Rate of convergence of Euler approximations of solution to mixed stochastic differential equation involving Brownian motion and fractional Brownian motion
aa r X i v : . [ m a t h . P R ] N ov RATE OF CONVERGENCE OF EULER APPROXIMATIONS OFSOLUTION TO MIXED STOCHASTIC DIFFERENTIALEQUATION INVOLVING BROWNIAN MOTION ANDFRACTIONAL BROWNIAN MOTION
YULIYA S. MISHURA AND GEORGIY M. SHEVCHENKO
Abstract.
We consider a mixed stochastic differential equation involvingboth standard Brownian motion and fractional Brownian motion with Hurstparameter
H > /
2. The mean-square rate of convergence of Euler approxi-mations of solution to this equation is obtained.
Introduction
The main object of this paper is the following mixed stochastic differential equa-tion involving independent Wiener process B and fractional Brownian motion B H with Hurst index H ∈ (1 / , X t = X + Z t a ( s, X s ) ds + Z t b ( s, X s ) dW s + Z t c ( X s ) dB Hs , t ∈ [0 , T ] , where the integral w.r.t. Wiener process is the standard Itˆo integral, and the inte-gral w.r.t. fBm is the forward stochastic integral. The questions of existence anduniqueness of solution for equations of such type were considered in [7, 9, 5, 11].Such mixed equations arise in different applied areas. In financial mathematics,for example, it is often natural to assume that the underlying random noise consistsof two parts: a “fundamental” part, describing the economical background for astock price, and a “trading” part, coming from the randomness inherent for thestock market. In this case the fundamental part of the noise should have a longmemory, while the second part is likely to be a white noise.Due to a wide area of applications of equation (1), it is important to considercertain numerical methods to solve it. We use here the most popular and probablythe simplest method of Euler approximations: one takes a uniform partition ofthe interval, where the equation is being solved, and replaces differentials by acorrespondent finite differences. There is a vast literature dedicated to numericalmethods for stochastic differential equations driven by the Wiener process, we referto classical monographs [8] and [6] for an overview of the subject. There are alsoseveral papers dealing with discrete time approximations for stochastic differentialequations with fractional Brownian motion, for example, [10, 12, 3].The main difficulty when considering equation (1) lies in the fact that the ma-chinery behind the two stochastic integrals is very different. The Itˆo integral istreated usually in a mean square sense, while the integral with respect to fractional Mathematics Subject Classification.
Key words and phrases.
Fractional Brownian motion, mixed stochastic differential equation,pathwise integral, Euler approximation.
Brownian motion is understood and controlled in a pathwise sense. The mixtureof two integrals makes things a lot harder, forcing us to consider very smooth coef-ficients and to make delicate estimates.The paper is organized as follows. In Section 1, we give basic fact about for-ward and Skorokhod integration with respect to fractional Brownian motion andformulate main hypotheses. In Section 2, we define Euler approximations of (1)and establish some uniform integrability results for them. Section 3 contains themain result about rate of convergence of Euler approximations for equation (1).Unsurprisingly, the rate of convergence appears to be equal to the worst of therates for corresponding “pure” equations, i.e. the mean-square distance betweentrue and approximate solutions is of order δ / ∨ δ H − , where δ is the mesh of thepartition. 1. Preliminaries
Fractional Brownian motion and stochastic integration.
