Some Properties of Solutions of Double Stochastic

Differential Equations

C. Tudor^{∗}^{†}and M. Tudor^{‡}

Abstract

The paper deals with double stochastic differential equations. Existence and uniqueness are obtained under a Hölder-type hypothesis.

The convergence in probability of the successive approximations in the Hölder norm and the existence of a weak solution for continuous coefficients are also proved.

Under an additional independence hypothesis, the mean square convergence of fractional step approximations is shown. This last result is used in order to deduce a comparison result and as a consequence the existence of a strong solution in the case when the ”double ”noise is additive.

Finally, a counterexample which shows that the comparison result does not work for double noise depending on the state is given.

1 Introduction

The multiple stochastic integral is introduced by Meyer [10] for martingales as differentials. The concept allows to consider the associated stochastic equations (see [18]). We refer to these equations as multiple stochastic equations. Such stochastic equations include the Itô equations, integro-differential and as well some classes of Volterra equations.

∗University of Bucharest and CIMAT, Department of Mathematics, 70109 Bucharest, Romania.

†To whom correspondence should be addressed.

‡Academy of Economical Studies, Department of Mathematics, 70167 Bucharest, Romania

In [18] we considered double stochastic equations under the Lipschitz condition and we addressed the problem of convergence of some approximation schemes. In the present paper we consider double Itô equations with the Lipschitz condition replaced by a Hölder-type one. We obtain the existence and uniqueness of the strong solution and, by using an exponential Doob type inequality for stochastic integrals that are not martingales, we deduce the convergence in probability (in the Hölder norm) of the successive approximations. By means of the standard procedure (Skorohod representation of the weak convergence) the existence of a weak solution is shown for continuous coefficients. Next, under an independence assumption, we show the convergence of the fractional step approximations to the unique solution of the equation. Such approximations are used if other more traditional methods are not easy to apply, especially for numerical treatment, qualitative properties (as positivity, comparison, etc.). The fractional step (or splitting up) method is used in Rascanu and Tudor [13] for finite dimensional Itô equations, in Belopol’skaya and Nagolkina [2], Bensoussan-Glowinsky and Rascanu [3], Goncharuk and Kotelenez [6], for different classes of stochastic partial equations. We also use the splitting up method in order to obtain a comparison result for double stochastic equations with additive ”double” noise and in particular we derive the existence of strong solution for continuous and bounded double drifts. Finally, we give a counterexample which shows that the comparison result fails if the double diffusion coefficient is state dependent.

2 Existence and Uniqueness of Strong Solutions

Let (Ω, F, P, (F_{t})_{0≤t≤T} ) be a complete filtered probability space and {W_{t}}_{0≤t≤T},

\left\{{ \atop W̃_{t}}\right\}_{0≤t≤T} be F_{t}-Brownian motions.

Define

C_{2}(t) = {(t_{1}, t_{2}) : 0 ≤ t_{1} ≤ t_{2} ≤ t} ,
C_{2}(a, b) = {(t_{1}, t_{2}) : a ≤ t_{1} ≤ t_{2} ≤ b} .

Let F (t, x), G(t, x) : [0, T]×R ⟶ R, F̃(t_{1}, t_{2}, x), G̃(t_{1}, t_{2}, x) : C_{2}(T)×R ⟶ R be measurable functions and ξ be a F_{0}-measurable random variable. Define LS = {α : R_{+} ⟶ R_{+} : α is strictly increasing, continuous, concave, and ∫_{0+}^{1} \frac{du}{α(u)} = ∞\} .

Obvious α_{1}(u) = Lu, L > 0, α_{2}(u) = x |log x|^{1−ε} , α_{3}(u) = x |log x| |log |log x||^{1−ε} , 0 < ε < 1, belong to LS and α_{2}, α_{3} are not Lipschitz. Also if α_{1}, α_{2} ∈ LS, c_{1}, c_{2} ≥ 0, c_{1} + c_{2} > 0, then c_{1}α_{1} + c_{2}α_{2} ∈ LS. We consider the following double stochastic differential equation

X_{t} = ξ + ∫_{0}^{t} F (s, X_{s})ds + ∫_{0}^{t} G(s, X_{s})dW_{s}+

∫_{C_{2}(t)} F̃(s_{1}, s_{2}, X_{s_{1}})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, X_{s_{1}})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T. (2.1)

We introduce the following hypotheses on the initial data and the coefficients of (2.1). (H_{1}) (Growth condition on F, G): there exists a constant K > 0 such that

|F (t, x)|^{2} + |G(t, x)|^{2} ≤ K (1 + |x|^{2}) , ∀t ∈ [0, T] , x ∈ R.

(H̃_{1}) (Growth condition on F̃, G̃): there exists a constant K̃ > 0 such that for all (s_{1}, s_{2}) ∈ C_{2}(T), x ∈ R,

|F̃(s_{1}, s_{2}, , x)|^{2} + |G̃(s_{1}, s_{2}, x)|^{2} ≤ K̃ (1 + |x|^{2}) .

(H_{2}) (Hölder-type condition on F, G) : there exist α ∈ LS such that for all t ∈ [0, T] , x, y ∈ R,

|F (t, x) − F (t, y)|^{2} + |G(t, x) − G(t, y)|^{2} ≤ α (|x − y|^{2}) .

(H̃_{2}) (Hölder-type condition on F̃, G̃) : there exist α̃ ∈ LS such that for all (s_{1}, s_{2}) ∈ C_{2}(T), x, y ∈ R,

|F̃(s_{1}, s_{2}, x) − F̃(s_{1}, s_{2}, y)|^{2} + |G̃(s_{1}, s_{2}, x) − G̃(s_{1}, s_{2}, y)|^{2} ≤ α̃ (|x − y|^{2}) .

(H_{3}) ξ ∈ L^{2} (Ω, F_{0}, P) . Next for a process U and α > 0 we define

∥U∥_{α} = sup_{s≠t} \frac{|U_{t} − U_{s}|}{|t − s|^{α}}, ∥U∥_{α} = ∥U∥_{∞} + ∥U∥_{α} .

