A FINITE ELEMENT METHOD FOR MARTINGALE-DRIVEN

STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS

ANDREA BARTH

Abstract. The main objective of this work is to describe a Galerkin ap-proximation for stochastic partial differential equations driven by square–integrable martingales. Error estimates in the semidiscrete case, where dis-cretization is only done in space, and in the fully discrete case are derived.Parabolic as well as transport equations are studied.

1. Introduction

Arguably the modern literature on Finite Elements Methods (FEM’s from thispoint on) can be traced back to the 1956 paper of Turner, Clough, Martin andTopp [26]. Several contemporary publications followed [1] [17], mostly under anEngineering scope. This should come as no surprise given the aerospace back-ground of the authors. It was not until the early 1970’s that research on FEM’sapplied to the estimation of solutions to partial differential equations picked upsteam. Here one finds, among others, the works of Fujita and Mizutani [11], Ushi-jima [27] and Zlamal [30] [31] on parabolic PDE’s, together with those of Ciarlet[8] and Nedoma [19] for elliptic problems. From this point on the stream of relatedpublications swelled in breadth and depth, from various forms of approximations(Galerkin, Riez–Galerkin, Lagrange–Galerkin, etc.) to different methodologies(energy methods, dynamic FEM’s, etc.) and a more focused treatment of specificproblems (notably Navier-Stokes equations). Although beyond the scope of thispaper, it should be mentioned that the theory of convergence of finite-elementsapproximations developed in parallel fashion.

Unsurprisingly, the (numerical) study of stochastic partial differential equations(SPDE’s from this point on) took more time to develop. It was not until the mid-to-late 1990’s that work such as that of Gyongy and Nualart [13], Yoo [29], Crisan,Gaines and Lyons [9], and Gaines [12] started breaking trail in this direction.However, none of these papers present a FEM approach. Albeit more complex toimplement, FEM’s do present the advantage of greater degrees of freedom whenchoosing a discretization of the space-time continuum. This can be particularlyuseful to design finite elements tailor made to the characteristics of a specificproblem. To our knowledge, the 2003 paper by Yan [28] was the first to employFEM’s to estimate solutions to SPDE’s, where error estimates for linear equations

Received 2009-10-12; Communicated by Hui-Hsiung Kuo.2000 Mathematics Subject Classification. 60H15; 60H35.Key words and phrases. Finite element method, stochastic partial differential equation, Levy

process, martingal.

355

Serials Publications www.serialspublications.com

Communications on Stochastic Analysis Vol. 4, No. 3 (2010) 355-375

356 ANDREA BARTH

with a Brownian motion as driving noise were studied. Similar estimates for someparabolic equations and Zakai’s equation were presented by Chow and Jiang in [6]and Chow, Jiang and Menaldi in [7] respectively. The reader is directed to [5] [10][21] [22] for a thorough treatment of Hilbert-space valued stochastic equations.

In this paper we use Galerkin approximations to estimate the solution of someSPDE’s driven by square–integrable martingales. Our main aim is to provide errorestimates in several cases. In a first instance we look into a parabolic equation.Namely, we analyze approximations of the solution to

dX(t) = AX(t)dt+G(X(t))dM(t), X(t0) = X0, (1.1)

where M is a square–integrable martingale defined on some probability space(Ω,F , (Ft)t≥0,P) that takes values in a Hilbert space U ; A denotes a second orderdifferential operator acting on a Hilbert space H , which is not necessarily equalto U . For t ∈ [0, T ], X(t) is a H-valued random process. Infinite dimensionalequations of this kind appear in financial Mathematics and Physics among otherdisciplines.

On the other hand we study some transport equations that appear, for instance,in the modeling of bond markets: The price of a zero–coupon bond at time t isgiven by P (t, θ), 0 ≤ t ≤ θ. This is an instrument that delivers 1 dollar at time θ.P (t, θ) is given by the equation

P (t, θ) = exp(−

∫ θ

t

f(t, s)ds),

and f(t, s), 0 ≤ t ≤ s, is the so-called forward rate. The latter depends on twoparameters: time and time to maturity. For a general introduction to interestrate models in infinite dimensions we refer the reader to [4]. The widely usedHeath-Jarrow-Morton approach to interest rate modeling assumes the forwardrate satisfies the equation

df(t, θ) = µ(t, θ)dt+ (σ(t, θ), dZ(t))U , θ ≥ t.

Here Z denotes a Levy process defined on a probability space (Ω,F , (F)t,P) thattakes values in a Hilbert space (U, (·, ·)U ), and µ(t, θ) and σ(t, θ) are predictableprocesses for each θ ≥ 0. A common practice in interest rate theory is to parame-terize forward rates by time to maturity. Arguably this approach was introducedby Musiela in [18]. This paper provides a link between stochastic partial differen-tial equations and interest rate theory. To embed our setting into such frameworkwe require the following definitions: For t, ξ ≥ 0 and u ∈ U let

r(t)(ξ) := f(t, t+ ξ), a(t)(ξ) := µ(t, t+ ξ) and (g(t)u)(ξ) := (σ(t, t+ ξ), u)U .

We may then write

r(t)(ξ) = r(0)(t+ ξ) +

∫ t

0

a(s)(t − s+ ξ)ds+

∫ t

0

g(s)(t− s+ ξ)dZ(s)

= S(t)r(0)(ξ) +

∫ t

0

S(t− s)a(s)(ξ)ds +

∫ t

0

S(t− s)g(s)(ξ)dZ(s),

FEM FOR MARTINGALE-DRIVEN SPDE’S 357

where S denotes the shift semigroup S(t)φ(ξ) = φ(ξ + t). The equation above isthe mild solution to the equation

dr(t) = (∂

∂ξr(t) + a(t))dt+ g(t)dZ(t), (1.2)

which is the well known Musiela parametrization. As is the case for models whoserandomness stems from finite dimensional Wiener processes, a has to fulfill aHJM–drift condition (see for details [21] [16] [3]). The latter is required to have anarbitrage–free model. If the volatility g depends on r, we obtain the HJM–Musielaequation

dr(t)(ξ) = (∂

∂ξr(t)(ξ) + F (t, r(t))(ξ))dt +G(t, r(t))(ξ)dZ(t).

