The existence and uniqueness of solution of stochastic differential equation driven by standard Brownian motion and fractional Brownian motion with Hurst parameter H belongs (3/4, 1) is established

Mishura, Y.

Posashkov, S.

Наукова електронна бібліотека періодичних видань НАН України (Vernadsky National Library of Ukraine)

Theory of Stochastic ProcessesVol.13 (29), no.1-2, 2007, pp.152-165YULIA MISHURA AND SERGIY POSASHKOVEXISTENCE AND UNIQUENESS OF SOLUTIONOF MIXED STOCHASTIC DIFFERENTIALEQUATION DRIVEN BY FRACTIONALBROWNIAN MOTION AND WIENER PROCESSThe existence and uniqueness of solution of stochastic differentialequation driven by standard Brownian motion and fractional Brow-nian motion with Hurst parameter H ∈ (3/4, 1) is established.1. IntroductionWe will consider the following stochastic differential equation with non-homogeneous coefficients, defined on the stochastic basis (Ω,F , (Ft, t ∈[0, T ]),P) :Xt = X0 +∫ t0a(s, Xs)ds +∫ t0b(s, Xs)dWs +∫ t0c(s, Xs)dBHs , t ∈ [0, T ],(1)where X0 is F0-measurable random variable, EX20 < ∞, W = (Wt, t ∈[0, T ]) is standard Brownian motion (sBm), BH = (BHt , t ∈ [0, T ]) is frac-tional Brownian motion (fBm) with Hurst parameter H ∈ (3/4, 1). Coef-ficients a, b, c : [0, T ] × R −→ R are deterministic functions, which satisfywell-known conditions of existence of integrals in the right-hand of (1),where the integral∫ t0c(s, Xs)dBHs is considered in pathwise sense.If b = 0 we obtain the equation, which contains only fBm. The condi-tions of existence and uniqueness of solution of such equations were shownin [1]. The case b(t, x) = bx and c(t, x) = cx was considered in [7].Using the similarity to sBm of process MH,εt := Vt + 1/εBHt , ε > 0, forH ∈ (3/4, 1), which was proven in [2], and the representation of processessimilar to sBm from [8], the existence and uniqueness of solution of auxiliarystochastic differential equationXNt = X0 +∫ t0a(s, XNs )dt +∫ t0b(s, XNs )dWt+∫ t0c(s, XNs )dBHs +1N∫ t0c(s, XNs )dVs, t ∈ [0, T ], (2)2000 Mathematics Subject Classifications. 60G15, 60H05, 60H10Key words and phrases. Stochastic differential equation, fractional Brownian motion152SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 153where V = (Vt, t ∈ [0, T ]) is another standard Brownian motion, indepen-dent on W and BH , for any N ∈ N, was proved in the paper [3].The main aim of present work is to find the solution of equation (1) asthe limit in some complete space of solutions of (2) when N tends to ∞.2. Main partLet the coefficients of equation (1) satisfy, in addition, the followingassumptions(A) There exists A > 0 such that |a(t, x)| ≤ A, |b(t, x)| ≤ A, |c(t, x)| ≤ A∀t ∈ [0, T ], ∀x ∈ R.