Electronic Journal of Qualitative Theory of Differential Equations Proc. 7th Coll. QTDE, 2004, No. 15 1-9; http://www.math.u-szeged.hu/ejqtde/

On Stability for Numerical Approximations of Stochastic Ordinary Differential Equations^{∗}

Rózsa Horváth Bokor ^{†}

Abstract

Stochastic ordinary differential equations (SODE) represent physical phenomena driven by stochastic processes. Like for deterministic differential equations, various numerical schemes are proposed for SODE (see references). We will consider several concepts of stability and connection between them.

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

Keywords. Stochastic ordinary differential equations, numerical methods, stability

1 Introduction

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

dX(t) = a(t, X(t))dt + ∑_{j=0}^{m} b_{j}(t, X(t))dW_{j}(t), for 0 ≤ t < +∞, X(t) ∈ R,  (1)

where

a : [0, +∞) × R → R, b_{j} : [0, +∞) × R → R, X(0) = x, (∈ R),

where W_{j}(t) is one-dimensional Brownian motion, 1 ≤ j ≤ m. Let F_{t} denote the increasing family of σ-algebras (filtration) generated by the Brownian motion W_{j}(s), s ≤ t. Details about this stochastic object and corresponding calculus can be found in [1, 6].

It is known that in deterministic models we consider parameters which are completely known, though in the original problem one often has insufficient information on parameter values. These may fluctuate due to some external or internal ’noise’, which is random- at least appears to be so. In such a way we move from deterministic problems to stochastic problems (stochastic ordinary differential equations (SODE)). Explicit solutions are not usually known for equation (1), so they must be solved numerically so we have to do a qualitative investigation on the boundedness and stability of their solutions. In the literature there are some works on that topic such as [2, 4, 5, 8].

In this paper, we shall be interested in obtaining stability properties for numerical approximations of strong solutions of SODE. The concepts of numerical stability in the quadratic mean-square sense, stochastical numerical stability, asymptotical numerical stability in the quadratic mean-square sense, asymptotical stochastical numerical stability are introduced. This paper is organized as follows: in Section 2 we define stability concepts for SODE, though in Section 3 we analyse stability properties for numerical approximations of SODE, Section 4 is devoted to the asymptotic stability properties for the same. In Section 5 we illustrate our results for geometric Brownian motion and Ornstein-Uhlenbeck process.

∗Research supported by the Central Research Found of the Hungarian Academy of Sciences (Grant No. T26176) and The Ministry of Science and Tecnology, Croatia (Grant. No O37011)

This paper is in final form and no version of it will be submitted for publication elsewhere.

†Department of Mathematics and Computing University of Veszprém, e–mail: hrozsa@almos.vein.hu

2 Stability for SODE

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

∥ · ∥ denotes an Eucledean norm in R^{d}, or this is equivalent to absolute value in R.

The following definitions are due to Hasminski:

Definition 2.1 The steady state solution X^{t_{0},0}(t) ≡ 0 of the equation (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 2.2 The steady state solution X^{t_{0},0}(t) ≡ 0 of the equation (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.

Notice, that in view of the 0-1 law the above probability is either equal to 1, or to 0.

For the general SODE (1) Kloeden and Platen (cf. [7]) gave the following definition:

Definition 2.3 The steady state solution X^{t_{0},0}(t) ≡ 0 of the equation (1) is stable in the pth 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 2.4 The steady state solution X^{t_{0},0}(t) ≡ 0 of the equation (1) is asymptotically stable in the pth mean if in addititon, there exists a δ = δ(t_{0}) such that

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

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

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

3 Numerical Stability

Doing numerics for SODE it means that we have to define discretization.

The most 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. The differences Δ_{n} = t_{n+1} − t_{n} are defined as step sizes.

We recall the definition of the general time discretization from [7]: for a given maximum step size δ ∈ (0, δ_{0}) we define a time discretization

(τ)_{δ} = {τ_{n} : n = 0, 1, …}

as a sequence of time instants {τ_{n} : n = 0, 1, …} which may be random, satisfying

0 = τ_{0} < τ_{1} < … < τ_{N} < … < +∞,
sup_{n}{τ_{n+1} − τ_{n}} ≤ δ,
n_{t} < +∞

w.p.1 for all t ∈ R^{+}, where τ_{n+1} is F_{τ_{n}}-measurable for each n = 0, 1, 2, … and n_{t} is defined as follows

n_{t} = max{n = 0, 1, …: τ_{n} ≤ t}.

