On drift parameter estimation in models with fractional Brownian motion
aa r X i v : . [ m a t h . P R ] D ec ON DRIFT PARAMETER ESTIMATION IN MODELS WITHFRACTIONAL BROWNIAN MOTION
Y. KOZACHENKO A , A. MELNIKOV B ∗ AND Y. MISHURA A Abstract.
We consider a stochastic differential equation involving standardand fractional Brownian motion with unknown drift parameter to be esti-mated. We investigate the standard maximum likelihood estimate of the driftparameter, two non-standard estimates and three estimates for the sequentialestimation. Model strong consistency and some other properties are proved.The linear model and Ornstein-Uhlenbeck model are studied in detail. Asan auxiliary result, an asymptotic behavior of the fractional derivative of thefractional Brownian motion is established. [2010]60G22; 60J65; 60H10; 62F05 Fractional Brownian motion, Brownian mo-tion, parameter estimation; stochastic differential equation; sequential estimation1.
Introduction
Modern mathematical statistics tends to shift away from the standard statisticalschemes based on independent random variables; besides, these days many statis-tical models are based on continuous time. Therefore, the corresponding statisticalproblems (e.g., parameter estimation) can be handled by methods of the theoryof stochastic processes in addition to the standard statistical methods. Statisticsfor stochastic processes is well-developed for diffusion processes and even for semi-martingales (see, for instance, [LipSh]) but is still developing for the processes withlong-range dependence. The latter is an integral part of stochastic processes, fea-turing a wide spectrum of applications applications in economics, physics, financeand other fields. The present paper is devoted to the parameter estimation in suchmodels involving fractional Brownian motion (fBm) with Hurst parameter
H > which is a well-known long-memory process. The paper also studies a mixed modelbased on both standard and fractional Brownian motion which turns out to bemore flexible. One of the reasons to consider such model comes from the modernmathematical finance where it it has become very popular to assume that the un-derlying random noise consists of two parts: the fundamental part, describing theeconomical background for the stock price, and the trading part, related to therandomness inherent to the stock market. In our case the fundamental part of thenoise has a long memory while the trading part is a white noise. ∗ Corresponding author. Email: [email protected] a Department of Probability, Statistics and Actuarial Mathematics, Mechanics and Mathe-matics Faculty, Kyiv Taras Shevchenko National University, Volodymyrska, 60, 01601 Kyiv ; b Department of Mathematical and Statistical Sciences, University of Alberta, 632 Central Aca-demic Building, Edmonton, AB T6G 2G1, Canada . A , A. MELNIKOV B ∗ AND Y. MISHURA A Statistical aspects of models involving fractional Brownian motion were studiedin many sources. One of the important problems in particular is the drift parame-ter estimation. In this regard, let us mention papers [HuNu] and [KlLeBr], wherethe fractional Ornstein-Uhlenbeck process with unknown drift parameter originallywas studied, books [Bish08], [Mish08] and [Prara] and the references therein, andpapers [BTT], [XZX], [XZZ], and [HuXZ], where the estimate was constructed viadiscrete observations. We shall also use the results for sequential estimates forsemimartingales from [MN88]. In the present paper we consider stochastic differen-tial equations involving fractional Brownian motion along with equations involvingboth standard and fractional Brownian motion. We derive the standard maximumlikelihood estimate and propose non-standard estimates for the unknown drift pa-rameter. Several non-standard estimates for the drift parameter were proposed in[HuNu] for the fractional Ornstein-Uhlenbeck process. We go a step ahead andpropose non-standard estimates for the drift parameter in a general stochastic dif-ferential equation involving fBm. For the models involving only fractional Brownianmotion, we compare properties of the estimates. In the mixed models the standardmaximum likelihood estimate does not exist but the non-standard estimate works.To formulate the conditions for strong consistency of the non-standard estimates,we need to investigate the asymptotic behavior of the fractional derivative of thefractional Brownian motion using the general growth results for Gaussian processes.The paper is organized as follows. In Section 2 we introduce the models andthe estimates: the maximum likelihood estimate, two non-standard estimates andthree sequential estimates. Asymptotic growth of the fractional derivative of fBmis established in Section 4. Section 5 contains the main results concerning thestrong consistency of all estimates and some additional properties of sequentialestimates. The linear model and Ornstein-Uhlenbeck model are studied in detail.We generalize the result of strong consistency of the drift parameter estimate in theOrnstein-Uhlenbeck model from [KlLeBr] to the model with variable coefficients.2.
Model description and preliminaries
Model description.
Let (Ω , F , F , P ) be a complete probability space withfiltration F = {F t , t ∈ R + } satisfying the standard assumptions. It is assumed thatall processes under consideration are adapted to filtration F . Definition 1.