The existence of solutions to Equation (1.2) is studied in [21]. The same type ofequation can be derived for energy forwards (see [2]), where the non–hedgeableunderlying is the price of electricity instead of the interest rate.

These transport equations can also be stated in the terminology of differentialoperators as

dX(t) = BX(t)dt+G(X(t))dM(t), X(t0) = X0. (1.3)

The essential difference between Equation (1.1) and the one above is that in theformer case the differential operator A generates an analytic semigroup, whereasin the latter B generates a C0-semigroup of contractions.

Given the rapid growth of the fixed income and electricity markets (among otherpossible applications), the simulation of SPDE’s with Levy–Noise is an interestingand current problem. We derive estimates for the mean–square error of a Galerkinapproximation for parabolic and hyperbolic equations.

The remainder of this paper is structured as follows: Section 2 contains a de-tailed analysis of Equation (1.1) and Equation (1.3). In Section 3 we introduce thediscretization schemes. Then we compute error estimates for the semidiscrete, i.e.discrete in space, and the discrete equations, i.e. discrete in time and space, forthe parabolic equation. We borrow some results for the deterministic equationsfrom [25]. Finally we derive error estimates for first order hyperbolic stochasticdifferential equations (stochastic transport equation), using results from [15].

2. Preliminaries

Throughout this paper (U, (·, ·)U ) will be a Hilbert space and (Ω,F , (Ft),P)will denote a filtered probability space satisfying the “usual conditions”. Thespace of all cadlag square integrable (Ft)–martingales taking values in U will berepresented by M2(U). The spatial covariance structure ofM ∈ M2(U) is impliedby a covariance operator Q ∈ L+

1 (U), the space of all symmetric, non-negative andnuclear operators. We restrict ourselves to the following class of square integrablemartingales

C := M ∈ M2(U) :∃Q ∈ L+1 such that

∀t ≥ s ≥ 0, 〈〈M,M〉〉t − 〈〈M,M〉〉s ≤ (t− s)Q.

358 ANDREA BARTH

It is a known result (see [24]) that if Q ∈ L+1 (U) there exists an orthonormal

basis en of U consisting of eigenvectors of Q. This leads to the representationQen = γnen, where γn is the eigenvalue corresponding to en. The square root ofQ is defined as

Q1

2x :=∑

n

(x, en)Uγ1

2

n en x ∈ U,

and Q− 1

2 will be the pseudo inverse to Q1

2 .Let (H, (·, ·)H) be the Hilbert space defined by H = Q

1

2 (U) endowed with the

inner product (x, y)H = (Q− 1

2 x,Q− 1

2 y)U for x, y ∈ H. We write LHS(H, H) torefer to the space of all Hilbert-Schmidt operators from H to some Hilbert spaceH . We use the following proposition repeatedly, its proof can be found in [21].Let P[0,T ] denote the σ–field of predictable sets in Ω × [0, T ].

Proposition 2.1. Let L2H,T (H) = L2(Ω × [0, T ],P[0,T ],Pdt;LHS(H, H)) be the

space of integrands. Then for every X ∈ L2H,T (H)

E‖

∫ t

0

X(s) dM(s)‖2H ≤ E

∫ t

0

‖X(s)‖2LHS(H,H) ds. (2.1)

As we mentioned previously, the two equations that concern us are

dX(t) = AX(t)dt+G(X(t))dM(t), X(0) = X0, (2.2)

where A generates an analytic semigroup (parabolic case) S on the Hilbert spaceH = L2(D) and D ⊂ R

d has a smooth boundary ∂D and

dX(t) = BX(t)dt+G(X(t))dM(t), X(0) = X0 (2.3)

where B generates a C0–semigroup (hyperbolic case) S on H . We denote by D(A)and D(B) the domains of the differential operators, and by D(G) the domain of G,which takes values in L(H, H). Hα denotes the Sobolev space of order α endowedwith the corresponding norm ‖ · ‖Hα for α > 0.

Let M ∈ M2(U), then Equations (2.2) and (2.3) are well defined if the followingconditions are satisfied:

(1) X0 is an F0-measurable random variable with values in H .(2) D(G) is dense in H .

(3) There exists a function b : R+ → R+ with∫ T

0 b2(t) dt <∞ for all T <∞,such that:(a) ‖S(t)G(x)‖LHS(H,H) ≤ b(t)(1 + ‖x‖H) for all t > 0 and x ∈ D(G).(b) ‖S(t)(G(x)−G(y))‖LHS(H,H) ≤ b(t)‖x− y‖H for all t > 0 and x, y ∈

D(G).(c) ‖AαG(x)‖LHS(H,H) ≤ C(‖x‖H2α) for α ∈ 0, 1/2, 1 and x ∈ H2α.(d) ‖BαG(x)‖LHS(H,H) ≤ C(‖x‖Hα) for α ∈ 0, 1 and x ∈ Hα.

Note that the properties above also hold if we take S(t) = I.

Definition 2.2. Let X0 be a F0-measurable, square integrable random variablewith values in H . A predictable process X : R+ ×Ω → H is called a mild solution

to Equation (2.2) or (2.3) if

supt∈[0,T ]

E|X(t)|2H <∞ ∀T ∈ (0,∞)

FEM FOR MARTINGALE-DRIVEN SPDE’S 359

and

X(t) = S(t)X0 +

∫ t

0

S(t− s)G(X(s))dM(s) ∀t > 0. (2.4)

The conditions above imply that the integral in Equation (2.4) is well defined.If E‖X0‖

2H2 < ∞, then the solution X(t), for t ∈ [0, T ], will be element of H2 as

well.