In this sectionwe give basic facts about the stochastic calculus for fractional Brownian motion. Amore extensive exposition can be found e.g. in [4, 1].Fractional Brownian motion (fBm) B H is by definition a centered Gaussian pro-cess with the covariance E (cid:2) B Ht B Hs (cid:3) = 12 (cid:16) t H + s H − | t − s | H (cid:17) , t, s ≥ . It has a version with almost surely κ -H¨older continuous paths for any κ < H .For H ∈ (1 / ,
1) (the case we consider here) it exhibits a property of long-rangedependence.Let L H [0 , T ] be the completion of the space of continuous functions with respectto the scalar product h f, g i H = Z Z [0 ,T ] f ( t ) g ( s ) ψ ( t, s ) ds dt, where ψ ( t, s ) = H (2 H − | t − s | H − . Denote also by k f k H = p h f, f i H thecorresponding norm.Now we recall the notion of stochastic derivative. Let infinitely differentiablefunction F : R n → R be bounded along with derivatives. For a smooth functional G = F ( B Ht , . . . , B Ht n ), where t , . . . , t n ∈ [0 , T ] the stochastic derivative is definedas D s G = n X k =1 F ′ x k ( B Ht , . . . , B Ht n )1I [0 ,t k ] ( s ) . The Sobolev space D , is the closure of the space of smooth functionals with respectto the norm k G k , = E (cid:2) G (cid:3) + E h k D G k H i . The Skorokhod, or divergence, stochastic integral is the adjoint to the stochasticderivative in the following sense. Let the domain dom δ of the divergence integralbe the space of random processes u ∈ L (Ω , L H [0 , T ]) such that E [ h D G, u i H ] ≤ C u k G k L (Ω)ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 3 for all G ∈ D , . Then the divergence integral δ ( u ) = Z T u t δB Ht is defined as the unique element of L (Ω) such that(2) E [ h D G, u i H ] = E [ Gδ ( u )]for all G ∈ D , . It is worth to remark that dom δ contains the space D , ( L [0 , T ])of processes such that k u k H ;1 , = E h k u k H i + Z Z Z Z [0 ,T ] E [ D t u v D s u z ] ψ ( t, s ) ψ ( v, z ) ds dt dz dv is finite. Moreover, for such processes(3) E (cid:2) δ ( u ) (cid:3) ≤ k u k H ;1 , . The forward integral with respect to fBm is defined as the uniform limit inprobability Z t u s dB Hs = lim ε → Z t u s B Hs + ε − B Hs ε ds, provided this limit exists. It is well-known (see e.g. [1]) that if for u ∈ D , ( L H [0 , T ]) Z Z [0 ,T ] | D s u t | ψ ( t, s ) ds dt < ∞ , then the forward integral exists and is equal to(4) Z T u t dB Ht = Z T u t δB Ht + Z Z [0 ,T ] D s u t ψ ( t, s ) ds dt. Assumptions.
The following hypotheses on the ingredients of equation (1)will be assumed throughout the paper.(A) The functions a and b are bounded together with their derivatives a ′ x , b ′ x : | a ( t, x ) | + | b ( t, x ) | + | a ′ x ( t, x ) | + | b ′ x ( t, x ) | ≤ K ;(B) the functions a and b are uniformly (2 H − | a ( t, x ) − a ( s, x ) | + | b ( t, x ) − b ( s, x ) | ≤ K | t − s | H − ;(C) the coefficient c is bounded together with its first and second derivativesand uniformly positive:0 ≤ c ( x ) + c ( x ) − + | c ′ ( x ) | + | c ′′ ( x ) | ≤ K. Here K is a constant independent of x , t and s ;(D) the Wiener process W and the fractional Brownian motion B H are inde-pendent.In what follows C will denote a generic constant, whose value might change fromline to line. To emphasize dependence on some variables, we will put them intosubscript. For a random process X we denote its increments by X t,s = X t − X s . YULIYA S. MISHURA AND GEORGIY M. SHEVCHENKO Euler approximations and auxiliary results
For N ≥ , T ] : { ν <ν < · · · < ν N = T, δ = T /N } , ν k = kδ. The Euler approximation for equation (1) is defined recursively as X δν k +1 = X δν k + a ( ν k , X δν k ) δ + b ( ν k , X δν k )∆ W k + c ( X δν k )∆ B Hk , where ∆ W k = W ν k +1 ,ν k , ∆ B Hk = B Hν k +1 ,ν k . The initial value of approximations is X δν = X .Set n δu = max { n : ν n ≤ u } , t δu = ν n δu , and define continuous interpolation by X δu = X δt δu + a ( t δu , X δt δu )( u − t δu ) + b ( t δu , X δt δu ) W u,t δu + c ( X δt δu ) B Hu,t δu , or, in the integral form,(5) X δu = X + Z u a ( t δs , X δt δs ) ds + Z u b ( t δs , X δt δs ) dW s + Z u c ( X δt δs ) dB Hs . The following lemma is a discrete analogue of the Gronwall inequality.
Lemma 2.1.
If a non-negative sequence { x n , n ≥ } satisfies x n +1 ≤ x n (1 + Kδ ) + Kδ.
Then x n ≤ ( x + 1) e Kδn . The following two lemmas are technical.
Lemma 2.2.
For s < ν n , n ≥ , one has (6) D s X δν n = c ( X δt δs ) n − Y k = n δs (cid:0) a ′ x ( ν k , X δν k ) δ + b ′ x ( ν k , X δν k )∆ W k + c ′ ( X δν k )∆ B Hk (cid:1) (the product is set to when the upper limit is smaller than the lower).Proof. Clearly, D s X δν n = 0 if ν n < s . Now observe that D s ∆ B Hn = 1I n = n δs . Hence,for n = n δs , we have D s X δν n = D s (cid:0) X δν n − + a ( ν n − , X δν n − ) δ + b ( ν n − , X δν n − )∆ W n − + c ( X δν n − )∆ B Hn − (cid:1) = c ( X δν n − ) D s ∆ B Hn − = c ( X δν n − ) . Further, for n > n δs we can write D s X δν n = D s (cid:0) X δν n − + a ( ν n − , X δν n − ) δ + b ( ν n − , X δν n − )∆ W n − + c ( X δν n − )∆ B Hn − (cid:1) = (cid:0) a ′ x ( ν n − , X δν n − ) δ + b ′ x ( ν n − , X δν n − )∆ W n − + c ′ ( X δν n − )∆ B Hn − (cid:1) D s X δν n − , and deduce (6) by induction. (cid:3) Lemma 2.3.
