler–Maruyama scheme, y TEM - y LSD , for small enough Δ such that TEM works.

Figure 4

Difference of (4.17) with (4.12) and (4.16) for the approximation of (4.1) with various step sizes.

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5 Conclusion and future work

In this paper, we study the asymptotic stability of the semi-discrete (SD) numerical method for the approximation of stochastic differential equations. Recently, we examined the order of ℒ 2 -convergence of the truncated SD method and showed that it can be arbitrarily close to 1 2 ; see [14]. We show that the truncated SD method is able to preserve the asymptotic stability of the underlying SDE. Motivated by a numerical example, we also propose a different SD scheme, where we actually approximate first the Lamperti transformation of the original SDE. This scheme preserves positivity (in this case) of the solution, has similar asymptotic properties as the other versions of the SD method and seems promising, since there is no need for an exponential calculation. We will study this numerical method and its properties in a forthcoming paper.