Mixed stochastic differential equations with long-range dependence: existence, uniqueness and convergence of solutions
aa r X i v : . [ m a t h . P R ] N ov Mixed stochastic differential equations with long-rangedependence: existence, uniqueness and convergence ofsolutions
Yuliya Mishura, Georgiy Shevchenko ∗ Kyiv National Taras Shevchenko University, Department of Mechanics and Mathematics,Volodymyrska 64, 01601 Kyiv, Ukraine
Abstract
For a mixed stochastic differential equation involving standard Brownian motionand an almost surely H¨older continuous process Z with H¨older exponent γ > /
2, we establish a new result on its unique solvability. We also establish anestimate for difference of solutions to such equations with different processes Z and deduce a corresponding limit theorem. As a by-product, we obtain a resulton existence of moments of a solution to a mixed equation under an assumptionthat Z has certain exponential moments. Keywords:
Mixed stochastic differential equation, pathwise integral,long-range dependence, fractional Brownian motion, stochastic differentialequation with random drift
Introduction
In this paper we study the following mixed stochastic differential equation: X t = X + Z t a ( s, X s ) ds + Z t b ( s, X s ) dW s + Z t c ( s, X s ) dZ s , t ∈ [0 , T ] , (1)where W is a standard Wiener process, and Z is an almost surely H¨older con-tinuous process with H¨older exponent γ > /
2. The processes W and Z can bedependent.The motivation to consider such equations comes, in particular, from finan-cial mathematics. When it is necessary to model randomness on a financialmarket, it is useful to distinguish between two main sources of this randomness.The first source is the stock exchange itself with thousands of agents. The noise ∗ Corresponding author
Email addresses: [email protected] (Yuliya Mishura), [email protected] (GeorgiyShevchenko)
Preprint submitted to Elsevier June 21, 2018 oming from this source can be assumed white and is best modeled by a Wienerprocess. The second source has the financial and economical background. Therandom noise coming from this source usually has a long range dependenceproperty, which can be modeled by a fractional Brownian motion B H with theHurst parameter H > / /
2. Most of long-range-dependent pro-cesses have one thing in common: they are H¨older continuous with exponentgreater than 1 /
2, and this is the reason to consider a rather general equation(1).Equation (1) with Z = B H , a fractional Brownian motion, was first con-sidered in [2], where existence and uniqueness of solution was proved for time-independent coefficients and zero drift. For inhomogeneous coefficients, uniquesolvability was established in [3] for H ∈ (3 / ,
1) and bounded coefficients, in[1] for any
H > /
2, but under the assumption that W and B H are indepen-dent, and in [5] for any H > /
2, but bounded coefficient b . In this paper wegeneralize the last result replacing the boundedness assumption by the lineargrowth.There is, however, an obstacle to use equation (1) in applications because it isvery hard to analyze with standard tools of stochastic analysis. The main reasonfor this is that the two stochastic integrals in (1) have very different nature. Theintegral with respect to the Wiener process is Itˆo integral, and it is best analyzedin a mean square sense, while the integral with respect to Z is understood in apathwise sense, and all estimates are pathwise with random constants. So thereis a need for good approximations for such equations. One way to approximateis to replace integrals by finite sums, this leads to Euler approximations. Forequation (1) such approximations were considered in [4], where a sharp estimatefor the rate of convergence was obtained. Another way is to replace process Z by a smooth process Z , transforming equation (1) into a usual Itˆo stochasticdifferential equation with random drift a ( s, x ) + c ( s, x ) Z ′ s . Since there is a well-developed theory for Itˆo equations, such smooth approximations may prove veryuseful in applications.The paper is organized as follows. In Section 1, we give basic facts aboutintegration with respect to fractional Brownian motion and formulate mainhypotheses. In Section 2, we establish auxiliary results. As a by-product, weobtain a result on existence of moments of a solution to a mixed equation underan assumption that Z has certain exponential moments, which is satisfied, forexample, by a fractional Brownian motion with Hurst parameter H > / Z and deduce a limit theorem for equation(1) from this estimate.