Definition 2.3. Let X0 be a F0-measurable, square integrable random variablewith values in H . A predictable process X : R+ ×Ω → H is called a weak solution

to (2.3) if

supt∈[0,T ]

E|X(t)|2H <∞ ∀T ∈ (0,∞)

and for all a ∈ D(B∗)

(a,X(t))H = (a,X0)H +

∫ t

0

(B∗a,X(s))Hds+

∫ t

0

(G∗(X(s))a, dM(s))H ∀t > 0.

(2.5)

Here D(B∗) denotes the domain of the adjoint operator B∗. It can be shownthat X is a mild solution if and only if it is a weak solution if the followingconditions are fulfilled: For a ∈ D(B∗) exists c(a) <∞ such that

‖G∗(x)a‖H ≤ c(a)(1 + ‖x‖H)

and

‖(G∗(x) −G∗(y))a‖H ≤ c(a)‖x− y‖H

for all x, y ∈ D(G).However, for the weak solution regularity can not be achieved so easily as for

the mild solution. For the time being we assume supt∈[0,T ] E‖X(t)‖2H2 <∞.

We quote below Theorem 6.13 from [20] since it is essential for most proofs inthis section.

Theorem 2.4. Let A be the infinitesimal generator of a semigroup T (t). If 0 ∈ρ(A) then

(1) T (t) : X → D(Aα) for every t > 0 and α ≥ 0.(2) For every x ∈ D(Aα) we have T (t)Aαx = AαT (t)x.(3) For every t, δ > 0 the operator AαT (t) is bounded and

‖AαT (t)‖ < Cαt−αe−δt.

(4) Let 0 < α ≤ 1 and x ∈ D(Aα) then

‖T (t)x− x‖ < Cαtα‖Aαx‖.

In the following sections we need a result for the regularity of the mild solutionwhen it is discretized in time.

Lemma 2.5. If X is the mild solution to (2.2) or (2.3), then we have for 0 ≤t1 ≤ t2 ≤ T and α ∈ 1, 2

E‖X(t2) −X(t1)‖2H ≤ C(t2 − t1)(E‖X0‖

2Hα + sup

s∈[0,t2]

E‖X(s)‖2Hα). (2.6)

360 ANDREA BARTH

Proof. To ease notation we write R as a proxy for operators A and B, both ofwhich generate a C0–semigroup. The general mild solution is given by

X(t) = S(t)X0 +

∫ t

0

S(t− s)G(X(s))dM(s).

The regularity is provided by

E‖X(t2) −X(t1)‖2H ≤ C(E‖(S(t2) − S(t1))X0‖

2H

+ E‖

∫ t1

0

(S(t2 − s) − S(t1 − s))G(X(s))dM(s)‖2H

+ E‖

∫ t2

t1

S(t2 − s)G(X(s))dM(s)‖2H).

Theorem 2.4, resp. the basic properties of a C0–semigroup, implies for the firstterm that

E‖(S(t2) − S(t1))X0‖2H = E‖S(t1)(S(t2 − t1) − I)X0‖

2H

≤ C(t2 − t1) E‖RX0‖2H

≤ C(t2 − t1) E‖X0‖2Hα .

To bound the second term we use Theorem 2.4 and Equation (2.1):

E‖

∫ t1

0

(S(t2 − s) − S(t1 − s))G(X(s))dM(s)‖2H

≤ E

∫ t1

0

‖R (S(t2 − s) − S(t1 − s))G(X(s))‖2LHS(H,H)ds

≤ E

∫ t1

0

‖S(t1 − s)R (S(t2 − t1) − I)G(X(s))‖2LHS(H,H)ds

≤ C(t2 − t1)E

∫ t1

0

‖RG(X(s))‖2LHS(H,H)ds

≤ C(t2 − t1)E

∫ t1

0

‖X(s)‖2Hαds

≤ C(t2 − t1) t1 sups∈[0,t1]

E‖X(s)‖2Hα .

The last term is dominated by

E‖

∫ t2

t1

S(t2 − s)G(X(s))dM(s)‖2H ≤ E

∫ t2

t1

‖S(t2 − s)RG(X(s))‖2LHS(H,H)ds

≤ E

∫ t2

t1

‖S(t2 − s)‖2‖RG(X(s))‖2LHS(H,H)ds

≤ C(t2 − t1) sups∈[t1,t2]

E‖X(s)‖2Hα .

FEM FOR MARTINGALE-DRIVEN SPDE’S 361

This concludes the proof, since

E‖X(t2) −X(t1)‖2H ≤ C(t2 − t1) E‖X0‖

2Hα + C(t2 − t1) sup

s∈[0,t1]

E‖X(s)‖2Hα

+ C(t2 − t1) sups∈[t1,t2]

E‖X(s)‖2Hα

≤ C(t2 − t1)(E‖X0‖2Hα + sup

s∈[0,t2]

E‖X(s)‖2Hα).

The case α = 1 refers here to the hyperbolic equation and α = 2 to the parabolicequation.

3. Finite Element Method for Parabolic SPDE’s

In this section we study the case A = ∆, the Laplace operator with Dirichletboundary conditions. This is done for notational simplicity. The results containedherein are valid for a broader set of second order differential operators. Namely,those that posses semidiscrete analogs of the form Ah = PhAPh. The notions ofAh and Ph are fully discussed below.

We follow the error estimates for the Galerkin approximation for non–smoothdata described in [25]. The solution operator for the SPDE has a smoothingproperty that results in a regular solution even if the initial data is not so. Wealso work in the space Hs(D) = D(A

s2 ) endowed with the norm ‖ · ‖Hs = ‖A

s2 · ‖

for s ∈ R as introduced in [25].