For any
M > it holds E " exp ( M N − X k =0 (cid:0) (∆ W k ) + (∆ B Hk ) (cid:1)) < C M for all N large enough with C M independent of N . ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 5
Proof.
Using independence of W and B H , we then can write E " exp ( M N − X k =0 (cid:0) (∆ W k ) + (∆ B Hk ) (cid:1)) = E " exp ( M N − X k =0 (∆ W k ) ) E " M exp ( N − X k =0 (∆ B Hk ) ) ≤ N − Y k =0 E (cid:2) exp (cid:8) M (∆ W k ) (cid:9)(cid:3) (cid:16) N − Y k =0 E (cid:2) exp (cid:8) M N (∆ B Hk ) (cid:9)(cid:3) (cid:17) /N = C (1 − M δ ) − N/ (1 − M N δ H ) − / = C (1 − M T /N ) − N/ (1 − M T H N − H ) − / , where the last equalities hold provided 2 M T /N <
M T H N − H <
1, whichis true for all N large enough. Observing that C (1 − M T /N ) − N/ (1 − M T H N − H ) − / → Ce MT , N → ∞ , we get the desired boundedness. (cid:3) Now we are ready to prove that the moments of Euler approximations as well asof their stochastic derivatives are uniformly bounded.
Lemma 2.4.
For any p > one has (7) E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p (cid:3) < C p for all s ∈ [0 , T ] , n ≤ N , with C p independent of δ .Proof. It is easy to see from (6) that the left-hand side of (7) is finite. Therefore,it suffices to establish boundedness only for N large enough.Introduce the following notation: a k = a ′ x ( ν k , X δν k ) , b k = b ′ x ( ν k , X δν k ) , c k = c ′ ( X δν k ) , Θ k = (cid:12)(cid:12) a k δ + b k ∆ W k + c k ∆ B Hk (cid:12)(cid:12) , γ k = | ∆ W k | + (cid:12)(cid:12) ∆ B Hk (cid:12)(cid:12) ,d k = a ( ν k , X δν k ) δ + b ( ν k , X δν k )∆ W k , ∆ k = d k + c ( X δν k )∆ B Hk = X δν k +1 ,ν k . Fix a small positive constant γ (its value will be specified later to satisfy ourneeds). Put A = {∀ k γ k ≤ γ } . E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p (cid:3) = E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p A (cid:3) + E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) . Step 1 . First we estimate E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) . Write E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) = X B E " c ( X δt δs ) p Y k / ∈ B Θ k γ k ≤ γ Y k ∈ B Θ k γ k >γ ! p , where the outer sum is taken over all non-empty B ⊂ (cid:8) n δs , n δs + 1 , . . . , n − (cid:9) .Observe that (cid:12)(cid:12) a k δ + b k ∆ W k + c k ∆ B Hk (cid:12)(cid:12) ≤ K ( δ + γ k ) , so this expression does not exceed 1 whenever δ < / (2 K ) and γ k ≤ γ < / (2 K ),and we can write Θ k γ k ≤ γ ≤ exp (cid:8) a k δ + b k ∆ W k + c k ∆ B Hk (cid:9) . YULIYA S. MISHURA AND GEORGIY M. SHEVCHENKO
