A SPLITTING UP METHOD FOR NONLINEAR PARABOLIC ITO EQUATIONS

Wilfried Grecksch ^{∗}

Abstract

A nonlinear parabolic Ito equation with multiplicative noise is approximated by a sequence of Ito equations with additive noise. The convergence is in the sence of probability one.

Keywords: Parabolic Ito equations, splitting up, multiplicative noise, additive noise

AMS subject classification: 60H15,35R60,93E11

1 Introduction

The idea of operator-splitting methods for operator equations with different operators consists in the approximation by a sequence of equations with only one operator. The technique of splitting up methods for deterministic partial differential equations has been used extensively by many authors. We refer here to [?]. In [?] the idea of splitting up is applied to the Zakai equation considering this equation as the sum of two operators. It is constructed a sequence of two equations, where the first equation is a linear deterministic partial differential equation and the second equation is a stochastic differential equation which has an explicit solution. The Zakai equation is an example of a homogeneous linear stochastic evolution equation over a triplet of rigged Hilbert spaces. In [?] a stochastic evolution equation is approximated by a sequence of elliptic stochastic equations and stochastic integrals. Here we approximate a stochastic evolution equation with multiplicative noise over a triplet of rigged Hilbert spaces by a sequence of stochastic evolution equations with additive noise where the additive noise is a separable computable stochastic integral. In the case of a linear drift term the approximation equations are explicite solvable.

∗Martin-Luther-Universität Halle-Wittenberg, Fachbereich Mathematik und Informatik, Institut für Optimierung und Stochastik, 06099 Halle/Saale, Germany, e-mail:

grecksch@mathematik.uni-halle.d400.de

2 A Stochastic Evolution Equation

Let (V, H, V^{∗}) be a triplet of rigged Hilbert spaces (see[?]). Also suppose that there are given:

- (Ω, F, P, F_{t}, w(t)) is a complete probability space equipped with a Filtration F_{t} and a Wiener process w(t) with values in a separable Hilbert space K and a positive self adjoint nuclear covariance operator Q.

- (Z(t), F_{t}) is H-valued continuous square integrable martingale.

- A, B are two functions defined on [0, T] × Ω × V such that A(t, v) ∈ V^{∗} and B(v, t) ∈ L_{Q}(K, H) for all t ∈ [0, T], v ∈ V with probability one where L_{Q}(K, H) is the space of all linear operators C : Q^{1/2}K → H such that CQ^{1/2} is a Hilbert-Schmidt-operator. A and B are progressively measurable.

Denote by (., .)_{H} the scalar procuct in H and by ⟨v^{∗}, v⟩ the value of the functional v^{∗} ∈ V^{∗} for v ∈ V. Assume the following hypotheses:

(A1) λ ∈ R^{1} → ⟨A(t, v_{1} + λv_{2}), v⟩ is continuous for all t ∈ [0, T], ω ∈ Ω and v, v_{1}, v_{2} ∈ V.

(A2) There are positive constants L, α such that 2⟨A(t, v_{1}) − A(t, v_{2}), v_{1} − v_{2}⟩ + α ∥ v_{1} − v_{2} ∥_{V}^{2} ≤ 0 and ∥ B(t, v_{1}) − B(t, v_{2}) ∥_{Q}≤ L ∥ v_{1} − v_{2} ∥_{H} for all t ∈ [0, T], ω ∈ Ω, v_{1}, v_{2} ∈ V where ∥ . ∥_{Q} is the Hilbert-Schmidt Norm of BQ^{1/2}.

(A3) Then are a positive constant β such that 2 < A(t, v), v > +β ∥ v ∥_{V}^{2} ≤ 0 and ∥ B(t, v) ∥_{Q}^{2}≤ L^{2}(1+ ∥ v ∥_{H}^{2}) for all t ∈ [0, T], ω ∈ Ω, v ∈ V.

(A4) There is a positive constant γ such that ∥ A(t, v) ∥_{V^{∗}}^{2}≤ γ^{2}(1+ ∥ v ∥_{V}^{2} ) for all t ∈ [0, T], ω ∈ Ω, v ∈ V.