1. Preliminaries
Let (Ω , F , {F t } t ∈ [0 ,T ] , P ) be a complete probability space equipped with afiltration satisfying standard assumptions, and { W t , t ∈ [0 , T ] } be a standard2 t -Wiener process. Let also { Z t , t ∈ [0 , T ] } be an F t -adapted stochastic process,which is almost surely H¨older continuous with exponent γ > /
2. We considera mixed stochastic differential equation (1). The integral w.r.t. Wiener process W is the standard Itˆo integral, and the integral w.r.t. Z is pathwise generalizedLebesgue–Stieltjes integral (see [6, 7]), which is defined as follows. Consider twocontinuous functions f and g , defined on some interval [ a, b ] ⊂ R . For α ∈ (0 , (cid:0) D αa + f (cid:1) ( x ) = 1Γ(1 − α ) (cid:18) f ( x )( x − a ) α + α Z xa f ( x ) − f ( u )( x − u ) α du (cid:19) ( a,b ) ( x ) , (cid:0) D − αb − g (cid:1) ( x ) = e − iπα Γ( α ) (cid:18) g ( x )( b − x ) − α + (1 − α ) Z bx g ( x ) − g ( u )( u − x ) − α du (cid:19) ( a,b ) ( x ) . Assume that D αa + f ∈ L [ a, b ] , D − αb − g b − ∈ L ∞ [ a, b ], where g b − ( x ) = g ( x ) − g ( b ). Under these assumptions, the generalized (fractional) Lebesgue-Stieltjesintegral R ba f ( x ) dg ( x ) is defined as Z ba f ( x ) dg ( x ) = e iπα Z ba (cid:0) D αa + f (cid:1) ( x ) (cid:0) D − αb − g b − (cid:1) ( x ) dx. (2)In view of this, we will consider the following norms for α ∈ (1 − H, / k f k ,α ; t = Z t k f k α ; s g ( t, s ) ds, k f k ∞ ,α ; t = sup s ∈ [0 ,t ] k f k α ; s , where g ( t, s ) = s − α + ( t − s ) − α − / and k f k α ; t = | f ( t ) | + Z t | f ( t ) − f ( s ) | ( t − s ) − − α ds. Also define a seminorm k f k ,α ; t = sup ≤ u
2. Auxiliary resultsLemma 2.1.
Let g : [0 , T ] → R be a γ -H¨older continuous function. Define for ε > g ε ( t ) = ε − R t ∨ t − ε g ( s ) ds . Then for α ∈ (1 − γ, k g − g ε k ,α ; T ≤ CK γ ( g ) ε γ + α − , where K γ ( g ) = sup ≤ s 0. Take any t, s ∈ [0 , T ]. For | t − s | ≥ ε | g ( t ) − g ε ( t ) − g ( s ) + g ε ( s ) | = ε − (cid:12)(cid:12)(cid:12)(cid:12)Z tt − ε (cid:0) g ( t ) − g ( u ) (cid:1) du − Z ss − ε (cid:0) g ( s ) − g ( v ) (cid:1) dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ K γ ( g ) ε − (cid:12)(cid:12)(cid:12)(cid:12)Z tt − ε ( t − u ) γ du (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z ss − ε ( s − v ) γ dv (cid:12)(cid:12)(cid:12)(cid:12) ≤ CK γ ( g ) ε γ ;for | t − s | < ε | g ( t ) − g ε ( t ) − g ( s ) + g ε ( s ) | ≤ | g ( t ) − g ( s ) | + ε − (cid:12)(cid:12)(cid:12)(cid:12)Z − ε (cid:0) g ( t + u ) − g ( s + u ) (cid:1) du (cid:12)(cid:12)(cid:12)(cid:12) ≤ CK γ ( g ) | t − s | γ , consequently | g ( t ) − g ε ( t ) − g ( s ) + g ε ( s ) | ≤ CK γ ( g ) (cid:0) ε ∧ | t − s | (cid:1) γ . (6)Now write k g − g ε k ,α ; T ≤ A ε + B ε , A ε = sup ≤ u Let g : [0 , T ] → R be a γ -H¨older continuous function with g (0) = 0 . There exists a sequence of continuously differentiable functions { g n , n ≥ } such that for any α ∈ (1 − γ, k g − g n k ,α ; T → , n → ∞ . One possiblechoice of such sequence is g n ( t ) = a − n R t ∨ t − a n g ( s ) ds , where a n ↓ , n → ∞ . Further throughout the paper there will be no ambiguity about α , so for thesake of shortness we will usually