Fractional Brownian motion (fBm) with Hurst index H ∈ (0 ,
1) isa Gaussian process B H = { B Ht , t ∈ R + } on (Ω , F , P ) featuring the properties(a) B H = 0;(b) EB Ht = 0 , t ∈ R + ;(c) EB Ht B Hs = ( t H + s H − | t − s | H ) , s, t ∈ R + .We consider the continuous modification of B H whose existence is guaranteedby the classical Kolmogorov theorem.To describe the statistical model, we need to introduce the pathwise integralsw.r.t. fBm. Consider two non-random functions f and g defined on some interval[ a, b ] ⊂ R + . Suppose also that the the following limits exist: f ( u +) := lim δ ↓ f ( u + δ ) and g ( u − ) := lim δ ↓ g ( u − δ ) , a ≤ u ≤ b . Let f a + ( x ) := ( f ( x ) − f ( a +)) ( a,b ) ( x ) , g b − ( x ) := ( g ( b − ) − g ( x )) ( a,b ) ( x ) . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION3
Suppose that f a + ∈ I αa + ( L p [ a, b ])) , g b − ∈ I − αb − ( L q [ a, b ])) for some p ≥ , q ≥ , /p + 1 /q ≤ , ≤ α ≤ . (For the standard notation and statements concerningfractional analysis, see [SMK]). Introduce the fractional derivatives( D αa + f a + )( x ) = 1Γ(1 − α ) (cid:16) f a + ( s )( s − a ) α + α Z sa f a + ( s ) − f a + ( u )( s − u ) α du (cid:17) ( a,b ) ( x )( D − αb − g b − )( x ) = e − i πα Γ( α ) (cid:16) g b − ( s )( b − s ) − α + (1 − α ) Z bs g b − ( s ) − g b − ( u )( s − u ) − α du (cid:17) ( a,b ) ( x ) . It is known that D αa + f a + ∈ L p [ a, b ] , D − αb − g b − ∈ L q [ a, b ] . Definition 2. ([Zah98], [Zah99]) Under above assumptions, the generalized (frac-tional) Lebesgue-Stieltjes integral R ba f ( x ) dg ( x ) is defined as Z ba f ( x ) dg ( x ) := e i πα Z ba ( D αa + f a + )( x )( D − αb − g b − )( x ) dx + f ( a +)( g ( b − ) − g ( a +)) , and for αp < Z ba f ( x ) dg ( x ) := e i πα Z ba ( D αa + f )( x )( D − αb − g b − )( x ) dx. As follows from [SMK], for any 1 − H < α < D − αb − B Hb − and D − αb − B Hb − ∈ L ∞ [ a, b ] for any 0 ≤ a < b. Therefore, for f ∈ I αa + ( L [ a, b ]) we can define the integral w.r.t. fBm in the following way. Definition 3. ([NuaR], [Zah98], [Zah99]) The integral with respect to fBm is de-fined as(1) Z ba f dB H := e i πα Z ba ( D αa + f )( x )( D − αb − B Hb − )( x ) dx. An evident estimate follows immediately from (1):(2) (cid:12)(cid:12)(cid:12) Z ba f dB H (cid:12)(cid:12)(cid:12) ≤ sup a ≤ x ≤ b | ( D − αb − B Hb − )( x ) | Z ba | ( D αa + f )( x ) | dx. Let us take a Wiener process W = { W t , t ∈ R + } on probability space (Ω , F , F , P ),possibly correlated with B H . Assume that H > and consider a one-dimensionalmixed stochastic differential equation involving both the Wiener process and thefractional Brownian motion(3) X t = x + θ Z t a ( s, X s ) ds + Z t b ( s, X s ) dB Hs + Z t c ( s, X s ) dW s , t ∈ R + , where x ∈ R is the initial value, θ is the unknown parameter to be estimated, thefirst integral in the right-hand side of (3) is the Lebesgue-Stieltjes integral, the sec-ond integral is the generalized Lebesgue-Stieltjes integral introduced in Definition3, and the third one is the Itˆo integral. From now on, we shall assume that thecoefficients of equation (3) satisfy the following assumptions on any interval [0 , T ]:( A ) Linear growth of a and b : for any s ∈ [0 , T ] and any x ∈ R | a ( s, x ) | + | b ( s, x ) | ≤ K (1 + | x | ) . ( A ) Lipschitz continuity of a, c in space : for any t ∈ [0 , T ] and x, y ∈ R | a ( t, x ) − a ( t, y ) | + | c ( t, x ) − c ( t, y ) | ≤ K | x − y | . Y. KOZACHENKO A , A. MELNIKOV B ∗ AND Y. MISHURA A ( A ) H¨older continuity in time : function b ( t, x ) is differentiable in x and thereexists β ∈ (1 − H,
1) such that for any s, t ∈ [0 , T ] and any x ∈ R | a ( s, x ) − a ( t, x ) | + | b ( s, x ) − b ( t, x ) | + | c ( s, x ) − c ( t, x ) | + | ∂ x b ( s, x ) − ∂ x b ( t, x ) | ≤ K | s − t | β . ( A ) Lipschitz continuity of ∂ x b in space : for any t ∈ [0 , T ] and any x, y ∈ R | ∂ x b ( t, x ) − ∂ x b ( t, y ) | ≤ K | x − y | . ( A ) Boundedness of c and ∂ x b : for any s ∈ [0 , T ] and x ∈ R | c ( s, x ) | + | ∂ x b ( s, x ) | ≤ K. Here K is a constant independent of x , y , s and t . For an arbitrary interval[0 , T ] , α > κ = ∧ β define the following norm: k f k ∞ ,α, [0 ,T ] = sup s ∈ [0 ,T ] (cid:18) | f ( s ) | + Z s | f ( s ) − f ( z ) | ( s − z ) − − α dz (cid:19) . It was proved in [MiSh] that under assumptions ( A ) − ( A ) there exists solution X = { X t , F t , t ∈ [0 , T ] } for equation (3) on any interval [0 , T ] which satisfies(4) k X k ∞ ,α, [0 ,T ] < ∞ a.s.for any α ∈ (1 − H, κ ). This solution is unique in the class of processes satisfying(4) for some α > − H . Remark . In case when components W and B H are independent, assumptions forthe coefficients can be relaxed, as it has been shown in [GuNu]. More specifically,coefficient c can be of linear growth, and ∂ x b can be H¨older continuous up to someorder less than 1.2.2. Construction of drift parameter estimates: the standard maximumlikelihood estimate.