(A5) E ∥ X_{0} ∥_{H}^{2}< ∞.

We define

dX(t) = A(t, X(t))dt + B(t, X(t))dw(t) + dZ(t), X(0) = X_{0}  (1)

as

(X(t), v)_{H} = (X_{0}, v)_{H} + ∫_{0}^{t}⟨A(t, X(s)), v⟩ds

+ ∫_{0}^{t}(v, B(s, X(s))dw(s))_{H} + (Z(t), v)_{H}

for all t ∈ [0, T] and v from a dense set M ⊂ V with probability one. The stochastic integral is defined in the sense of Ito. It follows from [4]:

Theorem 1 These are Ω′ ⊂ Ω with P (Ω′) = 1 and an adapted pathwise unique solution X(t) of (1) with following properties:

(a) X(t) is pathwise continuous in H.

(b) X(t) ∈ V for all t ∈ [0, T] with probability one.

(c) It holds (1) for all ω ∈ Ω′ and t ∈ [0, T].

(d)

E sup_{0≤t≤T} ∥ X(t) ∥_{H}^{2} +E ∫_{0}^{T} ∥ X(t) ∥_{V}^{2} dt
≤ C(E ∥ X_{0} ∥_{X}^{2} +E ∥ Z(T) ∥_{H}^{2} +D)

where C and D are fixed constants.

3 Main Result

Let O = t_{0}^{N} < t_{1}^{N} < ... < t_{N}^{N} = T (N = 2, 3, ...) be partitions of [0, T] with h_{N} := t_{n+1}^{N} − t_{n}^{N} → 0 for N → ∞ for all n. We write in the following for t_{n}^{N} shortly t_{n}. We consider the following approximation equations:

X^{0}(t) = X_{0} + ∫_{0}^{t} A(s, X^{0}(s))ds, t ∈ [0, t_{1}[  (2)

Z^{k}(t) = ∫_{t_{k}}^{t} B(s − h_{N}, X^{k−1}(s − h_{N}))dw(s),  (3)

X^{k}(t) = X^{k−1}(t_{k} − 0) + ∫_{t_{k}}^{t} A(s, X^{k}(s))ds + Z^{k}(t)  (4)
k = 1, ..., N − 1, t ∈ [t_{k}, t_{k+1}[.

Define X^{N−1}(T) := X^{N−1}(T −0). The problem (1) is for fixed ω a deterministic parabolic equation over (V, H, V^{∗}). The solution X^{0}(t) is F_{0}-measurable and has the properties (a), (b), (c) and Esup_{0≤t≤T} ∥ X^{0}(t) ∥_{H}^{2} +E ∫_{0}^{T} ∥ X^{0}(t) ∥_{V}^{2} ≤ CE ∥ X_{0} ∥_{H}^{2}. Z^{k}(t) is a H-valued continuous square integrable F_{t}-Martingale.

(4) is a parabolic Ito equation over (V, H, V^{∗}) with a known continuous H-valued martingale as additive noise. We see that the structure of (2), (3), (4) is easier as (1). Especially, we can solve (4) in the case of a linear nonrandom and time independently Operator A by

X^{k}(t) = T (t)X^{k−1}(t_{k} − 0) + ∫_{t_{k−1}}^{t} T (t − s)dZ^{k}(s)  (5)

where T is the semigroup generated by A with D(A) = {v ∈ V : Av ∈ H}. Further it holds X^{k}(t_{k} + 0) = X^{k−1}(t_{k} − 0). Subsequently the approximation is ”continuous”. We have the following approximation result:

Theorem 2 (1) If X(t) is the solution of (1) and X^{k}(t)(k = 1, ..., N − 1) are the solutions of (2), (3), (4) then

lim_{N→∞}[E ∥ X^{N−1}(T)−X(T) ∥_{H}^{2} + ∑_{k=0}^{N−1} E ∫_{t_{k}}^{t_{k+1}} ∥ X^{k}(t)−X(t) ∥_{V}^{2} dt] = 0.