abbreviate k f k t = k f k α ; t and k f k x,t = k f k x,α ; t ,where x ∈ { , , ∞} . Lemma 2.2. Under assumptions (4) and (5) k X k t ≤ C k Z k ,t (cid:18) Z t k X k s (cid:0) s − α + ( t − s ) − α (cid:1) ds (cid:19) + I b ( t ) , where I b ( t ) = (cid:13)(cid:13)R · b ( s, X s ) dW s (cid:13)(cid:13) t .Proof. Write k X k t ≤ | X | + I a ( t ) + I b ( t ) + I c ( t ), where I a ( t ) = (cid:13)(cid:13)R · a ( s, X s ) ds (cid:13)(cid:13) t , I c ( t ) = (cid:13)(cid:13)R · c ( s, X s ) dZ s (cid:13)(cid:13) t . Denote for shortness Λ = k Z k ,t .Estimate I a ( t ) ≤ C (cid:18) Z t | a ( s, X s ) | ds + Z t Z ts | a ( u, X u ) | du ( t − s ) − − α ds (cid:19) ≤ C (cid:18) Z t (cid:0) | X s | (cid:1) ds + Z t Z ts (cid:0) | X u | (cid:1) du ( t − s ) − − α ds (cid:19) ≤ C (cid:18) Z t | X s | ds + Z t | X u | ( t − u ) − α du (cid:19) ≤ C (cid:18) Z t k X k s ( t − s ) − α ds (cid:19) . Further, I c ( t ) ≤ I ′ c ( t ) + I ′′ c ( t ) , I ′ c ( t ) = (cid:12)(cid:12)(cid:12)(cid:12)Z t c ( s, X s ) dZ s (cid:12)(cid:12)(cid:12)(cid:12) ≤ C Λ Z t (cid:18)(cid:0) | X s | (cid:1) s − α + Z s | X s − X u | ( s − u ) − − α du (cid:19) ds ≤ C Λ (cid:18) Z t k X k s s − α ds (cid:19) ,I ′′ c ( t ) = Z t (cid:12)(cid:12)(cid:12)(cid:12)Z ts c ( u, X u ) dZ v (cid:12)(cid:12)(cid:12)(cid:12) ( t − s ) − − α ds ≤ C Λ Z t Z ts (cid:18) | X v | ( v − s ) − α + Z vs | X v − X z | ( v − z ) − − α dz (cid:19) dv ( t − s ) − − α ds ≤ C Λ (cid:18) Z t Z ts k X k v ( v − s ) − α dv ( t − s ) − − α ds (cid:19) = C Λ (cid:18) Z t k X k v Z v ( v − s ) − α ( t − s ) − − α ds dv (cid:19) ≤ C Λ (cid:18) Z t k X k v ( t − v ) − α dv (cid:19) . Combining these estimates, we get k X k t ≤ C Λ (cid:18) Z t k X k s (cid:0) s − α + ( t − s ) − α (cid:1) ds (cid:19) + I b ( t ) . Proposition 2.1. Under assumptions (4) , (5) and E h exp n a k Z k / (1 − α )0 ,T oi < ∞ , (7) all moments of X are bounded, moreover, E h k X k p ∞ ,T i < ∞ for all p > .Proof. By the generalized Gronwall lemma from [6] it follows from Lemma 2.2that k X k t ≤ C k Z k ,t sup s ∈ [0 ,t ] I b ( s ) exp n C k Z k / (1 − α )0 ,t o , whence k X k ∞ ,T ≤ C k Z k ,T sup s ∈ [0 ,T ] I b ( s ) exp n C k Z k / (1 − α )0 ,T o , whence the assertion follows, as all moments of sup s ∈ [0 ,T ] I b ( s ) are bounded dueto the Burkholder inequality and the boundedness of b . Remark . The assumption (7) might seem very restrictive. However, it istrue if Z is Gaussian and α < / α is possible iff γ > / / (1 − α ) < 2. Examples of such processes are fractionalBrownian motion with Hurst parameter H > / / N ≥ τ N = inf n t : k Z k ,t ≥ N o ∧ T and astopped process Z Nt = Z t ∧ τ N , denote by X N the solution of (1) with Z replacedby Z N . Lemma 2.3. Under assumptions (4) and (5) it holds E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,T i < C p,N with the constant C p,N independent of Z and K .Proof. First note that the finiteness of E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,T i can be deduced fromLemma 2.2 exactly the same way as Proposition 2.1.Second, it follows from Lemma 2.2 and the generalized Gronwall lemma [6]that (cid:13)(cid:13) X N (cid:13)(cid:13) t ≤ CN sup s ∈ [0 ,t ] I Nb ( s ) exp n CtN / (1 − α ) o ≤ C N sup s ∈ [0 ,t ] I Nb ( s ) , which implies (cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,t ≤ C N,p sup s ∈ [0 ,t ] (cid:0) I Nb ( s ) (cid:1) p . Write E " sup s ∈ [0 ,T ] (cid:0) I Nb ( t ) (cid:1) p ≤ I ′ b + I ′′ b , where, denoting b Nu = b ( u, X Nu ), I ′ b = E " sup s ∈ [0 ,t ] (cid:12)(cid:12)(cid:12)(cid:12)Z s b Nu dW u (cid:12)(cid:12)(cid:12)(cid:12) p ≤ C p E "(cid:18)Z t (cid:12)(cid:12) b Ns (cid:12)(cid:12) ds (cid:19) p/ ≤ C p E "(cid:18)Z t (1 + (cid:13)(cid:13) X N (cid:13)(cid:13) s ) ds (cid:19) p/ ≤ C p E "(cid:18)Z t (cid:13)(cid:13) X N (cid:13)(cid:13) s ds (cid:19) p/ ≤ C p (cid:18) Z t E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,s i ds (cid:19) ,I ′′ b = E " sup s ∈ [0 ,t ] (cid:18)Z s (cid:12)(cid:12)(cid:12)(cid:12)Z su b Nz dW z (cid:12)(cid:12)(cid:12)(cid:12) ( s − u ) − − α du (cid:19) p . Obviously, we can assume without loss of generality that p > / (1 − α ).It follows from the Garsia–Rodemich–Rumsey inequality that for arbitrary η ∈ (0 , / − α ), u, s ∈ [0 , t ] (cid:12)(cid:12)(cid:12)(cid:12)Z su b Nz dW z (cid:12)(cid:12)(cid:12)(cid:12) ≤ Cξ Nη ( t ) | s − u | / − η , ξ Nη ( t ) = Z t Z t (cid:12)(cid:12)R yx b Nv dW v (cid:12)(cid:12) /η | x − y | /η dx dy ! η/ . Setting η = 2 /p , we get ξ Nη ( t ) = Z t Z t (cid:12)(cid:12)R yx b Nv dW v (cid:12)(cid:12) p | x − y | p/ dx dy ! /p . I ′ b , we get E h(cid:0) ξ Nη ( t ) (cid:1) p i ≤ C p Z t Z t E (cid:2)(cid:12)(cid:12)R yx b Nv dW v (cid:12)(cid:12) p (cid:3) | x − y | p/ dx dy ≤ C p Z t Z y E (cid:20)(cid:16)R yx (1 + (cid:13)(cid:13) X N (cid:13)(cid:13) ∞ ,v ) dv (cid:17) p/ (cid:21) ( y − x ) p/ dx dy ≤ C p Z t Z y ( y − x ) p/ + E (cid:20)(cid:16)R yx (cid:13)(cid:13) X N (cid:13)(cid:13) ∞ ,v dv (cid:17) p/ (cid:21) ( y − x ) p/ dx dy ≤ C p Z t Z y ( y − x ) p/ + ( y − x ) p/ − R yx E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i dv ( y − x ) p/ dx dy ≤ C p (cid:18) Z t Z y Z yx E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i dv ( y − x ) − dv dx dy (cid:19) = C p (cid:18) Z t Z y E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i log yy − v dv dy (cid:19) ≤ C p (cid:18) Z t E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i dv (cid:19) . whence I ′′ b ≤ C E (cid:2) ξ Nη ( t ) p (cid:3) sup s ∈ [0 ,t ] (cid:18)Z t ( t − s ) − / − η − α ds (cid:19) p ≤ C p (cid:18) Z t E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i dv (cid:19) . Thus, we have the estimate (cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,t ≤ C N,p (cid:18) Z t E h(cid:13)(cid:13) X N (cid:13)(cid:13) p ∞ ,v i dv (cid:19) , from which we derive the desired statement with the help of the Gronwall lemma. 3. Existence of solution Now we prove existence and uniqueness of solution to equation (1) withoutassumption (5). As above, we define a stopped process Z Nt = Z t ∧ τ N , where τ N = inf n t : k Z k ,t ≥ N o ∧ T . Denote by X N the solution of (1) with Z replaced by Z N . Theorem 3.1. If the coefficients of equation (1) satisfy conditions (4) , then ithas a unique solution X such that k X k ∞ ,T < ∞ a.s. roof. For integer N ≥ R ≥ X N,R the solution of equation (1) withprocess Z