# Stochastic Volterra Equation Driven by Wiener Process and Fractional Brownian Motion

Authors: Zhi Wang; Litan Yan
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

For a mixed stochastic Volterra equation driven by Wiener process and fractional Brownian motion with Hurst parameterH>1/2, we prove an existence and uniqueness result for this equation under suitable assumptions.


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## Body

## 1. Introduction

In recent years, there has been considerable interest in studying fractional Brownian motion (fBm) due to some of its compact properties such as self-similarity, stationary increments, Hölder's continuity, and long-range dependence (whenH>1/2), and also due to its applications in various scientific areas including telecommunications, turbulence, image processing, finance, and other fields. It is a suitable generalization of standard Wiener process. Some surveys and complete literatures could be found in Biagini et al. [1], Hu [2], Mishura [3], Nualart [4], and Zähle [5]. The so-called fBm with index H∈(0,1) is a zero mean Gaussian process BH={BtH,t≥0} with the covariance:
(1)E[BtHBsH]=12[s2H+t2H-|t-s|2H].SinceBH is neither a semimartingale nor a Markov process unless H=1/2, the classical Itô theory is not available when dealing with it. There are several approaches to define an integral with respect to fBm. One possibility is Skorokhod or divergence integral introduced in the fractional Brownian setting in Decreusefond and Üstünel [6]. We will use a pathwise integral, defined first in Zähle [5], for fBm with H>1/2 as a Young integral. The aim of this paper is to study the following mixed stochastic Volterra equation on ℝd:
(2)Xt=X0+∫0t‍a(t,s,Xs)ds+∫0t‍b(t,s,Xs)dWs+∫0t‍c(t,s,Xs)dBsH,t∈[0,T],
where  a=(ai)d×1:[0,T]2×ℝd→ℝd, b=(bi,j)d×r:[0,T]2×ℝd→ℝd×ℝr, c=(ci,j)d×m:[0,T]2×ℝd→ℝd×ℝm, W is an r-dimensional standard Wiener process, and BH is an m-dimensional fBm with H>1/2. We assume that the processes W and BH are independent, the integral with respect to W is Itô type, and the integral with respect to BH is a pathwise Riemann-Stieltjes integral in the sense of Zähle [5].The mixed stochastic differential equation:(3)Xt=X0+∫0t‍a(s,Xs)ds+∫0t‍b(s,Xs)dWs+∫0t‍c(s,Xs)dBsH,t∈[0,T]
was first considered in Kubilius [7], where unique solvability was proved in one-dimensional case for time-independent coefficients and zero drift; that is, a=0. Later, in Zähle [8], existence of a solution to (3) was proved under less restrictive assumptions, but only locally, that is, up to a random time. In Guerra and Nualart [9], global existence and uniqueness of solution to (3) was established under the assumption that W and BH are independent. The latter result was obtained in Mishura and Shevchenko [10, 11] without the independence assumptions, and it was also shown in Mishura and Shevchenko [11] that all moments of the solution are finite for H>3/4. The motivation to consider such equations comes from some financial applications, where Wiener process as a model is inappropriate because of the lack of memory, and fBm with H>1/2 is too smooth. A model driven by both processes is free of such drawbacks. Stochastic Volterra equations are also studied by many authors (see, e.g., [12–16]). In this paper, we consider the mixed stochastic Volterra equations driven by both Wiener process and fBm, and also give an existence and uniqueness result for this equation under suitable assumptions.This paper is organized as follows. Section2 contains some preliminaries on function spaces and lists our assumptions on the coefficients of (2). It also states the main results in this section. In Section 3, we will give some estimates of fractional and Itô integrals, based on the fractional calculus. Finally, in Section 4, we will prove Theorem 1 using the classical Yamada and Watanabe theorem [17, 18] to check the pathwise uniqueness property and the existence of weak solutions of (2).