To start with, consider the case c ( t, x ) ≡ X t = x + θ Z t a ( s, X s ) ds + Z t b ( s, X s ) dB Hs , t ∈ R . Let assumptions ( A ) and ( A ) with c ≡ , T ], togetherwith the following assumptions:( A ′ ) Lipschitz continuity of a, b in space : for any t ∈ [0 , T ] and x, y ∈ R | a ( t, x ) − a ( t, y ) | + | b ( t, x ) − b ( t, y ) | ≤ K | x − y | , ( A ′ ) H¨older continuity of ∂ x b ( t, x ) in space: there exists such ρ ∈ (3 / − H, t ∈ [0 , T ] and x, y ∈ R | ∂ x b ( t, x ) − ∂ x b ( t, y ) | ≤ D | x − y | ρ , Then, according to [NuaR], solution for equation (5) exists on any interval [0 , T ]and is unique in the class of processes satisfying (4) for some α > − H .In addition, suppose that the following assumption holds:( B ) b ( t, X t ) = 0 , t ∈ [0 , T ] and a ( t,X t ) b ( t,X t ) is a.s. Lebesgue integrable on [0 , T ] forany T > N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION5
Denote ψ ( t, x ) = a ( t,x ) b ( t,x ) , ϕ ( t ) := ψ ( t, X t ). Also, let the kernel l H ( t, s ) = c H s − H ( t − s ) − H I {
Let ψ ( t, x ) ∈ C ( R + ) × C ( R ) . Then for any t > J ′ ( t ) = (2 − H ) C H ψ (0 , x ) t − H + Z t l H ( t, s ) ( ψ ′ t ( s, X s ) + θψ ′ x ( s, X s ) a ( s, X s )) ds − (cid:16) H − (cid:17) c H Z t s − − H ( t − s ) − H Z s (cid:16) ψ ′ t ( u, X u ) + θψ ′ x ( u, X u ) a ( u, X u ) (cid:17) duds +(2 − H ) c H t − H Z t s H − Z s u − H ( s − u ) − H ψ ′ x ( u, X u ) b ( u, X u ) dB Hu ds + c H t − Z t u − H ( t − u ) − H ψ ′ x ( u, X u ) b ( u, X u ) dB Hu , where C H = B ( − H, − H ) c H = (cid:16) Γ( − H )2 H Γ( H + )Γ(3 − H ) (cid:17) , and all of the involvedintegrals exist a.s.Remark . Suppose that ψ ( t, x ) ∈ C ( R + ) × C ( R ) and limit ς (0) = lim s → ς ( s )exists a.s., where ς ( s ) = s − H ϕ ( s ). In this case J ( t ) can be presented as J ( t ) = c H Z t ( t − s ) − H ς ( s ) ds = c H t − H − H ς (0) + c H Z t ( t − s ) − H − H ς ′ ( s ) ds, Y. KOZACHENKO A , A. MELNIKOV B ∗ AND Y. MISHURA A and J ′ ( t ) from (7) can be simplified to J ′ ( t ) = c H t − H ς (0) + Z t l H ( t, s ) (cid:16)(cid:16) − H (cid:17) s − ϕ ( s ) + ψ ′ t ( s, X s )+ θψ ′ x ( s, X s ) a ( s, X s ) (cid:17) ds + Z t l H ( t, s ) ψ ′ x ( s, X s ) b ( s, X s ) dB Hs . Same way as Z , processes J and J ′ are functionals of X . It is more convenientto consider process χ ( t ) = (2 − H ) − J ′ ( t ) t H − , so that Z t = (2 − H ) θ Z t χ ( s ) s − H ds + M Ht = θ Z t χ ( s ) d h M H i s + M Ht . Suppose that the following conditions hold:( B ) EI T := E R T χ s d h M H i s < ∞ for any T > B ) I ∞ := R ∞ χ s d h M H i s = ∞ a.s.Then we can consider the maximum likelihood estimate θ (1) T = R T χ s dZ s R T χ s d h M H i s = θ + R T χ s dM Hs R T χ s d h M H i s . Condition ( B ) ensures that process R t χ s dM Hs , t > B ) alongside with the law of large numbers for martingalesensure that R T χ s dM Hs R T χ s d h M H i s → T → ∞ . Summarizing, we arrive at thefollowing result ([Mish08]). Proposition 1.
Let ψ ( t, x ) ∈ C ( R + ) × C ( R ) and assumptions ( A ) , ( A ) , ( A ′ ) , ( A ′ ) and ( B ) – ( B ) hold. Then estimate θ (1) T is strongly consistent as T → ∞ . Construction of drift parameter estimates: two non-standard esti-mates.
In case when c = 0, it is possible to construct another estimate for pa-rameter θ , preserving the structure of the standard maximum likelihood estimate.Similar approach was applied in [HuNu] to the fractional Ornstein-Uhlenbeck pro-cess with constant coefficients. We shall use process Y to define the estimate as(8) θ (2) T = R T ϕ s dY s R T ϕ s ds = θ + R T ϕ s dB Hs R T ϕ s ds . Let us return to general equation (3) with non-zero c and construct the estimateof parameter θ . Suppose that the following assumption holds:( C ) c ( t, X t ) = 0 , t ∈ [0 , T ], a ( t,X t ) c ( t,X t ) is a.s. Lebesgue integrable on [0 , T ] for any T > R T b ( t,X t ) c ( t,X t ) dB Ht .Define functions ψ ( t, x ) = a ( t,x ) c ( t,x ) and ψ ( t, x ) = b ( t,x ) c ( t,x ) , processes ϕ i ( t ) = ψ i ( t, X t ) , i =1 , Y t = Z t b − ( s, X s ) dX s = θ Z t ϕ ( s ) ds + Z t ϕ ( s ) dB Hs + W t . Evidently, Y is a functional of X and is observable. Assume additionally that thegeneralized Lebesgue–Stieltjes integral R T ϕ ( t ) ϕ ( t ) dB Ht exists and( C ) for any T > E R T ϕ ( s ) ds < ∞ . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION7
Denote ϑ ( s ) = ϕ ( s ) ϕ ( s ). We can consider the following estimate of parameter θ :(9) θ (3) T = R T ϕ ( s ) dY s R T ϕ ( s ) ds = θ + R T ϑ ( s ) dB Hs R T ϕ ( s ) ds + R T ϕ ( s ) dW s R T ϕ ( s ) ds . Estimate θ (3) T preserves the traditional form of maximum likelihood estimates fordiffusion models. The right-hand side of (9) provides a stochastic representation of θ (3) T . We shall use it to investigate the strong consistency of this estimate.2.4. Construction of drift parameter estimates: sequential estimates.
Re-turn to model (5) and suppose that conditions ( B ) − ( B ) hold. For any h > τ ( h ) = inf { t > Z t χ s d h M i s = h } . Under conditions ( B ) − ( B ) we have τ ( h ) < ∞ a.s. and R τ ( h )0 χ s d h M i s = h . Thesequential maximum likelihood estimate has a form(10) θ (1) τ ( h ) = R τ ( h )0 χ s dZ s h = θ + R τ ( h )0 χ s dM Hs h . Sequential versions of estimates θ (2) T and θ (3) T have a form θ (2) τ ( h ) = θ + R τ ( h )0 ϕ s dB Hs h and θ (3) υ ( h ) = θ + R υ ( h )0 ϑ ( s ) dB Hs h + R υ ( h )0 ϕ ( s ) dW s h , where υ ( h ) = inf { t > Z t ϕ ( s ) ds = h } . To provide an exhaustive study of the introduced estimates, we will need anumber of auxiliary facts about Gaussian processes. These facts are presented inthe next section. Technical proofs may be found in Appendix.3.
Auxiliary results for Gaussian processes related to thefractional Brownian motion.
We start with the exponential maximal bound for a Gaussian process defined onan abstract pseudometric space, expressed in terms of the metric capacity of thisspace. This result is a particular case of the general theorem proved in [BulKoz],p. 100.
Lemma 2.
