Strong approximations for stochastic differential equations with boundary conditions

Marco Ferrante ^{∗} Dipartimento di Matematica

Universitá di Padova via Belzoni 7 35100 Padova, Italy

Arturo Kohatsu-Higa^{†}

Universitat Pompeu Fabra Departament d’Economia Balmes 132 08008 Barcelona, Spain

Marta Sanz-Solé ・^{†} Facultat de Matemá ・ iques Universitat de Barcelona Gran Via 585 08007 Barcelona, Spain

Abstract

We study the Euler approximation scheme for solutions of stochastic differential equations with boundary conditions in two different examples: (a) the one-dimensional case with linear boundary condition, (b) the multidimensional case with constant diffusion coefficient and general boundary condition. In both the cases the error is measured in the L^{p}- norm.

Keywords: Stochastic differential equations with boundary conditions, numerical approximations.

AMS 1990 SCI: 60H99, 34B10, 34B15, 65Nxx.

∗This paper was done while the author was visiting the University of Barcelona with a HCM fellowship of the European Union

†Supported by a grant of the DGICYT No. PB 930052

1 Introduction

This article deals with Stratonovich stochastic differential equations driven by a Wiener process

dX_{t} = σ(X_{t}) ∘ dW_{t} + b(X_{t}) dt, 0 ≤ t ≤ 1,  (1.1)

with a boundary condition of the form

h(X_{0}, X_{1}) = \overline{h}.  (1.2)

The solution {X_{t} , t ∈ [0, 1]}, whenever it exists, is a non-adapted stochastic process, because of (1.2). Our goal is to prove an Euler scheme for these kinds of equations in two different cases considered in Donati-Martin (1991) and Nualart and Pardoux (1991), respectively:

(a) The one-dimensional case with linear boundary condition;

(b) The multidimensional case with constant coefficient σ and a general boundary condition.

We prove L^{p}– convergence of the approximation scheme and obtain the rate of convergence (see Theorems 2 and 3).

Section 2 is devoted to the analysis of case (a). Here we assume h(X_{0}, X_{1}) = F_{0}X_{0} + F_{1}X_{1}, F_{0} F_{1} ≥ 0 and then, X_{t} = Φ_{t}(X_{0}), where {Φ_{t}(x) , t ∈ [0, 1]} is the stochastic flow associated with (1.1) and X_{0} is the unique solution to F_{0}X_{0} + F_{1}Φ_{1}(X_{0}) = h_{0}. Therefore, the approximation problem is related with the strong convergence of an Euler scheme for (1.1) with some given initial condition X_{0} studied in Ahn and Kohatsu-Higa (1995). Actually, we have to prove that X_{0} ∈ ⋂_{p≥1} L^{p} (this condition ensures the assumption

of Ahn and Kohatsu-Higa 1995, Theorem 3.1) and then provide an approximation scheme for X_{0} as well.

In Section 3 we study case (b). Following Nualart and Pardoux (1991), the solution X_{t} is obtained by composition of a linearized equation related with (1.1) and a bijective non-linear transformation T of the paths of the Wiener process. The main difficulty here is to find approximations for T and it is solved reducing the problem to one of approximation of stochastic differential equations where the initial condition is the fixed point of some mapping related with T. Along the paper all constants are denoted by C although they may be different from one expression to another.

2 The Stratonovich stochastic differential equation with linear boundary condition

Consider the stochastic Stratonovich differential equation with linear boundary condition

\{ \begin{matrix}dX_{t} = σ(X_{t}) ∘ dW_{t} + b(X_{t}) dt, & 0 ≤ t ≤ 1, \\ F_{0}X_{0} + F_{1}X_{1} = h_{0}. \end{matrix}  (2.1)

Here σ and b are real functions, F_{0}, F_{1}, h_{0} ∈ IR. Set b_{1} = b + \frac{1}{2}σσ′ and suppose that b_{1}, σ

are C^{4} functions with bounded derivatives and F_{0} F_{1} > 0. Then, there exists a unique continuous solution to (2.1) and moreover, this solution belongs to the space IL_{C}^{1,loc} (see Theorem 4.1 in Donati-Martin, 1991). More precisely, consider the stochastic flow associated with (2.1),

Φ_{t}(x) = x + ∫_{0}^{t} σ(Φ_{s}(x)) ∘ dW_{s} + ∫_{0}^{t} b(Φ_{s}(x)) ds,  (2.2)

and the function G(x) = F_{0}x + F_{1}Φ_{1}(x). There exists a unique measurable X_{0} : Ω → IR such that G(ω, X_{0}(ω)) = h_{0}. Furthermore, X_{t} := Φ_{t}(X_{0}) is the unique continuous solution of equation (2.1) in the space IL_{C}^{1,loc}.

Our aim is to present a strong Euler approximation scheme for (2.1). Taking into account the results of Ahn and Kohatsu-Higa (1995), the program consists in giving numerical approximations for X_{0} and then checking the assumptions of Theorem 3.1 in Ahn and Kohatsu-Higa (1995).

We introduce some notation. Let π = {0 = t_{0} < t_{1} < ⋯ < t_{m+1} = 1} be a partition of [0, 1]; as usually, we denote by ∥π∥ its mesh, that is, ∥π∥ = sup_{0≤k≤m} |t_{k+1} − t_{k}|. We define

η : [0, 1] → IR as follows,

η(t) = ∑_{k=0}^{m} t_{k} 1_{[t_{k},t_{k+1})}(t).