3.1. Error estimates in the semidiscrete case. Let Th be a family of trian-gulations of D, indexed by the maximal edge length h. For each Th we construct afinite element space Sh, consisting of piecewise continuous polynomials. Further-more, we assume Sh ⊂ H1

0 (D). The semidiscrete problem to (2.2) is then to findXh(t) ∈ Sh such that for t ∈ [0, T ]

dXh(t) = AhXh(t)dt+ PhG(Xh(t))dM(t), Xh(0) = PhX0. (3.1)

Here Ah : Sh → Sh denotes the discrete Laplacian operator defined by

(Ahφ, ψ) = −(∇φ,∇ψ) ∀φ, ψ ∈ Sh,

and Ph denotes the L2-projection on Sh. We write Sh(t) for the discrete analogueof the operator S(t). This object is formally introduced via Sh(t) = e−tAh . Thesemidiscrete mild solution to Equation (3.1) is:

Xh(t) = Sh(t)PhX0 +

∫ t

0

Sh(t− s)PhG(Xh(s))dM(s). (3.2)

In what follows we study the magnitude of the error that results from approx-imating Equation (2.4) by (3.1). For this we require the error estimates for thedeterministic case. The deterministic equation is given by

Y (t) = S(t)Y0

and the corresponding semidiscrete equation by

Yh(t) = Sh(t)PhY0.

The following error estimate is a consequence of Theorem 3.1 in [25].

362 ANDREA BARTH

Lemma 3.1. For ‖Y0‖2H2

<∞

‖Yh(t) − Y (t)‖H ≤ Ch2‖Y0‖H2 ∀t ≥ 0. (3.3)

For simplicity in what follows C absorbs any constants stemming from ourcomputations. The theorem below provides a bound for the mean square error inthe semidiscrete case.

Theorem 3.2. If E‖X0‖2H1

< ∞, then for t ∈ [0, T ] there exists C(t) < ∞ such

that

E‖Xh(t) −X(t)‖2H ≤ C(t)h2(E‖X0‖

2H1

+ sups∈[0,t]

E‖X(s)‖2H). (3.4)

Proof. The mild and the semidiscrete mild solutions to (2.4) are respectively

X(t) = S(t)X0 +

∫ t

0

S(t− s)G(X(s))dM(s)

and

Xh(t) = Sh(t)PhX0 +

∫ t

0

Sh(t− s)PhG(Xh(s))dM(s).

The mean square error satisfies

E‖Xh(t) −X(t)‖2H ≤ C (E‖(Sh(t)Ph − S(t))(t)X0‖

2H

+ E‖

∫ t

0

(Sh(t− s)Ph − S(t− s))G(X(s))dM(s)‖2H

+ E‖

∫ t

0

Sh(t− s)Ph(G(Xh(s)) −G(X(s)))dM(s)‖2H).

We analyze the terms on the right-hand side one by one. For the first one, itfollows from Lemma 3.1 that

E‖(Sh(t)Ph − S(t))X0‖2H ≤ C h2

E‖X0‖2H1.

To approximate the second term we need (2.1), which yields

E‖

∫ t

0

(Sh(t− s)Ph − S(t− s))G(X(s))dM(s)‖2H

≤ E

∫ t

0

‖(Sh(t− s)Ph − S(t− s))G(X(s))‖2LHS(H,H)ds

≤

∫ t

0

‖(Sh(t− s)Ph − S(t− s))‖2ds sups∈[0,t]

E‖G(X(s))‖2LHS(H,H)

≤ C h2 sups∈[0,t]

E‖X(s)‖2H .

FEM FOR MARTINGALE-DRIVEN SPDE’S 363

We have used the fact that for v ∈ H∫ t

0

‖(Sh(t− s)Ph − S(t− s))‖2ds =

∫ t

0

supv 6=0

‖(Sh(t− s)Ph − S(t− s))v‖2H

‖v‖2H

ds

≤ supv 6=0

∫ t

0 ‖(Sh(t− s)Ph − S(t− s))v‖2Hds

‖v‖2H

≤ Ch2.

This is a consequence of Lemma 3.7 from [25].Proposition 2.1 and the Lipschitz-type condition for G, imply for the third term

E‖

∫ t

0

Sh(t− s)Ph(G(Xh(s)) −G(X(s)))dM(s)‖2H

≤ E

∫ t

0

‖Sh(t− s)Ph(G(Xh(s)) −G(X(s)))‖2LHS(H,H)ds

≤ C

∫ t

0

‖Sh(t− s)Ph‖2E‖(Xh(s) −X(s))‖2

Hds

≤ C

∫ t

0

E‖(Xh(s) −X(s))‖2Hds.

Using Gronwall’s inequality we get

E‖Xh(t) −X(t)‖2H ≤ C(Ch2

E‖X0‖2H1

+ Ch2 sups∈[0,t]

E‖X(s)‖2H

+ C

∫ t

0

E‖(Xh(s) −X(s))‖2Hds)

≤ C(t)h2 (E‖X0‖2H1

+ sups∈[0,t]

E‖X(s)‖2H).

From the proof can be seen that a higher regularity of the initial condition,E‖X0‖

2H2

<∞, leads to optimal convergence of order O(h2).

3.2. Error estimates in the fully discrete case. We use the linearized back-ward Euler–Galerkin Method described in [25]. If one were to implement a non–linearized backward Euler scheme the result would not be satisfactory. This isdue to the fact that both backward and forward schemes are incompatible withthe Ito-Integral. As to have a discretization of [0, T ] we introduce the time stepstn = nk, with step size k. We also introduce

dXh(t)

dt≈Xn −Xn−1

k.

as the backward difference for the time derivative. Formally the discrete SPDE is

Xn = r(kAh)Xn−1 +

∫ tn

tn−1

r(kAh)PhG(Xn−1)dM(s) (3.5)

with initial condition X0 = PhX0. The rational function r, which is defined onthe spectrum of kAh, is used to approximate the semigroup Sh(t). In the case of

364 ANDREA BARTH

a backward Euler scheme r is given by r(λ) = (1 + λ)−1, and the approximationof Xh is

Xh(tn) ≈ Xn.

For the Crank–Nicolson scheme r(λ) = (1 − λ2 )(1 + λ

2 )−1 for the first term; how-

ever, in the stochastic integral r(λ) = (1 + λ2 )−1. When using this scheme the

approximation of Xh is given by

Xh(tn) ≈Xn +Xn−1

2.