(2) If N = 2^{m}(m = 1.2....) then lim_{N→∞} X^{N−1}(T) with probability 1 in H.

4 Proof of the Main Result

At first we prove some a priori estimates.

Lemma 1 There is a constant C_{1} > 0 with

E ∥ X^{k}(t) ∥_{H}^{2}≤ C_{1}  (6)

∑_{k=0}^{N−1} E ∫_{t_{k}}^{t_{k+1}} ∥ X^{k}(s) ∥_{V}^{2} ds ≤ C_{1}  (7)

for all t ∈ [0, T], k = 0, ..., N − 1, N = 1, 2, ...

Proof: From (2) and (A3) it follows

E ∥ X^{0}(t_{1} − 0) ∥_{H}^{2} +βE ∫_{0}^{t_{1}} ∥ X_{0}(s) ∥_{V}^{2} ds ≤ E ∥ X_{0} ∥_{H}^{2}  (8)

If t ∈ [0, T] then it exists k with t ∈ [t_{k}, t_{k+1}[. The Ito formula for the norm square [?] and the properties of the Ito integral give

E ∥ X^{k}(t) ∥_{H}^{2} = E ∥ X^{k−1}(t_{k−0} ∥_{H}^{2}
+ 2E ∫_{t_{k}}^{t}⟨A(s, X^{k}(s)), X^{k}(s)⟩ds  (9)
+ E ∫_{t_{k}}^{t} ∥ B(s − h_{N}, X^{k−1}(s − h_{N})) ∥_{Q}^{2} ds

E ∥ X^{j}(t_{j+1} − 0) ∥_{H}^{2} = E ∥ X^{j−1}(t_{j} − 0) ∥_{H}^{2}
+ 2E ∫_{t_{j}}^{t_{j+1}}⟨A(s, X^{k}(s)), X^{k}(s)⟩ds

+ E ∫_{t_{j}}^{t_{j}+1} ∥ B(s − h_{N}, X^{k−1}(s − h_{N})) ∥_{Q}^{2} ds
(j = 1, ..., k − 1).

Summing up from j = 1 to j = k − 1 and addition of (9) and applying the properties of A and B we have

E ∥ X^{k}(t) ∥_{H}^{2} +β ∑_{j=1}^{k−1} E ∫_{t_{j}}^{t_{j}+1} ∥ X^{j}(s) ∥_{V}^{2} ds + βE ∫_{t_{k}}^{t} ∥ X^{j}(s) ∥_{V}^{2} ds

≤ E ∥ X^{0}(t_{1} − 0) ∥_{H}^{2} + ∑_{j=1}^{k−1} E ∫_{t_{j−1}}^{t_{j}} L^{2}(1+ ∥ X^{j−1}(τ) ∥_{H}^{2})dτ (10)

+E ∫_{t_{k−1}}^{t−h_{N}} L^{2}(1+ ∥ X^{k−1}(τ) ∥_{H}^{2})dτ.

Subsequently

E ∥ X^{k}(t) ∥_{H}^{2} ≤ E ∥ X^{0}(t_{1} − 0) ∥_{H}^{2}
+ ∑_{j=1}^{k} E ∫_{t_{j−1}}^{t_{j}} L^{2}(1+ ∥ X^{j−1}(τ) ∥_{H}^{2} dτ

+ E ∫_{t_{k}}^{t} L^{2}(1+ ∥ X^{k}(τ) ∥_{H}^{2})dτ.

Introducing the function ϕ(τ) = E ∥ X^{j−1}(τ) ∥_{H}^{2} for τ ∈ [t_{j−1}, t_{j}[ (j = 1, ..., k + 1) so we obtain with Gronwall Lemma

ϕ(t) ≤ (E ∥ X^{0}(t_{1} − 0) ∥_{H}^{2} +L^{2}T)exp{L^{2}T },

that is, the inequality (6) holds. The inequality (7) follows from (10) with k = N−1, t = T.