Let T be a non-empty set, X = { X ( t ) , t ∈ T } be centered Gaussianprocess. Suppose that the pseudometric space ( T , ρ ) with pseudometric ρ ( t , s ) = (cid:0) E ( X ( t ) − X ( s )) (cid:1) is separable and process X is separable on this space. Also, let the following condi-tions hold: a := sup t ∈ T (cid:0) E | X ( t ) | (cid:1) < ∞ , Y. KOZACHENKO A , A. MELNIKOV B ∗ AND Y. MISHURA A and Z a (log N T ( u )) du < ∞ , where N T ( u ) is the number of elements in the minimal u -covering of space ( T , ρ ) .Then for any λ > and any θ ∈ (0 , the following inequality holds: E exp (cid:26) λ sup t ∈ T | X ( t ) | (cid:27) ≤ Q ( λ, θ ) , where Q ( λ, θ ) = exp ( λ a − θ ) + 2 λθ (1 − θ ) Z θa (log( N T ( u ))) du ) . Consider set T = { t = ( t , t ) ∈ R : 0 ≤ t ≤ t } supplied with the distance m ( t , s ) = | t − s | ∨ | t − s | . Assume random process X = { X ( t ) , t ∈ T } satisfies the following conditions.( D ) Process X is a centered Gaussian process on T , separable on metric space( T , m ).( D ) There exist β > , γ > C ( β, γ ) independent of X , t and s such that for any t , s ∈ T (11) (cid:0) E ( X ( t ) − X ( s )) (cid:1) ≤ C ( β, γ ) ( t ∨ s ) β ( m ( t , s )) γ . ( D ) There exist δ > C ( δ ) independent of X and t such thatfor any t ∈ T (12) (cid:0) E ( X ( t )) (cid:1) ≤ C ( δ ) t δ . Let us introduce the following notations. Let A ( t ) > , t ≥ A ( t ) → ∞ , t → ∞ . Consider an increasing sequence b = 0, b ℓ < b ℓ +1 , l ≥ b ℓ → ∞ , ℓ → ∞ . For δ ℓ = A ( b ℓ ) and κ > S ( δ ) = ∞ X ℓ =0 b δℓ +1 δ − ℓ , κ = κ (cid:18) βγ − δγ (cid:19) , B = C ( δ ) S ( δ ) ,C = C κ − S ( δ + κ ) and C = 2 − κ − κ γ ( C ( δ )) − κ γ ( C ( β, γ )) κ γ . Now we shall present the auxiliary exponential maximal bound for a Gaussianprocess defined on ( T , m ). Theorem 1.
Let { X ( t ) , t ∈ T } be a random process satisfying assumptions ( D ) − ( D ) . Let ≤ a < b , set T a,b = { t = ( t , t ) ∈ T : a ≤ t ≤ b, ≤ t ≤ t } . Thenfor any < θ < , λ > and < κ < ∧ γ the following inequality holds: E exp ( λ sup t ∈ T a,b | X ( t ) | ) ≤ e Q ( λ, θ ) , where e Q ( λ, θ ) = exp (cid:26) λ ( b δ C ( δ )) − θ ) + 2 λ − θ b δ + κ C θ κ γ κ (cid:27) . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION9
Proof.
It follows from (12) and (11) that(13) d := sup t ∈ T a,b (cid:0) E | X ( t ) | (cid:1) ≤ C ( δ ) b δ and(14) sup m ( t , s ) ≤ h, t , s ∈ T a,b (cid:0) E ( X ( t ) − X ( s )) (cid:1) ≤ σ ( h ) := C ( β, γ ) b β h γ . In turn, it follows from (14) that(15) N T a,b ( v ) ≤ (cid:18) b − a σ ( − ( v ) + 1 (cid:19) (cid:18) b σ ( − ( v ) + 1 (cid:19) ≤ ( C ( β, γ )) γ b βγ v γ + 1 ! . Define J ( θd ) := R θd (cid:0) log N T a,b ( u ) (cid:1) du. It follows from (15) that(16) J ( θd ) ≤ Z θd √ " log ( C ( β, γ )) γ b βγ v γ + 1 ! dv. For any 0 < κ ≤
1, log(1 + x ) = 1 κ log(1 + x ) κ ≤ x κ κ . Now, let κ ∈ (0 , ∧ γ ). Then it follows from (13) and (16) that J ( θd ) ≤ √ κ Z θd (( C ( β, γ )) γ b βγ ) κ (2 v γ ) κ dv = √ κ (1 − κ γ ) ( C ( β, γ )) γ b βγ ! κ ( θd ) − κ γ ≤ b δ + κ θ − κ γ κ C . Separability of X on ( T , m ) and relation (14) ensure separability of X on ( T , ρ )with ρ ( t , s ) = (cid:0) E ( X ( t ) − X ( s )) (cid:1) . Hence the statement of the theorem followsfrom Lemma 2. (cid:3) Now we are ready to state the general result concerning the asymptotic maximalgrowth of a Gaussian process defined on ( T , m ). Theorem 2.