## 2. Preliminaries and Main Results

Let(Ω,ℱ,P) be a complete probability space equipped with a filtration satisfying standard assumptions. Denote BH={BtH,t∈[0,T]} an m-dimensional fBm with H>1/2 and W={Wt,t∈[0,T]} an r-dimensional standard Wiener process, independent of BH. X0 is a d-dimensional random variable independent of (BH,W). For each t∈[0,T], we denote by ℱt the σ-field generated by the random variables {X0,BsH,Ws,s∈[0,t]} and the P-null sets. {𝒢t,t∈[0,T]} denotes the bigger filtrations such that {𝒢t} is right-continuous and 𝒢0 contains the P-null sets, X0, BH are 𝒢0-measurable, and W is a 𝒢t-Wiener process.Let us now introduce some function spaces for later use. Leting1/2<H<1  and α∈(1-H,1/2), for any measurable function f:[0,T]→ℝd, we introduce the following notation:
(4)∥f(t)∥α≔|f(t)|+∫0t‍|f(t)-f(s)|(t-s)α+1ds.
Denote by W0α,∞ the space of measurable function f:[0,T]→ℝd such that
(5)∥f∥α,∞≔supt∈[0,T]∥f(t)∥α<∞.
For any 0<λ≤1, let Cλ be space of λ-Hölder continuous functions f:[0,T]→ℝd such that
(6)∥f∥λ≔∥f∥∞+sup0≤s<t≤T|f(t)-f(s)|(t-s)λ<∞,
where ∥f∥∞≔supt∈[0,T]|f(t)|. Given any 0<ɛ<α, we have the following inclusions:
(7)Cα+ɛ⊂W0α,∞⊂Cα-ɛ.
Notice that both the fBm BH with H>1/2 and the standard Wiener process W have their trajectories in W0α,∞.In what follows, we will assume the following standard hypotheses. Throughout the paper, the symbolC will denote a generic constant, whose value is not significant and can change from one line to another. To emphasize its dependence on some parameters, we will put them into subscripts consider the following.  (H1)
a:[0,T]2×ℝd→ℝd is a measurable function such that there exists a0∈Lρ([0,T]2;ℝd) with ρ≥2, 0<μ≤1 and for all N≥0 there exists C>0 such that
(8)|a(t,s,x)-a(t,s,y)|≤C|x-y|,|a(t1,s,x)-a(t2,s,x)|≤C|t1-t2|μ,|a(t,s,x)|≤C(1+|x|),|a(t1,s,x1)-a(t1,s,x2)-a(t2,s,x1)+a(t2,s,x2)|≤C|t1-t2||x1-x2|,
for all x,y,x1,x2∈ℝd and s,t,t1,t2∈[0,T].(H2)
b:[0,T]2×ℝd→ℝd×ℝm is a measurable function and there exists a constant C>0 such that
(9)|b(t,s,x)-b(t,s,y)|≤C|x-y|,|b(t,s,x)|≤C(1+|x|),
for all x,y∈ℝd and s,t∈[0,T].(H3)
c:[0,T]2×ℝd→ℝd×ℝm is a measurable function. Moreover, there exist the derivatives ∂xc(t,s,x), ∂tc(t,s,x),  ∂x,t2c(t,s,x) and  constants 0<β, μ,δ≤1 and C>0 such that the following properties hold:
(10)|c(t,s,x)-c(t,s,y)|+|∂tc(t,s,x)-∂tc(t,s,y)|≤C|x-y|,|∂xic(t,s,x)-∂yic(t,s,y)|+|∂xi,t2c(t,s,x)-∂yi,t2c(t,s,y)|≤C|x-y|δ,|c(t1,s,x)-c(t2,s,x)|+|∂xic(t1,s,x)-∂xic(t2,s,x)|≤C|t1-t2|μ,|c(t,s1,x)-c(t,s2,x)|+|∂tc(t,s1,x)-∂tc(t,s2,x)|≤C|s1-s2|β,|∂xi,t2c(t,s1,x)-∂xi,t2c(t,s2,x)|+|∂xic(t,s1,x)-∂xic(t,s2,x)|≤C|s1-s2|β,
for all x,y∈ℝd, s,t,t1,t2,s1,s2∈[0,T], i=1,2,…,d.Denote byEW the conditional expectation given ℱ0, that is, given X0 and BH. A strong solution of (2) is a d-dimensional ℱt-adapted stochastic process {Xt,t∈[0,T]} such that a.s. the trajectories of X belong to W0α,∞ and ∫0T‍EW∥Xs∥α2ds<∞, which satisfies (2) a.s. The main result of our paper is the following theorem on the existence and uniqueness of a solution for (2).Theorem 1.