Theorem 2.1 Suppose (H_{1}) − (H_{3}), (H̃_{1}), (H̃_{2}) are satisfied. Let {X_{t}^{n}}_{t∈[0,T]} be the successive approximations defined by

X_{t}^{0} = ξ, X_{t}^{n+1} = ξ + ∫_{0}^{t} F (s, X_{s}^{n})ds + ∫_{0}^{t} G(s, X_{s}^{n})dW_{s}+

∫_{C_{2}(t)} F̃(s_{1}, s_{2}, X_{s_{1}}^{n})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, X_{s_{1}}^{n})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T.

(a) Then the equation (2.1) has a pathwise unique solution {X_{t}}_{t∈[0,T]} such that

E (sup_{t≤T} |X_{t}|^{2}) < ∞.  (2.2)

Moreover

lim_{n→∞} E (sup_{t≤T} |X_{t}^{n} − X_{t}|^{2}) = 0.  (2.3)

(b) In addition assume that one of the following two conditions is satisfied: (b_{1}) For some r > 1, E (|ξ|^{2r}) < ∞, either, (b_{2}) There exists α ∈ (0, 1) and a positive constant K_{1} such that

|G̃(s_{1}, t_{2}, x) − G̃(s_{1}, t_{1}, x)| ≤ K |t_{1} − t_{2}|^{α} for all s_{1} ≤ t_{i} ≤ T.  (2.4)

Then for every 0 < β < min(\frac{1}{2}, 2α) we have ∥X^{n} − X∥_{β} ⟶^{p}_{n→∞} 0.

Proof. (a) Existence. By standart computation it follows that

sup_{n} E (sup_{t≤T} |X_{t}^{n}|^{2}) < ∞.  (2.5)

Next from (H_{2}), (H̃_{2}) we deduce

g_{m,n}(t) := E (sup_{s≤t} |X_{s}^{m} − X_{s}^{n}|^{2}) ≤ ∫_{0}^{t} (α + α̃) (g_{m,n}(s))ds,

\overline{lim}_{m,n→∞}g_{m,n}(t) ≤ ∫_{0}^{t} (α + α̃) (\overline{lim}_{m,n→∞}g_{m,n}(s))ds,

where from \overline{lim}_{m,n→∞}g_{m,n}(t) = 0. Therefore there exists a continuous process {X_{t}}_{t∈[0,T]} such that

lim_{n→∞} E (sup_{t≤T} |X_{t}^{n} − X_{t}|^{2}) = 0.

It is clear that X is a solution of (2.1). Uniqueness. If {X_{t}}_{t∈[0,T]} , {Y_{t}}_{t∈[0,T]} are solutions of (2.1) and denote δ(t) = E (|X_{t} − Y_{t}|^{2}) then by (H_{2}), (H̃_{2}) we have

δ(t) ≤ ∫_{0}^{t} (α + α̃) (δ(s))ds,

and this implies δ(t) = 0.

(b) Assume (b_{1}) is satisfied. Then in a standard manner it follows

sup_{n} E (sup_{t≤T} |X_{t}^{n}|^{2r}) < ∞,

where from we get

E (|X_{t}^{n} − X_{s}^{n}|^{2r}) ≤ K_{2} (t − s)^{r} , E (|X_{t} − X_{s}|^{2r}) ≤ K_{2} (t − s)^{r} ,

and, if we define Y^{n} = X^{n} − X, the previuos inequalities imply

E (|Y_{t}^{n} − Y_{s}^{n}|^{2r}) ≤ K_{3} (t − s)^{r} .  (2.6)

As in [1], (2.6) and the Garcia-Rodemich-Rumsey criterion imply the result (see also Lemma 8.27 of [15]). Now assume that (b_{2}) hold. Choose η > 0 such that β + η < \frac{1}{2}. Taking into account the previous result from (a) all we need to show is that [Y^{n}]_{β} ⟶^{p} 0, or since for all μ, ε > 0,

P ([Y^{n}]_{β} > ε) ≤ P ([Y^{n}]_{β+μ} > εμ^{−η}) + 2P (∥X^{n} − X∥_{∞} > εμ^{β}) ,

to show that

lim_{M→∞} sup_{n} P ([Y^{n}]_{θ} > M) = 0, ∀0 < θ < min(\frac{1}{2}, 2α).

By using (2.2), (2.5) we obtain

P ([Y^{n}]_{θ} > M) ≤ P (∥X^{n}∥_{∞} > M_{1}) + P (∥X∥_{∞} > M_{1})+

P ([Y^{n}]_{θ} > M, ∥X^{n}∥_{∞} ≤ M_{1}, ∥X∥_{∞} ≤ M_{1}) ≤

\frac{K_{3}}{M_{1}^{2}} + P ([Y^{n}]_{θ} > M, ∥X^{n}∥_{∞} ≤ M_{1}, ∥X∥_{∞} ≤ M_{1}),

so we have to prove that

lim_{M→∞} sup_{n} P ([Y^{n}]_{θ} > M, ∥X^{n}∥_{∞} ≤ M_{1}, ∥X∥_{∞} ≤ M_{1}) = 0, ∀M_{1} > 0. (2.7)

Denote A = {∥X^{n}∥_{∞} ≤ M_{1}, ∥X∥_{∞} ≤ M_{1}} . Then we have

P ([Y^{n}]_{θ} > M, A) ≤ P ([∫_{0}^{.} (G(s, X_{s}^{n}) − G(s, X_{s})) dW_{s}]_{θ} > \frac{M − K_{4}\sqrt{1 + M_{1}^{2}}}{2} , A) +

P ([∫_{C_{2}(.)} (G̃(s_{1}, s_{2}, X_{s_{1}}^{n}) − G̃(s_{1}, s_{2}, X_{s_{1}})) dW̃_{s_{1}}dW̃_{s_{2}}]_{θ} > \frac{M − K_{4}\sqrt{1 + M_{1}^{2}}}{2} , A) ,

and hence (2.7) follows if we prove that

lim_{M→∞} sup_{n} P ([∫_{0}^{.} (G(s, X_{s}^{n}) − G(s, X_{s})) dW_{s}]_{θ} > M, A) = 0,  (2.8)

lim_{M→∞} sup_{n} P ([∫_{C_{2}(.)} (G̃(s_{1}, s_{2}, X_{s_{1}}^{n}) − G̃(s_{1}, s_{2}, X_{s_{1}})) dW̃_{s_{1}}dW̃_{s_{2}}]_{θ} > M, A) = 0.

