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 -convergence of the truncated SD method and showed that it can be arbitrarily close to ; see [I. S. Stamatiou and N. Halidias, Convergence rates of the semi-discrete method for stochastic differential equations, Theory Stoch. Process. 24 2019, 2, 89–100]. 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. Numerical simulations support our theoretical findings.
We study the following class of scalar stochastic differential equations (SDE):
where are measurable functions such that (1.1) has a unique solution and is independent of all . We assume that the coefficients , 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 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. , where the Euler method may explode in finite time.
The semi-discrete (SD) method, originally proposed in , is a strongly converging numerical method with the qualitative property of domain preservation; if for instance the solution process is nonnegative, then the approximation process is also nonnegative (see [6, 2, 4, 3, 5, 12, 13]). The -convergence of the truncated SD method, see , was recently shown to be arbitrarily close to .
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 . Let and be a complete probability space. Let also be a one-dimensional Wiener process adapted to the filtration . 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 and with and with the property and . Suppose the equidistant partition with step-size and assume that for every the SDE
with a.s., has a unique strong solution. The first and third variable in denote the discretized part of the original SDE. Now, we proceed to the construction of the truncated SD method. We define the truncated functions and by
for , where we set when , the strictly increasing function is such that
for every , and the strictly decreasing function satisfies
for a constant . Let
with a.s., admit a unique strong solution for every .
Let satisfy the following condition:
for all and , where is as in (2.4) and ϕ stands for f and g, respectively.
In the statement of the main result in , which we rewrite below, we use the compact form of the approximation process:
where when .
Theorem 2.2 (Order of strong convergence).
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 and . Moreover, to guarantee the asymptotic stability of (2.1), we use an assumption similar to [7, Assumption 5.1].
We assume the existence of a continuous non-decreasing function with and for all such that
for all and .
Theorem 3.2 (Asymptotic stability of underlying process).
for any .
where are the Wiener increments, is the step-size and stands for . We assume the following decomposition of for the above representation (3.2):
The following theorem shows that the truncated SD method is able to preserve the asymptotic stability property of the underlying SDE.
Theorem 3.3 (Asymptotic numeric stability).
Let the auxiliary function from (3.3) satisfy
for all and .
Proof of Theorem 3.3.
Let us first fix a . Set
Recall that for the conditional expectation holds
to find that , , is a martingale. Application of the nonnegative semi-martingale convergence theorem, cf. [10, Theorem 7, p. 139], implies
which in turn yields
By the property of the function , we get that
Assertion (3.5) follows. ∎
We would like to mention here that representation (3.2), which in turn defines decomposition (3.3) and consequently the function , depends heavily on the discretization (2.5), which enjoys the freedom of choice. Depending on the SDE at hand, we may discretize the coefficients in (2.5) in an additive or multiplicative way; cf. [1, 2, 3, 4, 5, 6, 12, 14]. We also present an example in the following section.
One of the main advantages of the SD method is the domain preservation. Therefore, even if we may require no weaker conditions than the existing conditions to guarantee the stability of the SD method, we prefer to use it in cases where the solution process of the original SDE stays at a specific domain a.s.
Using standard arguments, one may show that the solution process of SDE (4.1) is positive; see Section B. Assumption 3.1 holds with . 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 and in the following way:
Thus, (2.2) becomes
respectively, with 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,
Note that (2.3) holds with since
Therefore, in the notation of Theorem 2.2, and . Finally,
for any . Clearly, and
for any and . Therefore, we take . The truncated versions of the semi-discrete method (TSD) read
for , where
4.1 Asymptotic stability of truncated semi-discrete method
to see that
implying that we may choose in the following way:
so that condition (3.4) holds and therefore Theorem 3.3 applies. Note that and for any . We conclude that the truncated SD scheme (4.6) 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 (4.7) and get that
Set the last term of the above equality to , that is,
to see that
is an exponential martingale. Moreover,
implying that we may choose in the following way:
so that once more condition (3.4) holds and consequently Theorem 3.3 applies. We conclude that the truncated exponential SD scheme (4.7) preserves the asymptotic stability perfectly in the sense that a.s. for any .
4.3 Semi-discrete method and Lamperti transformation
Instead of approximating directly (4.1), 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 (4.1). In particular, consider . Itô’s formula implies the following dynamics for (see Section C):
with . Equation (4.9) is a Bernoulli-type equation with solution satisfying
Recall that when , for the solution process holds a.s., which implies a.s., which in turn suggests that we take the negative root of (4.10) as the solution. Therefore, we propose the following semi-discrete method for the approximation of (4.8):
which suggests for the approximation of (4.1):
We also examine another modification of the semi-discrete method, , where in each subinterval we do not need to solve a new differential equation, but only an algebraic equation. For consider
with . The solution of (4.13) satisfies
We propose the following version of the semi-discrete method for the approximation of (4.8):
which suggests for the approximation of (4.1):
4.4 Simulation paths
We present simulations for the numerical approximation of (4.1) with and compare with the truncated Euler–Maruyama method (TEM), which reads
for , where and . According to the results in , it is shown that method (4.17) is asymptotically stable for any . 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 .Figure 1
that is, we calculated the integrand in the lower limit of integration. The above inequality is of the order , with ; see Section D.Figure 3
We also present in Figure 4 the difference between the Lamperti schemes and the truncated Euler–Maruyama scheme, , for small enough Δ such that TEM works.Figure 4
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 -convergence of the truncated SD method and showed that it can be arbitrarily close to ; see . 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.
A Solution of linear SDEs in the narrow sense
Consider the following linear in the narrow sense SDE:
for , where are constants. Applying the Itô formula to the transformation , we obtain
B Positivity of (4.1)
In order to prove that a.s., we first show moment bounds of SDE (4.1).
Lemma B.1 (Uniform moment bounds for ).
The solution process of SDE (4.1) satisfies
for some and any integer p with .
Proof of Lemma B.1.
For all with , we have that
where the last inequality is valid for all p such that . Thus is bounded for all , since this clearly holds for all . Application of [11, Theorem 2.4.1] implies
for any , since . Using Itô’s formula on , with (in order to use Doob’s martingale inequality later), we have that
for any even p with , or , where . Taking the supremum and then expectations in the above inequality, we get
where in the last step we have used Doob’s martingale inequality to the diffusion term . ∎
Lemma B.2 (Positivity of ).
The solution process of SDE (4.1) is positive in the sense that a.s.
Proof of Lemma B.2.
Set the stopping time for some , with the convention that . Application of Itô’s formula on implies
Taking expectations in the above inequality and using the fact that , we get that
with C independent of R. Therefore,
We conclude that a.s. ∎
C Lamperti transformation of (4.1)
Applying Itô’s formula to the transformation , we obtain
or, for ,
D Stochastic integral approximation
We want to estimate the stochastic integral appearing in the proposed truncated semi-discrete method (4.6) for the approximation of SDE (4.1). In a similar way, we calculate the integral appearing in the exponential truncated semi-discrete scheme (4.4).
In the numerical simulations, we used the following relation:
We show the following estimation:
suggesting that the probability of the absolute difference of these two random variables being of order , with , approaches unity as Δ goes to zero. First, we write the difference of the two local martingales as
and then use the martingale inequality to get for any that
where in the last step we used the inequality for any . We apply the above inequality for , with to get (D.1).
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