Suppose that the coefficientsa,b, and c satisfy the assumptions (H1), (H2), and (H3) with β>1-H,δ>1/H-1, min{β,δ/(δ+1)}>1-μ, respectively. If  max{1-H,1-μ}<α<min{1/2,β,δ/(δ+1)} and ρ≤1/α, then (2) has a unique strong solution X.Notice that in all our results we can replace the fBmBH with an arbitrary stochastic process with Hölder continuous trajectories of order λ>1/2.


## 3. Integral Estimates

In this section, we will first briefly recall some basic definitions of fractional integrals and derivatives and define the integral with respect to fBm as a generalized Stieltjes integral (see, e.g., Nualart and Răşcanu [19] and Zähle [5]). We will also give some estimates of this integral.Letf∈L1(a,b) and let 0<α<1. Consider the left-sided and right-sided fractional Riemann-Liouville integrals of f of order α:
(11)Ia+αf(x)≔1Γ(α)∫ax‍f(t)(x-t)α-1dt,Ib-αf(x)≔1Γ(α)∫xb‍f(t)(t-x)α-1dt,
if the integrals exist for almost all x∈(a,b) respectively, where Γ(·) denotes the gamma function defined by
(12)Γ(z)=∫0∞‍tz-1e-tdt,z>0.
Let Ia+α(Lp) (resp., Ib-α(Lp)) be the image of Lp(a,b), by the operator Ia+α (resp., Ib-α). If f∈Ia+α(Lp) (resp., Ib-α(Lp)) then the Weyl derivatives of f:
(13)Da+αf(x)≔≔1Γ(1-α)(∫xb‍f(x)(x-a)αDa+αf(x)≔+aα+α∫ax‍f(x)-f(t)(x-t)α+1dt)1(a,b)(x),Db-αf(x)≔≔1Γ(1-α)(∫xb‍f(x)(b-x)αDa+αf(x)===≔+α∫xb‍f(x)-f(t)(t-x)α+1dt)1(a,b)(x)
are defined for almost all x∈(a,b) (the convergence of the integrals at the singularity t=x holds pointwise for almost all x∈(a,b) if p=1 and moreover in Lp-sense if 1<p<∞), respectively.Letf(a+)≔limɛ↘0f(a+ɛ) and let g(b-)≔limɛ↘0g(b-ɛ) (supposing that the limits exist and are finite) and define
(14)fa+(x)≔(f(x)-f(a+))1(a,b)(x),gb-(x)≔(g(x)-g(b-))1(a,b)(x).
Assuming that f(a+), g(a+), g(b-) exists, fa+∈Ia+α(Lp) and gb-∈Ib-1-α(Lp) for some p,q≥1, (1/p)+(1/q)≤1, 0<α<1, the generalized Stieltjes Integral is defined as
(15)∫ab‍fdg≔(-1)α∫ab‍Da+αfa+(x)Db-1-αgb-(x)dx+f(a+)(g(b-)-g(a+)).
The following properties hold.  (i)
Ifαp<1, under the preceding assumptions we have f∈Ia+α(Lp) and we can write
(16)∫ab‍fdg≔(-1)α∫ab‍Da+αfa+(x)Db-1-αgb-(x)dx.(ii)
Iff∈Cλ(a,b) and g∈Cμ(a,b) with λ+μ>1, then the generalized Stieltjes integral exists; it is given by (16) and coincides with the Riemann-Stieltjes integral.The linear spacesIa+α(Lp) are Banach spaces with respect to the norms:
(17)∥f∥Ia+α(Lp)=∥f∥Lp+∥Da+αf∥Lp~∥Da+αf∥Lp,
and the same is true for Ib-α(Lp). If αp<1 then the norms on Ia+α(Lp) and Ib-α(Lp) are equivalent and if a≤c<d≤b, then
(18)∫cd‍fdg=∫ab‍1(c,d)fdg.
Fix a parameter 0<α<1/2; denote by WT1-α,∞ the space of measurable functions g:[0,T]→ℝ such that
(19)∥g∥1-α,∞,T≔sup0<s<t<T(∫st‍|g(t)-g(s)|(t-s)1-α≔sup0<s<t<T===+∫st‍|g(y)-g(s)|(y-s)2-αdy)<∞.
Clearly,
(20)C1-α+ɛ⊂WT1-α,∞⊂C1-α,
for all ɛ>0. Moreover, if g belongs to WT1-α,∞, we define
(21)Λα(g)≔1Γ(1-α)sup0<s<t<T|(Dt-1-αgt-)(s)|≤1Γ(1-α)Γ(α)∥g∥1-α,∞,T<∞.
