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Existence, uniqueness and approximation of solutions to Carathéodory delay differential equations

In this paper we address the existence, uniqueness and approximation of solutions of delay differential equations (DDEs) with Carathéodory type right-hand side functions. We provide construction of randomized Euler scheme for DDEs and investigate its error. We also report results of numerical experiments.

• 2 publications
• 13 publications
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1. Introduction

We deal with approximation of solutions to the following delay differential equations (DDEs)

 (1.1) {x′(t)=f(t,x(t),x(t−τ)),t∈[0,(n+1)τ],x(t)=x0,t∈[−τ,0),

with the constant time-lag , fixed time horizon , , and . We assume that is integrable with respect to and (at least) continuous with respect to . Hence, we consider Carathéodory type conditions for .

Motivation of considering such DDEs comes, for example, from the problem of modeling switching systems with memory, see [hale1977theory], [Hale1993IntroductionTF]. Moreover, another inspiration follows from delayed differential equations with rough paths of the form

 (1.2) {dU(t)=a(U(t),U(t−τ))+dZ(t),t∈[0,(n+1)τ],U(t)=U0,t∈[−τ,0),

where is an integrable perturbation (which might be of stochastic nature). Then satisfies the (possibly random) DDE (1.1) with and , where we assume that for . In this case the function inherits from its low smoothness with respect to the variable . (The exemplary equation (1.2) is a generalization of the ODE with rough paths discussed in [RKYW2017].)

In the case of classical assumptions (such as -regularity of wrt all variables ) imposed on the right-hand side function errors for deterministic schemes have been established, for example, in the book [bellen1], which is the standard reference. See also [CZPMPP], where error of the Euler scheme has been investigated for some class of nonlinear DDEs under nonstandard assumptions, such as one-side Lipschitz condition and local Hölder continuity. In contrast, much less is known about the approximation of solutions of DDEs with less regular Carathéodory right-hand side function . In the case of Carathéodory ODEs we need to turn to randomized algorithms, such as randomized Euler scheme, since it is well-known that there is lack of convergence for deterministic algorithms, see [RKYW2017]. This behavior is inherited by DDEs, since ODEs form a subclass of DDEs. Hence, we define randomized version of the Euler scheme that is suitable for DDEs of the form (1.1).

While the randomized algorithms for ODEs have been widely investigated in the literature (see, for example, [bochacik2], [BGMP2021], [daun1], [hein_milla1], [jen_neuen1], [BK2006], [RKYW2017]), according to our best knowledge this is the first paper that defines randomized Euler scheme and rigorously investigates its error for (Carathéodory type) DDEs.

The main contributions of the paper are as follows:

• we show existence, uniqueness, and Hölder regularity of the solution to (1.1) when the right-hand side function is only integrable with respect to and satisfies local Lipschitz assumption with respect to (Theorem 3.1),

• we perform rigorous error analysis of the randomized Euler scheme applied to (1.1) when the right-hand side function is only integrable with respect to , satisfies global Lipschitz condition with respect to and it is globally Hölder continuous with respect to (Theorem 4.2),

• we report results of numerical experiments that show stable error behaviour as stated in Theorem 4.2.

In addition, as a consequence of Theorem 3.1 we establish almost sure convergence of the randomized Euler scheme, see Proposition 4.3.

We want to stress here that the techniques used when proving upper error bounds in Theorem 4.2 differ significantly comparing to that used in [bochacik2], [BGMP2021], [daun1], [hein_milla1], [jen_neuen1], [RKYW2017]

for randomized algorithms defined for ordinary differential equations. Mainly, due to the fact that DDEs have to be considered interval-by-interval we developed a suitable proof technique that is based on mathematical induction. In particular, suitable inductive assumptions have to be related with the Hölder continuity of

with respect to the (delayed) variable .

The structure of the article is as follows. Basic notions, definitions together with assumptions and definition of the randomized Euler scheme are given in Section 2. All Section 3 is devoted to the issue of existence and uniqueness of solutions of the Carathéodory type DDEs (1.1) in the case when is only integrable with respect to and satisfies local Lipschitz assumption with respect to . Section 4 contain proof of the main result of the paper (Theorem 3.1) that states upper bounds on the error of the randomized Euler scheme. In Section 5 we report results of numerical experiments. Finally, Appendix contains auiliary results for Carathéodory type ODEs that we use in the paper.

2. Preliminaries

By we mean the Euclidean norm in

. We consider a complete probability space

. For a random variable

we denote by , where .