(2.9)

On the set A we have

|G(s, X_{s}^{n}) − G(s, X_{s})| ≤ K_{5} (1 + ∥X^{n}∥_{∞} + ∥X∥_{∞}) ≤ K_{5} (1 + 2M_{1}) ,

and similarly

|G̃(s_{1}, s_{2}, X_{s_{1}}^{n}) − G̃(s_{1}, s_{2}, X_{s_{1}})| ≤ K_{5} (1 + 2M_{1}) .

¿From Lemma 10.2.2 of [8] it follows that on A,

[∫_{0}^{.} (G(s, X_{s}^{n}) − G(s, X_{s})) dW_{s}]_{θ} ≤ K_{6}ζ_{n},

where E (ζ_{n}1_{A}) ≤ K_{7}T^{2}. Therefore

P ([∫_{0}^{.} (G(s, X_{s}^{n}) − G(s, X_{s})) dW_{s}]_{θ} > M, A) ≤

P (A, ζ_{n} > \frac{M}{K_{6}} ) ≤ \frac{K_{6}E (ζ_{n})}{M} ≤ \frac{K_{8}}{M} ⟶_{M→∞} 0, uniformly in n.

Next define

I_{n}(t) = ∫_{0}^{t} (G̃(s_{1}, t, X_{s_{1}}^{n}) − G̃(s_{1}, t, X_{s_{1}})) dW̃_{s_{1}}.

Then

P ([∫_{C_{2}(.)} (G̃(s_{1}, s_{2}, X_{s_{1}}^{n}) − G̃(s_{1}, s_{2}, X_{s_{1}})) dW̃_{s_{1}}dW̃_{s_{2}}]_{θ} > M, A) ≤

P (A, ∥I_{n}∥_{∞} ≤ M_{2}, [∫_{0}^{.} I_{n}(s)dW̃_{s}]_{θ} > M) + P (∥I_{n}∥_{∞} > M_{2}, A) . (2.10)

Now from Theorem 2 of [11] or Theorem 2.4 of [12] it follows

P (∥I_{n}∥_{∞} > M_{2}, A) ≤ exp {−K_{8}(M_{1})M_{2}^{2}} ,  (2.11)

and from Lemma 10.2.2 of [8]

P (A, ∥I_{n}∥_{∞} ≤ M_{2}, [∫_{0}^{.} I_{n}(s)dW̃_{s}]_{θ} > M) ≤ \frac{K_{9}(M_{1}, M_{2})}{M}.  (2.12)

Finally from (2.10)-(2.12) we get (2.9). ■

Theorem 2.2 (Existence of weak solutions) Assume that (F, G, F̃, G̃)

satisfy the hypotheses (H_{1}), (H_{3}), (H̃_{1}). Moreover suppose that:

(i) For some r > 1, E (|ξ|^{2r}) < ∞. (ii) F, G, F̃, G̃ are continuous in all variables. (iii) There exists α ∈ (0, 1) and a positive constant K̄ such that

|G̃(s_{1}, t_{1}, x) − G̃(s_{1}, t_{2}, x)| ≤ K̄ |t_{1} − t_{2}|^{α} , ∀s_{1} ≤ t_{i} ≤ T, x ∈ R.

Then (2.1) has a weak solution.

We need the following convergence result for double stochastic integrals. Lemma 2.3. Let {V_{t}^{n}}_{t∈[0,T]} , {V_{t}}_{t∈[0,T]} be Brownian motions and {h_{n}(s_{1}, s_{2})}_{(s_{1},s_{2})∈C_{2}(T)} , {h(s_{1}, s_{2})}_{(s_{1},s_{2})∈C_{2}(T)} be predictable processes such that

E (∫_{C_{2}(T)} |h_{n}(s_{1}, s_{2})|^{2} ds_{1}ds_{2}) < ∞, E (∫_{C_{2}(T)} |h(s_{1}, s_{2})|^{2} ds_{1}ds_{2}) < ∞.

Assume that:

(1) For each t, V_{t}^{n} ⟶^{p} V_{t}.

(2) For each (s_{1}, s_{2}) ∈ C_{2}(T), h_{n}(s_{1}, s_{2}) ⟶^{p} h(s_{1}, s_{2}) and

lim_{C→∞} sup_{n} P (∥h_{n}∥_{∞} > C) = 0.  (2.13)

(3) There exists α ∈ (0, 2] and for every n there exists a nonnegative random variable ξ_{n} such that

lim_{C→∞} sup_{n} P (ξ_{n} > C) = 0,  (2.14)

∫_{0}^{t_{1}∧t_{2}} |h_{n}(s_{1}, t_{1}) − h_{n}(s_{1}, t_{2})|^{2} ds_{1} ≤ ξ_{n} |t_{1} − t_{2}|^{α} , ∀0 ≤ t_{i} ≤ T. (2.15)

(4) For each ε > 0, 0 ≤ t ≤ T,

lim_{h→0} lim_{n→∞} sup_{|s_{1}−s_{2}|≤h}_{0≤s_{i}≤t} P (|h_{n}(s_{1}, t) − h_{n}(s_{2}, t)| > ε) = 0.  (2.16)

Then

∫_{C_{2}(T)} h_{n}(s_{1}, s_{2})dV_{s_{1}}^{n}dV_{s_{2}}^{n} ⟶^{p} ∫_{C_{2}(T)} h(s_{1}, s_{2})dV_{s_{1}}dV_{s_{2}}.

