On Mean-Square and Asymptotic Stability for Numerical Approximations of Stochastic Ordinary

Differential Equations

Rózsa Horváth–Bokor^{∗} and Taketomo Mitsui^{†}

Abstract

This note tries to connect the stochastic mean-square stability [Saito and Mitsui, SIAM J. Numer.Anal., 33(1996), pp. 2254-2267] and the asymptotic stability [D.J. Higham, SIAM J. Numer. Anal., 38(2000), pp. 753-769]. Saito and Mitsui generalizes the deterministic A-stability for a stochastic differential equation test problem with multiplicative noise. For the test equation, we know that the asymptotic stability in the mean-square sense implies the stochastic asymptotic stability in large. We will prove the same property for numerical schemes.

A.M.S. Classification codes (1991): 60H10, 65C05, 65C20, 65U05

Keywords: Stochastic ordinary differential equations, numerical methods, mean square convergence

1 Introduction

We consider the scalar stochastic initial value problem (SIVP) for the Itô ordinary differential equations (SODE) given by :

dX(t) = a(t, X(t))dt + b(t, X(t))dW (t) for 0 ≤ t ≤ T and X(t) ∈ ℝ,  (1)

where functions are defined by

a : [0, +∞) × ℝ → ℝ, b : [0, +∞) × ℝ → ℝ and X(0) = X_{0},

and W (t) is the 1-dimensional Brownian motion. Let F_{t} denote the increasing family of σ-algebras (filtration) generated by the Brownian motion W (s), s ≤ t. Details about this stochastic object and corresponding calculus can be found in [1, 4].

We consider an Itô equation (1) with a steady solution X_{t} ≡ 0. This means that a(t, 0) = b(t, 0) = 0 holds. X^{t_{0},0}(t) means that X^{t_{0},0}(t_{0}) = 0.

The following definitions are due to Hasminski:

∗Department of Mathematics and Computing University of Veszprém, e–mail: hrozsa@almos.vein.hu †Graduate School of Information Science, Nagoya University, e-mail: tom.mitsui@cc.nagoya-u.ac.jp

Definition 1 The steady state solution X^{t_{0},0}(t) ≡ 0 of (1) is said to be stochastically stable if for any ϵ > 0 and t_{0} ≥ 0

lim_{x_{0}→0} P (sup_{t≥t_{0}} ∥ X^{t_{0},x_{0}}(t) ∥≥ ϵ) = 0.

Definition 2 The steady state solution X^{t_{0},0}(t) ≡ 0 of (1) is said to be stochastically asymptotically stable if, in addition to being stochastically stable,

lim_{x_{0}→0} P ( lim_{t→+∞} ∥ X^{t_{0},x_{0}}(t) ∥= 0) = 1.

Definition 3 The steady state solution X^{t_{0},0}(t) ≡ 0 of (1) is said to be stochastically asymptotically stable in the large, if moreover to the above two,

P ( lim_{t→+∞} ∥ X^{t_{0},x_{0}}(t) ∥= 0) = 1. for all x_{0}.

For the general SODE (1) Kloeden and Platen (cf. [5]) gave the following definition:

Definition 4 The steady state solution X^{t_{0},0}(t) ≡ 0 of (1) is stable in the p-th mean if for

∀ϵ > 0 ∀t_{0} > 0, ∃δ = δ(t_{0}, ϵ) > 0

such that

E ∥ X^{t_{0},x_{0}}(t) ∥^{p}< ϵ

for all t ≥ t_{0} and ∥ x_{0} ∥< δ.

Definition 5 The steady state solution X^{t_{0},0}(t) ≡ 0 of (1) is asymptotically stable in the p-th mean if in addititon, there exists a δ_{0} = δ(t_{0}) such that

lim_{t→+∞} E ∥ X^{t_{0},x_{0}}(t) ∥^{p}= 0 for all ∥ x_{0} ∥< δ_{0}.

The most frequently used case p = 2 is called the mean-square case and in the sequel we focus our investigation on the mean-square stability.

We suppose that the equation (1) has a unique, bounded strong solution X(t) in the mean-square sense.

Usual deterministic time discretization of a bounded time-interval [0, T], T > 0 is of the form 0 = t_{0} < t_{1} < … < t_{N} = T, where N is a natural number. We suppose that t_{0} = 0 and adopt an equidistant discretization with the step size Δ = t_{n+1} − t_{n}, n = 0, 1, …, N − 1.

In the sequel Y_{n} always denotes the approximation of X(t_{n}) using a given numerical scheme with the step size Δ.

2 Mean-square and asymptotic stabilities for the test equation

Consider the one-dimensional stochastic test equation

dX(t) = λX(t)dt + μX(t)dW (t) (t ≥ 0), X(0) = x_{0},  (2)

where λ and μ are complex numbers and we assume that x_{0} is a non zero deterministic value. The exact solution of (2) is

X(t) = exp ((λ − \frac{μ^{2}}{2} )t + μW (t)) x_{0},

which is sometimes called geometric Brownian motion. It has the second moment

E|X(t)|^{2} = exp((2ℜ(λ) + |μ|^{2})t)|x_{0}|^{2}.

For this test equation, the stationary solution X(t) ≡ 0 is stochastically asymptotically stable in large if the inequality

ℜ(λ − \frac{1}{2}μ^{2}) < 0

holds. Saito-Mitsui [6] showed that the zero solution of equation (2) is asymptotically mean-square stable if and only if

2ℜ(λ) + |μ|^{2} < 0,

as we can see the form of the second moment of X(t).