We also denote by W0α,1 the space of measurable functions f:[0,T]→ℝ such that
(22)∥f∥α,1≔∫0T‍|f(s)|sαds+∫0T‍∫0s‍|f(s)-f(y)|(s-y)1+αdyds<∞.
Then, if f∈W0α,1 and g∈WT1-α,∞, the integral ∫0t‍f(s)dgs exists for all t∈[0,T] and we have
(23)∫0t‍f(s)dgs=∫0T‍f(s)1(0,t)(s)dgs.
Furthermore, the following estimate holds:
(24)|∫0t‍f(s)dgs|≤Λα(g)∥f∥α,1.
Given f:[0,T]2→ℝ such that for any t∈[0,T], f(t,·)∈W0α,1, we can also consider the integral:
(25)Gt(f)=∫0t‍f(t,s)dgs=∫0T‍f(t,s)1(0,t)(s)dgs
and the estimate:
(26)|∫0t‍f(t,s)dgs|≤Λα(g)∥f(t,·)∥α,1.Now we will derive some useful estimates for the integrals involved in (2). Fix a parameter α∈(0,1/2). Consider first the ordinary Lesbesgue integral. Given f:[0,T]→ℝd a measurable function we define
(27)Fta(f)=∫0t‍a(t,s,f(s))ds.Proposition 2.
Assume thata satisfies (H1) with ρ=1/α and μ>α. Let f:[0,T]→ℝd be a measurable function. If f∈W0α,∞, then F·a(f)∈W0α,∞, for all t∈[0,T], one has
(28)∥Fta(f)∥α≤C(∫0t‍|f(s)|(t-s)αds+1).Proof.
It follows from Proposition 2.1 in Besalú and Rovira [15] and the growth assumption in (H1) that
(29)|Fta(f)|+∫0t‍|Fta(f)-Fsa(f)|(t-s)α+1ds≤C(∫0t‍|a(t,s,f(s))|(t-s)αds+1)≤C(∫0t‍1+|f(s)|(t-s)αds+1)≤C(∫0t‍|f(s)|(t-s)αds+1),
as required.Proposition 3.
Assume thata satisfies (H1) with P=1/α and μ>α. Let f,h:[0,T]→ℝd be measurable functions. If f,h∈W0α,∞, then for all t∈[0,T], one has
(30)∥Fta(f)-Fta(h)∥α≤C(∫0t‍∥f(s)-h(s)∥α(t-s)αds+1).Proof.
By Proposition 2.1 in Besalú and Rovira [15] and the growth assumption in (H1), one obtains
(31)∥Fta(f)-Fta(h)∥α≤C(∫0t‍|a(t,s,f(s))-a(t,s,h(s))|(t-s)αds+1)≤C(∫0t‍|f(s)-h(s)|(t-s)αds+1)≤C(∫0t‍∥f(s)-h(s)∥α(t-s)αds+1).Given a measurable functionf:[0,T]→ℝd, let us define
(32)Gtc(f)=∫0t‍c(t,s,f(s))dBsH.
Notice that if f∈W0α,∞, one has c(t,·,f(·))∈W0α,∞.Proposition 4.
Assume thatc satisfies (H3) with 1-μ<α<β, f∈W0α,∞, then Gtc(f)∈W0α,∞ for all t∈[0,T] and we have
(33)∥Gtc(f)∥α≤CΛα(BH)∥Gtc(f)∥α≤×∫0t‍((t-s)-2α+s-α)∥Gtc(f)∥α≤×(1+∥f(s)∥α)ds.Proof.
It follows from Proposition 3.1 in Besalú and Rovira [15] and the growth assumption in (H3) that
(34)∥Gtc(f)∥α≤CΛα(BH)∥Gtc(f)∥α≤×(∫0t‍((t-s)-2α+s-α)∥Gtc(f)∥α≤h×∥c(t,·,f(·))∥αds+1∫0t‍)∥Gtc(f)∥α≤CΛα(BH)∫0t‍((t-s)-2α+s-α)∥Gtc(f)∥α≤CΛα(BH)∫0t‍×(1+∥f(s)∥α)ds.Proposition 5.