Proof. Define

Z_{t}^{n} = ∫_{0}^{t} h_{n}(s_{1}, t)dV_{s_{1}}^{n}, Z_{t} = ∫_{0}^{t} h(s_{1}, t)dV_{s_{1}}.

Then

∫_{C_{2}(T)} h_{n}(s_{1}, s_{2})dV_{s_{1}}^{n}dV_{s_{2}}^{n} = ∫_{0}^{T} Z_{s}^{n}dV_{s}^{n},

∫_{C_{2}(T)} h(s_{1}, s_{2})dV_{s_{1}}dV_{s_{2}} = ∫_{0}^{T} Z_{s}dV_{s}.

By the convergence result for first order stochastic integrals due to Skorohod (see [14]-page 32) we have

Z_{t}^{n} ⟶^{p} Z_{t}, ∀0 ≤ t ≤ T.

Define A = {∥h_{n}∥_{∞} ≤ C_{1}, ξ_{n} ≤ C_{1}} . Then from (2.13)-(2.15) and Theorem 2 of [11] it follows

lim_{C→∞} sup_{n} P (∥Z^{n}∥_{∞} > C) ≤ P (∥Z^{n}∥_{∞} > C, A) +

P (∥h_{n}∥_{∞} > C_{1}) + P (ξ_{n} > C_{1}) ≤
exp {−γ(C_{1})C^{2}} + P (∥h_{n}∥_{∞} > C_{1}) + P (ξ_{n} > C_{1}) ⟶ 0,

as C → ∞, C_{1} → ∞.

Next Lemma 10.2.2 of [8] yields

|Z_{t}^{n} − Z_{s}^{n}| ≤ η_{n} |t − s|^{α} on A,  (2.17)

where

sup_{n} E (1_{A}η_{n}) = C_{2} < ∞.  (2.18)

Next by using (2.17), (2.18) we obtain

P (∥Z_{t}^{n} − Z_{s}^{n}∥ > ε) ≤ P (∥Z_{t}^{n} − Z_{s}^{n}∥ > ε, A)+P (∥h_{n}∥_{∞} > C_{1})+P (ξ_{n} > C_{1}) ≤ P (1_{A}η_{n} > ε |t − s|^{−α}) + P (∥h_{n}∥_{∞} > C_{1}) + P (ξ_{n} > C_{1}) ≤

\frac{|t − s|^{α}}{ε} E (1_{A}η_{n}) + P (∥h_{n}∥_{∞} > C_{1}) + P (ξ_{n} > C_{1}) ≤

C_{2} |t − s|^{α} + P (∥h_{n}∥_{∞} > C_{1}) + P (ξ_{n} > C_{1}) ,

so that

\overline{lim_{h→0}} sup_{n} sup_{|t−s|≤h}_{0≤s,t≤T} P (|Z_{t}^{n} − Z_{s}^{n}| > ε) ≤

sup_{n} P (∥h_{n}∥_{∞} > C_{1}) + sup_{n} P (ξ_{n} > C_{1}) ⟶ 0 as C_{1} → ∞.

Applying once again the result of Skorohod ( [14]-page 32) we obtain that

∫_{0}^{T} Z_{s}^{n}dV_{s}^{n} ⟶^{p} ∫_{0}^{T} Z_{s}dV_{s}. ■

Proof of Theorem 2.2. Choose a sequence (F_{n}, G_{n}, F̃_{n}, G̃_{n}) of continuous functions such that (i_{1}) (F_{n}, G_{n}, F_{n}, G̃_{n}) satisfy (H_{1}),(H̃_{1}) uniformly with respect to n.

(i_{2}) (F_{n}, G_{n}, F̃_{n}, G̃_{n}) are Lipschitz in the last variable. (i_{3}) For each s, s_{1}, s_{2},

F_{n}(s, x) ⟶ F (s, x), G_{n}(s, x) ⟶ G(s, x),
F̃_{n}(s_{1}, s_{2}, x) ⟶ F̃(s_{1}, s_{2}, x), G̃_{n}(s_{1}, s_{2}, x) ⟶ G̃(s_{1}, s_{2}, x),

uniformly for x in compact sets. Let (by Theorem 2.1) X_{t}^{n} be the strong solution of

X_{t}^{n} = ξ + ∫_{0}^{t} F_{n}(s, X_{s}^{n})ds + ∫_{0}^{t} G_{n}(s, X_{s}^{n})dW_{s}+

∫_{C_{2}(t)} F̃_{n}(s_{1}, s_{2}, X_{s_{1}}^{n})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃_{n}(s_{1}, s_{2}, X_{s_{1}}^{n})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T.

¿From (i), (H_{1}),(H̃_{1}) we deduce

sup_{n} E (|X_{t}^{n}|^{2r}) < ∞,

and this implies

E (|X_{t}^{n} − X_{s}^{n}|^{2r}) ≤ K_{1} |t − s|^{r} , ∀n, s, t.

Therefore on some probability space (Ω̂, F^{∧}, P̂) there exist (ξ̂_{n}, X̂^{n}, Ŵ^{n}, Ṽ^{n}) ,

(ξ̂, X̂, Ŵ , Ṽ ) such that:

(a) For every n, the laws of (ξ̂_{n}, X̂^{n}, Ŵ^{n}, Ṽ^{n}) and (ξ, X^{n}, W, Ŵ ) coincide.