Let us fix the horizon parameter . On the right-hand side function we impose the following assumptions:

1. [label=(A0),ref=(A0)]

2. for all ,

3. is Borel measurable for all ,

4. there exists such that and for all

 (2.1) ∥f(t,x,z)∥≤K(t)(1+∥x∥)(1+∥z∥),
5. for every compact set there exists such that and for all , ,

 (2.2) ∥f(t,x1,z)−f(t,x2,z)∥≤LU(t)(1+∥z∥)∥x1−x2∥.

In Section 3, under the assumptions 1-4, we investigate existence and uniqueness of solution for (1.1). Next, in Section 4 we investigate error of the randomized Euler scheme under slightly stronger assumptions. Namely, we impose global Lipschitz assumption on with respect to instead of its local version 4.

The mentioned above randomized Euler scheme is defined as follows. Fix the discretization parameter and set

 tjk=jτ+kh,k=0,1,…,N, j=0,1,…,n,

where

 (2.3) h=τN.

Note that for each the sequence provides uniform discretization of the subinterval . Let be an iid sequence of random variables, defined on the complete probability space , where every

. We set and then for , we take

 (2.4) yj0=yj−1N, (2.5) yjk+1=yjk+h⋅f(θjk+1,yjk,yj−1k),

where . As the output we obtain the sequence of

-valued random vectors

that provides a discrete approximation of the values . It is easy to see that each random vector , , , is measurable with respect to the -field generated by the following family of independent random variables

 (2.6) {θ01…,θ0N,…,θj−11,…,θj−1N,θj1,…,θjk}.

As the horizon parameter is fixed, the randomized Euler scheme uses evaluations of (with a constant in the ’’ notation that only depends on but not on ).

In Section 4 we provide upper bounds on the error

 (2.7) ∥∥max0≤i≤N∥x(tji)−yji∥∥∥Lp(Ω)

for .

3. Properties of solutions to Carathéodory DDEs

In this section we investigate the issue of existence and uniqueness of the solution of (1.1) under the assumptions 1-4.

In the sequel we use the following equivalent representation of the solution of (1.1

), that is very convenient when proving its properties and when estimating the error of the randomized Euler scheme. For

and it holds

 (3.1) x′(t+jτ)=f(t+jτ,x(t+jτ),x(t+(j−1)τ).

Hence, we take , and for we consider the following sequence of initial-value problems

 (3.2) {ϕ′j(t)=gj(t,ϕj(t)),t∈[0,τ],ϕj(0)=ϕj−1(τ),

with , . Then the solution of (1.1) can be written as

 (3.3) x(t)=n∑j=−1ϕj(t−jτ)⋅1[jτ,(j+1)τ](t),t∈[0,(n+1)τ].

We prove the following result about existence, uniqueness and Hölder regularity of the solution of the delay differential equation (1.1). We will use this theorem in the next section when proving error estimate for the randomized Euler algorithm. Since we were not able to find references in literature that are suitable for (1.1) under the assumptions 1-4, for the completeness and convenience of the reader we provide its justification.

Theorem 3.1.

Let , , and let satisfy assumptions 1-4. Then there exists a unique absolutely continuous solution of (1.1) such that for we have

 (3.4) sup0≤t≤τ∥ϕj(t)∥≤Kj

where and

 (3.5) Kj=(1+Kj−1)(1+∥K∥L1([jτ,(j+1)τ]))⋅exp((1+Kj−1)∥K∥L1([jτ,(j+1)τ])).

Moreover, if we additionally assume that for some the function in 3 satisfies

1. [label=(A5),ref=(A5)]

2. ,

then for all , it holds

 (3.6) ∥ϕj(t)−ϕj(s)∥≤(1+Kj−1)(1+Kj)∥K∥Lp([jτ,(j+1)τ])|t−s|1−1p.
Proof.

We proceed by induction. We start with the case when and consider the following initial-value problem

 (3.7) {ϕ′0(t)=g0(t,ϕ0(t)),t∈[0,τ],ϕ0(0)=x0,

with . Of course for all the function is continuous and for all the function is Borel measurable. Moreover, by (2.1) we have for all , and by (2.2) for every compact set in there exists a positive function such that for all , it holds

 (3.8) ∥g0(t,x)−g0(t,y)∥≤LU(t)(1+∥x0∥)∥x−y∥.

Therefore, by Lemma 7.1 we have that there exists a unique absolutely continuous solution of (3.7) that satisfies (3.4) with . In addition, if for some then by Lemma 7.1 we obtain that satisfies (3.6) for .