Let X = { X ( t ) , t ∈ T } satisfy assumptions ( D ) − ( D ) . Supposethat function A ( t ) is chosen in such a way that series S ( δ ) converges. In case when βγ − δγ > , assume additionally that there exists such < κ < that series S ( δ + κ ) converges with κ = κ (cid:16) βγ − δγ (cid:17) . Then there exists such random variable ξ > that on any ω ∈ Ω and for any t ∈ T | X ( t ) | ≤ A ( t ) ξ, and ξ satisfies the following assumption: ( D ) for any ε > (2 C + 1) γ γ + κ P { ξ > ε } ≤ − (cid:16) ε − ε κ γ + κ (2 C + 1) (cid:17) B . A , A. MELNIKOV B ∗ AND Y. MISHURA A Here the value of κ < γ is chosen to ensure the convergence of series S ( δ + κ ) incase when βγ − δγ > , and we set κ = ∧ γ in case when βγ − δγ ≤ .Proof. It is easy to check that(17) I := E exp (cid:26) λ sup t ∈ T | X ( t ) | A ( t ) (cid:27) ≤ E exp ( λ ∞ X ℓ =0 ( δ ℓ ) − sup t ∈ ( b ℓ ,b ℓ +1 ) | X ( t ) | ) . Let ℓ ≥ , r ℓ > P ∞ ℓ =0 1 r ℓ = 1. Then it follows from (17),Theorem 1 and H¨older inequality that for any θ ∈ (0 ,
1) and 0 < κ < ∧ γI ≤ ∞ Y ℓ =0 E exp ( λ r ℓ δ ℓ sup t ∈ ( b ℓ ,b ℓ +1 ) | X ( t ) | )! rℓ ≤ ∞ Y ℓ =0 (2 Q ℓ ( λ, θ )) rℓ = 2 ∞ Y ℓ =0 ( Q ℓ ( λ, θ )) rℓ , where Q ℓ ( λ, θ ) = exp (cid:26) λ r ℓ δ ℓ ( b δℓ C ( δ )) (1 − θ ) + 2 λr ℓ (1 − θ ) δ ℓ b δ + κ ℓ C θ κ γ κ (cid:27) . Therefore, if we take such value of κ < γ that series S ( δ + κ ) converges in casewhen 1 + βγ − δγ > κ = ∧ γ in case when 1 + βγ − δγ ≤
0, we obtain(18) I ≤ ( λ ( C ( δ )) − θ ) ∞ X ℓ =0 r ℓ ( b δℓ ) δ ℓ + 2 λC κ − S ( δ + κ )(1 − θ ) θ κ γ ) Now we can substitute r ℓ = S ( δ ) b − δℓ δ ℓ into (18): I ≤ ( λ ( S ( δ ) C ( δ )) − θ ) + 2 λC κ − S ( δ + κ )(1 − θ ) θ κ γ ) . Therefore,(19) E exp (cid:26) λ sup t ∈ T | X ( t ) | A ( t ) (cid:27) ≤ (cid:26) λ B + 2 λ ˆ C (cid:27) , where ˆ B = S ( δ ) C ( δ )1 − θ and ˆ C = C κ − S ( δ + κ )(1 − θ ) θ κ γ . It follows immediately from (19) that for any λ > , ε > P (cid:26) sup t ∈ T | X ( t ) | A ( t ) > ε (cid:27) ≤ exp {− λε } E exp (cid:26) λ sup t ∈ T | X ( t ) | A ( t ) (cid:27) ≤≤ (cid:26) λ B + 2 λ ˆ C − λε (cid:27) . If we minimize the right-hand side of (20) w.r.t. λ then we obtain that for any ε > C (21) P (cid:26) sup t ∈ T | X ( t ) | A ( t ) > ε (cid:27) ≤ ( − ( ε − C ) B ) = 2 exp ( − ( ε (1 − θ ) − θ − κ γ C ) B ) . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION11
Finally, we can insert θ = ε − γ γ + κ into (21) and derive that for ε > (2 C + 1) γ γ + κ P (cid:26) sup t ∈ T | X ( t ) | A ( t ) > ε (cid:27) ≤ ( − ( ε − ε κκ +2 γ (1 + 2 C )) B ) . Denote ξ := sup t ∈ T | X ( t ) | A ( t ) . Then ξ satisfies assumption ( D ), and on any ω ∈ Ω X ( t ) ≤ A ( t ) ξ, which concludes the proof. (cid:3) Theorem 3.
Let < H < , − H < α < , T = { t = ( t , t ) , ≤ t < t } , X ( t ) = B Ht − B Ht ( t − t ) − α + Z t t B Hu − B Ht ( u − t ) − α du. Then for any p > there exists random variable ξ = ξ ( p ) such that for any t ∈ T | X ( t ) | ≤ (( t H + α − (log( t )) p ) ∨ ξ ( p ) , where ξ ( p ) satisfies assumption ( D ) with some constants B and C . The proof of Theorem 3 is of a technical nature and therefore it is placed inAppendix. 4.
Main results
General results on strong consistency.
In this section we shall establishconditions for strong consistency of θ (2) T and θ (3) T . Theorem 4.
Let assumptions ( A ) , ( A ) , ( A ′ ) , ( A ′ ) ( B ) and ( B ) hold and letfunction ϕ satisfy the following assumption: ( B ) There exists such α > − H and p > that T H + α − (log T ) p R T | ( D α ϕ )( s ) | ds R T ϕ s ds → a.s. as T → ∞ . (22) Then estimate θ (2) T is correctly defined and strongly consistent as T → ∞ .Proof. We must prove that R T ϕ s dB Hs R T ϕ s ds → T → ∞ . According to (2), (cid:12)(cid:12)(cid:12) Z T ϕ s dB Hs (cid:12)(cid:12)(cid:12) ≤ sup ≤ t ≤ T | ( D − αT − B HT − )( t ) | Z T | ( D α ϕ )( s ) | ds. Furthermore, according to Theorem 3, for any p > ξ = ξ ( p ) independent of T such that for any T > ≤ t ≤ T | ( D − αT − B HT − )( t ) | ≤ ξ ( p ) T H + α − (log T ) p , which concludes the proof. (cid:3) Relation (22) ensures convergence R T ϕ s dB Hs R T ϕ s ds → ϕ is non-random and integral R T ϕ s dB Hs is a Wienerintegral w.r.t. the fractional Brownian motion, conditions for existence of thisintegral are simpler since assumption (22) can be simplified. A , A. MELNIKOV B ∗ AND Y. MISHURA A Theorem 5.