Here we give the definition of a time discrete approximation as in [7].

Definition 3.1 We shall say that a right continuous with left hand limits process Y = (Y (t), t ≥ 0) is a time discrete approximation of the solution of equation (1) with maximum step size δ ∈ (0, δ_{0}) if it is based on a time discretization (τ)_{δ} such that Y (τ_{n}) is F_{τ_{n}}− measurable and Y (τ_{n+1}) can be expressed as a function of Y (τ_{0}), Y (τ_{1}), …, Y (τ_{n}), τ_{0}, τ_{1}, …, τ_{n}, τ_{n+1} and a finite number r of F_{τ_{n+1}}− measurable random variables Z_{n,j} for j = 1, …, r and each n = 0, 1, …

In the sequel Y_{n}^{Δ} always denotes the approximation of X(t_{n}) using a given numerical scheme with maximum step size Δ. The most simplest form of the discretization is an equidistant discretization. We will omit the superscript Δ if there is no confusion.

The used label Const is independent of Δ and it may depend on T. In the sequel Const always denotes constants of this property.

Let the general multi-dimensional one-step scheme for a given time discretization be defined as

Y_{n+1} = Y_{n} + ψ(τ_{n}, Y_{n+1}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}).  (2)

Here Z_{n,1}, …, Z_{n,r} are stochastic variables depending on Δ and measurable with respect to F_{t_{n+1}} evaluated at points of the partition t_{m}, m = n, n + 1. Function ψ is defined by

ψ = Δ ∑_{j=0}^{1} β_{1,j}f̂_{n+j,1} + Δ^{2} ∑_{j=0}^{1} β_{2,j}f̂_{n+j,2} + … + Δ^{k} ∑_{j=0}^{1} β_{k,j}f̂_{n+j,k} +
Z_{n,1}β_{0,1}f_{n,1} + … + Z_{n,r}β_{0,r}f_{n,r},

where f̂_{n+j,i} = f̂_{i}(t_{n+j}, Y_{n+j}), f_{n+j,i} = f_{i}(t_{n+j}, Y_{n+j}), and functions f_{i}, f̂_{i} depend on functions a, b_{j}, j = 1, …, m and derivatives of them.

To this end let

ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}),

be defined implicitly by

ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) = ψ(t_{n}, ϕ + Y_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}).  (3)

The one-step formula (2) can then be written as

Y_{n+1} = Y_{n} + ϕ(t_{n}, Y_{n}, Δ_{n}, Z_{n,1}, …, Z_{n,r}).  (4)

We have several concepts of the stability for numerical approximations of SODE. The concept of stochastical numerical stability is defined in [7].

Definition 3.2 Let Y^{Δ} denotes a time discrete approximation (i. e. numerical solution) with an maximum step size Δ > 0, starting at time 0 at Y_{0}^{Δ}, Ŷ^{Δ} denotes the corresponding approximation (constructed using the same driving Brownian path) starting at time 0 at Ŷ_{0}^{δ}. We shall say that a time discrete Y^{Δ} is stochastically numerically stable for a given stochastic differential equation if for any finite interval [0, T] there exists a positive constant Δ_{0} such that for every ϵ > 0 and every Δ ∈ (0, Δ_{0})

lim_{∥Y_{0}^{Δ}−Ŷ_{0}^{Δ}∥→0} sup_{0≤t≤T} P (∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥≥ ϵ) = 0.

In paper [5] there was established the connections, which guarantee numerical stability, namely, the next theorem was proven.

Theorem 3.3 (Stochastical numerical stability) Suppose that the time discretization is equidistant with step size Δ and the increment function ϕ a one-step approximation

Y_{n+1} = Y_{n} + ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r})  (5)

of the SODE (1) satisfies

(E ∥ E(ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) ∥^{2})^{\frac{1}{2}}
≤ Const · (E(∥ Y_{n} − Ŷ_{n}∥^{2})^{\frac{1}{2}} Δ,  (6)

(E ∥ ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r}) −  (7)
− E(ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) + E(ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) ∥^{2})^{\frac{1}{2}}
≤ Const · (E ∥ Y_{n} − Ŷ_{n}∥^{2})^{\frac{1}{2}} Δ^{\frac{1}{2}} ,

where Y_{n} and Ŷ_{n} be two numerical approximation of the equation (1) of the form (8) and having initial values Y_{0} and Ŷ_{0}, respectively.