(b) For some subsequence (n_{k}) (for simplicity we assume n_{k} = k for all k ),

ξ̂_{n} ⟶^{a.s.} ξ, X̂^{n} ⟶^{a.s.} X̂, Ŵ^{n} ⟶^{a.s.} Ŵ , Ṽ^{n} ⟶^{a.s.} Ṽ ,

uniformly on every compact interval. If we define

F^{∧} _{t}^{n}= B (ξ̂_{n}, X̂_{s}^{n}, Ŵ_{s}^{n}, Ṽ_{s}^{n} : s ≤ t) , F^{∧}_{t}= B (ξ̂, X̂_{s}, Ŵ_{s}, Ṽ_{s} : s ≤ t) ,

then it is clear that Ŵ_{t}^{n}, Ṽ^{n}(resp. Ŵ_{t}, Ṽ) are F^{∧} _{t}^{n} (resp. F^{∧}_{t}) Brownian motions. Also we have (we omit the details)

X̂_{t}^{n} = ξ̂_{n} + ∫_{0}^{t} F_{n}(s, X̂_{s}^{n})ds + ∫_{0}^{t} G_{n}(s, X̂_{s}^{n})dŴ_{s}^{n}+

∫_{C_{2}(t)} F̃_{n}(s_{1}, s_{2}, X̂_{s_{1}}^{n})ds_{1}ds_{2}+∫_{C_{2}(t)} G̃_{n}(s_{1}, s_{2}, X̂_{s_{1}}^{n})dṼ_{s_{1}}^{n}dṼ_{s_{2}}^{n}, 0 ≤ t ≤ T. (2.19)

Next, it is easy to see that for each t,

∫_{0}^{t} F_{n}(s, X̂_{s}^{n})ds ⟶^{p} ∫_{0}^{t} F (s, X̂_{s})ds

∫_{C_{2}(t)} F̃_{n}(s_{1}, s_{2}, X̂_{s_{1}}^{n})ds_{1}ds_{2} ⟶^{p} ∫_{C_{2}(t)} F̃(s_{1}, s_{2}, X̂_{s_{1}})ds_{1}ds_{2},

and by [14], page 32,

∫_{0}^{t} G_{n}(s, X̂_{s}^{n})dŴ_{s}^{n} ⟶^{p} ∫_{0}^{t} G(s, X̂_{s})dŴ_{s}^{n}.

Finally by Lemma 2.3 we also have

∫_{C_{2}(t)} G̃_{n}(s_{1}, s_{2}, X̂_{s_{1}}^{n})dṼ_{s_{1}}^{n}dṼ_{s_{2}}^{n} ⟶^{p} ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, X̂_{s_{1}})dṼ_{s_{1}}dṼ_{s_{2}}.

It follows we can pass to the limit in (2.19) and obtain that

X̂_{t} = ξ̂+ ∫_{0}^{t} F (s, X̂_{s})ds + ∫_{0}^{t} G(s, X̂_{s})dŴ_{s}+

∫_{C_{2}(t)} F̃(s_{1}, s_{2}, X̂_{s_{1}})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, X̂_{s_{1}})dṼ_{s_{1}}dṼ_{s_{2}}, 0 ≤ t ≤ T,

i.e., X̂ is a weak solution of (2.1). ■

3 Fractional Step Method. A Comparison Result

In this section we assume that:

(i) The filtration (F_{t})_{0≤t≤T} is of the form F_{t} = F_{t} ∨ F̃_{t} and F_{t}, F̃_{t} are independent for all t ≥ 0. (ii) W_{t} is F_{t} -Brownian motion and W̃_{t} is F̃_{t}-Brownian motion. We introduce now the fractional step approximations associated to (2.1). Let Δ : 0 = t_{0} < t_{1} < ....t_{n} = T be a partition of [0, T] with the norm ∥Δ∥ = max_{1≤i≤n} (t_{i} − t_{i−1}) . We define the fractional step approximations as the processes {Z_{Δ} (t)}_{0≤t≤T} , {Y_{Δ} (t)}_{0≤t≤T} defined by the following recursive procedure: Z_{Δ} (0) = ξ and for t ∈ [t_{i−1}, t_{i}) , i ≥ 1,

\{ \begin{matrix}Z_{Δ} (t) & = & Z_{Δ} (t_{i−1}) + ∫_{t_{i−1}}^{t} F (s, Z_{Δ} (s))ds + ∫_{t_{i−1}}^{t} G(s, Z_{Δ} (s))dW_{s} \\ Y_{Δ} (t_{i−1}) & = & Z_{Δ} (t_{i}−) , \end{matrix}

(3.20)

\{ \begin{matrix}Y_{Δ} (t) & = & Y_{Δ} (t_{i−1}) + ∫_{t_{i−1}}^{t} ∫_{0}^{t_{i−1}} F̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))ds_{1}ds_{2} \\ & + & ∫_{C_{2}(t_{i−1},t)} F̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))ds_{1}ds_{2} \\ & + & ∫_{t_{i−1}}^{t} ∫_{0}^{t_{i−1}} G̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))dW̃_{s_{1}}dW̃_{s_{2}} \\ & + & ∫_{C_{2}(t_{i−1},t)} G̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))dW̃_{s_{1}}dW̃_{s_{2}} \\ Z_{Δ} (t_{i}) & = & Y_{Δ} (t_{i}−) . \end{matrix}  (3.21)

Lemma 3.1 (see [13]) Let 0 ≤ δ < T, b, c > 0, with 4cδ < 1 and f : [0, T] ⟶ R_{+} be such that

f(t + s) − f(t) ≤ (s + δ) (b + cf(t + s)) , ∀s, t.

Then

f(t) ≤ (f(0) + \frac{b}{c}) e^{4ct+1} − \frac{b}{c}.

Proposition 3.2 (a) For every t ∈ [t_{i−1}, t_{i}) , i ≥ 1,the fractional step approximations (Z_{Δ} (t) , Y_{Δ} (t)) is solution of the system of double stochastic equations

Z_{Δ} (t) = ξ + ∫_{0}^{t} F (s, Z_{Δ} (s))ds + ∫_{0}^{t} G(s, Z_{Δ} (s))dW_{s}
+ ∫_{C_{2}(t_{i−1})} F̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))ds_{1}ds_{2} + ∫_{C_{2}(t_{i−1})} G̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))dW̃_{s_{1}}dW̃_{s_{2}}
Y_{Δ} (t) = ξ + ∫_{0}^{t_{i}} F (s, Z_{Δ} (s))ds + ∫_{0}^{t_{i}} G(s, Z_{Δ} (s))dW_{s}
+ ∫_{C_{2}(t)} F̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))dW̃_{s_{1}}dW̃_{s_{2}},

(3.22)

and Z_{Δ} (t) (resp. Y_{Δ} (t)) is F_{tt_{i−1}} = F_{t} ∨ F̃_{t_{i−1}} (resp. F_{t_{i}t} = F_{t_{i}} ∨ F̃_{t} ) measurable.