The approximation of (2.2) is given by the solution to the equation,

\overline{Φ}_{t}(x) = x + ∫_{0}^{t} σ(\overline{Φ}_{η(s)}(x)) dW_{s} + ∫_{0}^{t} b_{1}(\overline{Φ}_{η(s)}(x)) ds.  (2.3)

Denote by \overline{X}_{0} a numerical approximation for X_{0} associated with π. We will prove

lim_{∥π∥→0} E \{ sup_{0≤t≤1} |Φ_{t}(X_{0}) − \overline{Φ}_{t}(\overline{X}_{0})|^{p}\} = 0, p ≥ 1

and we will obtain the rate of convergence.

Fix π with ∥π∥ = δ. Without loss of generality we can assume that π is a uniform partition. Let M > 0; Chebychev’s inequality yields

P \{ sup_{0≤k≤m} |W (Δ_{k})| > M\} ≤ \frac{1}{M^{q}} δ^{\frac{q}{2} −1}, q ≥ 2,  (2.4)

where W (Δ_{k}) denotes the increment W_{t_{k+1}} − W_{t_{k}}, k = 0, …, m. From (2.3) we obtain \overline{Φ}_{0}(x) = x,

\overline{Φ}_{t_{k+1}}(x) = \overline{Φ}_{t_{k}}(x) + σ (\overline{Φ}_{t_{k}}(x)) W (Δ_{k}) + b (\overline{Φ}_{t_{k}}(x)) δ  (2.5)

and, using a recursive argument, \overline{Φ}′_{0}(x) = 1,

\overline{Φ}′_{t_{k+1}}(x) = ∏_{i=0}^{k} (1 + σ′ (\overline{Φ}_{t_{i}}(x)) W (Δ_{k}) + b′ (\overline{Φ}_{t_{k}}(x)) δ) ,  (2.6)

k = 0, …, m.

Let C := ∥σ′∥_{∞} + ∥b′∥_{∞}. Choose δ, M > 0 satisfying δ ∨ M < \frac{1}{4C}. Then, on the set L_{M} = \{ sup_{0≤k≤m} |W (Δ_{k})| > M\}, \overline{Φ}′_{1}(x) is strictly positive and, consequently, if ω ∈ L_{M}, the function \overline{G}(x, ω) given by

\overline{G}(x, ω) = F_{0}x + F_{1}\overline{Φ}′_{1}(ω, x)

is monotone. We denote by X̃_{0}(ω) the unique solution to \overline{G}(x, ω) = h_{0}, ω ∈ L_{M}. Remark 2.1 The following procedure provides approximations for X̃_{0}(ω). Assume F_{0}F_{1} > 0. To simplify the notation we skip the dependence on ω. Fix X̃_{0}^{2,0} ≤ X̃_{0}^{1,0} such that \overline{G}(X̃_{0}^{2,0}) ≤ h_{0}, \overline{G}(X̃_{0}^{1,0}) ≥ h_{0}. We proceed inductively as follows. Let X̃_{0}^{2,i} ≤ X̃_{0}^{1,i} be such

that \overline{G}(X̃_{0}^{2,i}) ≤ h_{0}, \overline{G}(X̃_{0}^{1,i}) ≥ h_{0}, i ≥ 0. Consider \frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} . Then,

if \overline{G} (\frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} ) < h_{0} , set X̃_{0}^{2,i+1} = \frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} , X̃_{0}^{1,i+1} = X̃_{0}^{1,i},

if \overline{G} (\frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} ) > h_{0} , set X̃_{0}^{2,i+1} = X̃_{0}^{2,i} , X̃_{0}^{1,i+1} = \frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} ,

if \overline{G} (\frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} ) = h_{0} , set X̃_{0} = \frac{X̃_{0}^{2,i} + X̃_{0}^{1,i}}{2} .

Notice that |X̃_{0}^{2,i} − X̃_{0}^{1,i}| ≤ \frac{|X̃_{0}^{2,0} − X̃_{0}^{1,0}|}{2^{i−1}}, i ≥ 1.

Let \overline{X}_{0} = X̃_{0} 1_{L_{M}}. The integer M plays the role of a stability index. When the increments of the Brownian motion are too big, then a solution to \overline{G}(x, ω) = h_{0} may not exist. We can now state a fundamental ingredient in the proof of the main result of this section. Proposition 2.1 Assume that b_{1}, σ are C^{4} functions with bounded derivatives, F_{0} F_{1} > 0. Then X_{0} belongs to L^{p}(Ω) for any p ≥ 1. Proof: It suffices to check the following facts:

(a) P − lim_{δ→0} \overline{X}_{0} = X_{0} ,
(b) sup_{0<δ≤1} E |\overline{X}_{0}|^{p} < +∞.