For γ k > γ we estimate simply Θ k < exp { K ( δ + γ k ) } , therefore, E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) ≤ C p X B E " exp (X k / ∈ B ( a k δ + b k ∆ W k + c k ∆ B Hk ) ) Y k ∈ B Θ k γ k >γ ! p ≤ C p E " exp ( p X k / ∈ B ( Kδ + b k ∆ W k + K (cid:12)(cid:12) ∆ B Hk (cid:12)(cid:12) ) ) Y k ∈ B e pK ( δ + γ k ) γ k >γ ≤ C p E " exp ( p X k / ∈ B b k ∆ W k ) E " exp ( pK N − X k =0 (cid:12)(cid:12) ∆ B Hk (cid:12)(cid:12)) × E " Y k ∈ B e pKγ k γ k >γ / . By the standard properties of the stochastic integral with respect to W ,(8) E " exp ( p X k / ∈ B b k ∆ W k ) = E " exp (X k / ∈ B p b k δ/ ) ≤ exp (cid:8) p K N δ (cid:9) = exp (cid:8) p K T (cid:9) . Now estimate, using the H¨older inequality,(9) E " exp ( p X k / ∈ B c k ∆ B Hk ) ≤ E " exp ( pK N − X k =0 (cid:12)(cid:12) ∆ B Hk (cid:12)(cid:12)) ≤ N − Y k =0 E h e pKN | ∆ B Hk | i! /N ≤ E h e pKN | ∆ B H | i ≤ Ce p K N δ H = Ce p K T H N − H . Further, E " Y k ∈ B e pKγ k γ k >γ ≤ E " Y k ∈ B e pKγ k /γ γ k >γ ≤ E " exp ( pKγ N − X k =0 γ k ) E Y j ∈ B γ j >γ / ≤ C p,γ E Y j ∈ B γ j >γ / , where the last inequality hold for all N large enough thanks to Lemma 2.3. Toestimate the last expectation, recall that W and B H are independent and take first ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 7 the expectation with respect to W :(10) E Y j ∈ B γ j >γ ≤ E Y j ∈ B (cid:16)(cid:0)(cid:12)(cid:12) ∆ B Hj (cid:12)(cid:12) − γ (cid:1) δ − / (cid:17) ≤ n ( B ) Y j ∈ B E (cid:20) Φ (cid:16)(cid:0)(cid:12)(cid:12) ∆ B Hj (cid:12)(cid:12) − γ (cid:1) δ − / (cid:17) n ( B ) (cid:21) /n ( B ) ≤ n ( B ) E (cid:20) Φ (cid:16)(cid:0)(cid:12)(cid:12) ∆ B H (cid:12)(cid:12) − γ (cid:1) δ − / (cid:17) n ( B ) (cid:21) , where n ( B ) is the number of elements of B . We split the inner expectationinto parts where (cid:12)(cid:12) ∆ B H (cid:12)(cid:12) ≤ γ/ (cid:12)(cid:12) ∆ B H (cid:12)(cid:12) > γ/
2. For (cid:12)(cid:12) ∆ B H (cid:12)(cid:12) ≤ γ/ (cid:0)(cid:0)(cid:12)(cid:12) ∆ B H (cid:12)(cid:12) − γ (cid:1) δ − / (cid:1) ≤ Φ( − γδ − / / n ( B ) , also we have P ( (cid:12)(cid:12) ∆ B H (cid:12)(cid:12) > γ/ ≤ − γδ − H / E h Φ (cid:0) ( (cid:12)(cid:12) ∆ B H (cid:12)(cid:12) − γ ) δ − / (cid:1) n ( B ) i ≤ Φ( − γδ − / / n ( B ) + 2Φ( − γδ − H / ≤ e − γ δ − n ( B ) / + e − γ δ − H / ≤ e − C γ n ( B ) N + e − C γ N H . Plugging this into (10), we get E Y j ∈ B γ j >γ ≤ E Y j ∈ B (cid:0) ( (cid:12)(cid:12) ∆ B Hj (cid:12)(cid:12) − γ ) δ − / (cid:1) ≤ e C (1 − C γ N ) n ( B ) + e Cn ( B ) − C γ N H and combining this with (8) and (9), we arrive to E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) ≤ C p,γ e C p N − H X B e C (1 − C γ N ) n ( B ) + e − C γ N H X B e Cn ( B ) ! . For N large enough it holds C ( C γ N − ≥ C γ N (naturally, with different constants C γ in the left-hand and in the right-hand sides), so the first sum is bounded fromabove by X B e − C γ Nn ( B ) = (cid:0) e − C γ N (cid:1) n − n δs − ≤ (cid:0) e − C γ N (cid:1) N − (cid:8) N log (cid:0) e − C γ N (cid:1)(cid:9) − ≤ exp (cid:8) CN e − C γ N (cid:9) − ≤ exp (cid:8) C γ e − C γ N (cid:9) − ≤ C γ e − C γ N . Similarly, the second sum is bounded by e − C γ N H X B e Cn ( B ) = e − C γ N H (cid:0) e C (cid:1) n − n δs ≤ C γ e CN − C γ N H . Since H ∈ (1 / , E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) ≤ C p,γ e C p N − H (cid:16) e − C γ N + e CN − C γ N H (cid:17) ≤ C p,γ (cid:16) e − C p,γ N + e − C p,γ N − H (cid:17) YULIYA S. MISHURA AND GEORGIY M. SHEVCHENKO for all N large enough. This expression vanishes as N → ∞ , hence we get theboundedness of E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p Ω \ A (cid:3) . Step 2 . Now we turn to E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p A (cid:3) . If we take γ < K − / δ < K − / (cid:12)(cid:12) ∆ k (cid:12)(cid:12) < K − / A and (cid:12)(cid:12) c ( X δν k +1 ) − c ( X δν k ) (cid:12)(cid:12) < K − /
3. But c ( X δν k ) > K − ,so c ( X δν k +1 ) /c ( X δν k ) ∈ (1 / , / c ( X δν k +1 ) c ( X δν k ) = c ′ ( X δν k ) c ( X δν k ) ∆ k + R ′ k = c k ∆ B Hk + c k c ( X δν k ) d k + R ′ k , where | R ′ k | ≤ C ∆ k . Similarly, on A log Θ k = log (cid:0) a k δ + b k ∆ W k + c k ∆ B Hk (cid:1) = a k δ + b k ∆ W k + c k ∆ B Hk + R ′′ k with | R ′′ k | ≤ C ∆ k . Plugging into this formula the expression for c k ∆ B Hk from (11),we get Θ k = c ( X δν k +1 ) c ( X δν k ) exp { α k δ + β k ∆ W k + R k } , where R k = R ′′ k − R ′ k , α k = a k − c k a ( X δν k ) /c ( X δν k ), β k = b k − c k b ( X δν k ) /c ( X δν k ).Now we can estimate E (cid:2)(cid:12)(cid:12) D s X δν n (cid:12)(cid:12) p A (cid:3) = E c ( X δt δs ) p n − Y k = n δs Θ pk A = E c ( X δν n − ) p exp p n − X k = n δs ( α k δ + β k ∆ W k + R k ) ≤ C p E exp p n − X k = n δs β k ∆ W k E " exp ( p N − X k =0 | R k | ) / ≤ C p E exp p n − X k = n δs β k δ E " exp ( C p N − X k =0 ∆ k ) / ≤ C p E " exp ( C p N − X k =0 γ k ) ≤ C p , where the last holds for all N large enough due to Lemma 2.3. This completes theproof. (cid:3) Lemma 2.5.
For any p > one has (12) E (cid:2)(cid:12)(cid:12) X δt (cid:12)(cid:12) p (cid:3) ≤ C p for all t ∈ [0 , T ] .Moreover, (13) E h(cid:12)(cid:12)(cid:12) X δt − X δt δt (cid:12)(cid:12)(cid:12) p i ≤ C p δ p/ for any t ∈ [0 , T ] . ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 9
Proof.
It is enough to prove this for p = 2 m , m ∈ N . We first prove (12) for t = ν n ,using an induction by m .Start with m = 1.Denote a n = a ( ν n , X δν n ) , b n = b ( ν n , X δν n ) , c n = c ( X δν n ) and write for δ ∈ (0 , / E h(cid:0) X δν n +1 (cid:1) i ≤ E h(cid:0) X δν n + b n ∆ W n + c n ∆ B Hn (cid:1) i (1 − δ ) − + 2 E h(cid:0) a n δ (cid:1) i δ − ≤ (cid:16) E h(cid:0) X δν n (cid:1) i + E (cid:2) ( b n ∆ W n ) (cid:3) + E (cid:2) ( c n ∆ B Hn ) (cid:3) +2 E (cid:2) X δν n b n ∆ W n (cid:3) + 2 E (cid:2) b n c n ∆ W n ∆ B Hn (cid:3) + 2 E (cid:2) X δν n ∆ B Hn (cid:3) (cid:17) e δ + Cδ ≤ (cid:16) E h(cid:0) X δν n (cid:1) i + Cδ + Cδ H + 2 E (cid:2) X δν n ∆ B Hn (cid:3) (cid:17) e δ + Cδ ≤ E h(cid:0) X δν n (cid:1) i e δ + E (cid:2) X δν n ∆ B Hn (cid:3) e δ + Cδ.
By (2) and (7), we can write E (cid:2) X δν n ∆ B Hn (cid:3) = α H Z ν n Z ν n +1 ν n E (cid:2) D s X δν n (cid:3) ( t − s ) H − dt ds ≤ C Z ν n +1 ν n ( t − ν n ) H − dt ≤ Cδ.