(B) There exists L > 0 such that(a(t, x)−a(t, y))2 +(b(t, x)−b(t, y))2 +(c(t, x)−c(t, y))2 ≤ L2(x−y)2,for ∀t ∈ [0, T ], ∀x, y ∈ R.(C) The function c(t, x) is differentiable by c, and there exist constantsB > 0 and β ∈ (1 − H, 1) such that ∀s, t ∈ [0, T ], ∀x ∈ R|c(s, x) − c(t, x)| + |∂xc(s, x) − ∂xc(t, x)| ≤ Bs − t|β.(D) Holder continuity of ∂xc(t, x) in x :|∂xc(t, x) − ∂xc(t, y)| ≤ D|x − y|ρ,for ∀t ∈ [0, T ], ∀x, y ∈ R with some parameter ρ ∈ (3/2 − H, 1).Note that this conditions imply the conditions from [3] of existence anduniqueness of solution of equation (2).Let consider for any 1−H < α < min(1/2, β, ρ−1/2) the space of BesovtypeWα([0, T ]) := {Y = Yt(ω) : (t, ω) ∈ [0, T ] × Ω, ||Y ||α < ∞} (3)with the norm||Y ||α := supt∈[0,T ](E(Yt)2 + E(∫ t0|Yt − Ys|(t − s)1+α ds)2), (4)and prove that the solution of SDE (2) belongs to this space for any N ∈ N.We shall denote different constants as C if it is unimportant for statedresults.From [5] we have for pathwise integral∫ t0f(s)dBHs the representation∫ t0f(s)dBHs =∫ t0Dα0+f(s)D1−αt− BHt−(s)ds,154 YULIA MISHURA AND SERGIY POSASHKOVtherefore this integral can be estimated as∣∣∣∣∫ t0f(s)dBHs∣∣∣∣ ≤ Ct(ω) (∫ t0|f(s)|sαds +∫ t0∫ r0|f(r) − f(u)|(r − u)1+α dudr), (5)whereCt(ω) = sup0≤u≤s≤t|D1−αs− BHs−(u)|. (6)The existence of all moments E|Ct(ω)|p for any p ≥ 1 has also been provedin [1].Lemma 1. The process |Ct(ω)| is dominated by some continuous process.Proof. To prove this statement we estimate |D1−αt− BHt−(s)|:|D1−αt− BHt−(s)| ≤1Γ(α)( |BHt − BHs |(t − s)1−α + (1 − α)∫ ts|BHs − BHu |(s − u)2−α du).Using Garsia - Rodemich - Rumsey inequality from [1] that has a form|f(t) − f(s)|p ≤ Cλ,p|t − s|λp−1∫ t0∫ t0|f(x) − f(y)|p|x − y|λp+1 dxdy (7)with f ∈ C[0, T ], t, s ∈ [0, T ], p ≥ 1 and λ > p−1, we obtain|BHt − BHs | ≤ Cλ,p|t − s|λ−1/p(∫ t0∫ t0|BHu − BHv |p|u − v|λp+1 dudv)1/p.Let λ = H − ε/2, p = 2/ε for 0 < ε < H + α − 1, then|BHt − BHs | ≤ Cε|t − s|H−ε(∫ t0∫ t0|BHu − BHv |2/ε|u − v|2H/ε dudv)ε/2,note that 1 − α < H − ε. Denoteψt,ε =(∫ t0∫ t0|BHu − BHv |2/ε|u − v|2H/ε dudv)ε/2, (8)The process ψt,ε is continuous and nondecreasing in t, and Eψqt,ε < ∞ forany q > 0 and any t ∈ [0, T ] [1].So,|BHt − BHs |(t − s)1−α ≤ CH,ε(t − s)H−ε−1+αψt,ε ≤ Cψt,ε,and∫ ts|BHu − BHs |(u − s)2−α du ≤ CH,ε∫ ts(u − s)H−ε−2+αψt,εdu≤ CH,ε(t − s)H+α−1−εψt,ε ≤ Cψt,ε,SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 155where H + α − 1 − ε > 0.So sup0≤u≤s≤t|D1−αs− BHs−(u)| ≤ Cψt,ε, and the statement of lemma followsfrom continuity and strictly increasing property of ψt,ε.Introduce the random variableC(ω) := sup0≤t≤TCt(ω). (9)Note that C(ω) ≤ ψT,ε and E|C(ω)|q < ∞ for any q ≥ 1.First of all we prove the Hölder continuity of the solution of equation(2).Theorem 2. For any δ ∈ (0, 1/2) the solution of equation (2) is Höldercontinuous with parameter 1/2 − δ.Proof. At first, establish Hölder properties of the integral {∫ t0bsdWs, t ∈[0, T ]}, where bs is a predictable bounded process. For any 0 < δ < 1/4 putp = 2δ, θ = 1/2 − δ/2 in Garsia - Rodemich - Rumsey inequality (7). Then∣∣∣∣∫ tsbudWu∣∣∣∣ ≤ Cδ|t − s|1/2−δξbt,δ,whereξbt,δ :=(∫ t0∫ t0| ∫ yxbudWu|p|x − y|θp+1 dxdy)1/p=(∫ t0∫ t0| ∫ yxbudWu|2/δ|x − y|1/δ dxdy)δ/2(10)and for any q > p from Hölder and Burkholder inequalitiesE(ξbt,δ)q = E(∫ t0∫ t0| ∫ yxbudWu|2/δ|x − y|1/δ dxdy)qδ/2≤(∫ t0∫ t0E| ∫ yxbudWu|qdxdy|x − y|q/2)tqδ−2≤ Cq(∫ t0∫ t0| ∫ yxb2udu|q/2dxdy|x − y|q/2)tqδ−2 ≤ Θt,q.So, the process ξbt,δ from (10) is continuous, strictly increasing and has themoments of any order.156 YULIA MISHURA AND SERGIY POSASHKOVNow consider |XNr − XNz | for 0 < z < r < T :|XNr −XNz | ≤ |∫ rza(u, Xu)du|+ |∫ rzb(u, Xu)dWu|+ 1N|∫ rzc(u, Xu)dVu|+ |∫ rzc(u, Xu)dBHu | ≤ A(r − z) + Cξbr,δ|r − z|1/2−δ +CNξcr,δ|r − z|1/2−δ+ Cr(ω)∫ rz|c(u, XNu )|duuα+ Cr(ω)∫ rz∫ uz|c(u, XNu ) − c(v, XNv )|(u − v)1+α dvdu≤ C ′r(ω)(r − z)1/2−δ + C ′r(ω)∫ rz∫ uz|XNu − XNv |(u − v)1+α dvdu,whereC ′t(ω) := C(ψt,ε ∨ ξbt,δ ∨ ξct,δ ∨ 1), (11)ψt,ε is defined by (8), C′t(ω) ≤ C ′T (ω) and C ′T (ω) has the moments of anyorder.Therefore, for δ < 1/2 − αφr,s :=∫ rs|XNr − XNz |(r − z)1+α dz ≤ C′r(ω)∫ rs(r − z)−1/2−δ−αdz+ C ′r(ω)∫ rs1(r − z)1+α∫ rz∫ uz|XNu − XNv |(u − v)1+α dvdudz= C ′r(ω)(r − s)1/2−α−δ + C ′r(ω)∫ rs(r − u)−α∫ us|XNu − XNv |(u − v)1+α dvdu≤ C ′r(ω)(r − u)1/2−α−δ + C ′r(ω)∫ rs(r − u)−αφu,sdu.From modified Gronwall inequality (Lemma 7.6 [1])φr,s ≤ C ′r(ω)(r − s)1/2−α−δ exp[C ′r(ω)11−α ].Return to |XNr − XNz |:|XNr − XNz | ≤ C ′r(ω)(r − z)1/2−δ+ C ′r(ω) exp[C′r(ω)11−α ]∫ rz(v − z)1/2−α−δdv ≤ C̃r(ω)(r − z)1/2−δ, (12)where C̃r(ω) = C′r(ω) exp[C′r(ω)11−α ], and the theorem is proved for 0 < δ <1/2 − α consequently for 0 < δ < 1/2.Introduce the random variableC̃(ω) := sup0≤t≤TC̃t(ω), (13)SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 157it also has moments of any order.Now prove that the solution of (2) belongs