Let us first show (a). We have

(X_{0} − X̃_{0}) 1_{L_{M}} = − \frac{F_{1} (Φ_{1}(X_{0}) − \overline{Φ}_{1}(X_{0}))}{F_{0} + F_{1}\overline{Φ}′_{1}(Z_{0})} 1_{L_{M}} ,  (2.7)

with Z_{0} a random point between X_{0}(ω) and X̃_{0}(ω), ω ∈ L_{M}. Indeed, for ω ∈ L_{M},

F_{0}X̃_{0}(ω) + F_{1}\overline{Φ}_{1}(ω, X̃_{0}(ω)) = h_{0}.  (2.8)

Consequently,

\begin{aligned}P \{|X_{0} − \overline{X_{0}}| > η\} ≤ & P \{(|X_{0} − X̃_{0}| > η) ⋂ L_{M}\} \\ & + P \{L_{M}^{c}\} , η > 0. \end{aligned} (2.9)

Let k ≥ 0 be such that P {|X_{0}| > k} ≤ δ . Then, (2.8), (2.7) and (2.4) with q = 4 yield

P \{|X_{0} − \overline{X_{0}}| > η\} ≤ P \{|Φ_{1}(X_{0}) − \overline{Φ}_{1}(X_{0})| > η′, |X_{0}| ≤ k\} + 2δ

≤ P \{ sup_{|x|≤k} |Φ_{1}(x) − \overline{Φ}_{1}(x)| > η′\} + 2δ,

(2.10)

with η′ = η |\frac{F_{0}}{F_{1}} |.

By lemmas 3.2 and 3.3 in Ahn and Kohatsu-Higa (1995), the right-hand side of (2.10) is bounded by \frac{1}{η^{′p}} C δ^{\frac{p}{2}} + 2δ and, consequently,

lim_{δ→0} P \{|X_{0} − \overline{X_{0}}| > η\} = 0, η > 0.

We now show (b). The identity (2.8) implies

X̃_{0} 1_{L_{M}} = \frac{h_{0} − F_{1} \overline{Φ}_{1}(0)}{F_{0} + F_{1}\overline{Φ}′_{1}(Z̃_{0})} 1_{L_{M}} ,  (2.11)

with Z̃_{0} a random point lying between 0 and X̃_{0}(ω), ω ∈ L_{M}. Next we check

sup_{0≤δ≤1} sup_{0≤k≤m+1} E |\overline{Φ}_{t_{k}}(x)|^{p} ≤ C (1 + |x|^{p}) , p ≥ 1. (2.12)

Indeed, the linear growth of the coefficients implies

E |\overline{Φ}_{t}(x)|^{p} ≤ C (|x|^{p} + 1 + ∫_{0}^{t} E |\overline{Φ}_{η(s)}(x)|^{p} ds) .

Consequently

sup_{0≤t≤T} E |\overline{Φ}_{t}(x)|^{p} ≤ C (|x|^{p} + 1 + ∫_{0}^{T} sup_{0≤r≤s} E |\overline{Φ}_{r}(x)|^{p} ds) T ∈ [0, 1]

and Gronwall’s lemma yields (2.12).

Then, (2.11) and (2.12) imply

sup_{0<δ≤1} E |\overline{X}_{0}|^{p} = sup_{0<δ≤1} E \{|\frac{h_{0} − F_{1} \overline{Φ}_{1}(0)}{F_{0} + F_{1}\overline{Φ}′_{1}(Z̃_{0})}|^{p} 1_{L_{M}} \}

≤ |F_{0}|^{−p} (|h_{0}|^{p} + |F_{1}|^{p} sup_{0<δ≤1} E |\overline{Φ}_{1}(0)|^{p}) ≤ C.

The proof of the Proposition is complete.

◯

We can now state the result which motivated this section.

Suppose that the assumptions of Proposition 2.1 are satisfied. Let π = {0 = t_{0} < t_{1} < ⋯ < t_{m+1} = 1} be a partition with ∥π∥ ≤ δ. Then

E \{ sup_{0≤t≤1} |Φ_{t}(X_{0}) − \overline{Φ}_{t}(\overline{X}_{0})|^{p}\} ≤ C δ^{\frac{p}{2}} ,  (2.13)

for some constant C independent of δ. Proof: Clearly,

E \{ sup_{0≤t≤1} |Φ_{t}(X_{0}) − \overline{Φ}_{t}(\overline{X}_{0})|^{p}\} ≤ C (a(δ) + b(δ)) ,  (2.14)

with

a(δ) = E \{ sup_{0≤t≤1} |Φ_{t}(X_{0}) − \overline{Φ}_{t}(X_{0})|^{p}\} ,

b(δ) = E \{ sup_{0≤t≤1} |\overline{Φ}_{t}(X_{0}) − \overline{Φ}_{t}(\overline{X}_{0})|^{p}\} .

Proposition 2.1 and the analogue of Theorem 3.1 in Ahn and Kohatsu-Higa (1995) for L^{p} norms yield

a(δ) ≤ C δ^{\frac{p}{2}} .  (2.15)

For b(δ) we have

b(δ) ≤ C c(δ) d(δ) ,

where

\begin{matrix}c(δ)^{2} = E \{ sup_{0≤t≤1} |\overline{Φ}′_{t}(η_{0})|^{2p}\} , \\ d(δ)^{2} = E |X_{0} − \overline{X}_{0}|^{2p} , \end{matrix}

with η_{0} a random point between X_{0} and \overline{X}_{0}. In Lemma 2.1 we prove that c(δ) is finite. By the definition of \overline{X}_{0},

E |X_{0} − \overline{X}_{0}|^{p} ≤ C (e(δ) + f(δ)),

with

\begin{matrix}e(δ) = E \{ |X_{0}|^{p} 1_{L_{M}^{c}} \} , \\ f(δ) = E \{ |X_{0} − X̃_{0}|^{p} 1_{L_{M}} \} . \end{matrix}

Schwarz’s inequality, Proposition 2.1 and the estimate (2.4) with q = 2p + 2 yield

e(δ) ≤ (E |X_{0}|^{2p})^{\frac{1}{2}} (P (L_{M}^{c}))^{\frac{1}{2}} ≤ C δ^{\frac{p}{2}} .  (2.16)

Finally, since on L_{M}, \overline{Φ}′_{1}(x) is positive, using (2.7) and Theorem 3.1 in Ahn and Kohatsu-Higa (1995) we obtain

f(δ) ≤ |\frac{F_{1}}{F_{0}} | E |Φ_{1}(X_{0}) − \overline{Φ}_{1}(X_{0})|^{p} ≤ C δ^{\frac{p}{2}} .  (2.17)

The estimates (2.16) and (2.17) yield b(δ) ≤ Cδ^{\frac{p}{2}} . This result, together with (2.15) and (2.14) conclude the proof.