For both schemes Xh has to be approximated via Xn−1 in the stochastic integral,since in an Ito-integral the integrand can only depend on the lower bound of thedomain of integration.

As before we borrow a result for the error approximation in the fully discretecase for deterministic equations (Theorem 7.8 in [25]):

The deterministic equation is given as before by

Y (t) = S(t)Y0,

the semidiscrete equation by

Yh(t) = Sh(t)PhY0.

and the corresponding discrete equation by

Y n = r(kAh)Y n−1

Lemma 3.3. For ‖Y0‖2H2

<∞

‖Y n − Y (tn)‖H ≤ C(k + h2)‖Y0‖H2 ∀t ≥ 0. (3.6)

With Lemma 2.5 in hand we can prove the main result in this section, namelythe mean square error estimate for the fully discrete case:

Theorem 3.4. If E‖X0‖2H2

< ∞, then for tn ∈ [0, T ] there exists C(tn) < ∞such that

E‖Xn −X(tn)‖2H ≤ C(tn)(k +

h4

k)(E‖X0‖

2H2

+ sups∈[t0,tn]

E‖X(s)‖2H1

). (3.7)

Proof. For all tn ∈ [0, T ] the mild solution to Equation (2.2) is given by

X(tn) = S(tn)X0 +

∫ tn

0

S(tn − s)G(X(s))dM(s).

We iterate (3.5) and obtain

Xn = r(kAh)Xn−1 +

∫ tn

tn−1

r(kAh)PhG(Xn−1)dM(s)

= r(kAh)nPhX0 +

n∑

j=1

∫ tj

tj−1

r(kAh)n−j+1PhG(Xj−1)dM(s).

FEM FOR MARTINGALE-DRIVEN SPDE’S 365

The error satisfies

E‖Xn −X(tn)‖2H ≤ C(E‖(r(kAh)nPh − S(tn))X0‖

2H

+ E‖

n∑

j=1

∫ tj

tj−1

r(kAh)n−j+1Ph(G(Xj−1) −G(X(tj−1)))dM(s)‖2H

+ E‖

n∑

j=1

∫ tj

tj−1

r(kAh)n−j+1Ph(G(X(tj−1)) −G(X(s)))dM(s)‖2H

+ E‖

n∑

j=1

∫ tj

tj−1

(r(kAh)n−j+1Ph − S(tn − tj−1))G(X(s))dM(s)‖2H

+ E‖

n∑

j=1

∫ tj

tj−1

(S(tn − tj−1) − S(tn − s))G(X(s))dM(s)‖2H).

Lemma 3.3 implies for the first term

E‖(r(kAh)nPh − S(tn))X0‖2H ≤ C(k + h2)2 E‖X0‖

2H2

≤ C(k2 + h4) E‖X0‖2H2.

For the other four terms we need Proposition 2.1. Proposition 1.12 in [10] guar-anties that we can interchange the sum with the norm, since all the cross termsare zero. It is here that it is crucial that the integrands depend exclusively onthe lower end of the domain of integration. Dependence on the upper limit ofintegration would lead to non–convergence, since n would appear as a factor. Toget an estimate for the second term we use the Lipschitz-type condition for G:

E‖n

∑

j=1

∫ tj

tj−1

r(kAh)n−j+1Ph(G(Xj−1) −G(X(tj−1)))dM(s)‖2H

≤ E

n∑

j=1

∫ tj

tj−1

‖r(kAh)n−j+1Ph(G(Xj−1) −G(X(tj−1)))‖2LHS(H,H)ds

≤ kE

n∑

j=1

‖r(kAh)n−j+1Ph(G(Xj−1) −G(X(tj−1)))‖2LHS(H,H)

≤ k

n∑

j=1

‖r(kAh)n−j+1Ph‖2

E‖Xj−1 −X(tj−1)‖2H

≤ Ck

n∑

j=1

E‖Xj−1 −X(tj−1)‖2H .

Here, as well as for the next term we need the stability of the approximatedsolution operator En

k = r(kAh)nPh, i.e. ‖Enk ‖ ≤ C for n ≥ 1. Hence, we calculate

using Lemma 2.5, on the regularity of the mild solution, and the Lipschitz-type

366 ANDREA BARTH

condition for G:

E‖

n∑

j=1

∫ tj

tj−1

r(kAh)n−j+1Ph(G(X(tj−1)) −G(X(s)))dM(s)‖2H

≤ C E

n∑

j=1

∫ tj

tj−1

‖r(kAh)n−j+1Ph(G(X(tj−1)) −G(X(s)))‖2LHS(H,H)ds

≤ C

n∑

j=1

∫ tj

tj−1

‖r(kAh)n−j+1Ph‖2

E‖X(tj−1) −X(s)‖2Hds

≤ C

n∑

j=1

∫ tj

tj−1

(s− tj−1) ds (E‖X0‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H)

≤ C(tn) k (E‖X0‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H).

Next we use Theorem 7.7 from [25] to obtain, in analogous fashion to the semidis-crete case,

k

n∑

j=1

‖r(kAh)n−j+1Ph − S(tn − tj−1)‖2

= k

n∑

j=1

supv 6=0

‖(r(kAh)n−j+1Ph − S(tn−j+1))v‖2H

‖v‖2H

≤ k

n∑

j=1

supv 6=0

C(h2 t−1n−j+1 + k t−1

n−j+1)2‖v‖2

H

‖v‖2H

≤ C(h2 + k)2k

n∑

j=1

1

((n− j + 1)k)2

≤ C(h2 + k)21

k

n∑

j=1

1

j2

≤ C(k +h4

k).

It follows from the above that

E‖

n∑

j=1

∫ tj

tj−1

(r(kAh)n−j+1Ph − S(tn − tj−1))G(X(s))dM(s)‖2H

≤ E

n∑

j=1

∫ tj

tj−1

‖(r(kAh)n−j+1Ph − S(tn − tj−1))G(X(s))‖2LHS(H,H)ds

≤ C kn

∑

j=1

‖(r(kAh)n−j+1Ph − S(tn − tj−1))‖2 sup

s∈[t0,tn]

E‖X(s)‖2H

≤ C(k +h4

k) sup

s∈[t0,tn]

E‖X(s)‖2H .