Assume thatc satisfies (H3) with 1-μ<α<β, f,h∈W0α,∞, then for all t∈[0,T] one has
(35)∥Gtc(f)-Gtc(h)∥α≤CΛα(BH)∫0t‍((t-s)-2α+s-α)∥Gtc(f)-Gtc(h)∥α≤CΛα(BH)hh×(1+Δf(s)+Δh(s))∥Gtc(f)-Gtc(h)∥α≤CΛα(BH)hh×∥f(s)-h(s)∥αds,
where
(36)Δf=sups∈[0,T]∫0s‍|f(s)-f(u)|δ(s-u)α+1du.Proof.
Following Proposition 3.1 in Besalú and Rovira [15] and the assumptions in (H3), we get that
(37)∥Gtc(f)-Gtc(h)∥α≤CΛα(BH)[∫0t‍((t-s)-2α+s-α)×∥c(t,·,f(·))-c(t,·,h(·))∥αds+1∫0t‍]≤CΛα(BH)[∫0t‍((t-s)-2α+s-α)≤CΛα(BH)hh×(∫0t‍|c(t,s,f(s))-c(t,s,h(s))|≤CΛα(BH)hhhhhh+∫0s‍(∫0s‍|c(t,s,f(s))-c(t,s,h(s))≤CΛα(BH)hhhhhhhhhhhhh-c(t,u,f(u))+c(t,u,h(u))|≤CΛα(BH)hhhhhhhhhhhhh×((s-u)α+1)-1∫0s‍)du∫0t‍)∫0t‍ds+1].
By Lemma A.1 of Besalú and Rovira [15], the hypothesis (H3) imply that for all xi,∈ℝ, i=1,…,4, t,s1,s2∈[0,T], one has
(38)|c(t,s1,x1)-c(t,s2,x2)-c(t,s1,x3)+c(t,s2,x4)|≤C(+|x1-x3|(|x1-x2|δ+|x3-x4|δ)|x1-x2-x3+x4|+|x1-x3||s2-s1|β+|x1-x3|(|x1-x2|δ+|x3-x4|δ)).
Thus, we obtain that
(39)∥Gtc(f)-Gtc(h)∥α≤CΛα(BH)[∫0t‍((t-s)-2α+s-α)(1+Δf(s)+Δh(s))≤CΛα(BH)hh×(∫0s‍|f(s)-h(s)|≤CΛα(BH)hhhhhh+∫0s‍(((s-u)α+1)-1|f(s)-h(s)-f(u)+h(u)|≤CΛα(BH)hhhhhhhhhh×((s-u)α+1)-1)du∫0t‍)ds+1∫0t‍]≤CΛα(BH)∫0t‍((t-s)-2α+s-α)(1+Δf(s)+Δh(s))×∥f(s)-h(s)∥αds.
This completes the proof.Finally, we will consider the estimate of Itô integral based on the Itô calculus. Letf:[0,T]→ℝd be a measurable function and define
(40)Gtb(f)=∫0t‍b(t,s,f(s))dWs.Proposition 6.
Letf:[0,T]→ℝd be a 𝒢t-adapted stochastic process. If f∈W0α,∞ and ∫0T‍EW∥f(s)∥α2ds<∞ a.s., then for all t∈[0,T] a.e., one has
(41)EW∥Gtb(f)∥α2≤C∫0t‍(t-s)-(1/2)-α[1+EW∥f(s)∥α2]ds.Proof.
It follows from Lemma 3.7 in Guerra and Nualart [9] and the growth assumption in (H2).Proposition 7.
Letf,h:[0,T]→ℝd be 𝒢t-adapted stochastic processes. If f,h∈W0α,∞ and ∫0T‍EW∥ϕ(s)∥α2ds<∞, ϕ∈{f,h} a.s., then for all t∈[0,T] a.e., one has
(42)EW∥Gtb(f)-Gtb(h)∥α2≤C∫0t‍(t-s)-(1/2)-αEW|f(s)-h(s)|2ds.Proof.
By Lemma 3.7 in Guerra and Nualart [9] and the Lipschitz assumption in (H2), one has
(43)EW∥Gtb(f)-Gtb(h)∥α2≤C∫0t‍(t-s)-(1/2)-αEW|b(t,s,f(s))-b(t,s,h(s))|2ds≤C∫0t‍(t-s)-(1/2)-αEW|f(s)-h(s)|2ds,
which completes the proof.