◯

The following lemma has been used in the proof of the preceding theorem.

Lemma 2.1 For any random variable X,

sup_{0<δ≤1} E \{ sup_{0≤t≤1} |\overline{Φ}′_{t}(X)|^{p}\} < ∞, p ≥ 2.

Proof: Taking derivatives in (2.3) we obtain

E \{ sup_{0≤t≤1} |\overline{Φ}′_{t}(X)|^{p}\} = E \{ sup_{0≤k≤m} sup_{t∈[t_{k},t_{k+1})} |\overline{Φ}′_{t}(X)|^{p}\}

≤ C (E \{ sup_{0≤k≤m} |\overline{Φ}′_{t_{k}}(X)|^{p}\}

+ E \{ sup_{0≤k≤m} sup_{t∈[t_{k},t_{k+1})} |\overline{Φ}′_{t_{k}}(X)|^{p} (|W_{t} − W_{t_{k}}|^{p} + δ^{p})\}) .

The identity (2.6) yields

\begin{aligned}E \{ sup_{0≤k≤m} |\overline{Φ}′_{t_{k}}(X)|^{p}\} & ≤ C E \{ ∏_{i=0}^{m} (1 + |W (Δ_{i})|^{p} + δ^{p} )\} \\ & ≤ C (1 + δ^{\frac{p}{2}} )^{\frac{1}{δ}} . \end{aligned}

Notice that the right-hand side of the preceding inequality is bounded uniformly in δ ∈ (0, 1].

In addition, Doob’s maximal inequality implies

E \{ sup_{0≤k≤m} sup_{t∈[t_{k},t_{k+1})} |W_{t} − W_{t_{k}}|^{p}\} ≤ C δ^{\frac{p}{2} −1}.

This ends the proof of the lemma.

◯

3 Multidimensional stochastic differential equations with general boundary conditions.

In this section we consider the stochastic differential equation

dX_{t} + f(X_{t}) dt = B dW_{t}, t ∈ [0, 1],  (3.1)

with boundary condition

h(X_{0}, X_{1}) = \overline{h}.  (3.2)

We assume that {X_{t} , t ∈ [0, 1]} is IR^{d}- valued, {W_{t} , t ∈ [0, 1]} is a IR^{k}- valued Brownian motion, f : IR^{d} → IR^{d} takes the form

f(x) = Ax + Bf̃(x),

where A is a d × d matrix, f̃: IR^{d} → IR^{k} is measurable and locally bounded and B is a d × k matrix. Finally h : IR^{2d} → IR^{d} and \overline{h} ∈ IR^{d}. Without loss of generality we may assume k ≤ d. In Nualart and Pardoux (1991) a theorem on existence and uniqueness of solution for this kind of equations has been established. More explicitly, let C_{0} ([0, 1]; IR^{k}) be the set of contin-

uous, IR^{d}- valued functions vanishing at 0; set F = {∫_{0}^{1} e^{At}B dφ(t) ; φ ∈ C ([0, 1]; IR^{k})},

where the integrals are defined using integration by parts. Assume

(H.1) For any z ∈ F the equation h (y, e^{−A}(y + z)) = \overline{h} has a unique solution y = g(z).

In order to find the solution of (3.1), (3.2) we consider the linear equation

dY_{t} + AY_{t} dt = B dW_{t}, t ∈ [0, 1],

with boundary condition (3.2). This equation has a unique solution given by

Y_{t} = e^{−At} [g (∫_{0}^{1} e^{As}B dW_{s}) + ∫_{0}^{t} e^{As}B dW_{s}] .  (3.3)

Let Σ = \{ξ ∈ C ([0, 1]; IR^{k}) , ξ_{t} − ξ_{0} + ∫_{0}^{t} Aξ_{s} ds ∈ ImB, 0 ≤ t ≤ 1, h(ξ_{0}, ξ_{1}) = \overline{h}\}. There

exists a bijection ψ : C_{0} ([0, 1]; IR^{k}) → Σ such that Y_{t} = (ψ(W))_{t}. Finally we define the mapping T : C_{0} ([0, 1]; IR^{k}) → C_{0} ([0, 1]; IR^{k}) by

T (θ) = θ + ∫_{0}^{·} f̃((ψ(θ))_{s}) ds.  (3.4)

(Nualart and Pardoux, 1991) Assume T is a bijection and (H.1). Then equation (3.1) with boundary condition (3.2) possess a unique solution given by

X = ψ (T^{−1}(W)) .  (3.5)

As in the preceding section our purpose is to provide numerical approximations for the process X defined by (3.5). The difficulty here comes from the fact that we do not have an explicit expression for the paths T^{−1}(W). Indeed, set θ = T^{−1}(W) then (3.4) and (3.3) yield

