BoI. Soc. Esp. Mat. ApI. n° 51(2010), 83-91

DISCRETE APPROXIMATION OF STOCHASTIC DIFFERENTIAL EQUATIONS

RAPHAEL KRUSE*

*Department of Mathematics, Bielefeld University, P.O. Box 100131, 33501 Bielefeld, Germany

rkruse@math.uni-bielefeld.de

Abstract

It is shown how stochastic Ito-Taylor schemes for stochastic ordinary differential equations can be embedded into standard concepts of consistency, stability and convergence. An appropriate choice of function spaces and norms, in particular a stochastic generalization of Spijker's norm (1968), leads to two-sided estimates for the strong error of convergence under the usual assumptions.

Key words: BODE, stochastic differential equations, !to- Taylor schemes, discrete approximation, bistability, two-sided error estimates, stochastic Bpijker norm

AMS subject classifications: 65C20, 65C30, 65J15, 65L20, 65L'l0.

1 Introduction

The invention of Ito-Taylor schemes was a major breakthrough in numerical analysis of stochastic ordinary differential equations (SODEs). We refer to the pioneering book [7] and the influential monographs [9] and [10].

In this paper we show how the strong convergence theory of these schemes can be embedded into the standard framework of consistency, stability and convergence as it is formulated in abstract terms in the theory of discrete approximations (see [14]). Moreover, by a special choice of norms, namely a stochastic version of the deterministic Spijker norm (see [12],[13],[6, Ch.III.8]), we are able to derive two-sided estimates for the strong convergence error.

While our notion of consistency and (numerical) stability goes back to the work of F. Stummel [14] there already exist other concepts in the literature. One can find notions of consistency and local truncation errors in the books [7, 9, 10]. We refer to [3] for a discussion. Other authors, who have considered the question of stability, are for instance [2, 4].

*supported by CRC 701 'Spectral Analysis and Topological Structures in Mathematics'.

83

84 R. Kruse

To be more precise, we deal with the numerical approximation of JRd-valued stochastic processes X, which satisfy an ordinary Ito stochastic differential equation of the form

m

dX(t) = bO(t, X(t))dt + L bk(t, X(t))dWk(t), t E [0, T], k=l (1)

X(O) = Xo.

We assume that the initial value Xo has finite second moment. By W k ,

k = 1, ... , m, we denote real and pairwise independent standard Brownian motions and we also assume that the drift and diffusion coefficient functions bk : [0, T] x JRd ---7 JRd fulfill the usual global Lipschitz and linear growth conditions such that (1) has a unique solution [1].

Note that the corresponding integral form of the SODE (1) has the representation

Ito-Taylor schemes are based on an iterated application of Ito's formula on the integrands of (2), provided that all appearing integrals and derivatives exist. Again, we refer to the books [7, 9, 10] for a rigorous derivation.

Let M be the set of all multi-indices a = (jl, ... , jz), lEN, ji E {O, ... , m}, i = 1, ... , l. By £(a) E Nand n(a) E N we denote the length of a E M and the number of zeros in a E M respectively. For, E {:g: : n E N} consider the finite set of multi-indices (c.f. [7])

~ = { a EM: 1 ::; £(a) + n(a) ::; 2, or£(a) = n(a) =, + ~} .

For a time grid ° = to < tl < ... < tN = T with (for simplicity) equidistant step size h = rr, N EN, the Ito-Taylor scheme of order, is given by

Xh(to) = X o,

Xh(tk) = Xh(tk-d + L:aEA fa(tk-l, Xh(tk-d)Ia,k, k ?: 1, , (3)

with the iterated (stochastic) integrals

(4)

where a = (j 1, ... ,j z) and dWo ( s) = ds. For the same a the coefficient function fa : [0, T] X JRd ---7JRd is defined by

(5)

Discrete approximation of SODEs 85

where f : [0, T] x ]Rd ---7 ]Rd is the projection with respect to the second coordinate, i.e. f(t, x) = x, and the Lk are differential operators of the form

8 d 8 1 d m 82 L O = _ + "bO,i_ + _ ""bk,ibk,j __ _

8t L.J 8x 2 L.J L.J 8x8x·' i=l ~ i,j=l k=l ~ J

d

L k = "bk,j~ k L.J 8x.' = 1, ... , m. j=l J

Example 1 If we choose I = ~ then the set A~ just consists of all multi­

indices of length 1, i. e. Al = {( 0), (1), ... (m)}, and the coefficient functions 2

fa simplify to the drift and diffusion coefficient functions of the SODE (1), i.e. f(k) = bk for k = 0, ... , m. Since I(o),k = hand I(j),k = Wj(tk) - Wj(tk-d, the

ltd-Taylor scheme of order I = ~ is the well-known Euler-Maruyama scheme. One also easily checks that the choice I = 1 leads to the Milstein method.