Then by Lemma 2.1 E h(cid:0) X δν n (cid:1) i ≤ X e Cδn ≤ Ce CδN ≤ C, as required.Now let m ≥ l ≤ m E h(cid:0) X δt (cid:1) l i ≤ C l . In the further estimates constants may depend on m , but not on n .Observe that by the Jensen inequality for δ < a + b ) m ≤ (1 − δ ) − m a m + δ − m b m , whence(14) ( a + b ) m ≤ a m (1 + C m δ ) + C m b m δ − m , therefore E h(cid:0) X δν n +1 (cid:1) m i ≤ E h(cid:0) X δν n + b n ∆ W n + c n ∆ B Hn (cid:1) m i (1 + Cδ )+ E h(cid:0) a n δ (cid:1) m i δ − m ≤ E h(cid:0) X δν n + b n ∆ W n + c n ∆ B Hn (cid:1) m i (1 + Cδ ) + Cδ.
Expand the power in the first term and consider a generic term of this expansion(without a coefficient): E h(cid:0) X δν n (cid:1) m − i − k ( b n ∆ W n ) i ( c n ∆ B Hn ) k i = E h(cid:0) X δν n (cid:1) m − i − k b in c kn E (cid:2) (∆ W n ) i (∆ B Hn ) k | F ν n (cid:3)i = E h(cid:0) X δν n (cid:1) m − i − k b in c kn E (cid:2) (∆ W n ) i (cid:3) E (cid:2) (∆ B Hn ) k | F ν n (cid:3)i = E h(cid:0) X δν n (cid:1) m − i − k b in c kn (∆ B Hn ) k i i !2 i/ ( i/ δ i/ i even . Thus, we can write E h(cid:0) X δν n + b n ∆ W n + c n ∆ B Hn (cid:1) m i = m X k =0 m X j =0 (cid:18) mk, j, m − k − j (cid:19) E h(cid:0) X δν n (cid:1) m − j − k b jn c kn (∆ B Hn ) k i (2 j )!2 j j ! δ j , where (cid:18) a + b + ca, b, c (cid:19) = ( a + b + c )! a ! b ! c !is a trinomial coefficient.For k = 0, j ≥
1, the terms of this sum are bounded by Cδ by the inductionhypothesis and boundedness of b n , c n .Further, for k ≥ E h(cid:0) X δν n (cid:1) m − j − k b jn c kn (∆ B Hn ) k i ≤ C E h(cid:12)(cid:12) X δν n (cid:12)(cid:12) m − j − k (cid:12)(cid:12) ∆ B Hn (cid:12)(cid:12) k i ≤ C E h(cid:0) X δν n (cid:1) m i /λ E h(cid:12)(cid:12) ∆ B Hn (cid:12)(cid:12) kη i /η ≤ C (cid:16) E h(cid:0) X δν n (cid:1) m i (cid:17) δ H ≤ C (cid:16) E h(cid:0) X δν n (cid:1) m i (cid:17) δ, where λ = 2 m/ (2 m − j − k ), η = λ/ ( λ − (cid:16) E h(cid:0) X δν n (cid:1) m i (cid:17) /λ ≤ E (cid:2)(cid:0) X δν n (cid:1)(cid:3) m . Now estimate the term with k = 1, j = 0, using formula (2): E h(cid:0) X δν n (cid:1) m − c n ∆ B Hn i = Z ν n Z ν n +1 ν n E h D s (cid:16)(cid:0) X δν n (cid:1) m − c ( X δν n ) (cid:17)i ψ ( t, s ) dt ds = Z ν n Z ν n +1 ν n E h(cid:16) (2 m − (cid:0) X δν n (cid:1) m − c ( X δν n ) + (cid:0) X δν n (cid:1) m − c ′ ( X δν n ) (cid:17) D s X δν n i × H (2 H − t − s ) H − dt ds ≤ C (cid:16) E h(cid:0) X δν n (cid:1) m i (cid:17) Z ν n +1 ν n ( t − ν n ) H − dt ≤ C (cid:16) E h(cid:0) X δν n (cid:1) m i (cid:17) δ. Here, as above we have used the H¨older inequality, inequality (15) and boundednessof moments of the stochastic derivative. The terms with k = 1, j ≥ E h(cid:0) X δν n +1 (cid:1) m i ≤ E h(cid:0) X δν n (cid:1) m i (1 + Cδ ) + Cδ,
ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 11 so by Lemma 2.1 we get the desired boundedness.Now write for s ∈ [ ν n , ν n +1 ) E h(cid:12)(cid:12) X δs − X δν n (cid:12)(cid:12) p i ≤ C p (cid:16) E [ | a n ( s − ν n ) | p ] + E [ | b n W s,ν n ) | p ] + E h(cid:12)(cid:12) c n B Hs,ν n (cid:12)(cid:12) p i (cid:17) ≤ C p (cid:0) ( s − ν n ) p + ( s − ν n ) p/ + ( s − ν n ) pH (cid:1) ≤ C p ( s − ν n ) p/ , which gives (13) and together with (12) for t = ν n implies (12) for all t ∈ [0 , T ]. (cid:3) Rate of convergence
Now we are ready to prove the main result about the mean-square rate of con-vergence of Euler approximations.