Note that since the inequality ℜ(2λ − μ^{2}) ≤ 2ℜ(λ) + |μ|^{2} is always valid, the asymptotic stability in the mean-square sense implies the stochastic asymptotic stability in large.

As considered in [2] and [6] the analytical theory is not applicable to numerical schemes because it is impossible to carry out a numerical scheme until all the sample paths (2) diminish to zero if ℜ(λ) > 0 and ℜ(λ − \frac{1}{2}μ^{2}) < 0.

Generally, in the vector case, when we apply a numerical scheme (Y_{n}) to the equation (2) and take the mean-square norm, we obtain a one-step difference equation of the form

E ∥ Y_{n+1} ∥^{2}= R(\overline{Δ}, k)E ∥ Y_{n} ∥^{2},  (3)

where \overline{Δ} = Δλ and k = −μ^{2}/λ.

Saito and Mitsui in their work [6] called the function R(\overline{Δ}, k) as the stability function of the scheme. In this case E ∥ Y_{n} ∥^{2}→ 0 as n → +∞ iff

|R(\overline{Δ}, k)| < 1.

They gave in their work [6] the following definition:

Definition 6 The scheme is said to be MS-stable for those values of \overline{Δ} and k satisfying

|R(\overline{Δ}, k)| < 1.  (4)

The set R given by

R = \{(\overline{Δ}, k) : |R(\overline{Δ}, k)| < 1\}

is analogously called the domain of MS-stability of the scheme.

Higham [3] introduces the new concept of stability. Definition 7 When a numerical scheme is applied to the stohastically asymptotically stable in large as equation (2) and generating the sequence (Y_{n}), it is said to be numerically asymptotically stable in large (or simply, asymptotically stable) if

lim_{n→+∞} ∥ Y_{n} ∥= 0  (5)

with probability 1. We can establish a relationship between two stability concepts as follows. Theorem 1 The MS-stable schemes satisfying (4) are numerically asymptotically stable in large. In the proof of Theorem we will use the statement of the following Lemma. Lemma 1

lim_{n→+∞} ∥ Y_{n} ∥= 0 (a.s)

iff

lim_{n→+∞} P (∪_{k=n}^{∞}{∥ Y_{k} ∥≥ ϵ}) = 0 for all ϵ > 0.  (6)

Proof of Lemma. We prove this statement by defining for n ∈ ℕ and ϵ > 0 the events

A_{n}(ϵ) = {∥ Y_{n} ∥≥ ϵ}

and

A(ϵ) = \overline{lim_{n}}A_{n}(ϵ) = ∩_{n=1}^{+∞} ∪_{k=n}^{+∞} A_{k}(ϵ).

We notice that since ∪_{k=n}^{+∞}A_{k}(ϵ) ↓ A(ϵ) as n → +∞, the equality P (A(ϵ)) = lim_{n→+∞}P (∪_{k=n}^{+∞}A_{k}(ϵ)) holds.

Now we define an event

D = {ω ∈ Ω; ∥ Y_{n} ∥ does not converge to 0 as n → +∞}.

Then

D = ∪_{ϵ>0}A(ϵ) = ∪_{m=1}^{+∞}A (\frac{1}{m}) ,

because of A(ϵ_{1}) ⊂ A(ϵ_{2}) for ϵ_{1} > ϵ_{2}.

The above equalities imply that lim_{n→+∞} ∥ Y_{n} ∥= 0 with probability 1 that is equivalent to

P (D) = 0. It holds only if and only if P (A(ϵ)) = 0 for all ϵ > 0 , which is equivalent to

lim_{n→+∞} P (∪_{k=n}^{∞}{∥ Y_{k} ∥≥ ϵ}) = 0 for all ϵ > 0.  (7)

Using this result we can prove the statement of the Theorem.

Proof of Theorem.

We estimate the right-hand side of the inequality (7)

P (∪_{l=n}^{∞}\{∥ Y_{l} ∥≥ ϵ\}) ≤ ∑_{l=n}^{∞} P (∥ Y_{l} ∥≥ ϵ) ≤ ∑_{l=n}^{∞} \frac{E ∥ Y_{l} ∥^{2}}{ϵ^{2}}

≤ ∑_{l=n}^{∞} \frac{R(\overline{Δ}, k)^{l}E ∥ Y_{0} ∥^{2}}{ϵ^{2}} ≤ \frac{E ∥ Y_{0} ∥^{2}}{ϵ^{2}} ∑_{l=n}^{∞} R(\overline{Δ}, k)^{l}

tends to 0 as n → +∞ due to the condition (4) for all ϵ > 0. Now we attain

lim_{n→∞} ∥ Y_{n} ∥= 0 (a.s).

By this we have the same conclusion as for exact solution: all MS-stable schemes are asymptotically stable in large.

References

[1] L. Arnold, Stochastic Differential Equations, Wiley, New York, 1974.

[2] K. Burrage, P. M. Burrage and T. Mitsui, Numerical Solutions of Stochastic Differential Equations– Implementation and Stability Issues –,J. Comput. Appl. Math. 125, No. 1-2(2000),171-182. -

[3] D. J. Higham, Mean-Square and Asymptotic Stability of Numerical Methods for Stochastic Ordinary Differential Equations,J. Numer. Anal. 38 (2000), 753-769.

[4] I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, (Berlin Heidelberg New York, 1988).

[5] P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, Springer-Verlag, (Berlin Heidelberg New York, 1992).

[6] Y. Saito and T. Mitsui, Stability Analysis of Numerical Schemes for Stochastic Differential Equations, SIAM J. Numer. Anal. 33 (1996), 2254–2267.