It is well-known (see for example [7, 9, 10D that the Ito-Taylor scheme of order I converges at least with order I in the strong sense, i.e. there exists a constant C > 0, independent of the step size h, such that

(6)

where X is the analytic solution to (1) and X h denotes the numerical solution. Note that [7, 9, 10] use an even stronger norm, where max occurs inside the expectation. It is an open problem whether our approach can handle this norm as well.

In order to embed the Ito-Taylor scheme into the discrete approximation framework, we will write the equations (3) as Ah(Xh ) = Rh with a suitable operator Ah and right-hand side R h. We use the norm

(7)

and the following generalization of Spijker's norm

(8)

Here II . IIL2(0) denotes the L 2-norm of random variables. The key to our two-sided error estimate is the following bistability inequality

In the following section we show how the Ito-Taylor scheme fits into the discrete approximation theory. In Section 3 we give a precise formulation of our main result together with all assumptions.

86 R. Kruse

2 Writing Ito-Taylor schemes as discrete approximations

In the discrete approximation theory the concepts of consistency, (numerical) stability and convergence are defined in a very general way. Our notions of bistability and of the local truncation error are directly related to the abstract framework invented by F. Stummel [14]. We present the basic ideas behind Stummel's theory in this section. Simultaneously we embed the Ito-Taylor scheme into the framework.

The starting point of the discrete approximation theory is an equation of the form A(X) = Y. Here, the operator A : E --7 F is a mapping between two sets E and F. For a given Y E F our aim is to find a discrete approximation of the solution X. To this end we assume the existence of two sequences of metric spaces (Eh)hEI and (Fh)hEI and operators Ah : Eh --7 Fh, hE I, for some index set I. With the help of two sequences of restriction operators r~ : E --7 Eh and rf : F --7 Fh, for h E I, the discrete spaces Eh and Fh are connected to the original spaces E and F respectively. Figure 1 visualizes the setting.

Figure 1: Visualisation of the discrete approximation theory

By solving equations of the form Ah(Xh) = rfY we obtain a sequence of discrete approximations (Xh)hEI. Now, the theory of F. Stummel answers the questions, in which sense and under which conditions the sequence (Xh)hEI converges to the solution X. Let us first show how the SODE (1) and the Ito-Taylor scheme (3) can be embedded into figure l.

Since the existence of a unique solution X to (1) is guaranteed by our assumptions we consider the trivial operator

A: E--7F

X f-7 A(X) (10)

where E := {X} and F := {Y = (Xo,O)} are singletons (with the second component of Y being the stochastic process which is P-a.s. equal to ° E ll~d) and the operator A is given by

A(X) = ( X(O),

(X(t) - X(O) - J~bO(s, X(s))ds - 2::;;'=1 J~bk(s, X(s))dWk(s)) ) . 05ct5cT

Discrete approximation of SODEs 87

In order to define the discrete metric spaces we denote the time grid by Th := {t i = ih I i = 0, ... , N}. As our underlying discrete space we consider the set 9h := 9( Th, L2(0, F, P; JRd)) of all adapted and L2 (O)-valued grid functions, that is, for Zh E 9h, the random variables Zh(t i ) are square-integrable and Fti-measurable random variables for all ti E Th. Here (Ft)tE[O,Tj denotes the filtration which is generated by the Wiener processes W k , k = 1, ... ,m. Now, we choose the metric spaces Eh and Fh to be the vector space 9 h endowed with the metric induced by the norm

and the stochastic version of Spijker's norm

respectively. Note that Eh and Fh are Banach spaces. Next, define the two sequences of restriction operators

rE. h .

rF. h .

Finally, for h > 0, we introduce the operator

by the relationship

i = 0, i = 1, .. . N.

(11)

(12)

(13)

(14)

(15)

for 1 ::; i ::; N. Under the assumption that all Ito-Taylor coefficient functions fa satisfy a linear growth condition, [Ah(Xh)](t i ) is an adapted and mean-square integrable random variable. Therefore, Ah maps Eh into Fh. See Section 3 for a complete statement of all assumptions.