Then a one-step approximation (Y) is stochastically numerically stable.

Namely, it means that the conditions (9) and (10) imply continuous dependence of initial values in probability.

The next Definition is given in [2].

Definition 3.4 Let Y^{Δ} denotes a time discrete approximation (i. e. numerical solution) with a maximum step size Δ > 0, starting at time 0 at Y_{0}^{Δ}, Ŷ^{Δ} denotes the corresponding approximation (constructed using the same driving Brownian path) starting at time 0 at Ŷ_{0}^{Δ}. We shall say that a time discrete Y^{Δ} is numerically stable in the quadratic mean-square sense for a given stochastic differential equation if for any finite interval [0, T] there exists a positive constant Δ_{0} such that for every ϵ > 0 there is a δ = δ(ϵ, Δ_{0}) and every Δ ∈ (0, Δ_{0})

E ∥ Y_{0}^{Δ} − Ŷ_{0}^{Δ} ∥^{2}≤ δ

then

E ∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥^{2}≤ ϵ

holds for every positive integers n_{t} ≤ N.

This means that numerical approximations continuously depend on initial values in mean-square sense.

Remark 3.5 If we examine our proof of the Theorem 3.6 given in paper [5], we notice that (using the same notions as there) there was proven the next estimation

D(t) = sup_{0≤s≤t} E(∥ Y_{n_{s}}^{Δ} − Ŷ_{n_{s}}^{Δ} ∥^{2}) ≤ ConstE ∥ Y_{0}^{Δ} − Ŷ_{0}^{Δ} ∥^{2}, ∀t, 0 ≤ t ≤ T.

Now, using the definition above we notice that we actually prove the numerical stability in the quadratic mean-square sense.

Theorem 3.6 (Numerical stability in mean-square sense) Suppose that the time discretization is equidistant with step size Δ and the increment function ϕ a one-step approximation

Y_{n+1} = Y_{n} + ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r})  (8)

of the SODE (1) satisfies

(E ∥ E(ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) ∥^{2})^{\frac{1}{2}}
≤ Const · (E(∥ Y_{n} − Ŷ_{n}∥^{2})^{\frac{1}{2}} Δ,  (9)

(E ∥ ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r}) −  (10)
− E(ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) + E(ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}}) ∥^{2})^{\frac{1}{2}}
≤ Const · (E ∥ Y_{n} − Ŷ_{n}∥^{2})^{\frac{1}{2}} Δ^{\frac{1}{2}} ,

where Y_{n} and Ŷ_{n} be two numerical approximations of the equation (1) of the form (8) and having initial values Y_{0} and Ŷ_{0}, respectively.

Then a one-step approximation (Y) is numerically stable in the quadratic mean-square sense.

4 Asymptotical Numerical Stability

As in deterministic case, the numerical stability in the quadratic mean-square sense for one-step numerical approximations does not tell us how to pick an appropriate step size Δ. We will investigate some other concepts because of that.

We notice the propagated error of a numerically stable scheme which is theoretically still under control, may in fact, become so unrealistically large as to make approximation useless for some practical purposes.

As an example in the case of the simulation of the first exit times we do not know the appropriate time interval in advance. Because of that we need to control the error propagation for an arbitrarily large interval, namely, for an interval [t_{0}, +∞). The concept of asymptotic stochastic numerical stability is defined in [7].

Definition 4.1 We shall say that a time discrete approximation Y^{Δ} is asymptotically stochastically numerically stable for a given stochastic ordinary differential equation if it is stochastically numerically stable and there exists a positive constant Δ_{a} such that for each ϵ > 0 and Δ ∈ (0, Δ_{a})

lim_{∥Y_{0}^{Δ}−Ŷ_{0}^{Δ}∥→0} lim_{T →+∞} P ( sup_{0≤t≤T} ∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥≥ ϵ) = 0,

where we have used the same notation as in Definition (3.2).

