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A note on the asymptotic stability of the Semi-Discrete method for Stochastic Differential Equations

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 Stamatiou, Halidias (2019), Convergence rates of the Semi-Discrete method for stochastic differential equations, Theory of Stochastic Processes, 24(40). 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, using the Lamperti transformation to the original SDE, which we call Lamperti semi-discrete (LSD). Numerical simulations support our theoretical findings.

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1. Introduction

We study the following class of scalar stochastic differential equations (SDEs),

(1)

where are measurable functions such that (1) has a unique solution and is independent of all We assume that SDE (1) has non-autonomous coefficients, i.e. depend explicitly on SDEs of the type (1) rarely have explicit solutions, therefore the need for numerical approximations for simulations of the solution process is apparent. In the case of nonlinear drift and diffusion coefficients classical methods may fail to strongly approximate (in the mean-square sense) the solution of (1), c.f. [1], where the Euler method may explode in finite time.

In this direction, we study the semi-discrete (SD) method originally proposed in [2] and further investigated in [3], [4], [5], [6], [7] and recently in [8] and [9]. 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 beginning of the subinterval so as to obtain explicitly solved SDEs. Of course the way of freezing (discretization) is not unique.

The SD method is a fixed-time step explicit numerical method which strongly converges to the exact solution and also preserves the domain of the solution; if for instance the solution process is nonnegative then the approximation process is also nonnegative. The -convergence of the truncated SD method, see [10], was recently shown to be arbitrarily close to

Our main goal is to further examine qualitative properties of the SD method relevant with 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 to the positive, and is given in our main result, Theorem 4. In Section 2 we give all the necessary information about the truncated version of the semi-discrete method; the way of construction of the numerical scheme and some useful properties, whereas 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, which we call Lamperti semi-discrete (LSD). Numerical simulations support our theoretical findings. Finally, Section 5 contains concluding remarks.

2. Setting and Assumptions

Throughout, let and

be a complete probability space, meaning that the filtration

satisfies the usual conditions, i.e. is right continuous and includes all -null sets. Let be a one-dimensional Wiener process adapted to the filtration Consider SDE (1), which we rewrite here in its integral form

(2)

which admits a unique strong solution. In particular, we assume the existence of a predictable stochastic process such that ([11, Def. 2.1]),

and

Assumption 1.

Let be such that where satisfy the following condition

for any such that where the quantity depends on and denotes the maximum of

Let us now recall the SD scheme. Consider the equidistant partition and We assume that for every the following SDE

(3)

with a.s., has a unique strong solution.

In order to compare with the exact solution

which is a continuous time process, we consider the following interpolation process of the semi-discrete approximation, in a compact form,

(4)

where when Process (4) has jumps at nodes The first and third variable in denote the discretized part of the original SDE. We observe from (4) that in order to solve for , we have to solve an SDE and not an algebraic equation. The choice and reproduces the classical Euler scheme.

In the case of superlinear coefficients the numerical scheme (4) converges to the true solution of SDE (2) and this is stated in the following, cf. [3],

Theorem 1 (Strong convergence).

Suppose Assumption 1 holds and (3) has a unique strong solution for every where Let also

for some and Then the semi-discrete numerical scheme (4) converges to the true solution of (2) in the -sense, that is

(5)

Relation (5) does not reveal the order of convergence. We choose a strictly increasing function such that for every

(6)

The inverse function of denoted by maps to Moreover, we choose a strictly decreasing function and a constant such that

(7)

Now, we are ready to define the truncated versions of Let and defined by

(8)

for where we set when

It follows that the truncated functions are bounded in the following way for a given step-size

(9)

for all

For the equidistant partition of with consider now the following SDE

(10)

with a.s. We assume that (10) admits a unique strong solution for every and rewrite it in compact form,

(11)
Assumption 2.

Let the truncated versions of satisfy the following condition

for all and where is as in (7).

Let us also assume that the coefficients of the original SDE satisfy the Khasminskii-type condition.

Assumption 3.

We assume the existence of constants and such that and

for all .

A well-known result follows (see e.g. [11]) when the SDE (2) satisfies the local Lipschitz condition plus the Khasminskii-type condition.

Lemma 1.

Under Assumptions 1 (for the coefficients ) and 3 the SDE (2) has a unique global solution and for all there exists a constant such that

Theorem 2 (Order of strong convergence).