to space (3) with norm (4)for all N ∈ N.Theorem 3. Under assumptions (A) - (D) the solution of equation (2)belongs to space of Besov type (3) with norm (4) for all N ∈ N.Proof. To prove the statement of this theorem we estimateE(XNt )2 + E(∫ t0|XNt − XNs |(t − s)1+α ds)2=: A1(t) + A2(t). (14)At first, for E(XNt )2 we haveE(XNt )2 ≤ 5E(X0)2 + 5E(∫ t0a(s, XNs )ds)2 + 5E(∫ t0b(s, XNs )dWs)2+ 5E(∫ t0c(s, XNs )dBHs )2 + 5E(1N∫ t0c(s, XNs )dVs)2.EvidentlyE(∫ t0a(s, XNs )ds)2≤ A2T 2,E(∫ t0b(s, XNs )dWs)2≤ A2T,E(1N∫ t0c(s, XNs )dVs)2≤ A2TN2≤ A2T .Further, using the estimate (5), (12) and with the help of random variablesdefined by (9), (13), E(∫ t0c(s, XNs )dBHs)2can be estimated asE(∫ t0c(s, XNs )dBHs)2≤ E(C2(ω)(∫ t0c(s, XNs )sαds +∫ t0∫ s0|c(s, XNs ) − c(u, XNu )|(s − u)1+α duds)2)≤ CE(C2(ω)(t∫ t0A2s2αds+(∫ t0∫ s0B(s − u)β + LC̃(ω)(s − u)1/2−δ(s − u)1+α duds)2⎞⎠⎞⎠≤ C(A2t2−2αEC2(ω)+B2EC2(ω)t2(1−α+β) +L2E(C̃2(ω)C2(ω))T 3−2α−2δ)).158 YULIA MISHURA AND SERGIY POSASHKOVSo, we haveA1(t) ≤ C(A2T 2 + 2A2T + A2T 2−2αEC2(ω)+ B2EC2(ω)T 2(1−α+β) + L2E(C̃2(ω)C2(ω))T 3−2α−2δ) < ∞. (15)Consider now A2(t). We have thatA2(t) ≤ 4E(∫ t0| ∫ tsa(u, XNu )du|(t − s)1+α ds)2+ 4E(∫ t0| ∫ tsb(u, XNu )dWu|(t − s)1+α ds)2+ 4N−2E(∫ t0| ∫ tsc(u, XNu )dVu|(t − s)1+α ds)2+ 4E(∫ t0| ∫ tsc(u, XNu )dBHu |(t − s)1+α ds)2.Evidently,E(∫ t0| ∫ tsa(u, Xu)du|(t − s)1+α ds)2≤ CA2t2−2α.Now, let γ ∈ (α, 1/2), thenE(∫ t0| ∫ tsb(u, Xu)dWu|(t − s)1+α ds)2≤ Ct1−2γ∫ t0E| ∫ tsb(u, Xu)dWu|2(t − s)2+2α−2γ ds≤ Ct1−2γ∫ t0∫ tsb2(u, Xu)du(t − s)2+2α−2γ ds ≤ Ct1−2γ∫ t0A2(t − s)1+2α−2γ ds≤ CA2t1−2α,and similarlyE(∫ t0| ∫ tsc(u, Xu)dVu|(t − s)1+α ds)2≤ CA2t1−2α.SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 159Now we estimate E(∫ t0| ts c(u,Xu)dBHu |(t−s)1+α ds)2.E(∫ t0| ∫ tsc(u, Xu)dBHu |(t − s)1+α ds)2≤ E⎛⎝C(ω) ∫ t0∫ ts|c(u,Xu)|(u−s)α du +∫ ts∫ us|c(u,xNu )−c(r,XNr )|(u−r)1+α drdu(t − s)1+α ds⎞⎠2≤ E⎛⎝C(ω) ∫ t0∫ ts|c(u,Xu)|(u−s)α du +∫ ts∫ usB(u−r)β+LC(ω)(u−r)1/2−δ(u−r)1+α drdu(t − s)1+α ds⎞⎠2≤ E(C(ω)∫ t0A(t − s)1−α + B(t − s)1+β−α + LC̃(ω)(t − s)3/2−δ−α(t − s)1+α ds)2≤ C(A2t2−4αEC2(ω) + B2t2+2β−4αEC2(ω) + L2t3−2δ−4αEC2(ω)C̃2(ω)).Therefore A2(t) satisfies the inequalityA2(t) ≤ C(A2T 2−2α + A2T 1−2α + A2T 1−2α+(A2T 2−4αEC2(ω)+B2T 2+2β−4αEC2(ω)+L2T 3−2δ−4αEC2(ω)C̃2(ω)) < ∞.