Let assumptions ( A ) , ( A ) , ( A ′ ) , ( A ′ ) ( B ) and ( B ) hold and letfunction ϕ be non-random and satisfy the following assumption: ( B ) There exists such p > that lim sup T →∞ T H − p R T ϕ ( t ) dt < ∞ . Then estimate θ (2) T is strongly consistent as T → ∞ .Proof. It follows from [MMV] and the H¨older inequality that for any r > E (cid:12)(cid:12)(cid:12) Z T ϕ ( s ) dB Hs (cid:12)(cid:12)(cid:12) r ≤ C ( H, r ) || ϕ || rL H [0 ,T ] ≤ C ( H, r ) || ϕ || rL [0 ,T ] T ( H − ) r . Denote F T = | R T ϕ ( t ) dB Ht | R T ϕ ( t ) dt . Also, for any N > ε > A N = n F N > ε o . Then P ( A N ) ≤ ε − r E | R N ϕ ( s ) dB Hs | r ( R N ϕ ( t ) dt ) r ≤ C ( H, r ) || ϕ || rL H [0 ,N ] || ϕ || rL [0 ,N ] ≤ C ( H, r ) N ( H − ) r || ϕ || rL [0 ,N ] . Under condition ( B ) we have P ( A N ) ≤ C ( H, r, p ) N − rp . If r > p , then it followsimmediately from the Borel-Cantelli lemma that series P P ( A N ) converges, whence F N → N → ∞ . Now estimate the residual R N = sup T ∈ [ N,N +1] (cid:12)(cid:12)(cid:12) F T − F N | . Evidently, R N ≤ sup T ∈ [ N,N +1] (cid:12)(cid:12)(cid:12) R TN ϕ ( t ) dB Ht R T ϕ ( t ) dt (cid:12)(cid:12)(cid:12) + F N , and it is sufficient to estimate R N = sup T ∈ [ N,N +1] (cid:12)(cid:12)(cid:12) R TN ϕ ( t ) dB Ht R T ϕ ( t ) dt (cid:12)(cid:12)(cid:12) ≤ sup T ∈ [ N,N +1] (cid:12)(cid:12)(cid:12) R TN ϕ ( t ) dB Ht (cid:12)(cid:12)(cid:12)R N ϕ ( t ) dt := R N . According to Theorem 1.10.3 from [Mish08] and the H¨older inequality, E (cid:16) sup T ∈ [ N,N +1] (cid:12)(cid:12)(cid:12) Z TN ϕ ( t ) dB Ht (cid:12)(cid:12)(cid:12)(cid:17) r ≤ C ( H, r ) || ϕ || rL H [ N,N +1] ≤ C ( H, r ) || ϕ || rL [ N,N +1] . Now we can use condition ( B ) to conclude that for any ε > P ( R N > ε ) ≤ C ( H, r ) ε − r || ϕ || rL [ N,N +1] || ϕ || rL [0 ,N ] ≤ C ( H, r ) ε − r || ϕ || − rL [0 ,N ] ≤ C ( H, r ) ε − r N − r (2 H − p ) . We can set r > H − p and apply the Borel-Cantelli lemma again. Then we obtainthat R N → N →
0, which means that θ (2) T is strongly consistent. (cid:3) Theorem 6.
Let assumptions ( C ) and ( C ) hold, and, in addition, ( C ) R T ϕ ( s ) ds = ∞ a.s. N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION13 ( C ) There exist such α > − H and p > that T H + α − (log T ) p R T | ( D α ϑ )( s ) | ds R T ϕ ( s ) ds → a.s. as T → ∞ . (23) Then estimate θ (3) T is strongly consistent as T → ∞ .Proof. The last term in the right-hand side of (9) tends to zero under condition( C ). The proof of convergence of the second term repeats the proof of Theorem4. (cid:3) Similarly to Theorem 5, conditions stated in Theorem 6 can be simplified in casewhen function ϑ is non-random. Theorem 7.
Let assumptions ( C ) and ( C ) hold. Then, if functions ϕ and ϕ are non-random, function ϕ satisfies condition ( B ) , function ϕ is bounded, thenestimate θ (3) T is strongly consistent as T → ∞ . Now we shall take a look at the properties of sequential estimates.
Theorem 8. ( a ) Let assumptions ( B ) − ( B ) hold. Then estimate θ (1) τ ( h ) isunbiased, efficient, strongly consistent, E ( θ (1) τ ( h ) − θ ) = h , and for anyestimate of the form θ τ = R τ χ s dZ s R τ χ s d h M H i s = θ + R τ χ s dM Hs R τ χ s d h M H i s with τ < ∞ a.s. and E R τ χ s d h M H i s ≤ h we have that E ( θ (1) τ ( h ) − θ ) ≤ E ( θ τ − θ ) . ( b ) Let function ϕ be separated from zero, | ϕ ( s ) | ≥ c > a.s. and satisfy theassumption: for some − H < α < and p > R τ ( h )0 | ( D α ϕ )( s ) | ds ( τ ( h )) − α − H − p → a.s.as h → ∞ . Then estimate θ (2) τ ( h ) is strongly consistent. ( c ) Let function ϕ be separated from zero, | ϕ ( s ) | ≥ c > a.s. and let function ϑ satisfy the assumption: for some − H < α < and p > R υ ( h )0 | ( D α ϑ )( s ) | ds ( υ ( h )) − α − H − p → a.s.as h → ∞ . Then estimate θ (3) υ ( h ) is strongly consistent. ( d ) Let function ϑ be non-random, bounded and positive, ϕ be separated fromzero. Then estimate θ (3) υ ( h ) is consistent in the following sense: for any p > , E (cid:12)(cid:12)(cid:12) θ − θ (3) υ ( h ) (cid:12)(cid:12)(cid:12) p → as h → ∞ .Proof. (a) Process R τ ( h )0 χ s dM Hs is a square-integrable martingale which impliesthat estimate θ (1) τ ( h ) is unbiased. Besides, the results from [LipSh], Chapter 17, canbe applied to (10) directly, therefore estimate θ (1) τ ( h ) is efficient, E ( θ (1) τ ( h ) − θ ) = h , A , A. MELNIKOV B ∗ AND Y. MISHURA A and for any estimate of the form θ τ = R τ χ s dZ s R τ χ s d h M H i s = θ + R τ χ s dM Hs R τ χ s d h M H i s with τ < ∞ a.s. and E R τ χ s d h M H i s ≤ h we have that E ( θ (1) τ ( h ) − θ ) ≤ E ( θ τ − θ ) . Strongconsistency is also evident.(b) We have that | R τ ( h )0 ϕ ( s ) dB Hs | ≤ ( τ ( h )) H + α − p R τ ( h )0 | ( D α ϕ )( s ) | ds . It issufficient to note that h = R τ ( h )0 ϕ s ds ≥ c τ ( h ). The proof of statement (c) is nowevident.(d) It was proved in [Mish08] that in case of non-random bounded positive func-tion 0 ≤ ϑ ( s ) ≤ ϑ ∗ , for any stopping time υ (cid:16) E (cid:16) sup ≤ t ≤ υ (cid:12)(cid:12)(cid:12) Z t ϑ ( s ) dB Hs (cid:12)(cid:12)(cid:12)(cid:17) p (cid:17) p ≤ C ( H, p ) ϑ ∗ (cid:16) Eυ pH (cid:17) p . Furthermore, same as before, | ϕ ( s ) | ≥ c and h = R υ ( h )0 ϕ ( s ) ds ≥ c υ ( h ). Theseinequalities together with the Burkholder-Gundy inequality yield E (cid:12)(cid:12)(cid:12) θ − θ (3) υ ( h ) (cid:12)(cid:12)(cid:12) p ≤ C ( H, p ) (cid:16) ϑ ∗ c h H − + h − p (cid:17) → h → ∞ . (cid:3) Remark . Another proof of statement (a) is contained in [Prara]. Assumptions(24)and (25) hold, for example, for bounded and Lipschitz functions ϕ and ϑ correspond-ingly.4.2. Linear models and strong consistency.