## 4. Proof of the Theorem1

To prove Theorem1, we will use the classical Yamada and Watanabe theorem [17, 18], which asserts that pathwise uniqueness and existence of weak solutions imply the existence of a strong solution. A weak solution of (2) in this paper is a triple (Ω,ℱ,P), (X,BH,W), {𝒢t,t∈[0,T]} such that:(i)
(Ω,ℱ,P) is a complete probability space, {𝒢t} is right-continuous, and 𝒢0 contains the P-null sets,(ii)
BH is a fBm which is 𝒢0-measurable,(iii)
W is a 𝒢t-r-dimensional Wiener process,(iv)
X is 𝒢t-adapted and has the trajectories belong to W0α,∞ and ∫0T‍EW∥Xs∥α2ds<∞ a.s,(v)
(X,BH,W) satisfies (2) a.s.The pathwise uniqueness property holds for (2) means that if (X,BH,W) and (Y,BH,W) are two weak solutions, defined on the same probability space (Ω,ℱ,P) with the same filtration {𝒢t} and X0=Y0 a.s., then X=Y.Letf∈W0α,∞. By Propositions 2.2 and 3.2 in Besalú and Rovira [15], the sample paths of the integral processes Gta(f) and Gtc(f) are continuously differentiable and η-Hölder continuous of order η<1-α, respectively. Therefore, if X is a weak solution of (2), then the trajectories of X are η-Hölder continuous for all η<1/2. Consequently, the pathwise uniqueness property holds for (2). It is exactly the same result as that of  Theorem 4.4 given in Guerra and Nualart [9]. So it remains to prove that the existence of weak solutions for (2).Let us now introduce the Euler approximation for (2). Take a sequence of partitions
(44)0=t0n<t1n<⋯<tin<⋯<tnn=T
of the interval [0,T] such that
(45)sup0≤i≤n-1|ti+1n-tin|⟶0
as n→∞. Defined as Xt0=X0 and for all n≥1,
(46)Xtn=X0+∫0t‍a(t,vn(s),Xvn(s)n)ds+∫0t‍b(t,vn(s),Xvn(s)n)dWs+∫0t‍c(t,vn(s),Xvn(s)n)dBsH,
where vn(t)=max{tin:tin≤t}. We first prove the tightness of the law of the sequence {Xn(t)} in the space C0η of η-Hölder continuous functions, with η<1/2, such that
(47)limɛ→0sup0<|t-s|<ɛ|f(t)-f(s)|(t-s)η=0.
Using the criterion given by Hamadouche [20], is sufficient to prove the following lemma.Lemma 8.
For anyN≥1, there exists a random variable RN>0, depending on X0 and BH, such that
(48)EW[|Xtn-Xsn|2N]≤RN|t-s|N
for all s,t∈[0,T] and n∈ℕ.Proof.
Firstly, we will show that there exists a random variableKN>0 such that(49)EW∥Xtn∥α2N≤KN
for all t∈[0,T] and n∈ℕ. In fact, from (46), we have
(50)EW∥Xtn∥α2N≤CN{|X0|2N+EW∥∫0t‍a(t,vn(s),Xvn(s)n)ds∥α2NEW∥Xtn∥α2N≤CNh+EW∥∫0t‍b(t,vn(s),Xvn(s)n)dWs∥α2NEW∥Xtn∥α2N≤CNh+EW∥∫0t‍c(t,vn(s),Xvn(s)n)dBsH∥α2N}EW∥Xtn∥α2N≔CN(|X0|2N+A1+A2+A3).
Using the estimate (28) and Hölder inequality, we get that
(51)A1≤CNEW(∫0t‍|Xvn(s)n|(t-s)αds+1)2N≤CNEW(∫0t‍|Xvn(s)n|2ds)N+CN≤CNEW(∫0t‍|Xvn(s)n|2Nds)+CN.
And also
(52)A2≤CNEW|∫0t‍b(t,vn(s),Xvn(s)n)dWs|2N+CNEW(∫0t‍|∫st‍b(t,vn(u),Xvn(s)n)dWu|(t-s)α+1ds)2N≔A21+A22.
By Burkhölder-Davis-Gundy and Hölder inequalities, we obtain
(53)A21≤CNEW[∫0t‍|b(t,vn(s),Xvn(s)n)|2Nds]≤CN∫0t‍[1+EW|Xvn(s)n|2N]ds.