Theorem 3.1.
Euler approximations (5) for the solution of equation (1) satisfy E h(cid:0) X t − X δt (cid:1) i ≤ C ( δ + δ H − ) . Proof.
Define ψ ( x ) = R x c ( z ) − dz . It is clear that K − | x − y | ≤ | ψ ( x ) − ψ ( y ) | ≤ K | x − y | . Write by the Itˆo formula ψ ( X t ) = ψ ( X ) + Z t (cid:16) α ( s, X s ) ds + β ( s, X s ) dW s (cid:17) + B Ht , where α ( s, x ) = a ( s, x ) c ( x ) − b ( s, x ) c ′ ( x )2 c ( x ) , β ( s, x ) = b ( s, x ) c ( x ) . Similarly, ψ ( X δt ) = ψ ( X δ ) + Z t (cid:16) α ( s, X δs ) ds + β ( s, X δs ) dW s (cid:17) + B Ht − G δt , where G δt = Z t (cid:20) c − ( X δs ) (cid:16)(cid:0) a ( s, X δs ) − a ( t δs , X δt δs ) (cid:1) ds + (cid:0) b ( s, X δs ) − b ( t δs , X δt δs ) (cid:1) dW s (cid:17) + c − ( X δs ) (cid:0) c ( X δs ) − c ( X δt δs ) (cid:1) dB Hs + c ′ ( X δs )2 c ( X δs ) (cid:0) b ( s, X δs ) − b ( t δs , X δt δs ) (cid:1) ds (cid:21) =: Z t (cid:0) a δs ds + b δs dW s + c δs dB Hs + d δs ds (cid:1) . Estimate E "(cid:18)Z t a δs ds (cid:19) ≤ C Z t E (cid:2) ( a δs ) (cid:3) ds ≤ C Z t E h c ( X δs ) − (cid:0) a ( s, X δt δs ) − a ( t δs , X δt δs ) (cid:1) + (cid:0) a ( s, X δs ) − a ( s, X δt δs ) (cid:1) i ds ≤ C (cid:18) δ H − + Z t E h(cid:0) X δs,t δs (cid:1) i ds (cid:19) ≤ Cδ H − . Similarly, E "(cid:18)Z t d δs ds (cid:19) ≤ C Z t E (cid:2) ( d δs ) (cid:3) ds ≤ Cδ H − . and using the Itˆo isometry, E "(cid:18)Z t b δs dW s (cid:19) = Z t E (cid:2) ( b δs ) (cid:3) ds ≤ Cδ H − . Further, by (4) Z t c δs dB Hs = Z t c δs δB Hs + Z Z [0 ,t ] D s c δu ψ ( s, u ) ds du =: I ′ ( δ, t ) + I ′′ ( δ, t ) . By the chain rule for the stochastic derivative, D u c δs = D u (cid:16) c − ( X δs ) (cid:0) c ( X δs ) − c ( X δt δs ) (cid:1)(cid:17) = c ′ ( X δs ) c ( X δs ) (cid:0) c ( X δs ) − c ( X δt δs ) (cid:1) D u X δs + c − ( X δs ) (cid:0) c ′ ( X δs ) D u X δs − c ′ ( X δt δs ) D u X δt δs (cid:1) = c ′ ( X δs ) c ( X δs ) (cid:0) c ( X δs ) − c ( X δt δs ) (cid:1) D u X δs + c − ( X δs ) (cid:0) c ′ ( X δs ) − c ′ ( X δt δs ) (cid:1) D u X δs + c − ( X δs ) c ′ ( X δt δs ) D u X δs,t δs (cid:1) =: D ( u, s ) + D ( u, s ) + D ( u, s ) . Now E (cid:2) D ( u, s ) (cid:3) ≤ C E h(cid:0) c ( X δs ) − c ( X δt δs ) (cid:1) ( D u X δs ) i ≤ C E h(cid:0) X δs − X δt δs (cid:1) i ≤ Cδ.
Similarly, E (cid:2) D ( u, s ) (cid:3) ≤ Cδ.