FEM FOR MARTINGALE-DRIVEN SPDE’S 367

Finally, for the last term we use Theorem 2.4 to get

E‖

n∑

j=1

∫ tj

tj−1

(S(tn − tj−1) − S(tn − s))G(X(s))dM(s)‖2H

≤ E

n∑

j=1

∫ tj

tj−1

‖S(tn − s)(S(s− tj−1) − I)G(X(s))‖2LHS(H,H)ds

≤ E

n∑

j=1

∫ tj

tj−1

C(s− tj−1)2α‖AαG(X(s))‖2

LHS(H,H)ds

≤ Cn

∑

j=1

∫ tj

tj−1

(s− tj−1)2αds sup

s∈[t0,tn]

E‖X(s)‖2H2α

≤ C

n∑

j=1

(tj − tj−1)2α(tj − tj−1) sup

s∈[t0,tn]

E‖X(s)‖2H2α

≤ C nk k2α sups∈[t0,tn]

E‖X(s)‖2H2α .

We use the discrete form of Gronwall’s inequality and set α = 12 to obtain

E‖Xn −X(tn)‖2H ≤ C(tn)(k2 + h4) E‖X0‖

2H2

+ C(tn)(k +h4

k) sup

s∈[t0,tn]

E‖X(s)‖2H

+ C(tn)k (E‖X0‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H1

+ sups∈[t0,tn]

E‖X(s)‖2H)

+ C

n∑

j=1

∫ tj

tj−1

E‖Xj −X(tj)‖2Hds

≤ C(tn)(k +h4

k)(E‖X0‖

2H2

+ sups∈[t0,tn]

E‖X(s)‖2H1

).

4. Finite Element Method for Hyperbolic SPDE’s

In this section we look into the approximation of the transport equation withmultiplicative noise, which is of the form

dX(t) = BX(t)dt+G(X(t))dM(t), X(0) = X0. (4.1)

Here the first order differential operator B generates the C0–semigroup of con-tractions S, G : H → L(U,H) is a linear map and M denotes a square integrablemartingale as before. As mentioned in the introduction, S is usually a shift semi-group. For a given vector field b, the first order differential operator is definedas

Bφ(x) :=d

∑

i=1

bi(x)φxi(x).

We require for technical reasons that supx∈D |div b(x)| < ∞. Let the domainD possess a piecewise smooth boundary ∂D of class C1. The weak solution to

368 ANDREA BARTH

Equation (4.1) is a process X that satisfies

(φ,X(t)) = (φ,X(0)) +

∫ t

0

(B∗φ,X(s))ds+

∫ t

0

(G∗(X(s))φ, dM(s)), (4.2)

for any φ ∈ D(B∗) and for all t ∈ [0, T ]. We assume that the weak solution satisfiessupt∈[0,T ] E‖X(t)‖2

H2 <∞. The inflow boundary is the set

∂D− := x ∈ ∂D : b(x) · n(x) > 0,

where n(x) denotes the exterior normal to ∂D at x. For convenience we imposethe Dirichlet boundary condition

X = 0 on ∂D−.

This particular structure has to be taken into consideration when defining thefinite dimensional spaces for the approximation: S−

h ⊂ H1(D) is a family offinite element spaces consisting of piecewise continuous polynomials with respectto the family of triangulations Th of D, which vanish on the inflow boundary.

In contrast to the parabolic case, in this section we derive error estimates byuse of the weak solution. For all φ ∈ H1(D) and ψ ∈ H we define the bilinearform

a1(φ, ψ) := (Bφ,ψ).

Integrating by parts we obtain

a1(φ, φ) =1

2(−

∫

D

(div b)φ2dx+

∫

∂D\∂D−

b · nφ2dx) ∀φ ∈ S−h .

The integral over ∂D\∂D− is negative given the conditions on the inflow boundary.

We let D = x ∈ D : div b(x) < 0 and we define µ := supD |div b|.

4.1. Error estimates in the semidiscrete case. The semidiscrete problem isto find Xh(t) ∈ S−

h such that for all φ ∈ S−h and t ∈ [0, T ]

d(Xh(t), φ) = a1(Xh(t), φ)dt + (G∗(Xh(t))φ, dM(t))H. (4.3)

To investigate the stability of this approximation we set φ = Xh, which yields byuse of the Ito formula for square integrable Hilbert space valued martingales

E(Xh(t), Xh(t))H = E(Xh(0), Xh(0))H + E

∫ t

0

a1(Xh(s−), Xh(s))ds

+ E

∫ t

0

(G∗(Xh(s−))Xh(s), dM(s))H + E

∫ t

0

d[M,M ]s.

Here the last term is, by the assumptions on the covariance process ofM , bounded.It follows from the fact thatXh is cadlag that it has at most countably many jumps.The latter, together with our assumption on the divergence of b, implies that

E

∫ t

0

a1(Xh(s−), Xh(s))ds ≤µ

2E

∫ t

0

‖Xh(s)‖2Hds.

Using Proposition 2.1, Jensen’s inequality and the fact that

‖G∗(Xh(s))Xh(s−)‖LHS(H,R) ≤ ‖G(Xh(s))‖LHS(H,H)‖Xh(s−)‖H

FEM FOR MARTINGALE-DRIVEN SPDE’S 369

we may write

E

∫ t

0

(G∗(Xh(s))Xh(s), dM(s))H ≤ C E

∫ t

0

‖Xh(s)‖2Hds.