(b) There exist δ_{0} > 0 and constants γ_{i} = γ_{i} (E(|ξ|^{2}), T, K, K̃) < ∞, i = 1, 2, 3, such that for every ∥Δ∥ ≤ δ_{0},

E ( sup_{0≤t≤T} |Z_{Δ} (t)|^{2}) ≤ γ_{1},  (3.23)

E ( sup_{0≤t≤T} |Y_{Δ} (t)|^{2}) ≤ γ_{2},  (3.24)
sup_{0≤t≤T} E (|Z_{Δ} (t) − Y_{Δ} (t)|^{2}) ≤ γ_{3} ∥Δ∥ .  (3.25)

Proof. Define for t ∈ [t_{i−1}, t_{i})

f_{Δ}(t) = C_{1} {E (|ξ|^{2}) + ∫_{0}^{t_{i}} E (|F (s, Z_{Δ} (s))|^{2} + |G (s, Z_{Δ} (s))|^{2}) ds+

∫_{C_{2}(t_{i})} E (|F̃ (s_{1}, s_{2}, Y_{Δ} (s_{1}))|^{2} + |G̃ (s_{1}, s_{2}, Y_{Δ} (s_{1}))|^{2}) ds_{1}ds_{2}} ,

where C_{1} is a constant sufficiently large, which do not depent on Δ.

Of course we have

E ( sup_{0≤s≤t} |Z_{Δ} (s)|^{2}) ≤ f_{Δ}(t), E ( sup_{0≤s≤t} |Y_{Δ} (s)|^{2}) ≤ f_{Δ}(t).  (3.26)

Now let t ∈ [t_{i−1}, t_{i}) , t + s ∈ [t_{j−1}, t_{j}) , i ≤ j. If i = j then f_{Δ}(t + s) − f_{Δ}(t) = 0. If i < j then by (H_{1}), (H̃_{1}) and (3.26) we have for some constants a_{1}, a_{2}, a_{3} > 0, which are independent of Δ,

f_{Δ}(t + s) − f_{Δ}(t) ≤ ∫_{t_{i}}^{t_{j}} (a_{1} + a_{2}f_{Δ}(t_{j−1}) ds ≤

∫_{C_{2}(t_{j})\C_{2}(t_{i})} (a_{1} + a_{2}f_{Δ}(t_{j−1}) ds_{1}ds_{2} ≤
a_{3} (a_{1} + a_{2}f_{Δ}(t + s) (t_{j} − t_{i} + λ (C_{2}(t_{j})\C_{2}(t_{i}))) ≤
a_{3} (a_{1} + a_{2}f_{Δ}(t + s) (s + ∥Δ∥) .

Then by Lemma 3.1 we get (3.23), (3.24). Next from (3.22), \binom{}{H̃_{1}} , \binom{}{H̃_{2}} and (3.23), (3.24) it follows easily (3.25). ■

Theorem 3.3 Assume that (H_{1})−(H_{3}) , \binom{}{H̃_{1}} , \binom{}{H̃_{2}} are satisfied. If \{X_{t}\}_{0≤t≤T} is the unique solution of (2.1) then

lim_{∥Δ∥⟶0} sup_{0≤t≤T} E (|Z_{Δ} (t) − X_{t}|^{2}) = 0,  (3.27)

lim_{∥Δ∥⟶0} sup_{0≤t≤T} E (|Y_{Δ} (t) − X_{t}|^{2}) = 0.  (3.28)

Proof. Define

F̃(s_{1}, s_{2}, Z_{Δ}, Y_{Δ}) = \{ \begin{matrix}F̃(s_{1}, s_{2}, Y_{Δ}(s_{1}) − F̃(s_{1}, s_{2}, Z_{Δ}(s_{1}), & (s_{1}, s_{2}) ∈ C_{2}(t_{i−1}) \\ −F̃(s_{1}, s_{2}, Z_{Δ}(s_{1}), & (s_{1}, s_{2}) ∈ C_{2}(t_{i−1}, t_{i}) ∪ R_{Δ}, \end{matrix}

and G̃(s_{1}, s_{2}, Z_{Δ}, Y_{Δ}) similarly, where R_{Δ} = [0, t_{i−1}) × [t_{i−1}, t_{i}) . Then we have the equality

Z_{Δ} (t) = ∫_{0}^{t} F (s, Z_{Δ} (s))ds + ∫_{0}^{t} G(s, Z_{Δ} (s))dW_{s}+

∫_{C_{2}(t)} F̃(s_{1}, s_{2}, Z_{Δ}(s_{1}))ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, Y_{Δ}(s_{1}))dW̃_{s_{1}}dW̃_{s_{2}} + γ_{Δ}(t),

(3.29)

where

γ_{Δ}(t) = ∫_{C_{2}(t)} F̃(s_{1}, s_{2}, Z_{Δ}, Y_{Δ})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2}, Z_{Δ}, Y_{Δ})dW̃_{s_{1}}dW̃_{s_{2}}.

¿From (2.1) and (3.29) it follows

E (|Z_{Δ} (t) − X_{t}|^{2}) ≤ C_{2} {E (|γ_{Δ}(t)|^{2}) + ∫_{0}^{t} (α + α̃) (E (|Z_{Δ} (s) − X_{s}|^{2})) ds}

(3.30)

and (H_{1}), (H̃_{1}), (3.23)-(3.25) yield for ∥Δ∥ ≤ δ_{0},

sup_{t≤T} E (|γ_{Δ}(t)|^{2}) ≤ C_{3} ∥Δ∥ .