θ_{t} = W_{t} − ∫_{0}^{t} f̃(e^{−As} [g (∫_{0}^{1} e^{Au}B dθ_{u}) + ∫_{0}^{s} e^{Au}B dθ_{u}]) ds.  (3.6)

This equation shows that θ depends on its past as well as its future. Therefore the approximation of θ does not seem easy to obtain from (3.6). Instead, we propose the following

approach: Set φ_{t} = ∫_{0}^{t} e^{Au}B dθ_{u}. Then,

\begin{aligned}X_{t} & = e^{−At} [g (∫_{0}^{1} e^{Au}B dθ_{u}) + ∫_{0}^{t} e^{Au}B dθ_{u}] \\ & = e^{−At} [g (φ_{1}) + φ_{t}] . \end{aligned}  (3.7)

From (3.6) we easily obtain φ_{t} = ξ_{t} + u_{t} with

ξ_{t} = ∫_{0}^{t} e^{As}B dW_{s},  (3.8)

u_{t} = − ∫_{0}^{t} e^{As}Bf̃(e^{−As} [g (ξ_{1} + u_{1}) + (ξ_{s} + u_{s})]) ds.  (3.9)

Fix y ∈ IR and assume

u_{t}(y) = − ∫_{0}^{t} e^{As}Bf̃(e^{−As} [g (ξ_{1} + y) + ξ_{s} + u_{s}(y)]) ds  (3.10)

has a unique solution. Moreover, suppose that the mapping y ↦ u_{1}(y) has a unique fixed point y^{∗}. Then, clearly

φ_{t} = ξ_{t} + u_{t}(y^{∗}).  (3.11)

Hence, our approximations for X_{t} will be constructed from approximations of ξ_{t} and approximations of the flow u_{t}(y^{∗}). We first establish a result ensuring that T is bijective. This question is also related with the existence of a unique fixed point for y ↦ u_{1}(y). Given a d × k matrix M we denote by |M| its norm, that is, |M| = sup_{|x|=1} |Mx|.

Proposition 3.1 Assume f̃ and g are Lipschitz functions with Lipschitz constant Lf̃ and L_{g}, respectively. If

K (Lf̃, L_{g}) = L_{g} [exp [(e^{2|A|} − 1) \frac{|B|}{2|A|} Lf̃] − 1] < 1,  (3.12)

the mapping T defined by (3.4) is bijective.

Proof: The assumptions on f̃, g imply the existence of a unique continuous solution {u_{t}(y), t ∈ [0, 1]} to (3.10). Then, following Nualart and Pardoux (1991, Proposition 2.5) the proof reduces to show that y ↦ u_{1}(y) has a unique fixed point. Let y_{1}, y_{2} ∈ IR^{d}, t ∈ [0, 1]. The Lipschitz property of f̃ and g yield

\begin{aligned}|u_{t}(y_{1}) − u_{t}(y_{2})| & ≤ |B| Lf̃ L_{g} \frac{e^{2|A|t} − 1}{2|A|} |y_{1} − y_{2}| \\ & + |B| Lf̃ ∫_{0}^{t} e^{2|A|s} |u_{s}(y_{1}) − u_{s}(y_{2})| ds. \end{aligned}  (3.13)

Lemma 3.1 (Gronwall’s lemma), with y_{t} = |u_{t}(y_{1}) − u_{t}(y_{2})|, f(t) = |B| Lf̃ L_{g}

×\frac{e^{2|A|} − 1}{2|A|} |y_{1} − y_{2}| and g(t) = |B| Lf̃ e^{2|A|t}, implies

|u_{t}(y_{1}) − u_{t}(y_{2})| ≤ L_{g} [exp [(e^{2|A|t} − 1) \frac{|B|}{2|A|} Lf̃] − 1] |y_{1} − y_{2}|.  (3.14)

The assumption (3.12) together with (3.14) for t = 1 imply that the mapping y ↦ u_{1}(y) is a strict contraction. This completes the proof.

◯

Remark 3.1 If d = 1, condition (3.12) can be replaced by

L_{g} [exp (|B| Lf̃) − 1] < 1,  (3.15)

which is weaker. Indeed, in this case

|u_{t}(y_{1}) − u_{t}(y_{2})| ≤ |B| Lf̃ L_{g} t |y_{1} − y_{2}| + |B| Lf̃ ∫_{0}^{t} |u_{s}(y_{1}) − u_{s}(y_{2})| ds.

Hence, by Gronwall’s lemma

|u_{t}(y_{1}) − u_{t}(y_{2})| ≤ L_{g} [exp (|B| Lf̃) − 1] |y_{1} − y_{2}| .

Remark 3.2 If f is linear then f̃ = 0 and therefore (3.12) is automatically satisfied.