(16)At last, the statement of our theorem follows from inequalities (15) and(16).Introduce for any R > 1 the stopping time τR byτR := inf{t : C ′t(ω) ≥ R} ∧ T, (17)where C ′t(ω) is defined by (11). For any ω ∈ Ω τR = T , for any R > R(ω).Define the processes {XNt∧τR , N ∈ N, t ∈ [0, T ]} as the solutions of equa-tion (2) stopped an the moment τR, and prove that they are fundamentalin the norm (4) of the space (3).Theorem 4. Under assumptions (A) - (D) the sequence {XNt∧τR , N ≥ 1, t ∈[0, T ]} of solutions of equations (2) is fundamental in the norm (4).Proof. ConsiderE(XNt∧τR − XMt∧τR)2 + E(∫ t0|XNt∧τR − XMt∧τR − XNs∧τR + XMs∧τR |(t − s)1+α ds)2= E(XNt∧τR − XMt∧τR)2 + E(∫ t∧τR0|XNt∧τR − XMt∧τR − XNs + XMs |(t − s)1+α ds)2=: AN,M1 (t) + AN,M2 (t). (18)160 YULIA MISHURA AND SERGIY POSASHKOVFirst, for AN,M1 (t) we haveAN,M1 (t) ≤ 4E(∫ t∧τR0(a(s, XNs ) − a(s, XMs ))ds)2+ 4E(∫ t∧τR0(b(s, XNs ) − b(s, XMs ))dWs)2+ 4E(∫ t∧τR0(c(s, XNs ) − c(s, XMs ))dBHs)2+ 4E(∫ t∧τR0(c(s, XNs )N− c(s, XMs )M)dVs)2=: 4(I1 + I2 + I3 + I4).ThenI1 ≤ CTL2∫ t0E(XNs∧τR − XMs∧τR)2ds,I2 ≤ CL2∫ t0E(XNs∧τR − XMs∧τR)2ds,I4 ≤ CA2T (N−2 + M−2).Now we are in position to estimate I3 :I3 = E(∫ t∧τR0(c(s, XNs ) − c(s, XMs ))dBHs)2≤ R2E(∫ t∧τR0|c(s, XNs ) − c(s, XMs )|sαds+∫ t∧τR0∫ s0|c(s, XNs ) − c(s, XMs ) − c(u, XNu ) + c(u, XMu )|(s − u)1+α duds)2≤ 2R2(E(∫ t∧τR0|c(s, XNs ) − c(s, XMs )|sαds)2+ E(∫ t∧τR0∫ s0|c(s, XNs ) − c(s, XMs ) − c(u, XNu ) + c(u, XMu )|(s − u)1+α duds)2)= 2R2(I4 + I5).Further,I4 ≤ CL2T 1−2αE∫ t∧τR0(XNs − XMs )2ds = CL2T 1−2α∫ t0(AN,M1 (s))2ds.SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 161Using Lemma 7.1 [1] we estimate I5 asI5 ≤ E(∫ t∧τR0∫ s0A|XNs − XMs − XNu + XMu |(s − u)1+α duds+∫ t∧τR0∫ s0AB|XNs − XMs |(s − u)β(s − u)1+α duds+∫ t∧τR0∫ s0D|XNs − XMs |(|XNs − XNu |ρ + |XMs − XMu |ρ)(s − u)1+α duds)2≤ 3E(∫ t∧τR0∫ s0A|XNs − XMs − XNu + XMu |(s − u)1+α duds)2+ 3E(∫ t∧τR0∫ s0AB|XNs − XMs |(s − u)β(s − u)1+α duds)2+ 3E(∫ t∧τR0∫ s0D|XNs − XMs |(|XNs − XNu |ρ + |XMs − XMu |ρ)(s − u)1+α duds)2= 3(I6 + I7 + I8),whereI6 ≤ CTA2∫ t0E(∫ s∧τR0|XNs − XMs − XNu + XMu |(s − u)1+α du)2ds,I7 ≤ CTA2∫ t0s2(β−α)E(|XNs∧τR − XMs∧τR|)2 ds,I8 ≤ E(∫ t∧τR0∫ s0B|XNs − XMs |(2R(s − u)ρ(1/2−δ))(s − u)1+α duds)2≤ CTD2R2∫ t0sρ−2ρδ−2αE(|XNs∧τR − XMs∧τR|)2 ds,where we choose δ in such a way that ρ− 2ρδ − 2α > 0. It is possible sinceα < ρ − 1/2 so ρ − 2α > 1/2 − α > 0. At last,I5 ≤ C∫ t0E(∫ s∧τR0|XNs − XMs − XNu + XMu |(s − u)1+α du)2ds+ C∫ t0s2(β−α)E(|XNs∧τR − XMs∧τR |)2 ds+ CR2∫ t0sρ−2ρδ−2αE(|XNs∧τR − XMs∧τR|)2 ds,162 YULIA MISHURA AND SERGIY POSASHKOVandAN,M1 (t) ≤ CR2∫ t0AN,M1 (s)ds + CR2∫ t0AN,M2 (s)ds+ C(N−2 + M−2). (19)Return to AN,M2 (t). It admits the following estimateAN,M2 (t) ≤ C⎛⎝E (∫ t∧τR0∫ t∧τRs(a(u, XNu ) − a(u, XMu ))du(t − s)1+α ds)2+ E(∫ t∧τR0∫ t∧τRs(b(u, XNu ) − b(u, XMu ))dWu(t − s)1+α ds)2+ E(∫ t∧τR0∫ t∧τRs(c(u, XNu ) − c(u, XMu ))dBHu(t − s)1+α ds)2+ E(∫ t∧τR0∫ t∧τRs( c(u,XNu )N− c(u,XMu )M)dVu(t − s)1+α ds)2⎞⎠≤ C(I9 + I10 + I11 + I12).I9 ≤ CT 1−2γE∫ t∧τR0(t − s) ∫ t∧τR0L2|XNu − XMu |2du(t − s)2+2α−2γ ds≤ CT 1−2α∫ t0E(XNs∧τR − XMs∧τR)2ds ≤ CT 1−2α∫ t0AN,M1 (s)ds,I10 ≤ CT 1−2γ∫ t0∫ tsE|XNu∧τR − XMu∧τR|2du(t − s)2+2α−2γ ds≤ CT 1−2γ∫ t0AN,M1 (s)(t − s)1+2α−2γ ds,where we choose γ in such a way that α < γ < 1/2.For I12 we haveI12 ≤ CT 1−2α(N−2 + M−2).SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 163Now consider I11 :I11 ≤ CR2T 1−2γ⎛⎜⎝E ∫ t∧τR0(∫ t∧τRsc(u,XNu )−c(u,XMu )(u−s)α du)2(t − s)2+2α−2γ ds+ E∫ t∧τR0(∫ t∧τRs∫ us|c(u,XNu )−c(u,XMu )−c(v,XNv )+c(v,XMv )|(u−v)1+α dvdu)2(t − s)2+2α−2γ ds⎞⎟⎠=: CR2T 1−2γ(I12 + I13).I12 ≤ C∫ t0(t − s) ∫ tsE(XNu∧τr−XMu∧τr)2(u−s)2α du(t − s)2+2α−2γ≤ C∫ t0AN,M1 (s)(t − s)1+2α−2γ ds.I13 ≤ C⎛⎜⎝E ∫ t∧τR0(∫ t∧τRs∫ usL|XNu −XMu −XNv +XMv |(u−v)1+α dvdu)2(t − s)2+2α−2γ ds+ E∫ t∧τR0(∫ t∧τRs∫ usD|XNu −XMu |(u−v)β(u−v)1+α dvdu)2(t − s)2+2α−2γ ds+ E∫ t∧τR0(∫ t∧τRs∫ usB|XNu −XMu |(|XNu −XNv |ρ+|XMu −XMv |ρ)(u−v)1+α dvdu)2(t − s)2+2α−2γ ds⎞⎟⎠=: C(I14 + I15 + I16).I14 ≤ CT 2γ−2α∫ t0E(∫ s∧τR0|XNs − XMs − XNu + XMu |(s − u)1+α du)2ds= CA2T 2γ−2α∫ t0AN,M2 (s)ds,I15 ≤ C∫ t0E(∫ t∧τRs|XNu − XMu |(u − s)β−α)2(t − s)2+2α−2γ ds≤ C∫ t0(t − s)1+2β−2α ∫ tsE(XNu∧τR − XMu∧τR)2du(t − s)2+2α−2γ ds≤ CT 2β+2γ−4α∫ t0AN,M1 (s)ds,164 YULIA MISHURA AND SERGIY POSASHKOVnote that α < β.I16 ≤ CR2E∫ t∧τR0(∫ t∧τRs∫ us|XNu −XMu |(u−v)ρ(1/2−δ)(u−v)1+α dvdu)2(t − s)2+2α−2γ ds,where we chose 0 < δ < 1/2−α/ρ, note that α < ρ− 1/2. Similarly to I15,I16 ≤ CT 2γ−2α∫ t0AN,M1 (s)ds.Therefore we haveI13 ≤ CR2(∫ t0AN,M1 (s)ds +∫ t0AN,M2 (s)ds).HenceI11 ≤ CR4(∫ t0AN,M1 (s)(t − s)1+2α−2γ ds +∫ t0AN,M2 (s)ds).At last,AN,M2 (t) ≤ CR4(∫ t0AN,M1 (s)(t − s)1+2α−2γds +∫ t0AN,M2 (s)ds)+ C(N−2 + M−2). (20)From (19) and (20) we obtain that the sum AN,M1 (t) + AN,M2 (t) admits thesame estimate as AN,M2 (t), i.e.AN,M1 (t) + AN,M2 (t) ≤ CR4(∫ t0AN,M1 (s)(t − s)1+2α−2γds +∫ t0AN,M2 (s)ds)+ C(N−2 + M−2), (21)and from modified Gronwall lemma [1]AN,M1 (t) + AN,M2 (t)≤ CR4(N−2 + M−2) exp{t(CR4)1/(2γ−2α)}, (22)taking, for example, γ := (1/2 + α)/2. As choosing N, M → 0 see thatright-hand side of (22) tends to zero whence the proof follows.Theorem 5. The SDE (1) has the solution on interval [0, T ], and thissolution is unique.SOLUTION OF STOCHASTIC DIFFERENTIAL EQUATION 165Proof. Since the space (3) is complete, and from Theorem 4 we can defineXt∧τR := limN→∞XNt∧τR , (23)where the limit is taken in space Wα[0, T ] (in particular, we have that thelimit exists in L2(Ω× [0, T ])).Using the similar estimates an Theorem 4, wecan prove that Xt∧τR is the unique solution of the original equation (1) oninterval [0, τR].From definition (17) of τR we have τR1 ≤ τR2 for R1 ≤ R2. So XτR1and XτR2 coincide a.s. on interval [0, τR1 ]. Where R → ∞ we obtain theexistence and uniqueness of solution of SDE (1) on interval [0, T ].3. ConclusionSo, we proved the existence and uniqueness of solution of stochasticdifferential equation driven by standard and fractional Brownian motions(1).Bibliography1. Nualart, D., Răşcanu Differential equations driven by fractional Brownianmotion, Collect. Math., 53, (2000), 55–81.2. Cheridito, P. Regularizing fractional Brownian motion with a view towardsstock price modeling. PhD thesis, Zurich (2001).3. Mishura, Yu., Posashkov, S., Existence and uniqueness of solution of sto-chastic differential equation driven by fractional Brownian motion with sta-bilizing term Theory Prob. Math. Stat, 76, (2007).4. Zähle, M. On the link between fractional and stochastic calculcus. In:Grauel, H., Gundlach, M. (eds.) Stochastic Dynamics, Springer, (1999),305–325.5. Zähle, M. Integration with respect to fractal functions and stochastic calcu-lus. I. Probab. Theory Rel. Fields, 111,(1998), 333–374.6. Zähle, M. Integraton with respect to fractal functions and stochastic calcu-lus. II. Math. Nachr., 225,(2001), 145–183.7. Krvavych, Yu., Mishura, Yu., Exponential formula and Girsanov theoremfor mixed semilinear stochastic differential equatios, in: Mathematical Fi-nance (Trends in Mathematics), Basel, Birkhäuser, (2001), 230–39.8. Hitsuda, M. Representation of Gaussian processes equivalent to Wienerprocess, Osaka Journal of Mathematics, 5, (1968), 299–312.Department of Probability Theory and Mathematical Statistics,Kyiv National Taras Shevchenko University, Kyiv, UkraineE-mail address: myus@univ.kiev.uaDepartment of Probability Theory and Mathematical Statistics,Kyiv National Taras Shevchenko University, Kyiv, UkraineE-mail address: corlagon@univ.kiev.ua

Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process

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Existence and uniqueness of solution of mixed stochastic differential equation driven by fractional Brownian motion and wiener process

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