Definition 4.2 We shall say that a time discrete approximation Y^{Δ} is asymptotically numerically stable in the quadratic mean-square sense for a given stochastic ordinary differential equation if it is numerically stable in the quadratic mean-square sense and there exists a positive constant Δ_{a} such that for each ϵ > 0 and Δ ∈ (0, Δ_{a})

lim_{E∥Y_{0}^{Δ}−Ŷ_{0}^{Δ}∥^{2}→0} lim_{T →+∞} E( sup_{0≤t≤T} ∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥^{2}) = 0,

where we have used the same notation as in Definition (3.2).

Theorem 4.3 A time discrete approximation Y^{Δ} which is asymptotically numerically stable in the quadratic mean-square sense is asymptotically stochastically numerically stable.

Proof of Theorem Using Chebysev inequality we get

P ( sup_{0≤t≤T} ∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥≥ ϵ) ≤ \frac{E(sup_{0≤t≤T} ∥ Y_{n_{t}}^{Δ} − Ŷ_{n_{t}}^{Δ} ∥^{2})}{ϵ^{2}}.

and using the definition for asymptotical numerical stability in quadratic mean-square sense we will easily prove asymptotical stochastical numerical stability property. □

Here we give a fundamental theorem on the asymptotic numerical stability in the quadratic meansquare sense:

Theorem 4.4 (Asymptotic numerical stability in the quadratic mean-square sense) Suppose that the time discretization is equidistant with step size Δ and the increment function ϕ of a one-step approximation

Y_{n+1} = Y_{n} + ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r})  (11)

of the SODE (1) satisfies

E(ϕ(t_{n}, x, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, y, Δ, Z_{n,1}, …, Z_{n,r})|F_{t_{n}})
= C_{1}(Δ) · (x − y),  (12)
E ∥ ϕ(t_{n}, x, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, y, Δ, Z_{n,1}, …, Z_{n,r}) ∥^{2}  (13)
≤ C_{2}(Δ)· ∥ x − y∥^{2},

where x and y are real numbers from R and C_{1}(Δ) and C_{2}(Δ) real continuous functions depending on Δ, and it may depend on x and y. If there exists Δ_{a} > 0 such that for all Δ ∈ (0, Δ_{a}) the next inequality

0 < 1 + 2C_{1}(Δ) + C_{2}(Δ) < 1,

is valid, then a one-step approximation (Y) is asymptotically numerically stable in the quadratic meansquare sense.

Proof of Theorem In the proof of the theorem we use the next theorem with a_{n}(Δ) = (1 + 2C_{1}(Δ) + C_{2}(Δ))^{n}. □

Remark 4.5 We notice that C_{2}(Δ) must be positive constant but for the validity of the condition above we need that C_{1}(Δ) be negative constant.

Theorem 4.6 Suppose that the time discretization is equidistant with step size Δ. Let Y denotes a time discrete approximation (i. e. numerical solution) starting at time 0 at Y_{0}, Ŷ denotes the corresponding approximation (constructed using the same driving Brownian path) starting at time 0 at Ŷ_{0}. If there exists Δ_{0} such that for every Δ ∈ (0, Δ_{0})

E ∥ Y_{n} − Ŷ_{n} ∥^{2}≤ a_{n}(Δ)E ∥ Y_{0} − Ŷ_{0} ∥^{2},

for every n = 0, 1, 2, …, where a_{n}(Δ) is continuous and positive function of Δ such that ∑_{n=0}^{+∞} a_{n}(Δ) = b(Δ) converges and {a_{n}(Δ) : n = 0, 1, 2, …; Δ ∈ (0, Δ_{0})} is bounded then a time discrete approximation Y is asymptotically numerically stable in the quadratic mean-square sense.

Proof of Theorem We notice the validity of the next relation:

E( sup_{0≤t≤T} ∥ Y_{n_{t}} − Ŷ_{n_{t}} ∥^{2}) ≤ ∑_{n=0}^{n_{T}} E(∥ Y_{n} − Ŷ_{n} ∥^{2})

Now, we take limit as T → +∞, so

lim_{E∥Y_{0}−Ŷ_{0}∥^{2}→0} lim_{T →+∞} E( sup_{0≤t≤T} ∥ Y_{n_{t}} − Ŷ_{n_{t}} ∥^{2}) ≤

≤ lim_{E∥Y_{0}−Ŷ_{0}∥^{2}→0} ∑_{n=0}^{+∞} E(∥ Y_{n} − Ŷ_{n} ∥)^{2})

≤ lim_{E∥Y_{0}−Ŷ_{0}∥^{2}→0} ∑_{n=0}^{+∞} a_{n}(Δ)E(∥ Y_{0} − Ŷ_{0} ∥)^{2}.