The constant C is the modulus of continuity of G. With the two previous inequal-ities in hand we get

E‖Xh(t)‖2H ≤ E‖Xh(0)‖2

H + (C +µ

2) E

∫ t

0

‖Xh(s)‖2Hds

so we obtain with Gronwall’s inequality

E‖Xh(t)‖2H ≤ CE‖Xh(0)‖2

H . (4.4)

We now show an error estimate for the semidiscrete case. In order to do so we needan interpolation operator, which is defined on the space of continuous functionsand takes values in the finite element space S−

h . For v ∈ C0(D) we define

Ih(v) :=

n∑

i=1

v(xi)φi,

where n denotes the dimension of the finite dimensional space and φini=1 is the

corresponding shape function basis. Notice that Ih(v)(xi) = v(xi) at all the nodesxi, i = 1, . . . , n.

Theorem 4.1. Let X and Xh be the solutions of Equation 4.2 and Equation 4.3.

Then, if E‖X0 − Ih(X0)‖ ≤ ChE‖X0‖H1 , we have

E‖Xh(t) −X(t)‖2 ≤ Ch2(E‖X(0)‖2H1 + sup

s∈[0,t]

E‖X(s)‖2H2)

Proof. We follow closely the proof of the deterministic case as presented in [15]or [23]. Since the error estimate in the deterministic case does not depend on thesource term, the same will be the case in the stochastic setting. We write

Xh −X = (Xh − IhX) + (IhX −X) =: η + ξ

We only concern ourselves with η, since by assumption

E‖ξ(t)‖2H ≤ Ch2

E‖X(t)‖2H1 .

By the definitions of the weak solutions for X and Xh, we have that for all φ ∈ S−h ,

η ∈ S−h satisfies

(η(t), φ) − (η(0), φ) − a1(

∫ t

0

η(s)ds, φ) = −(w, φ).

We have used the shorthand

w = ξ(t) − ξ(0) −

∫ t

0

Bξ(s)ds+

∫ t

0

(G(X(s)) − PhG(Xh(s)))dM(s).

370 ANDREA BARTH

If Xh(0) = Ih(X(0)), then η(0) = 0 and Equation (4.4) implies

E‖η(t)‖2H ≤ C (E‖η(0)‖2

H + E‖ξ(t)‖2H + E‖ξ(0)‖2

H + E

∫ t

0

‖Bξ(s)‖2Hds

+ E‖

∫ t

0

(G(X(s)) − PhG(Xh(s)))dM(s)‖2H).

For the terms involving ξ we write

E‖ξ(t)‖2H + E‖ξ(0)‖2

H + E

∫ t

0

‖Bξ(s)‖2Hds

≤ Ch2 (E‖X(0)‖2H1 + E‖X(t)‖2

H1 + E

∫ t

0

‖X(s)‖2H2ds)

≤ C(t)h2 (E‖X(0)‖2H1 + sup

s∈[0,t]

E‖X(s)‖2H1 + sup

s∈[0,t]

E‖X(s)‖2H2).

Theorem 1.1 in [25] and the Lipschitz-type condition onG as well as Equation (2.1)give:

E‖

∫ t

0

(G(X(s)) − PhG(Xh(s)))dM(s)‖2H

≤ E‖

∫ t

0

(G(X(s)) − PhG(X(s)))dM(s)‖2H

+ E‖

∫ t

0

(PhG(X(s)) − PhG(Xh(s)))dM(s)‖2H

≤ E

∫ t

0

‖(1 − Ph)‖2‖X(s)‖2Hds+

∫ t

0

E‖X(s) −Xh(s)‖2Hds

≤ Ch2 sups∈[0,t]

E‖X(s)‖2H1 +

∫ t

0

E‖X(s) −Xh(s)‖2Hds.

We conclude by applying Gronwall’s inequality, since

E‖Xh(t) −X(t)‖2 ≤ C (E‖η(t)‖2 + E‖ξ(t)‖2)

≤ C(t)h2 (E‖X(0)‖2H1 + sup

s∈[0,t]

E‖X(t)‖2H1 + sup

s∈[0,t]

E‖X(s)‖2H2).

Theorem 4.1 does not provide a result of optimal convergence order. For smoothinitial conditions and smooth solutions the Galerkin approximation might producereasonable results. However, if we encounter non–smooth solutions oscillationsmay occur at the boundary (in both the deterministic and stochastic settings).A Petrov–Galerkin approximation, which broadly speaking consists of adding anartificial diffusion, may increase the order of convergence. In particular the stream-

line diffusion method converges with order O(h3/2). The effect of the diffusion is

to dampen the oscillations along the characteristics near the boundary, even inthe presence of non–smooth solutions. The results concerning convergence of thestreamline diffusion method presented in [14] [15] can be naturally extended tothe stochastic setting by proceeding as we did above.

FEM FOR MARTINGALE-DRIVEN SPDE’S 371

4.2. Error estimates in the fully discrete case. In order to show convergencein the fully discrete case we employ the linearized backward Euler scheme asintroduced in Section 3. The fully discrete problem is to find X i for i = 1, . . . , nsuch that

(Xn, φ)H = (X0, φ)H + k

n∑

i=1

a1(Xi−1, φ)H +

n∑

i=1

∫ ti

ti−1

(G∗(X i−1)φ, dM(s))H.

(4.5)

To show stability we choose, as in the semidiscrete case, φ = X

to obtain

E(Xn, Xn)H ≤ E(X0, X0)H + k

n∑

i=1

a1(Xi−1, X i−1)H

+ E

n∑

i=1

∫ ti

ti−1

(G∗(X i−1)X i−1, dM(s))H.

In analogous fashion to Section 4.1 we get

E‖Xn‖2H ≤ E‖X0‖2

H + (µ

2+ C)k

n∑

i=1

E‖X i−1‖2H

≤ C(tn) E‖X0‖2H ,

where k stands for the time step. Note that the discrete version of Gronwall’sinequality produces a factor n. The latter times k yields the dependence of C ontn. The mean square error estimate in the fully discrete case reads:

Theorem 4.2. Let X and X

be the respective solutions to Equations 4.2 and 4.5.

If E‖X0 − Ih(X0)‖ ≤ ChE‖X0‖H1 , then

E‖Xn −X(tn)‖2H ≤ C(tn)(h2 + k) (E‖X(0)‖2

H2 + sups∈[t0,tn]

E‖X(s)‖2H2).