Since the Ito-Taylor schemes are explicit, the operators Ah are bijective, i.e. there exists a unique solution X h to the equation Ah(Xh) = Zh for all Zh E Fh. In particular, the Ito-Taylor approximation X h to (1) is equivalently written as the solution to the equation Ah(Xh) = rry.

Next, we introduce our notion of consistency, bistability and convergence.

88 R. Kruse

Definition 1 Consider a one-step method given by a sequence of operators (Ah) h. The method is called consistent of order I > 0, if there exists a constant C > 0 and an upper step size bound h > 0, such that the estimate

IIAh(r~ X) - rf A(X)II-1,h S; Chi (16)

holds for all grids Th with h S; h, where X denotes the analytic solution of (1).

The left hand side of (16) is called local truncation error or consistency error. Therefore, a one-step method is consistent if the diagram in Figure 1 commutes up to an error of order I, that is rf 0 A ~ Ah 0 r~ for h small enough.

The second ingredient in the convergence theory is the concept of (numerical) stability. In [14] F. Stummel introduces the stronger notion of bistability and he proves that bistability of a numerical method can be characterized by the equicontinuity of the operators (Ah)h and (Ah 1)h. In this sense the following definition is a sufficient condition for Stummel's notion of bistability.

Definition 2 A one-step method defined by operators (Ah)h is called bistable, if there exist constants C1, C2 > 0 and an upper step size bound h > 0 such that the operators Ah are bijective and the estimate

holds for all Zh, Zh E Eh and for grids Th with h < h.

Finally, we define the error of convergence in terms of the norm 11·llo,h, the space Eh and the restriction operators r~.

Definition 3 A one-step method is called convergent of order I > 0 if there exist an upper step size bound h > 0 and a constant C > 0 such that the corresponding operators Ah are bijective and

IIXh - r~ Xllo S; Chi (17)

for all h S; h. Here X h denotes the solution to Ah(Xh) = rfY.

3 Main result

In this section we give a precise formulation of the underlying assumptions and our main result.

(AI) The initial value Xo is an Fo-measurable and JRd-valued random variable satisfying JE( IXo 12) < 00.

(A2) For all a E ~ there exists a constant La > 0 such that

for all x, y E JRd and t E [0, T].

Discrete approximation of SODEs 89

(A3) For a given order, the Ito-Taylor expansion of X(t) with respect to ~ exists for all t E [0, Tj.

(A4) For all a E B(~) we have

faT lE (lfa(s,X(s))12) ds < 00.

The first two assumptions are used, for example, in [1] to assure the existence and uniqueness of the solution X on [0, Tj, such that X(t) is mean-square integrable for all t E [0, Tj. The assumption (A2) also assures that the operators Ah are well-defined and bistable. In (A3) we assume that the Ito-Taylor expansion exists up to a given order ,. Assumption (A4) is needed in order to prove the consistency of the Ito-Taylor schemes. There we use the notation of the remainder set B (~) of the Ito-Taylor expansion which is given by

B(~) = {a = (jl,12, ... ,jz) EM: (12, ... ,jz) E~} eM

(c.f. [7]). Now we formulate our main result, which is proven in [8].

Theorem 1 Let the assumptions (Al)-(A4) hold for, E {~I n EN}. Then the Jt6- Taylor scheme of order, is

(i) consistent of order"

(ii) bistable with respect to the norms II . Ilo,h and II . II-l,h, (iii) convergent of order ,.

Moreover, there exists h > ° such that the two-sided error estimate

C1llAh(r~ X) - rfYII-l,h ::; Ilr~ X - Xhllo,h ::; C21IAh(r~ X) - rfYII-l,h

holds for all grids Th with I hi ::; h.

Remark 1 Theorem 1 also holds for implicit methods like the stochastic theta method [3J and for stochastic multi-step methods [8].

Remark 2 The two-sided error estimate in Theorem 1 can be used to discuss the optimal order of convergence of the Jt6- Taylor methods. J. M. C. Clark and R. J. Cameron [5J constructed the example

X(O) = (~) , (18)

to show that, in general, the maximum order of convergence is equal to ~ if the numerical method, like the Euler-Maruyama scheme, uses only the increments Wk(ti) - Wk(ti_d of the driving Wiener processes. For this example the local

trunction error of the Euler-Maruyama is exactly computed to be J ~Th. Hence,

the strong error of convergence is bounded from below by a term of order, = ~. A suitable generalization of this example gives corresponding results for the

higher order schemes [8].

90 R. Kruse

References

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