Fix a partition π = {0 = t_{0} < t_{1} < ⋯ < t_{m+1} = 1}. In the sequel we will use the same notation as in the preceding section and assume that the hypotheses of Proposition 3.1 are satisfied. Associated with π we define \overline{ξ}_{t}, \overline{u}_{t}(y) by

\begin{matrix}\overline{ξ}_{t} = ∫_{0}^{t} e^{Aη(s)} B dW_{s}, \\ \overline{u}_{t}(y) = − ∫_{0}^{t} e^{Aη(s)}Bf̃(e^{−Aη(s)} [g (\overline{ξ}_{1} + y) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(y)]) ds. \end{matrix}

Using similar arguments as in the proof of Proposition 3.1 we prove that y ↦ \overline{u}_{1}(y) has a unique fixed point, say \overline{y}^{∗}. Indeed, the analogue of (3.13) is

\begin{aligned}|\overline{u}_{t}(y_{1}) − \overline{u}_{t}(y_{2})| & ≤ |B| Lf̃ L_{g} |y_{1} − y_{2}| ∫_{0}^{t} e^{2|A|η(s)} ds \\ & + |B| Lf̃ ∫_{0}^{t} e^{2|A|η(s)} |\overline{u}_{η(s)}(y_{1}) − \overline{u}_{η(s)}(y_{2})| ds \end{aligned}

and consequently, since e^{2|A|η(s)} ≤ e^{2|A|s},

\begin{matrix}sup_{0≤s≤t} |\overline{u}_{s}(y_{1}) − \overline{u}_{s}(y_{2})| ≤ |B| Lf̃ L_{g} |y_{1} − y_{2}| \frac{e^{2|A|t} − 1}{2|A|} \\ + |B| Lf̃ ∫_{0}^{t} e^{2|A|s} sup_{0≤r≤s} |\overline{u}_{r}(y_{1}) − \overline{u}_{r}(y_{2})| ds. \end{matrix}

Then, Gronwall’s lemma yields

sup_{0≤s≤t} |\overline{u}_{s}(y_{1}) − \overline{u}_{s}(y_{2})| ≤ L_{g} [exp [(e^{2|A|t} − 1) \frac{|B|}{2|A|}Lf̃] − 1] |y_{1} − y_{2}|.  (3.16)

Set \overline{φ}_{t} = \overline{ξ}_{t} + \overline{u}_{t} (\overline{y}^{∗}). We can now state the main result of this section.

Assume that the assumptions of Proposition 3.1 are satisfied.. Fix a partition π of [0, 1] and let

\overline{X}_{t} = e^{−At} [g (\overline{φ}_{1}) + \overline{φ}_{t}] .

Then

E \{ sup_{0≤t≤1} |\overline{X}_{t} − X_{t}|^{p} \} ≤ C δ^{p}, p ≥ 2,  (3.17)

where δ := ∥π∥ and C is a constant independent of δ. Proof: Taking into account (3.7), the proof of (3.17) reduces to check

E \{ sup_{0≤t≤1} |ξ_{t} − \overline{ξ}_{t}|^{p} \} ≤ C δ^{p},  (3.18)

E \{ sup_{0≤t≤1} |u_{t} (y^{∗}) − \overline{u}_{t} (\overline{y}^{∗})|^{p} \} ≤ C δ^{p},  (3.19)

p ≥ 2. Inequality (3.18) follows easily from Doob’s maximal inequality and Burkholder’s inequality.

In order to check (3.19) we decompose |u_{t} (y^{∗}) − \overline{u}_{t} (\overline{y}^{∗})| as follows,

|u_{t} (y^{∗}) − \overline{u}_{t} (\overline{y}^{∗})| ≤ ∑_{i=1}^{3} |u_{t}^{i}| ,

with

\begin{aligned}u_{t}^{1} = & ∫_{0}^{t} e^{As}B (f̃(e^{−As} [g (ξ_{1} + y^{∗}) + ξ_{s} + u_{s}(y^{∗})]) \\ & − f̃(e^{−As} [g (ξ_{1} + y^{∗}) + ξ_{s} + u_{η(s)}(y^{∗})])) ds, \end{aligned}

\begin{aligned}u_{t}^{2} = & ∫_{0}^{t} e^{As}B (f̃(e^{−As} [g (ξ_{1} + y^{∗}) + ξ_{s} + u_{η(s)}(y^{∗})]) \\ & − f̃(e^{−As} [g (\overline{ξ}_{1} + \overline{y}^{∗}) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(\overline{y}^{∗})])) ds, \end{aligned}

\begin{aligned}u_{t}^{3} = & ∫_{0}^{t} \{e^{As}Bf̃(e^{−As} [g (\overline{ξ}_{1} + \overline{y}^{∗}) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(\overline{y}^{∗})]) \\ & − e^{Aη(s)}Bf̃(e^{−Aη(s)} [g (\overline{ξ}_{1} + \overline{y}^{∗}) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(\overline{y}^{∗})])\} ds. \end{aligned}

First we prove

E { sup_{0≤t≤1} |u_{t}^{1}|^{p} } ≤ C δ^{p}, p ≥ 2.  (3.20)

Indeed,

sup_{0≤t≤1} |u_{t}^{1}| ≤ C ∫_{0}^{1} |u_{s}(y^{∗}) − u_{η(s)}(y^{∗})| ds,

for some positive constant C independent of π. Using (3.26) and (3.28) we can show

sup_{0≤t≤1} E |u_{t}(y^{∗}) − u_{η(t)}(y^{∗})|^{p} ≤ C δ^{p}.