Proof. The proof follows the same train of thought as the one for the semidiscretecase. Once again we split Xn −X(tn) and treat the summands separately.

Xn −X(tn) = (Xn − IhX(tn)) + (IhX(tn) −X(tn)) =: ηn + ξn.

The fact that

E‖ξn‖2H ≤ Ch2

E‖X(tn)‖2H1

follows by assumption. The definitions of the weak solutions to Xn and X(tn)allow us to write

(ηn, φ) − (η0, φ) − kn

∑

i=1

a1(ηi−1, φ) = −(w, φ)

372 ANDREA BARTH

for ηn ∈ S−h and φ ∈ S−

h , and w is given by

w = ξn − ξ0 − k

n∑

i=1

Bξi−1 −

n∑

i=1

∫ ti

ti−1

(BX(ti−1) −BX(s))ds

−

n∑

i=1

∫ ti

ti−1

(G(X i−1) −G(X(ti−1)))dM(s)

−

n∑

i=1

∫ ti

ti−1

(G(X(ti−1)) −G(X(s)))dM(s).

As before η(0) = 0, hence

E‖ηn‖2H ≤ E‖ξn‖2

H + E‖ξ0‖2H + E‖k

n∑

i=1

Bξi−1‖2H

+ E‖n

∑

i=1

∫ ti

ti−1

(BX(ti−1) −BX(s))ds‖2H

+ E‖

n∑

i=1

∫ ti

ti−1

(G(X(ti−1)) −G(X(s)))dM(s)‖2H

+ E‖

n∑

i=1

∫ ti

ti−1

(G(X i−1) −G(X(ti−1)))dM(s)‖2H .

The first three terms on the right hand side can be approximated as follows:

E‖ξn‖2H + E‖ξ0‖2

H + E‖k

n∑

i=1

Bξi−1‖2H

≤ Ch2E‖X(0)‖2

H1 + Ch2E‖X(tn)‖2

H1 + C(tn)h2 sups∈[t0,tn]

E‖X(s)‖2H2 .

Lemma 2.5 implies for the fourth term that

E‖

n∑

i=1

∫ ti

ti−1

(BX(ti−1) −BX(s))ds‖2H ≤ C

n∑

i=1

∫ ti

ti−1

E‖X(ti−1) −X(s)‖2H1ds

≤ C(tn)k (E‖X(0)‖2H2 + sup

s∈[t0,tn]

E‖X(s)‖2H2).

We use the same Lemma, plus the Lipschitz condition for G and Equation (2.1)in the approximation of the fifth term:

E‖

n∑

i=1

∫ ti

ti−1

(G(X(ti−1)) −G(X(s)))dM(s)‖2H

≤ C

n∑

i=1

∫ ti

ti−1

E‖X(ti−1) −X(s)‖2Hds

≤ C(tn)k (E‖X(0)‖2H1 + sup

s∈[t0,tn]

E‖X(s)‖2H1).

FEM FOR MARTINGALE-DRIVEN SPDE’S 373

The Lipschitz condition for G and Equation (2.1) provide an approximation of thelast term on the right hand side of the estimate for E‖ηn‖2

H , namely

E‖n

∑

i=1

∫ ti

ti−1

(G(X i−1) −G(X(ti−1)))dM(s)‖2H

≤ C

n∑

i=1

∫ ti

ti−1

E‖X i−1 −X(ti−1)‖2Hds

≤ Ckn

∑

i=1

E‖X i−1 −X(ti−1)‖2H .

The proof concludes with the use of the discrete version of Gronwall’s inequality,since

E‖Xn −X(tn)‖2H ≤ E‖ηn‖2

H + E‖ξn‖2H

≤ C(tn)h2 (E‖X(0)‖2H1 + sup

s∈[t0,tn]

E‖X(s)‖2H2)

+ Ck (E‖X(0)‖2H2 + sup

s∈[t0,tn]

E‖X(s)‖2H2)

+ Ck

n∑

i=1

E‖X i−1 −X(ti−1)‖2H

≤ C(tn)(h2 + k) (E‖X(0)‖2H2 + sup

s∈[t0,tn]

E‖X(s)‖2H2).

Theorem 4.2 does not provide convergence of optimal order, which has the samecause as the sub-optimality of the semidiscrete approximation. In the deterministiccase, the error approximations in space are of the same magnitude than those intime. However, the Euler–Maruyama scheme we used for the approximation ofthe SPDE has a maximum convergence of order ∆t = k. For higher convergenceone has to introduce a Milstein scheme, which also takes into account higher orderterms of the Ito–Taylor approximation. Another option is to treat time with aGalerkin approximation as well.

In the hyperbolic case, convergence of the fully discrete equation is not coupledin time and space, i.e. convergence in space is independent of convergence intime, whereas in the parabolic case our results indicate that refinements shouldbe made simultaneously in both variables. This is not surprising, since in contrastto a parabolic equation, in a transport equation the variables can be treated in asomehow indistinct manner.

Simulation of the discrete equations bears some different task. The squareintegrable martingale has to be discretized. One could approximate the noise bytruncating the series

M(t) =

∞∑

i=1

√

λi mi(t) ei,

where ei are the eigenvectors of the covariance operator with the correspondingeigenvalues λi and mi a sequence of independent Levy processes. If the eigenvalues

374 ANDREA BARTH

converge fast enough, the error by this approximation is much smaller than theoverall order of convergence. However, this question is open and subject to furtherresearch.

Acknowledgment. The author wishes to express her thanks to Fred EspenBenth, Annika Lang, Santiago Moreno and Jurgen Potthoff for all the helpfulcomments and fruitful discussions. Further the author is very grateful to thereferee whose comments led to the improvement of the paper.

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Andrea Barth: Centre of Mathematics for Applications (CMA), University of Oslo,N–0316 Oslo, Norway

E-mail address: andrea.barth@cma.uio.no