I. Consider the linear version ofmodel (5): dX t = θa ( t ) X t dt + b ( t ) X t dB Ht , where a and b are locally bounded non-random measurable functions. In this casesolution X exists, is unique and can be presented in the integral form X t = x + θ Z t a ( s ) X s ds + Z t b ( s ) X s dB Hs = x exp n θ Z t a ( s ) ds + Z t b ( s ) dB Hs o . Suppose that function b is non-zero and note that in this model ϕ ( t ) = a ( t ) b ( t ) . Suppose that ϕ ( t ) is also locally bounded and consider maximum likelihood estimate θ (1) T . According to (6), to guarantee existence of process J ′ , we have to assume thatthe fractional derivative of order − H for function ς ( s ) := ϕ ( s ) s − H exists and isintegrable. The sufficient conditions for the existence of fractional derivatives canbe found in [SMK]. One of these conditions states:( B ) Functions ϕ and ς are differentiable and their derivatives are locally inte-grable.So, the maximum likelihood estimate does not exist for an arbitrary locally boundedfunction ϕ . Suppose that condition ( B ) holds and limit ς = lim s → ς ( s ) exists.In this case, according to Lemma 1 and Remark 2, process J ′ admits both of the N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION15 following representations: J ′ ( t ) = (2 − H ) C H ϕ (0) t − H + Z t l H ( t, s ) ϕ ′ ( s ) ds − (cid:16) H − (cid:17) c H Z t s − − H ( t − s ) − H Z s ϕ ′ ( u ) duds = c H ς t − H + c H Z t ( t − s ) − H ς ′ ( s ) ds, and assuming ( B ) also holds true, the estimate θ (1) T is strongly consistent. Let usformulate some simple conditions sufficient for the strong consistency. The proof isobvious and therefore is omitted. Lemma 3.
If function ϕ is non-random, locally bounded, satisfies ( B ) , limit ς (0) exists and one of the following assumptions hold: (a) function ϕ is not identically zero and ϕ ′ is non-negative and non-decreasing; (b) derivative ς ′ preserves the sign and is separated from zero; (c) derivative ς ′ is non-decreasing and has a non-zero limit,then the estimate θ (1) T is strongly consistent as T → ∞ .Example . : Let the coefficients are constant, a ( s ) = a = 0 and b ( s ) = b = 0, thenthe estimate has a form θ (1) T = θ + bM HT aC H T − H and is strongly consistent. In thiscase assumption (a) holds. In addition, power functions ϕ ( s ) = s ρ are appropriatefor ρ > H −
1: this can be verified directly from (6).Let us now apply estimate θ (2) T to the same model. It has a form (8). We can useTheorem 5 directly and under assumption ( B ) estimate θ (2) T is strongly consistent.Note that we do not need any assumptions on the smoothness of ϕ , which is a clearadvantage of θ (2) T . We shall consider two more examples. Example . : If the coefficients are constant, a ( s ) = a = 0 and b ( s ) = b = 0, thenthe estimate has a form θ (2) T = θ + bB HT aT . We can refer to Theorem 5 and concludethat θ (2) T is strongly consistent. Alternatively, we can use Remark 5 which statesthat | B HT | ≤ ξT H (log T ) p for any p > ξ , therefore B HT T → T → ∞ . In this case both estimates θ (2) T and θ (2) T are stronglyconsistent and E ( θ − θ (1) T ) = γ T H − a C H has the same asymptotic behavior as E ( θ − θ (2) T ) = γ T H − a . Example . : If non-random functions ϕ and ς are bounded on some fixed interval[0 , t ] but ς is sufficiently irregular on this interval and has no fractional derivativeof order − H or higher then we can not even calculate J ′ ( t ) on this intervaland the maximum likelihood estimate does not exist. However, if we assume that ϕ ( t ) ∼ t H − ρ at infinity with some ρ >
0, then assumption ( B ) holds and estimate θ (2) T is strongly consistent as T → ∞ . In this sense estimate θ (2) T is more flexible.II. Consider a mixed linear model of the form(26) dX t = X t ( θa ( t ) dt + b ( t ) dB Ht + c ( t ) dW t ) , A , A. MELNIKOV B ∗ AND Y. MISHURA A where a , b and c are non-random measurable functions. Assume that they arelocally bounded. In this case solution X for equation (26) exists, is unique and canbe presented in the integral form X t = x exp n θ Z t a ( s ) ds + Z t b ( s ) dB Hs + Z t c ( s ) dW s − Z t c ( s ) ds o . In what follows assume that c ( s ) = 0. We have that ϕ ( t ) = a ( t ) c ( t ) and ϕ ( t ) = b ( t ) c ( t ) .Estimate θ (3) T has a form(27) θ (3) T = R T ϕ ( s ) dY s R T ϕ ( s ) ds = θ + R T ϕ ( s ) ϕ ( s ) dB Hs R T ϕ ( s ) ds + R T ϕ ( s ) dW s R T ϕ ( s ) ds . In accordance with Theorem 7, assume that function ϕ satisfies ( B ) and ϕ isbounded. Then estimate θ (3) T is strongly consistent. Evidently, these assumptionshold for the constant coefficients.4.3. The fractional Ornstein-Uhlenbeck model and strong consistency.
I.Consider the fractional Ornstein-Uhlenbeck, or Vasicek, model with non-constantcoefficients. It has a form dX t = θ ( a ( t ) X t + b ( t )) dt + γ ( t ) dB Ht , t ≥ , where a , b and γ are non-random measurable functions. Suppose they are locallybounded and γ = γ ( t ) >
0. The solution for this equation is a Gaussian processand has a form X t = e θA ( t ) (cid:16) x + θ Z t b ( s ) e − θA ( s ) ds + Z t γ ( s ) e − θA ( s ) dB Hs (cid:17) := E ( t ) + G ( t ) , where A ( t ) = R t a ( s ) ds , E ( t ) = e θA ( t ) (cid:16) x + θ R t b ( s ) e − θA ( s ) ds (cid:17) is a non-randomfunction, G ( t ) = e θA ( t ) R t γ ( s ) e − θA ( s ) dB Hs is a Gaussian process with zero mean.Denote c ( t ) = a ( t ) γ ( t ) , d ( t ) = b ( t ) γ ( t ) . Now we shall state the conditions for strongconsistency of the maximum likelihood estimate.