Similarly, we get
(54)A22≤CN(∫0t‍1(t-s)(2N/(2N-1))(α+(1/2)-(((1/2)+α)/2N))ds)2N-1×EW(∫0t‍|∫st‍b(t,vn(u),Xvn(s)n)dWu|2N(t-s)α+N+(1/2)ds)≤CN∫0t‍(t-s)-(3/2)-α×EW[∫st‍|b(t,vn(u),Xvn(s)n)|2Ndu]ds.
Now applying Fubini’s theorem and using the growth assumption in (H2), we have
(55)A22≤CN∫0t‍(t-u)-(1/2)-α(1+EW|Xvn(s)n|2N)du.
As a consequence,
(56)A2≤CN∫0t‍(t-u)-(1/2)-αEW|Xvn(s)n|2Ndu+CN.
Further, applying the estimate (33) and Hölder inequality, we get
(57)A3≤CNΛα(BH)2N×EW[∫0t‍((t-s)-2α+s-α)×(∥c(t,vn(s),Xvn(s)n)∥α)ds∫0t‍]2N≤CNΛα(BH)2N∫0t‍((t-s)-2α+s-α)×[1+EW∥Xvn(s)n∥α2N]ds.
Combining these estimates, we have
(58)EW∥Xtn∥α2N≤CN|X0|2N+CN(Λα(BH)2N+1)×∫0t‍((t-s)-(1/2)-α+s-α)×EW∥Xvn(s)n∥α2Nds.
Hence,
(59)sup0≤s≤tEW∥Xsn∥α2N≤CN|X0|2N+CN(Λα(BH)2N+1)×∫0t‍((t-s)-(1/2)-α+s-α)×(sup0≤u≤sEW∥Xun∥α2N)ds.
Therefore, by the generalized Gronwall lemma (Lemma 7.6 in Nualart and Răşcanu [19]), the estimate (49) is satisfied.
So it remains to prove that the estimate (48) is satisfied. Indeed, from (46), we have
(60)EW|Xtn-Xsn|2N≤CN{EW|∫st‍a(t,vn(u),Xvn(s)n)du|2N+EW|∫st‍b(t,vn(u),Xvn(s)n)dWu|2N+EW|∫st‍c(t,vn(u),Xvn(s)n)dBuH|2N}:=B1+B2+B3.
It follows from the growth assumption in (H1), the estimate (49), and Hölder inequality that
(61)B1≤CN(t-s)2N-1×∫st‍EW|a(t,vn(u),Xvn(s)n)|2Ndu≤RN(t-s)2N.
By Burkhölder-Davis-Gundy and Hölder inequalities and using the estimate (49), we obtain
(62)B2≤CN(t-s)N-1×EW[∫st‍|b(t,vn(u),Xvn(s)n)|2Ndu]≤RN(t-s)N.
Finally, using the estimates (24) and (49), we get
(63)B3≤CNEWΛα(BH)2N(t-s)2N(1-α)+2α-1×∫st‍∥c(t,vn(u),Xvn(s)n)∥α2N(u-s)2αdu≤RN(t-s)2N(1-α)+2α-1×EW∫st‍1+∥Xvn(s)n∥α2N(u-s)2αdu≤RN(t-s)N.
Summing up, we deduce the desired result.LetPn=P∘Xn, n≥0, be the sequence of probability measure induced by Xn on C0η, then the sequence is tight (a similar result to Proposition 5.2 in Guerra and Nualart [9]).Now, we can state that there exists a weak solution for (2), which follows the same computations to those given in Theorem 5.3 of Guerra and Nualart [9].Theorem 9 (existence of weak solution).
Suppose that the coefficientsa, b, and c satisfy the assumptions (H1), (H2), and (H3) with β>1-H, δ>(1/H)-1,  min{β,δ/(δ+1)}>1-μ, respectively. If  max{1-H,1-μ}<α<min{1/2,β,δ/(δ+1)} and ρ≤1/α, then (2) has a unique weak solution.Therefore, by the classical Yamada and Watanabe theorem [17, 18], Theorem 1 will be an immediate consequence of the pathwise uniqueness property and the existence of weak solutions (Theorem 9).



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*Source: 579013-2013-11-17.xml*