Consequently, (3.20) holds. For u_{t}^{2} we have

\begin{aligned}|u_{t}^{2}| & ≤ ∫_{0}^{t} e^{2|A|s}|B|Lf̃(L_{g} (|ξ_{1} − \overline{ξ}_{1}| + |y^{∗} − \overline{y}^{∗}|) \\ & + |ξ_{s} − \overline{ξ}_{η(s)}| + |u_{η(s)}(y^{∗}) − \overline{u}_{η(s)}(\overline{y}^{∗})|) ds. \end{aligned}

Therefore

E \{ sup_{0≤s≤t} |u_{s}^{2}|^{p}\} ≤ |B|Lf̃L_{g} \frac{e^{2|A|t} − 1}{2|A|} (E |ξ_{1} − \overline{ξ}_{1}|^{p} + |y^{∗} − \overline{y}^{∗}|^{p})

+ ∫_{0}^{t} e^{2|A|s}|B|Lf̃ E |ξ_{s} − \overline{ξ}_{η(s)}|^{p} ds

+ ∫_{0}^{t} e^{2|A|s}|B|Lf̃ E \{ sup_{0≤r≤s} |u_{r}(y^{∗}) − \overline{u}_{r}(\overline{y}^{∗})|\} ds.

Clearly,

E |ξ_{1} − \overline{ξ}_{1}|^{p} + sup_{0≤s≤1} E |ξ_{s} − \overline{ξ}_{η(s)}|^{p} ≤ C δ^{p},

with C independent of δ. Consequently,

E \{ sup_{0≤s≤t} |u_{s}^{2}|^{p}\} ≤ C_{1} δ^{p} + |B|Lf̃L_{g} \frac{e^{2|A|t} − 1}{2|A|} |y^{∗} − \overline{y}^{∗}|^{p}

+ ∫_{0}^{t} e^{2|A|s}|B|Lf̃ E \{ sup_{0≤r≤s} |u_{r}(y^{∗}) − \overline{u}_{r}(\overline{y}^{∗})|^{p}\} ds.

(3.21)

Finally, we deal with u_{t}^{3} and we obtain

\begin{aligned}|u_{t}^{3}| & ≤ ∫_{0}^{t} |e^{As} − e^{Aη(s)}| |B| |f̃(e^{−As} [g (\overline{ξ}_{1} + \overline{y}^{∗}) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(\overline{y}^{∗})])| ds \\ & + ∫_{0}^{t} e^{|A|s}|B|Lf̃ |e^{−As} − e^{−Aη(s)}| |g (\overline{ξ}_{1} + \overline{y}^{∗}) + \overline{ξ}_{η(s)} + \overline{u}_{η(s)}(\overline{y}^{∗})| ds. \end{aligned}

Clearly, E |\overline{ξ}_{1}|^{p} ≤ C_{1}, sup_{0≤s≤1} E |\overline{ξ}_{η(s)}|^{p} ≤ C_{2} for some positive constants C_{1}, C_{2} not

depending on δ. Hence, the Lipschitz property of the mapping t ↦ e^{At}, t ↦ e^{−At} and the results stated in Remark 3.3 yield

E { sup_{0≤t≤1} |u_{t}^{3}|^{p} } ≤ C δ^{p}, p ≥ 2,  (3.22)

for some constant C independent of δ. Thus the estimates (3.20) to (3.22) imply

E \{ sup_{0≤s≤t} |u_{s}(y^{∗}) − \overline{u}_{s}(\overline{y}^{∗})|^{p}\} ≤ Cδ^{p} + |B|Lf̃L_{g} \frac{e^{2|A|t} − 1}{2|A|} |y^{∗} − \overline{y}^{∗}|^{p}

+ ∫_{0}^{t} e^{2|A|s}|B|Lf̃ E \{ sup_{0≤r≤s} |u_{r}(y^{∗}) − \overline{u}_{r}(\overline{y}^{∗})|^{p}\} ds.

Lemma 3.1 with

y(t) = E \{ sup_{0≤s≤t} |u_{s}(y^{∗}) − \overline{u}_{s}(\overline{y}^{∗})|^{p}\} ,

f(t) = Cδ^{p} + |B|Lf̃L_{g} \frac{e^{2|A|t} − 1}{2|A|} |y^{∗} − \overline{y}^{∗}|^{p} ,

g(t) = e^{2|A|t}|B|Lf̃, yields

\begin{matrix}E \{ sup_{0≤s≤t} |u_{s}(y^{∗}) − \overline{u}_{s}(\overline{y}^{∗})|^{p}\} ≤ Cδ^{p} \\ + L_{g} [exp [(e^{2|A|t} − 1) \frac{|B|}{2|A|} Lf̃] − 1] |y^{∗} − \overline{y}^{∗}|^{p} . \end{matrix}  (3.23)

In particular, for t = 1

|y^{∗} − \overline{y}^{∗}|^{p} ≤ C (1 − K (Lf̃, L_{g}))^{−1} δ^{p},  (3.24)

because u_{1}(y^{∗}) = y^{∗} and \overline{u}_{1}(\overline{y}^{∗}) = \overline{y}^{∗}. The estimate (3.23) and (3.24) yield (3.19) and finish the proof of the theorem.

◯

Let \overline{y}^{∗,m} be an approximation of the fixed point \overline{y}^{∗} obtained at step m ≥ 1 through the bisection method. From (3.30) we get

|\overline{y}^{∗} − \overline{y}^{∗,m}| ≤ 2^{−(dm−1)} (1 − K (Lf̃, L_{g}))^{−1} \overline{τ}.  (3.25)

Set

\overline{X}_{t}^{m} = e^{−At} [g (\overline{φ}_{1}^{m}) + \overline{φ}_{t}^{m}] ,

with \overline{φ}_{t}^{m} = \overline{ξ}_{t} + \overline{u}_{t} (\overline{y}^{∗,m}). Then

E \{ sup_{0≤t≤1} |X_{t} − \overline{X}_{t}^{m}|^{p} \} ≤ C (δ^{p} + 2^{−p(dm−1)}) .