Let us now assume that for some there exists a unique absolutely continuous solution of

 (3.9) {ϕ′j(t)=gj(t,ϕj(t)),t∈[0,τ],ϕj(0)=ϕj−1(τ),

where , and that satisfies (3.4) with (3.6), if for some . We consider the following initial-value problem

 (3.10) {ϕ′j+1(t)=gj+1(t,ϕj+1(t)),t∈[0,τ],ϕj+1(0)=ϕj(τ),

with . Since is continuous on , it is straightforward to see that for all the function is continuous and for all the function is Borel measurable. Moreover, by (3.4) for all

 ∥gj+1(t,x)∥≤K(t+(j+1)τ)(1+∥ϕj(t)∥)(1+∥x∥) (3.11) ≤K(t+(j+1)τ)(1+Kj)(1+∥x∥),

where

 (3.12) τ∫0K(t+(j+1)τ)dt=(j+2)τ∫(j+1)τK(t)dt≤(n+1)τ∫0K(t)dt<+∞.

Furthermore, by (2.2) and (3.4) for every compact set in there exists a positive function such that for all , it holds

 (3.13) ∥gj+1(t,x)−gj+1(t,y)∥≤LU(t+(j+1)τ)(1+Kj)∥x−y∥,

where

 (3.14) τ∫0LU(t+(j+1)τ)dt=(j+2)τ∫(j+1)τLU(t)dt≤(n+1)τ∫0LU(t)dt<+∞.

Hence, by Lemma 7.1 there exists a unique absolutely continuous solution of (3.10). By the inductive assumption and Lemma 7.1 we get that

 sup0≤t≤τ∥ϕj+1(t)∥≤(∥ϕj(τ)∥+(1+Kj)τ∫0K(t+(j+1)τ)dt) ×exp((1+Kj)τ∫0K(t+(j+1)τ)dt) ≤(1+Kj)(1+∥K∥L1([(j+1)τ,(j+2)τ]))exp((1+Kj)∥K∥L1([(j+1)τ,(j+2)τ]))=Kj+1,

and

 (3.15) ∥ϕj+1(t)−ϕj+1(s)∥≤¯K|t−s|1−1p,

where

 ¯K=(1+Kj)(τ∫0|K(t+(j+1)τ)|pdt)1/p ×(1+(∥ϕj(τ)∥+(1+Kj)τ∫0K(t+(j+1)τ)dt)exp((1+Kj)τ∫0K(t+(j+1)τ)dt)) ≤(1+Kj)∥K∥Lp([(j+1)τ,(j+2)τ])(1+Kj+1).

This ends the inductive proof. ∎

Remark 3.2.

Theorem 3.1 can be applied, for example, to the function

 (3.16) f(t,x,z)=K(t)⋅cos(x2)⋅|z|α,(t,x,z)∈[0,(n+1)τ]×R×R,

where is any function from and .

4. Error of the randomized Euler scheme

In this section we perform detailed error analysis for the randomized Euler. As mentioned in Serction 1, for the error analysis we impose global Lipschtz assumption on with respect to together with global Hölder condition with respect to . Namely, instead of 3 and 4, we assume

1. [label=(A3’),ref=(A3’)]

2. there exist , and such that and for all

 (4.1) ∥f(t,0,0)∥≤¯K(t),

and for all ,

 (4.2) ∥f(t,x1,z1)−f(t,x2,z2)∥≤L(t)(∥x1−x2∥+∥z1−z2∥α).
Remark 4.1.

Note that the assumptions 1, 2, 1 are stronger than the assumptions 1-4. To see that note that if satisfies 1, 2, 1 then we get for all and that

 (4.3) ∥f(t,x,z)∥≤(¯K(t)+L(t))(1+∥x∥)(1+∥z∥),

and

 (4.4) ∥f(t,x1,z)−f(t,x2,z)∥≤L(t)∥x1−x2∥,

since and for all . Hence, the assumptions 1-4 are satisfied with , for any compact set , and under the assumptions 1, 2, 1 the thesis of Theorem 3.1 holds.

The main result of this section is as follows.

Theorem 4.2.

Let , , , and let satisfy the assumptions 1, 2, 1 for some and . There exist such that for all and it holds

 (4.5) ∥∥max0≤i≤N∥x(tji)−yji∥∥∥Lp(Ω)≤Cjh12αj.

In particular, if then for

 (4.6) ∥∥max0≤i≤N∥x(tji)−yji∥∥∥Lp(Ω)≤Cjh1/2.
Proof.

In the proof we use the following auxiliary notation: and . Then is uniformly distributed in and is uniformly distributed in .

We start with and consider the initial-vale problem (3.7). We define the auxiliary randomized Euler scheme as

 (4.7) ¯y00=y00=y−1N=x0, (4.8) ¯y0k+1=¯y0k+h⋅g0(θ0k+1,¯y0k), k=0,1,…,N−1.