Especially it follows from (6)

E ∥ X^{k}(t_{k+1} − 0) ∥^{2}≤ C_{1}.  (11)

Let (h_{n}) be a complete orthonormal system in H with h_{n} ∈ V. (e_{n}) denotes a complete orthonormal system in K. Consider the projection operators Π_{n} and π_{n} from V^{∗} to the finite dimensional spaces generated by {h_{1}, ..., h_{n}} and {e_{1}, ..., e_{n}}. Introduce for real processes U_{n}^{i}(t) the processes U_{n}(t) = ∑_{i=1}^{n} U_{n}^{i}(t)h_{i}. The Galerkin approximations of (1) are defined by

dU_{n}(t) = Π_{n}A(t, U_{n}(t))dt + Π_{n}B(t, U_{n}(t))dπ_{n}w(t)  (12)
U_{n}(0) = Π_{n}X_{0}.

It is easy to prove that there is a positive constant C_{2} (independently from n) such that we have for all t ∈ [0, T] and h > 0 with t − h ≥ 0

E ∥ U_{n}(t) − U_{n}(t − h) ∥_{H}^{2}≤ C_{2}h.

Consequently it also holds

E ∥ X(t) − X(t − h) ∥_{H}^{2}≤ C_{2}h  (13)

since U_{n}(t) is convergent to X(t) in the sequare mean in H for all t ∈ [0, T] ([4]).

We now can prove the first statement in Theorem 2.

By Ito formula for norm square, the property (A2) and the formulas (1), (4) we obtain

E ∥ X^{k}(t_{k+1} − 0) − X(t_{k+1}) ∥_{H}^{2}≤ E ∥ X^{k−1}(t_{k} − 0) − X(t_{k}) ∥_{H}^{2}
−αE ∫_{t_{k}}^{t_{k+1}} ∥ X^{k}(s) − X(s) ∥_{V}^{2} ds  (14)
+L^{2}E ∫_{t_{k}}^{t_{k+1}} ∥ X^{k−1}(s − h_{N}) − X(s) ∥_{H}^{2} ds.

Obviously it holds

E ∫_{t_{k}}^{t_{k+1}} ∥ X^{k−1}(s − h_{N}) − X(s) ∥_{H}^{2} ds
≤ 2E ∫_{t_{k}}^{t_{k+1}} ∥ X^{k−1}(s − h_{N}) − X(s − h_{N}) ∥_{H}^{2} ds
+2E ∫_{t_{k}}^{t_{k+1}} ∥ X(s) − X(s − h_{N}) ∥_{H}^{2} ds.

Therefore it follows from (13) for (14)

E ∥ X^{k}(t_{k+1} − 0) − X(t_{k+1}) ∥_{H}^{2}≤ E ∥ X^{k−1}(t_{k} − 0) − X(t_{k}) ∥_{H}^{2}
+2C_{2}h_{N}^{2} + 2E ∫_{t_{k−1}}^{t_{k}} ∥ X^{k−1}(τ) − X(τ) ∥_{H}^{2} dτ.

Summing up from k = 1 to k = N − 1 and adding of E ∫_{t_{N−1}}^{t_{N}} ∥ X^{N−1}(τ) − X(τ) ∥_{H}^{2} dτ on the righthand side then we have

E ∥ X^{N−1}(t_{N} − 0) − X(t_{N}) ∥_{H}^{2}≤ E ∥ X^{0}(t_{1} − 0) − X(t_{1}) ∥_{H}^{2} +2C_{2}h_{N}

+2 ∑_{k=1}^{N} ∫_{t_{k−1}}^{t_{k}} E ∥ X^{k−1}(τ) − X(τ) ∥_{H}^{2} dτ

If we introduce the function ψ(τ) = E ∥ X^{k−1}(τ)−X(τ) ∥_{H}^{2} for τ ∈ [t_{k−1}, t_{k}[ (k = 1, ..., N) the Gronwall Lemma gives

ψ(t_{N}) ≤ (E ∥ X^{0}(t_{1} − 0) − X(t_{1}) ∥_{H}^{2} +2C_{2}h_{N})exp{2T }.