Further, for u ≤ t δs X δs,t δs = D u X δt δs (cid:16) a ′ x ( t δs , X δt δs )( s − t δs ) + b ′ x ( t δs , X δt δs ) W s,t δs + c ′ ( X δt δs ) B Hs,t δs (cid:17) and E (cid:2) D ( u, s ) (cid:3) ≤ E h ( D u X δt δs ) i / × E (cid:20)(cid:16) a ′ x ( t δs , X δt δs )( s − t δs ) + b ′ x ( t δs , X δt δs ) W s,t δs + c ′ ( X δt δs ) B Hs,t δs (cid:17) (cid:21) / ≤ Cδ ;for u ∈ [ t δs , s ) D ( u, s ) = c ( X δt δs )and D ( u, s ) = 0 for u > s . Thus E (cid:2) I ′′ ( δ, t ) (cid:3) ≤ C Z Z [0 ,t ] E h D ( u, s ) + D ( u, s ) + D ( u, s ) [0 ,t δs ] ( u ) i ψ ( s, u ) du ds + C E Z t Z st δs (cid:12)(cid:12)(cid:12) c ( X δt δs ) (cid:12)(cid:12)(cid:12) ψ ( s, u ) du ds ! ≤ C δ + (cid:18)Z t (cid:12)(cid:12) s − t δs (cid:12)(cid:12) H − ds (cid:19) ! ≤ C ( δ + δ H − ) . ATE OF CONVERGENCE OF EULER APPROXIMATIONS OF MIXED SDE 13
By (3) E (cid:2) I ′ ( δ, t ) (cid:3) ≤ Z [0 ,t ] E (cid:2) c δs c δu (cid:3) ψ ( s, u ) ds du + Z [0 ,t ] E (cid:2) D u c δv D s c δz (cid:3) ψ ( s, u ) du dv ds dz. The first term is estimated by Cδ using that E (cid:2) c δs c δu (cid:3) ≤ E (cid:2) ( c δu ) (cid:3) + E (cid:2) ( c δs ) (cid:3) ≤ C E h(cid:0) X δs,t δs (cid:1) i + E h(cid:0) X δu,t δu (cid:1) i ≤ Cδ.
In the second, we write D u c δv = D ( u, v ) + D ( u, v ) + D ( u, v ) and similarly for D s c δz . For D , D we use the Cauchy inequality and the above estimates to geta bound of Cδ . This also works for D when u / ∈ [ t δv , v ) and s / ∈ [ t δz , z ). Tworemaining terms for D are similar, take e.g. Z t Z t Z t Z vt δv E (cid:2) D u c δv D s c δz (cid:3) | s − u | H − | z − v | H − du dv ds dz ≤ Cδ H − Z t Z t Z t E (cid:2)(cid:12)(cid:12) D s c δz (cid:12)(cid:12)(cid:3) | z − v | H − dv ds dz. Again, if s / ∈ [ t δz , z ), the integral can be estimated by Cδ / ; for s ∈ [ t δz , z ) we get δ H − . Ultimately, E (cid:2) I ′ ( δ, t ) (cid:3) ≤ C ( δ + δ H − )and adding this to the previous estimates, we get E (cid:2) ( G δt ) (cid:3) ≤ C ( δ + δ H − ) . So we can write E h(cid:0) ψ ( X t ) − ψ ( X δt ) (cid:1) i ≤ C (cid:18)Z t E h(cid:0) α ( s, X s ) − α ( s, X δs ) (cid:1) + (cid:0) β ( s, X s ) − β ( s, X δs ) (cid:1) i ds + E (cid:2) ( G δt ) (cid:3)(cid:19) ≤ C Z t E h(cid:0) X s − X δs (cid:1) i ds + C ( δ + δ H − ) ≤ C Z t E h(cid:0) ψ ( X s ) − ψ ( X δs ) (cid:1) i ds + C ( δ + δ H − ) . By the Gronwall lemma, E h(cid:0) ψ ( X t ) − ψ ( X δt ) (cid:1) i ≤ C ( δ + δ H − ) , hence E h(cid:0) X t − X δt (cid:1) i ≤ C ( δ + δ H − ) , as required. (cid:3) Remark . The obtained estimate can also be written as (cid:0) E h(cid:0) X t − X δt (cid:1) i (cid:1) / ≤ C ( δ / ∨ δ H − ), so the mean-square rate of convergence for the mixed equation isthe worst of the two rates for “pure” stochastic differential equation with Brownianmotion, Cδ / , and with fractional Brownian motion, Cδ H − . As long as theseestimates for pure equations are sharp (see [6, 12]), we get that in our case theestimate is sharp as well. An interesting observation is that the value of the Hurst index where the rateof convergence changes is H = 3 /
4. From [2] it is known that the measure inducedby the mixture of Brownian motion and independent fractional Brownian motion isequivalent to the Wiener measure iff
H > /
4. So in this case it is perhaps naturalto expect that the rate of convergence of Euler approximations is the same as forBrownian motion, and this is exactly what we see here.
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