replaced by Z N and coefficient b replaced by b ∧ ( K ( R +1)) ∨ ( − K ( R +1)) (we will call it an ( N, R )-equation). Let also τ N,R = inf n t : (cid:12)(cid:12)(cid:12) X N,Rt (cid:12)(cid:12)(cid:12) ≥ R o ∧ T . We argue that X N,R ′ t = X N,R ′′ t a.s. for t < τ N,R ′ ∧ τ N,R ′′ .For brevity define Y t,s = Y t − Y s and denote h ( t, s ) = ( t − s ) − − α , 1I t =1I t<τ N,R ′ ∧ τ N,R ′′ . All the constants in this step may depend on N and R ′ , R ′′ .Write( X N,R ′ t − X N,R ′′ t )1I t = (cid:18)Z t a ∆ ( s ) ds + Z t b ∆ ( s ) dW s + Z t c ∆ ( s ) dZ Ns (cid:19) t =: ( I a ( t ) + I b ( t ) + I c ( t )) 1I t , (8)where d ∆ ( s ) := d ( s, X N,R ′ ) − d ( s, X N,R ′′ ) , d ∈ { a, b, c } . Due to our hypotheses, | d ∆ ( s ) | ≤ C (cid:12)(cid:12) X N,R ′ s − X N,R ′′ s (cid:12)(cid:12) .Define ∆ t = R t (cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13) s s g ( t, s ) ds . Write∆ t ≤ I ′ a + I ′′ a + I ′ b + I ′′ b + I ′ c + I ′′ c ) , where I ′ d = R t I d ( s ) s g ( t, s ) ds , I ′′ d = R t (cid:0) R s |I d ( s ) −I d ( u ) | h ( s, u ) du (cid:1) s g ( t, s ) ds , d ∈ { a, b, c } .By the Cauchy–Schwarz inequality, we can write I a ( s ) s ≤ C Z s (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) u du ≤ C Z s (cid:13)(cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13)(cid:13) u u du, (9)therefore I ′ a ≤ C Z t ∆ s g ( t, s ) ds. Similarly, I ′′ a ≤ C Z t (cid:18)Z s Z su (cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) dv h ( s, u ) du (cid:19) s g ( t, s ) ds ≤ C Z t (cid:18) Z s (cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) v ( s − v ) − α dv (cid:19) g ( t, s ) ds ≤ C Z t Z s (cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) v ( s − v ) − α dv g ( t, s ) ds ≤ C Z t ∆ s g ( t, s ) ds. Further, by (3), for s ≤ t I c ( s ) s ≤ CN (cid:20) Z s (cid:18) | c ∆ ( u ) | u − α + Z u | c ∆ ( u ) − c ∆ ( z ) | h ( u, z ) dz (cid:19) du (cid:21) s ≤ C "(cid:18) Z s | c ∆ ( u ) | u − α du (cid:19) + (cid:18) Z s Z u | c ∆ ( u ) − c ∆ ( z ) | h ( u, z ) dz du (cid:19) s =: C ( J ′ c + J ′′ c ) . I a , J ′ c ≤ C Z s (cid:13)(cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13)(cid:13) u u u − α du. By Lemma 7.1 of Nualart and R˘a¸scanu (2002), the hypotheses (A)–(D) implythat for any t , t , x , . . . , x | c ( t , x ) − c ( t , x ) − c ( t , x ) + c ( t , x ) | ≤ K | x − x − x + x | + K | x − x | | t − t | β + K | x − x | ( | x − x | + | x − x | ) . (10)Therefore, | c ∆ ( u ) − c ∆ ( z ) | = (cid:12)(cid:12) c ( u, X N,R ′ u ) − c ( z, X N,R ′ z ) − c ( u, X N,R ′′ u ) + c ( z, X N,R ′′ z ) (cid:12)(cid:12) ≤ C (cid:18)(cid:12)(cid:12) X N,R ′ u,z − X N,R ′′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) ( u − z ) β + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12)(cid:16)(cid:12)(cid:12) X N,R ′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′′ u,z (cid:12)(cid:12)(cid:17)(cid:19) . Thus, we have J ′′ c ≤ C (cid:20) Z s Z u (cid:18)(cid:12)(cid:12) X N,R ′ u,z − X N,R ′′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) ( u − z ) β + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12)(cid:16)(cid:12)(cid:12) X N,R ′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′′ u,z (cid:12)(cid:12)(cid:17)(cid:19) h ( u, z ) dz u du (cid:21) ≤ C ( H + H ) , where H = (cid:18) Z s Z u (cid:16)(cid:12)(cid:12) X N,R ′ u,z − X N,R ′′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) ( u − z ) β (cid:17) h ( u, z ) dz u du (cid:19) ≤ C Z s (cid:18)(cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13) u u + (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) u u β − α (cid:19) du ≤ C Z s (cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13) u u du,H = (cid:18)Z s (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) Z u (cid:16)(cid:12)(cid:12) X N,R ′ u,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′′ u,z (cid:12)(cid:12)(cid:17) h ( u, z ) dz u du (cid:19) ≤ C (cid:18) Z s (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12)(cid:0)(cid:13)(cid:13) X N,R ′ (cid:13)(cid:13) ∞ ,u + (cid:13)(cid:13) X N,R ′′ (cid:13)(cid:13) ∞ ,u (cid:1) u du (cid:19) ≤ C ( R ′ + R ′′ ) Z s (cid:12)(cid:12) X N,R ′ u − X N,R ′′ u (cid:12)(cid:12) u du ≤ C Z s (cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13) u u du. It follows that I c ( s ) ≤ C Z s (cid:13)(cid:13)(cid:13) X N,R ′ − X N,R ′′ (cid:13)(cid:13)(cid:13) u u u − α du. (11)Consequently, I ′ c ≤ C Z t ∆ s g ( t, s ) ds. I ′′ c ≤ N Z t (cid:18) Z s Z su (cid:18) | c ∆ ( v ) | ( v − u ) − α + Z vu | c ∆ ( v ) − c ∆ ( z ) | h ( v, z ) dz (cid:19) dv h ( s, u ) du (cid:19) g ( t, s )1I s ds ≤ C Z t (cid:18) Z s (cid:18) | c ∆ ( v ) | ( s − v ) − α + Z v | c ∆ ( v ) − c ∆ ( z ) | h ( v, z )( s − z ) − α dz (cid:19) dv (cid:19) g ( t, s )1I s ds ≤ C Z t (cid:18) Z s (cid:18)(cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) ( s − v ) − α + Z v (cid:12)(cid:12) X N,R ′ v,z − X N,R ′′ v,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) ( v − z ) β + (cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12)(cid:0)(cid:12)(cid:12) X N,R ′ v,z (cid:12)(cid:12) + (cid:12)(cid:12) X N,R ′′ v,z (cid:12)(cid:12)(cid:1)! h ( v, z )( s − z ) − α dz v dv (cid:21) g ( t, s ) ds ≤ C Z t (cid:20) Z s (cid:18)(cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12)(cid:0) ( s − v ) − α + ( s − v ) β − α + ( R ′ + R ′′ )( s − v ) − α (cid:1) + Z v (cid:12)(cid:12) X N,R ′ v,z − X N,R ′′ v,z (cid:12)(cid:12) h ( v, z ) dz ( s − v ) − α (cid:19) v dv (cid:21) g ( t, s ) ds ≤ C Z t ∆ s g ( t, s ) ds. Summing up and taking expectations, we arrive to E [ I ′ a + I ′′ a + I ′ c + I ′′ c ] ≤ C Z t E [∆ s ] g ( t, s ) ds. (12)Now turn to I ′ b and I ′′ b . E (cid:2) I b ( s ) s (cid:3) = E "(cid:18) Z s b ∆ ( u ) dW u (cid:19) s ≤ Z s E (cid:2) b ∆ ( u ) u (cid:3) du ≤ C Z s E h ( X N,R ′ u − X N,R ′′ u ) u i du, (13)whence E [ I ′ b ] ≤ Z t E [∆ s ] g ( t, s ) ds. Further, E [ I ′′ b ] = Z t E "(cid:18) Z s (cid:12)(cid:12)(cid:12)(cid:12)Z su b ∆ ( v ) dW v (cid:12)(cid:12)(cid:12)(cid:12) ( s − u ) − − α du (cid:19) s g ( t, s ) ds ≤ Z t Z s E "(cid:18) Z su b ∆ ( v ) dW v (cid:19) s ( s − u ) − / − α du Z s ( s − u ) − / − α du g ( t, s ) ds ≤ C Z t Z s Z su E h(cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) v i dv ( s − u ) − / − α du g ( t, s ) ds ≤ C Z t Z s E h(cid:12)(cid:12) X N,R ′ v − X N,R ′′ v (cid:12)(cid:12) v