¿From (3.30) and Gronwall’s lemma we deduce

g(t) := \overline{lim_{∥Δ∥→0}} sup_{s≤t} E (|Z_{Δ} (s) − X_{s}|^{2}) ≤ C_{4} ∫_{0}^{t} (α + α̃) (E (g(s))) ds,

and hence g(t) ≡ 0. ■

Definition 3.4. Let (F_{i}, G_{i})_{i=1,2} satisfy (H_{1}), (H_{2}), \binom{}{H̃_{1}} , \binom{}{H̃_{2}} .We say that the comparison principle hold for (F_{i}, G_{i})_{i=1,2} if given two initial conditions ξ_{i}, i = 1, 2, ( ξ_{i} are F_{0}-measurable) with ξ_{1} ≤ ξ_{2} a.s. and strong solutions {X_{i}(t)} of the Itô equations

X_{i}(t) = ξ_{i} + ∫_{0}^{t} F_{i}(s, X_{i}(s))ds + ∫_{0}^{t} G_{i}(s, X_{i}(s))dW_{s}, 0 ≤ t ≤ T,

then X_{1}(t) ≤ X_{2}(t) a.s. for every t ≥ 0.

Theorem 3.5 (comparison). Let (ξ_{i}, F_{i}, G_{i})_{i=1,2} satisfy the hypotheses (H_{1})−

(H_{3}), \binom{}{H̃_{1}} , \binom{}{H̃_{2}} and let G̃(s_{1}, s_{2}) : C_{2}(T) ⟶ R be a measurable and bounded function. Let {X_{i}(t)} be the unique solution of the double stochastic equation with additive double noise

X_{i}(t) = ξ_{i} + ∫_{0}^{t} F_{i}(s, X_{i}(s))ds + ∫_{0}^{t} G_{i}(s, X_{i}(s))dW_{s}+

∫_{C_{2}(t)} F̃_{i}(s_{1}, s_{2}, X_{i}(s_{1}))ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T. (3.31)

Moreover suppose that:

(1) ξ_{1} ≤ ξ_{2} a.s..

(2) The comparison principle hold for (F_{i}, G_{i})_{i=1,2} .

(3) F̃_{1}(s_{1}, s_{2}, x) ≤ F̃_{2}(s_{1}, s_{2}, y) if x ≤ y, for all s_{i}. Then X_{1}(t) ≤ X_{2}(t) a.s. for every t ≥ 0.

Proof. By induction over the intervals [t_{i−1}, t_{i}) , i = 1, n, it is easily seen that the fractional step approximations satisfy

Z_{1,Δ}(t) ≤ Z_{2,Δ}(t), Y_{1,Δ}(t) ≤ Y_{2,Δ}(t), P -as for all t,

and passing to the limit (by Theorem 3.2) we obtain the conclusion. ■

Remark 3.6 The comparison result given by the previous Theorem is more useful if for the first order Itô equation the comparison principle holds only for markovian coefficients and not for functional ones. Comparison results for Itô equations can be found in [5] (and the references therein), [7], [9]. It should be interesting to extend the Theorem 3.5 to double stochastic Volterra equations, by using comparison results for first order stochastic Volterra equations as proven in [?], [17].

Theorem 3.7 Let (ξ, F, G) satisfy the hypotheses (H_{1}) − (H_{3}), \binom{}{H̃_{1}} , \binom{}{H̃_{2}}

and let F̃(s_{1}, s_{2}, x) : C_{2}(T) × R ⟶ R be bounded and continuous and G̃(s_{1}, s_{2}) : C_{2}(T) ⟶ R be a measurable and bounded function. Then the double stochastic equation

X_{t} = ξ + ∫_{0}^{t} F (s, X_{s})ds + ∫_{0}^{t} G(s, X_{s})dW_{s}+

∫_{C_{2}(t)} F̃(s_{1}, s_{2}, X_{s_{1}})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T, (3.32)

has a strong solution.

Proof. Choose a sequence F̃_{n}(s_{1}, s_{2}, x) of continuous, bounded and Lipschitz in x, which converges decreasing to F̃(s_{1}, s_{2}, x). Let X_{t}^{n} be the strong solutions of

X_{t}^{n} = ξ + ∫_{0}^{t} F (s, X_{s}^{n})ds + ∫_{0}^{t} G(s, X_{s}^{n})dW_{s}+

∫_{C_{2}(t)} F̃_{n}(s_{1}, s_{2}, X_{s_{1}}^{n})ds_{1}ds_{2} + ∫_{C_{2}(t)} G̃(s_{1}, s_{2})dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T.

It is clear that the comparison principle hold for (F, G) (see for example [7]). Thenb by Theorem 3.5 we have

X_{t}^{1} ≥ X_{t}^{2} ≥ ... ≥ X_{t}^{n} ≥ ..... P -a.s for all t ≥ 0.

It is easily seen that lim_{n→∞} X_{t}^{n} = X_{t} is a solution of (3.32). ■

Next we give an example (Lemma 3.9) which shows that the comparison principle of Theorem 3.5 is not satisfied if the diffusion coefficient of the double stochastic integral depends on the state variable. For f ∈ L_{loc}^{2}(R_{+}) and 0 ≤ t ≤ T we define

Z_{t}^{f} = exp \{∫_{0}^{t} f(s)dW_{s} − \frac{1}{2} ∫_{0}^{t} |f(s)|^{2} ds\} ,

Φ_{t}^{f} = exp {−2 ∫_{0}^{t} f(s)dW_{s}} .