Using the conditions of the Theorem we get

lim_{E∥Y_{0}−Ŷ_{0}∥^{2}→0} E(∥ Y_{0} − Ŷ_{0} ∥)^{2}( ∑_{n=0}^{+∞} a_{n}(Δ)) = 0.

We need to prove the numerical stability in the quadratic mean-square sense of the time discrete approximation (Y) under this condition.

Using the well-known relations from the convergence of the ∑_{n=0}^{+∞} a_{n}(Δ) we get that lim_{n→+∞} a_{n}(Δ) = 0 and the existence of the constant K > 0 such that |a_{n}(Δ)| < K for all n = 0, 1, 2, . . .. We estimate

sup_{0≤t≤T} E(∥ Y_{n_{t}} − Ŷ_{n_{t}} ∥^{2}) ≤ E(∥ Y_{0} − Ŷ_{0} ∥)^{2}{ sup_{0≤t≤T} a_{n_{t}}(Δ)}
≤ K · E(∥ Y_{0} − Ŷ_{0} ∥)^{2}.

For every ϵ > 0 there is a δ = δ(ϵ, Δ_{0}) and every Δ ∈ (0, Δ_{0})

E ∥ Y_{0} − Ŷ_{0} ∥^{2}≤ δ

then

E ∥ Y_{n_{t}} − Ŷ_{n_{t}} ∥^{2}≤ ϵ

holds for every positive integers n_{t} ≤ N.

Now the numerical stability in the quadratic mean-square sense for (Y) is proved. □

5 Examples

Example 5.1 As an example (cf. eg. [9]), let us consider the one-dimensional stochastic equation

dX(t) = νX(t)dt + μX(t)dW (t), X(0) = x_{0}, t ≥ 0,  (14)

where ν and μ are complex numbers.

The exact solution of (14) is

X(t) = exp((ν − \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((2Re(ν) + |μ|^{2})t)|x_{0}|^{2}.

Schurz [9] and Saito and Mitsui [8] showed that the zero solution of the equation (14) is asymptotically mean-square stable if and only if

2Re(ν) + |μ|^{2} < 0,

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

Generally, in the vector case, when we apply a numerical scheme (Y_{n}) to the equation (14) 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},  (15)

where \overline{Δ} = Δν and k = \frac{−μ^{2}}{ν} .

Y. Saito and T. Mitsui in their work [8] 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 the next definition in their work [8]:

Definition 5.2 The scheme is said to be MS-stable for those values of \overline{Δ} and k satisfying

|R(\overline{Δ}, k)| < 1.  (16)

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.

Consider the recursive form of one-step schemes given in [8]

Y_{n+1} = Y_{n} + ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}).

We notice that in that case the increment function is linear, so

ϕ(t_{n}, Y_{n}, Δ, Z_{n,1}, …, Z_{n,r}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r}) = ϕ(t_{n}, Y_{n} − Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r}).

It means

Y_{n+1} − Ŷ_{n+1} = Y_{n} − Ŷ_{n} + ϕ(t_{n}, Y_{n} − Ŷ_{n}, Δ, Z_{n,1}, …, Z_{n,r}).

Then the next equation

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

is valid, where the function R(\overline{Δ}, k) is the same as in the case of MS-stability. The condition |R(\overline{Δ}, k)| < 1 for MS-stability using theorem 4.6 guarantees the asymptotical numerical stability in quadratic meansquare sense. It means that any MS-stable one-step approximation is asymptotical numerical stable in quadratic mean-square sense, which implies asymptotical stochastical numerical stability.

Example 5.3 As we know from the theory of ODE, the numerical stability conditions of one-step methods for deterministic differential equations does not tell us how to pick an appropriate step size Δ. Because of that we consider a class of test equations. These are the complex-valued linear differential equations

\frac{dx}{dt} = λx,

with λ = λ_{r} + iλ_{i}.