Since and , for all we have that . Moreover, by (A1), (A2) and (A3’) we have that is Borel measurable, and for all and

 ∥g0(t,x)∥≤K(t)(1+∥x0∥)(1+∥x∥), (4.9) ∥g0(t,x)−g0(t,y)∥≤L(t)∥x−y∥,

where , as stated in Remark 4.1. Hence, by Theorem 3.1 and by using analogous arguments as in the proof of Theorem 4.3 in [RKYW2017] we get

 ∥∥max0≤i≤N∥ϕ0(t0i)−y0i∥∥∥Lp(Ω)≤21−1pexp((2τ)p−1p∥L∥pLp([0,τ])) ×(1+K−1)(1+K0)∥K∥Lp([0,τ])(2Cpτp−22p+τ1−1p∥L∥Lp([0,τ]))h1/2=C0h1/2,

where does not depend on . Since we get (4.5) for .

Let us now assume that there exists for which there exists such that for all

 (4.10) ∥∥max0≤i≤N∥ϕl(t0i)−yli∥∥∥Lp(Ω)≤Clh12αl.

We consider the following initial-value problem

 (4.11) {ϕ′l+1(t)=gl+1(t,ϕl+1(t)),t∈[0,τ],ϕl+1(0)=ϕl(τ),

with . Recall that by (A1), (A2), (A3’), Theorem 3.1, and Remark 4.1 the function is Borel measurable and for all , it satisfies

 (4.12) ∥gl+1(t,x)∥≤K(t+(l+1)τ)(1+Kl)(1+∥x∥),
 (4.13) ∥gl+1(t,x)−gl+1(t,y)∥≤L(t+(l+1)τ)∥x−y∥.

We define the auxiliary randomized Euler scheme as follows

 (4.14) ¯yl+10=yl+10=ylN, (4.15) ¯yl+1k+1=¯yl+1k+h⋅gl+1(αl+1k⋅h,¯yl+1k), k=0,1,…,N−1.

From the definition it follows that , , , is measurable with respect to the -filed generated by (2.6), so as . Moreover, approximates at , however is not implementable. We use only in order to estimate the error (4.5) of , since it holds

 (4.16) ∥∥max0≤i≤N∥ϕl+1(t0i)−yl+1i∥∥∥Lp(Ω) ≤∥∥max0≤i≤N∥ϕl+1(t0i)−¯yl+1i∥∥∥Lp(Ω) +∥∥max0≤i≤N∥¯yl+1i−yl+1i∥∥∥Lp(Ω).

Firstly, we estimate . For we get

 ¯yl+1k−yl+1k =k∑j=1(¯yl+1j−¯yl+1j−1)−k∑j=1(yl+1j−yl+1j−1) =hk∑j=1(f(θl+1j,¯yl+1j−1,ϕl(αl+1j−1⋅h))−f(θl+1j,yl+1j−1,ylj−1)),

which gives

 ∥¯yl+1k−yl+1k∥≤hk∑j=1L(θl+1j)⋅∥¯yl+1j−1−yl+1j−1∥+hk∑j=1L(θl+1j)⋅∥ϕl(αl+1j−1⋅h)−ylj−1∥α.

Note that the random variables and are not independent. However, by Theorem 3.1 we get

 ∥ϕl(αl+1j−1⋅h)−ylj−1∥ ≤∥ϕl(αl+1j−1⋅h)−ϕl(t0j−1)∥+∥ϕl(t0j−1)−ylj−1∥ ≤(1+Kl−1)(1+Kl)∥K∥Lp([lτ,(l+1)τ])⋅h1−1p+∥ϕl(t0j−1)−ylj−1∥.

Therefore for

 ∥¯yl+1k−yl+1k∥ ≤hk∑j=1L(θl+1j)⋅max0≤i≤j−1∥¯yl+1i−yl+1i∥ +h(k∑j=1L(θl+1j))⋅(cl+1hα(1−1p)+max0≤i≤N∥ϕl(t0i)−yli∥α).

Since and due to the fact that the random variables , are independent, we get

 E(max0≤i≤k∥¯yl+1k−yl+1k∥p)≤cphpE[k∑j=1L(θl+1j)⋅max0≤i≤j−1∥¯yl+1i−yl+1i∥]p+~cpE[hk∑j=1L(θl+1j)]p⋅(~cl+1hα(p−1)+E[max0≤i≤N∥ϕl(t0i)−yli∥αp]).

By Jensen inequality and (4.10) we have

 (4.17) E[max0≤i≤N