i ( s − v ) − / − α dv g ( t, s ) ds ≤ C Z t E [∆ s ] g ( t, s ) ds. E [∆ t ] ≤ C Z t E [∆ s ] g ( t, s ) ds, whence we get ∆ s = 0 a.s., which implies X N,R ′ t = X N,R ′′ t for t < τ N ∧ τ N,R ′ ∧ τ N,R ′ .This implies, in particular, that τ N,R ′′ ≥ τ N,R ′ a.s. On the other hand,almost surely τ N,R = T for all R large enough. Indeed, assuming the contrary,for some t ∈ [0 , T ) P ( ∀ R ≥ τ N,R < T ) = c > E (cid:2)(cid:13)(cid:13) X R,N (cid:13)(cid:13) ∞ (cid:3) ≥ cR ,contradicting Lemma 2.3.It follows that there exists a process (cid:8) X Nt , t ∈ [0 , T ] (cid:9) such that for each R ≥ t ≤ τ N,R X Nt = X N,Rt . Hence, it is evident that X N solves (1) with Z replaced by Z N .Since τ N increases with N and eventually equals T , we have that there existsa process which solves initial equation (1).Exactly as above, one can argue that any solution to (1) is a solution to( N, R )-equation for t < τ N ∧ τ N,R , which gives uniqueness. 4. Limit theorem Let coefficients of (1) satisfy (4), and let X be its unique solution. Let also X be the solution to stochastic differential equation X t = X + Z t a ( s, X s ) ds + Z t b ( s, X s ) dW s + Z t c ( s, X s ) dZ s , (14)where Z is a γ -H¨older continuous process.As above, for Y ∈ (cid:8) Z, Z (cid:9) define a stopped process Y Nt = Y t ∧ τ N , where τ N =inf n t : k Y k ,t ≥ N o ∧ T , and let X N and X N be the solutions to correspondingequations. Denote A N,Rt = (cid:26)(cid:13)(cid:13) X N (cid:13)(cid:13) ∞ ,t + (cid:13)(cid:13)(cid:13) X N (cid:13)(cid:13)(cid:13) ∞ ,t ≤ R (cid:27) . Lemma 4.1. Under assumptions (4) , E (cid:20)(cid:13)(cid:13)(cid:13) X N − X N (cid:13)(cid:13)(cid:13) ,T B N,RT (cid:21) ≤ C N,R E (cid:20)(cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T (cid:21) with the constant C N,R independent of Z , Z .Proof. We will use the same notation as in the proof of Theorem 3.1, exceptnow 1I t = 1I A N,Rt .Denote ∆ t = R t (cid:13)(cid:13)(cid:13) X N − X N (cid:13)(cid:13)(cid:13) s s g ( t, s ) ds . Exactly as in the proof of Theo-rem 3.1 it can be shown that E [∆ t ] ≤ C (cid:18) C N,R Z t E [∆ s ] g ( t, s ) ds + E [ I ′ Z ] + E [ I ′′ Z ] (cid:19) , (15)12here I ′ Z = Z t I Z ( s ) g ( t, s )1I s ds, I ′′ Z = Z t (cid:18)Z s |I Z ( s ) − I Z ( u ) | h ( s, u ) du (cid:19) g ( t, s )1I s ds, I Z ( t ) = Z t c ( s, X s ) d ( Z s − Z s ) . By (3), on A N,Rt I Z ( s ) ≤ C (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T (cid:18) Z s (cid:18) (cid:12)(cid:12)(cid:12) c ( u, X Nu ) (cid:12)(cid:12)(cid:12) u − α + Z u (cid:12)(cid:12)(cid:12) c ( v, X Nv ) − c ( u, X Nu ) (cid:12)(cid:12)(cid:12) h ( u, v ) dv (cid:19) du (cid:19) ≤ C (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T (cid:18) Z s (cid:18)(cid:0) (cid:12)(cid:12) X u (cid:12)(cid:12) (cid:1) u − α + Z u (cid:0) ( u − v ) β + (cid:12)(cid:12) X u − X v (cid:12)(cid:12) (cid:1) h ( u, v ) dv (cid:19) du (cid:19) ≤ C (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T Z t (cid:0) (cid:13)(cid:13) X (cid:13)(cid:13) ∞ ,s (cid:1) s g ( t, s ) ds ≤ CR (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T . (16)Hence, I ′ Z ≤ CN (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T . Similarly, I ′′ Z ≤ C (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T Z t (cid:20) Z s Z su (cid:18) (cid:12)(cid:12) c ( v, X v ) (cid:12)(cid:12) ( v − u ) − α + Z vu (cid:12)(cid:12) c ( v, X v ) − c ( z, X z ) (cid:12)(cid:12) h ( v, z ) dz (cid:19) dv h ( s, u ) du (cid:21) g ( t, s )1I s ds ≤ C (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T Z t (cid:20) Z s Z su k X k ∞ ,v ( v − u ) − α dv h ( s, u ) du (cid:21) g ( t, s )1I s ds ≤ CR (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T . Summing these estimates with (15), we obtain E [∆ t ] ≤ C N,R (cid:18) (cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T + Z t E [∆ s ] g ( t, s ) ds (cid:19) , whence we get the statement by the generalized Gronwall lemma.The proof of the following fact uses the Burkholder inequality and the sameideas as before, so we skip it. Corollary 4.1. For N > the estimate holds E " sup t ∈ [0 ,T ] (cid:12)(cid:12) X − X (cid:12)(cid:12) A N,RT ≤ C N E (cid:20)(cid:13)(cid:13)(cid:13) Z N − Z N (cid:13)(cid:13)(cid:13) ,T A N,RT (cid:21) with the constant C N independent of Z , Z . { Z n , n ≥ } be a sequence of γ -H¨older continuous processes. Considera sequence of stochastic differential equations X nt = X + Z t a ( s, X ns ) ds + Z t b ( s, X ns ) dW s + Z t c ( s, X ns ) dZ ns , t ∈ [0 , T ] . (17) Theorem 4.1. Assume that k Z − Z n k ,T → in probability. Then X nt → X t in probability uniformly in t .Proof. Let B n,Nt = n k X k ∞ ,t + k X n k ∞ ,t + k Z k ,t + k Z n k ,t ≤ N o , ∆ n = sup t ∈ [0 ,T ] | X nt − X t | .For ε > P (∆ n > ε ) ≤ P (cid:16) { ∆ n > ε } ∩ B n,NT (cid:17) + P (Ω \ B n,NT ) . From the assumption it is easy to see that E h k Z − Z n k ,T B N,nt i → n → ∞ .Then by (4.1) we have for any ε > P (cid:16) { ∆ n > ε } ∩ B n,NT (cid:17) → , n → ∞ . So lim sup n →∞ P (∆ n > ε ) ≤ lim sup n →∞ P (Ω \ B n,NT ) . (18)We know that Λ T ( Z ) < ∞ a.s., so by assumption, k Z n k ,T are bounded inprobability uniformly in n . Therefore by Lemma 2.3, k X n k ∞ ,T are boundedin probability uniformly in n and k X k ∞ ,T is finite a.s. Consequently, P (Ω \ B n,NT ) → N → ∞ uniformly in n . Thus, we conclude the proof by sending N → ∞ in (18). Remark . Under the assumptions of Theorem 4.1 we have also the conver-gence in probability k X − X n k ,T → n → ∞ . References [1] Guerra, J., Nualart, D., 2008. Stochastic differential equations driven byfractional Brownian motion and standard Brownian motion. Stoch. Anal.Appl. 26 (5), 1053–1075.[2] Kubilius, K., 2002. The existence and uniqueness of the solution of an inte-gral equation driven by a p -semimartingale of special type. Stochastic Pro-cess. Appl. 98 (2), 289–315.[3] Mishura, Y., 2008. Stochastic calculus for fractional Brownian motion andrelated processes. Springer, Berlin.144] Mishura, Y. S., Shevchenko, G. M., 2011. Rate of convergence of Euler ap-proximations of solution to mixed stochastic differential equation involvingBrownian motion and fractional Brownian motion. Random Oper. Stoch.Equ. 20 (4), 387–406.[5] Mishura, Y. S., Shevchenko, G. M., 2011. Stochastic differential equationinvolving Wiener process and fractional Brownian motion with Hurst index H > //