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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).
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 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.
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).
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
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.
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) |
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