Usually, the one-step numerical approximations with Re(λ) < 0 in the recursive form

Y_{n+1} = G(λΔ)Y_{n}

is given.

We shall call the set of complex number λΔ with Re(λ) < 0 and |G(λΔ)| < 1 the region of absolute stability of the scheme.

Definition 5.4 We shall say that a numerical method is A-stable if its region of absolute stability contains all of the left half of the complex plane, that is all λΔ with Re(λ) < 0 and Δ > 0.

In stochastic case, we will consider the same complex-valued linear differential equations with an additive noise

dX(t) = λX(t)dt + dW (t),  (17)

where the parameter λ is complex number with real part Re(λ) < 0 and W is a real-valued standard Wiener process.

We have almost the same recursive form as in the above case

Y_{n+1} = G(λΔ)Y_{n} + Z_{n},  (18)

for n = 0, 1, 2, … where G is a mapping of the complex plane C into itself and Z_{0}, Z_{1}, … are random variable which do not depend on λ or Y_{0}, Y_{1}, …

We have the same definition of A-stability as in deterministic case, namely

Definition 5.5 We shall call the set of complex numbers λΔ with Re(λ) < 0 and |G(λΔ)| < 1 the region of absolute stability of the scheme (18).

If this region coincides with the left half of the complex plane, we say the scheme is A-stable.

The general recursive one-step approximation for a stochastic differential equation

dX(t) = λX(t)dt + dW (t)

with a real constant λ, λ < 0 has an increment function

ϕ(t_{n}, Y_{n}, Δ, Z_{n}) = G(λΔ)Y_{n} − Y_{n} + Z_{n},

with E(Z_{n}) = 0 and E(Z_{n}^{2}) = Δ.

Now we estimate the conditions

E(ϕ(t_{n}, Y_{n}, Δ, Z_{n}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n})|F_{t_{n}}) = (G(λΔ) − 1)(Y_{n} − Ŷ_{n}),
E ∥ ϕ(t_{n}, Y_{n}, Δ, Z_{n}) − ϕ(t_{n}, Ŷ_{n}, Δ, Z_{n}) ∥^{2}= (G(λΔ) − 1)^{2} · E ∥ Y_{n} − Ŷ_{n}∥^{2}.

Using this conditions we get

E ∥ Y_{n+1} − Ŷ_{n+1} ∥^{2}= (1 + 2(G(λΔ) − 1) + (G(λΔ) − 1)^{2})(E ∥ Y_{n} − Ŷ_{n} ∥^{2}),

and now we get the condition

1 + 2(G(λΔ) − 1) + (G(λΔ) − 1)^{2} < 1,

which implies |G(λΔ)|^{2} < 1, and it means that |G(λΔ)| < 1. This condition is the same as in the case of A-stability. Our conclusion is that we have the condition which guarantees A-stability and implies asymptotical numerical stability in quadratic mean-square sense and asymptotical stochastical numerical stability.

In the simpliest case, for the Euler-Maruyama scheme for the test stochastic ordinary differential equation

dX_{t} = aX_{t}dt + dW (t),

we get the condition

0 < (1 + 2aΔ + a^{2}Δ^{2}) < 1,

which gives us two condition, namely a < 0 and Δ < \frac{−2}{a}. It means that for an asymptotical numerical stability in quadratic mean-square sense we have the same bounds on step size as for A-stability for Eulerscheme and A-stability for Euler-Maruyama scheme.

References

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

[2] C.T.H. Baker and E. Buckwar, Numerical analysis of explicit one-step methods for stochastic delay differential equations, LMS J. Comput. Math. 3, pp. 315-335, 2000.

[3] E. Harrier, G. Wanner, Solving Ordinary Differential Equations II, Springer-Verlag, 1991.

[4] R. Horváth-Bokor, Convergence and Stability Properties for Numerical Approximations of Stochastic Ordinary Differential Equations, doctoral thesis, 2000.

[5] R. Horváth-Bokor, Stochastically stable one-step approximations of solutions of stochastic ordinary differential equations, Applied Numerical Mathematics 44/3 (2003),299–312.

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

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

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

[9] H. Schurz, Stability, stationarity and boundedness of some numerical methods for stochastic differential equations and applications, Logos-Verlag, (1997)

(Received September 30, 2003)