Indeed, (3.16) and (3.22) ensure

\begin{matrix}E \{ sup_{0≤t≤1} |\overline{u}_{t}(\overline{y}^{∗}) − \overline{u}_{t}(\overline{y}^{∗,m})|^{p}\} ≤ [K (Lf̃, L_{g})]^{p} (1 − K (Lf̃, L_{g}))^{−1} \\ ×E |\overline{τ}|^{p} 2^{−p(dm−1)}. \end{matrix}

We end this section with some technical tools that have been used along the proofs.

Lemma 3.1 Let f : [0, 1] → IR, g : [0, 1] → IR_{+} be continuous mappings. Consider a continuous function y : [0, 1] → IR satisfying

y(t) ≤ f(t) + ∫_{0}^{t} g(s)y(s) ds, t ∈ [0, 1].

Then,

y(t) ≤ f(t) + ∫_{0}^{t} f(s)g(s) exp (∫_{s}^{t} g(u) du) ds, t ∈ [0, 1].

Lemma 3.2 Under the assumptions of Proposition 3.1, the unique fixed point y^{∗} of the mapping y ↦ u_{1}(y) , with {u_{t}(y), t ∈ [0, 1]} defined by (3.10), satisfies the following property:

There exists a random variable τ possessing moments of any order such that

|y^{∗}| ≤ (1 − K (Lf̃, L_{g}))^{−1} τ,  (3.26)

where K (Lf̃, L_{g}) is the constant given in (3.12). Proof: The Lipschitz property of the coefficients f̃, g and equation (3.10) yield

|u_{t}(y)| ≤ h_{1}(t) + h_{2}(t) + h_{3}(t) + ∫_{0}^{t} Lf̃|B|e^{2|A|s} |u_{s}(y)| ds,

with

h_{1}(t) = |B| |f(0)| \frac{e^{|A|t} − 1}{|A|},

h_{2}(t) = ∫_{0}^{t} Lf̃ |B| e^{2|A|s} [|g(0)| + L_{g}|ξ_{1}| + |ξ_{s}|] ds,

h_{3}(t) = Lf̃ |B| L_{g} \frac{e^{2|A|t} − 1}{2|A|}|y|.

Thus, Lemma 3.1 with f(t) = ∑_{i=1}^{3} h_{i}(t), g(t) = Lf̃|B|e^{2|A|t}, y(t) = |u_{t}(y)| ensures

\begin{aligned}|u_{t}(y)| & ≤ ∑_{i=1}^{3} h_{i}(t) + ∫_{0}^{t} (h_{1}(s) + h_{2}(s)) Lf̃|B|e^{2|A|s} exp (Lf̃|B|\frac{e^{2|A|t} − e^{2|A|s}}{2|A|} ) ds \\ & + |y| ∫_{0}^{t} L2 ; f̃L_{g}^{2}|B|\frac{e^{2|A|s} − 1}{2|A|}e^{2|A|s} exp (Lf̃|B|\frac{e^{2|A|t} − e^{2|A|s}}{2|A|} ) ds. \end{aligned}

Let

\begin{aligned}τ(t) & = h_{1}(t) + h_{2}(t) \\ & + ∫_{0}^{t} (h_{1}(s) + h_{2}(s)) Lf̃|B|e^{2|A|s} exp (Lf̃|B|\frac{e^{2|A|t} − e^{2|A|s}}{2|A|} ) ds. \end{aligned}  (3.27)

Notice that τ(t) is a random variable in ⋂_{p≥1} L^{p}(Ω), t ∈ [0, 1]. Then,

|u_{t}(y)| ≤ τ(t) + K (Lf̃, L_{g}) |y|.  (3.28)

The estimate (3.26) follows from (3.28) taking t = 1 and setting τ = τ(1).

◯

Remark 3.3 With parallel arguments the following results can also be proved

There exists a random variable \overline{τ}(t) in ⋂_{p≥1} L^{p}(Ω), t ∈ [0, 1], such that

|\overline{u}_{t}(y)| ≤ \overline{τ}(t) + K (Lf̃, L_{g}) |y|.  (3.29)
The unique fixed point \overline{y}^{∗} for the mapping y ↦ \overline{u}_{1}(y) satisfies
|\overline{y}^{∗}| ≤ (1 − K (Lf̃, L_{g}))^{−1} \overline{τ}.  (3.30)
where \overline{τ} = \overline{τ}(1).

Remark 3.4 For d = 1, equation (3.1) is a particular case of the Stratonovich equation in (2.1). The boundary condition (3.2) satisfying (H.1) includes the boundary condition F_{0}X_{0} + F_{1}X_{1} = h_{0}, F_{0}F_{1} > 0. However the restriction (3.12) between the Lipschitz constant of f̃ and the boundary condition is not necessary with the approach of Section 2.

References

[1] H. Ahn, A. Kohatsu-Higa ,The Euler scheme for anticipating stochastic differential equations, to appear in Stochastics and Stochastics Reports, (1995).

[2] C. Donati-Martin, Equations différentielles stochastiques dans IR avec conditions aux bords, Stochastics and Stochastics Reports 35, (1991), 143-173.

[3] D. Nualart, E. Pardoux , Boundary value problems for stochastic differential equations, The Annals of Prob., 19, no.3, (1991), 1118-1144.