Lemma 3.8 (a) The unique solution of the linear double stochastic equation

X_{t}^{f} = 1 + ∫_{C_{2}(t)} f(s_{1})f(s_{2})X_{s_{1}}^{f}dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T,  (3.33)

is given by

X_{t}^{f} = \frac{1}{2}Z_{t}^{f} (1 + Φ_{t}^{f}) = \frac{1}{2} (Z_{t}^{f} + Z_{t}^{(−f)}) .  (3.34)

(b) If (H_{t}) is a continuous semimartingale, then the unique solution of the double stochastic equation

X_{t} = H_{t} + ∫_{C_{2}(t)} f(s_{1})f(s_{2})X_{s_{1}}dW̃_{s_{1}}dW̃_{s_{2}}, 0 ≤ t ≤ T,  (3.35)

is given by

X_{t} = X_{t}^{f}H_{0} + X_{t}^{f} ∫_{0}^{t} \frac{(X_{s}^{f}dH_{s} − f(s)d [H, W̃ ]_{s})}{(|X_{s}^{f} |^{2} − |Y_{s}^{f} |^{2})^{2}}

−Y_{t}^{f} ∫_{0}^{t} \frac{Y_{s}^{f}dH_{s}}{(|X_{s}^{f} |^{2} − |Y_{s}^{f} |^{2})^{2}} ,  (3.36)
Y_{t}^{f} = X_{t}^{f} − 1,  (3.37)

where [., .] denotes the quadratic variation. Proof. (a) We have

∫_{C_{2}(t)} f(s_{1})f(s_{2})X_{s_{1}}^{f}dW̃_{s_{1}}dW̃_{s_{2}} = ∫_{0}^{t} f(s_{2}) (∫_{0}^{s_{2}} f(s_{1})X_{s_{1}}^{f}dW̃_{s_{1}}) dW̃_{s_{2}} =

\frac{1}{2} ∫_{0}^{t} f(s_{2}) [∫_{0}^{s_{2}} f(s_{1}) (Z_{s_{1}}^{f} + Z_{s_{1}}^{(−f)}) dW̃_{s_{1}}] dW̃_{s_{2}} =

\frac{1}{2} ∫_{0}^{t} f(s_{2}) (Z_{s_{2}}^{f} − 1 − Z_{s_{1}}^{(−f)} + 1) dW̃_{s_{2}} =

\frac{1}{2} (Z_{t}^{f} − 1 + Z_{s_{1}}^{(−f)} − 1) = X_{t}^{f} − 1.

(b) The previous computation shows that Y_{t}^{f} = ∫_{0}^{t} f(s_{1})X_{s_{1}}^{f}dW̃_{s_{1}} = X_{t}^{f} − 1.

Define

f̃(t) = ( \begin{matrix}f(t) & 0 \\ 0 & f(t) \end{matrix} ) , V_{t} = ( \begin{matrix}X_{t}^{f} & Y_{t}^{f} \\ Y_{t}^{f} & X_{t}^{f} \end{matrix} ) ,

U_{t}^{T} = (X_{t}^{f}, Y_{t}^{f}) , X̃_{t}^{T} = (X_{t}, Y_{t}) , H̃_{t}^{T} = (H_{t}, 0) .

Then

U_{t} = e_{1} + ∫_{0}^{t} f̃(s)U_{s}dW̃_{s},
X̃_{t} = H̃_{t} + ∫_{0}^{t} f̃(s)X̃_{s}dW̃_{s}.

By Theorem 4 of [19] it follows that

X̃_{t} = V_{t}H̃_{0} + V_{t} ∫_{0}^{t} V_{s}^{−1} (X_{s}^{f}dH_{s} − f(s)d [H, W̃ ]_{s})

−Y_{t}^{f} ∫_{0}^{t} \frac{Y_{s}^{f}dH_{s}}{(|X_{s}^{f} |^{2} − |Y_{s}^{f} |^{2})^{2}} ,

which implies (3.36). ■ We consider the following ”pure” double stochastic equations

X_{1}(t) = 1 + ∫_{C_{2}(t)} X_{1}(s_{1})dW̃_{s_{1}}dW̃_{s_{2}},

X_{2}(t) = 1 + \frac{t^{2}}{2} + ∫_{C_{2}(t)} X_{2}(s_{1})dW̃_{s_{1}}dW̃_{s_{2}}.

By Lemma 3.8 with f ≡ 1, H_{t} = \frac{t^{2}}{2} we have

X_{1}(t) = \frac{1}{2} (Z_{t}^{1} + Z_{t}^{(−1)}) , X_{2}(t) = X_{1}(t) + R(t),

R(t) = ∫_{0}^{t} \frac{2X^{1}(s)ds}{(2X^{1}(s) − 1)^{2}} +

X_{1}(t) ∫_{0}^{t} \frac{sds}{(2X^{1}(s) − 1)^{2}} − ∫_{0}^{t} \frac{sds}{(2X^{1}(s) − 1)^{2}} .

Lemma 3.9 We have that

lim_{t⟶∞} P (R(t) < 0) = 1,

Therefore the comparison principle is not satisfied for the solutions X_{1}, X_{2}(Note that in the present situation ξ_{1} = ξ_{2} = 1, F̃_{1} = 0 < F̃_{2} = 1).

Proof. It is known that

lim_{t⟶∞} Z_{t}^{1} = lim_{t⟶∞} Z_{t}^{(−1)} = 0,

and hence lim_{t⟶∞} X_{1}(t) = 0. Define

δ(t) = [∫_{0}^{t} \frac{sX_{1}(s)ds}{(2X^{1}(s) − 1)^{2}} ] [∫_{0}^{t} \frac{sds}{(2X^{1}(s) − 1)^{2}} ]^{−1} .

Then

P (R(t) < 0) = P (1 − X^{1}(t) > δ(t)) a.s.,

and all we need to show is that lim_{t⟶∞} δ(t) = 0 a.s. Obviously

lim_{t⟶∞} ∫_{0}^{t} \frac{sds}{(2X^{1}(s) − 1)^{2}} = ∞ a.s.

Assume that

lim_{t⟶∞} ∫_{0}^{t} \frac{sds}{(2X^{1}(s) − 1)^{2}} = σ.

If σ < ∞ then lim_{t⟶∞} δ(t) = 0 and if σ = 0 tha by l’Hôspital rule

lim_{t⟶∞} δ(t) = lim_{t⟶∞} X_{1}(t) = 0. ■

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