From (1), (2), the Ito formula for the norm square, the properties (A2), (A3) we obtain

E ∥ X^{0}(t_{1} − 0) − X(t_{1}) ∥_{H}^{2}≤ E ∫_{0}^{t_{1}} L^{2}(1+ ∥ X(s) ∥_{H}^{2} ds)

and from the statement (d) of Theorem 2 we deduce that there is a constant C_{3} > 0 with ψ(t_{N}) ≤ C_{3}h_{N} → 0 for N → ∞. So we have proved the first part of Theorem 2. We now consider N = N_{m} = 2^{m} and denote by X_{m}^{k}(t) the process X^{k}(t). If we pass over to m+1 we introduce the Processes Y_{m+1}^{0}(t) = Y_{m+1}^{1}(t) = X_{m}^{0}(t), Y_{m+1}^{2}(t) = Y_{m+3}^{3}(t) = X_{m}^{1}(t),..., Y_{m+1}^{N_{m+1}−2} = Y_{m+1}^{N_{m+1}−1}(t) = X_{m}^{N_{m}−1}(t)

and set Y^{∼}_{m+1} (t) = Y_{m+1}^{n−1}(t) for t ∈ [t_{n−1}^{m+1}, t_{n}^{m+1}[ (n = 1, ..., N_{m+1}), Y^{∼}_{m+1} (T) =

Y_{m+1}^{N_{m+1}−1}(t_{N} − 0), X^{∼}_{m+1} (t) = X_{m+1}^{n}(t) for t ∈ [t_{n−1}^{m+1}, t_{n}^{m+1}[ (n = 1, ..., N_{m+1})

and X^{∼}_{m+1} (T) = X_{m+1}^{N_{m+1}−1}(t_{N} − 0). If we consider with the equation (4) the difference X_{m+1}^{k}(t) − Y_{m+1}^{k}(t) then we obtain by summing up from k = 1 to N_{m+1} − 1

X^{∼}_{m+1} (T)− Y^{∼}_{m+1} (T) = ∫_{0}^{T} (A(s, X^{∼}_{m+1} (s)) − A(s, Y^{∼}_{m+1} (s)))ds

+ ∫_{0}^{T} {I_{0,t_{Nm−1}^{m}[}(s)(B(s − h_{N}, X^{∼}_{m+1} (s)) − B(s − h_{N}, Y^{∼}_{m+1} (s)))dw(s)

with B(s − h_{N}, .) = 0 for S ∈ [0, h_{N}[. Using again the Ito formula for norm square and the properties (A2), (A3) and (6) so we see

E ∥X^{∼}_{m+1} (T)− Y^{∼}_{m+1} (T) ∥_{H}^{2}

≤ L^{2}E ∫_{0}^{T} ∥X^{∼}_{m+1} (s)− Y^{∼}_{m+1} (s) ∥_{H}^{2} +\frac{1}{2^{m+1}} 2C_{1}L^{2}.

Applying the Gronwall Lemma then we get

E ∥X^{∼}_{m+1} (T)− Y^{∼}_{m+1} (T) ∥_{H}^{2}
≤ \frac{T}{2^{m+1}} (L^{2}(1 + C_{1}))exp\{L^{2}T \}
=: C_{4} \frac{1}{2^{m+1}} .

Therefore it holds

P \{ ∑_{m+1}^{∞} ∥X^{∼}_{m+1} (T)− Y^{∼}_{m+1} (T) ∥_{H}> \frac{1}{m^{2}} \}

≤ C_{4} · ∑_{m=1}^{∞} \frac{m^{4}}{2^{m+1}} < ∞.

Subsequently, it exists with probability 1 the limit lim_{m→∞}X_{m}^{N_{m}−1}(T). This limit is X(T) too. So we have proved the Theorem 2 completely.

References

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