Suppose Assumption 2 and Assumption 3 hold and (10) has a unique strong solution for every where for some Let and define for

where and are such that (7) holds. Then the semi-discrete numerical scheme (11) converges to the true solution of (2) in the -sense with order arbitrarily close to that is

(12)

3. Asymptotic Stability

Now we are ready to study the ability of the truncated SD method to preserve the asymptotic stability of (2). For that reason we also assume that and Moreover, to guarantee the asymptotic stability of (2) we use an assumption similar to [12, Assumption 5.1].

Assumption 4.

We assume the existence of a continuous non-decreasing function with and for all such that

(13)

for all and

Now, we state a result without proof concerning the asymptotic stability of (2), see also [12, Theorem 5.2] where autonomous coefficients are assumed.

Theorem 3 (asymptotic stability of underlying process).

Let Assumption 4 hold. Then the solution process of SDE (2) is asymptotically stable, that is

(14)

for any

Recall equation (10) which defines the truncated SD numerical scheme. We rewrite our proposed scheme, that is the solution of (10) at the discrete points in the following way

(15)

where are the Wiener increments, is the step-size and stands for We assume the following decomposition of for the above representation (15),

(16)

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

Theorem 4 (asymptotic numeric stability).

Let the auxiliary function from (16) satisfy

(17)

for any where has the same properties as in (13) with . Let also Assumption 4 hold.

Then the solution of the truncated SD method (15) is numerically asymptotically stable, that is

(18)

for all and .

Proof of Theorem 4.

Let us first fix a Denote

Then combining (15), (16) and (17) we get

where Recalling that implies that is a martingale. Application of the nonnegative semi-martingale convergence theorem, c.f. [13, Theorem 7, p.139], implies

which in turn

By the property of the function we get that

Assertion (18) follows. ∎

4. Example

We will use the numerical example of [12, Example 5.4], that is we consider an autonomous SDE of the form (2) with and with initial condition , that is,

(19)

Using standard arguments one may show that the solution process of SDE (19) is positive, see Appendix B. Assumption 4 holds with therefore by Theorem 3 SDE (19) is almost surely asymptotically stable. The classical Euler Maruyama method is not able to reproduce this asymptotic stability, see [12, 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 and in the following way

thus (3) becomes

(20)

and

(21)

respectively, with a.s. SDEs (20) and (21) are linear equations ( (20) is linear in the narrow sense and is known as Langevin equation) with variable coefficients which admit a unique strong solution, c.f. [14, Chapter 4.4] and Appendix A. In particular,

(22)

and

(23)

Note that (6) holds with since

Therefore, in the notation of Theorem 2, and Finally, for any Clearly and

for any and Therefore we take The truncated versions of the semi-discrete method (TSD) read,

(24)

and

(25)

for where

and therefore

4.1. Asymptotic stability of truncated Semi-Discrete method

Now, we compute taking the square of (24) and making some rearrangements to show that it admits representation (16).

Denote and set

to see that Moreover

implying that we may choose in the following way

so that condition (17) holds and therefore Theorem 4 applies. Note that and for any We conclude that the truncated SD scheme (24) preserves the asymptotic stability perfectly in the sense that a.s. for any

4.2. Asymptotic stability of exponential truncated Semi-Discrete method

We examine We take the square of (25) and get that

Set the last term of the above equality to that is

to see that since is an exponential martingale.

Moreover

implying that we may choose in the following way

so that once more condition (17) holds and consequently Theorem 4 applies. We conclude that the truncated exponential SD scheme (25) preserves the asymptotic stability perfectly in the sense that a.s. for any

4.3. Semi-Discrete method and Lampreti transformation

Instead of approximating directly (19) we first study a transformation of it, which produces a new SDE with constant diffusion coefficient; in other words we use the Lamperti transformation of (19). In particular, consider The Itô formula implies the following dynamics for see Appendix C,

(26)

Let and

(27)

with (27) is a Bernoulli type equation with solution satisfying

(28)

Recall that when the solution process a.s. which implies a.s. which in turn suggests that we take the negative root of (28) as the solution Therefore we propose the following semi-discrete method for the approximation of (26),

(29)

which suggests the Lamperti semi-discrete method for the approximation of (19)

(30)

4.4. Simulation Paths

We present simulations for the numerical approximation of (19) with and compare with the truncated Euler Maruyama method (TEM), which reads

(31)

for where