Theorem 9.
Let functions a , c , d and γ satisfy the following assumptions: ( B ) − a ≤ a ( s ) ≤ − a < , − c ≤ c ( s ) ≤ − c < , < γ ≤ γ ( s ) ≤ γ ,functions c and d are continuously differentiable, c ′ is bounded, c ′ ( s ) ≥ and c ′ ( s ) → as s → ∞ .Then estimate θ (1) T is strongly consistent as T → ∞ .Proof. We shall check the conditions of Proposition 1. Obviously, ψ ( t, x ) = c ( t ) x + d ( t ) ∈ C ( R + ) × C ( R ) and J ( t ) = Z t l H ( t, s )( d ( s ) + c ( s ) E ( s )) ds + Z t l H ( t, s ) c ( s ) G ( s ) ds := F ( t ) + H ( t ) . Furthermore, assumptions ( A ), ( A ), ( A ′ ), ( A ′ ) and ( B ) hold. Note that thetrajectories of process G are a.s. H¨older up to order H , whencelim s → s − H c ( s ) G ( s ) = 0 . Therefore J ′ ( t ) = F ′ ( t ) + H ′ ( t ) = F ′ ( t ) + Z t l H ( t, s ) f ( s ) G ( s ) ds + Z t l H ( t, s ) c ( s ) γ ( s ) dB Hs , N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION17 where f ( s ) = (cid:16) − H (cid:17) s − c ( s ) + c ′ ( s ) + θa ( s ) c ( s ). Evidently, J ′ t is Gaussian pro-cess with mean and variance that are bounded on any bounded interval. There-fore, condition ( B ) holds. As for condition ( B ), we must verify that I ∞ = R ∞ ( J ′ t ) t H − dt = ∞ a.s. For any λ > T ( λ ) = E exp {− λI T } = E exp {− λ Z T ( J ′ t ) t H − dt } and Θ ∞ ( λ ) = E exp {− λI ∞ } = E exp {− λ Z ∞ ( J ′ t ) t H − dt } , so that Θ ∞ ( λ ) = lim T →∞ Θ T ( λ ). Evidently, Z T ( J ′ t ) t H − dt ≥ T − (cid:16) Z T J ′ t t H − dt (cid:17) , whence Θ T ( λ ) ≤ Θ (1) T ( λ ) := E exp n − λT (cid:16) Z T J ′ t t H − dt (cid:17) o . Random variable R T J ′ t t H − dt is Gaussian with mean M ( T ) and variance σ ( T ),say. Note that for a Gaussian random variable ξ = m + σN (0 ,
1) we can easilycalculate(28) E exp {− aξ } = (cid:16) aσ + 1 (cid:17) − exp n − am aσ + 1 o . This value attains its maximum at the point m = 0. Hence, it is sufficient to provethat lim T →∞ Θ (2) T ( λ ) := lim T →∞ E exp n − λT (cid:16) Z T H ′ t t H − dt (cid:17) o = 0 . However, it follows from (28) that Θ (2) T ( λ ) = (cid:16) λσ T T + 1 (cid:17) − , therefore to provethe strong consistency of the maximum likelihood estimate θ (1) T , we only need toanalyze the asymptotic behavior of σ T . More specifically, we need to prove that σ T T → ∞ as T → ∞ . In what follows we apply the following formulae from [NVV99]and [MMV] for Wiener integrals w.r.t. the fractional Brownian motion E Z t g ( s ) dB Hs Z t h ( s ) dB Hs = H (2 H − Z t Z t g ( s ) h ( s ) | s − s | H − ds ds ≤ C ( H ) || g || L H [0 ,t ] || h || L H [0 ,t ] . (a) Let θ <
0. Divide R T H ′ t t H − dt into two parts: R T H ′ t t H − dt = H (1) T + H (2) T ,where H (1) T = Z T t H − Z t l H ( t, s ) f ( s ) G ( s ) dsdt and H (2) T = Z T t H − Z t l H ( t, s ) c ( s ) γ ( s ) dB Hs dt. A , A. MELNIKOV B ∗ AND Y. MISHURA A Since functions c and γ are bounded from below and from above,(29) E (cid:0) H (2) T (cid:1) = C ( H ) Z T Z T ( t t ) H − Z t Z t Π i =1 , l H ( t i , s i )( − c ( s i )) γ ( s i ) ×| s − s | H − ds ds dt dt ≍ C ( H ) Z T Z T ( t t ) H − × Z t Z t Π i =1 , l H ( t i , s i ) | s − s | H − ds ds dt dt ≍ C ( H ) T as T → ∞ .Consider the behavior of f . Under assumption ( B ) terms s − c ( s ) + c ′ ( s ) vanishat infinity, θa ( s ) c ( s ) is negative and separated from zero. Therefore, there exist C i > i = 1 , s > − C ≤ f ( s ) ≤ − C for all s > s . Bounded-ness of f implies that E ( H (1) T ) has the same asymptotic behavior as(30) Z Ts Z Ts ( t t ) H − Z t s Z t s (Π i =1 , l H ( t i , s i )( − f ( s i ))) × (cid:16) Z s s Z s s γ ( u ) γ ( u ) exp n θ (cid:16) Z s u + Z s u (cid:17) a ( v ) dv o | u − u | H − du du (cid:17) ds ds dt dt ≥ C ( H ) Z Ts Z Ts ( t t ) H − Z t s Z t s (Π i =1 , l H ( t i , s i )) × (cid:16) Z s s Z s s | u − u | H − du du (cid:17) ds ds dt dt ≍ C ( H ) T . Relations (29) and (30) mean that the asymptotic behavior of σ T is σ T ≍ C ( H ) T and σ T T → ∞ as T → ∞ .(b) Let θ >
0. This case is more involved. The asymptotic behavior of E ( H (2) T ) is the same as before, C ( H ) T , since it does not depend on θ . As for E ( H (1) T ) ,denote K ′ t = R t l H ( t, s ) f ( s ) G ( s ) ds , then H (1) T = Z T K ′ t t H − dt = T H − K T − (cid:16) H − (cid:17) Z T t H − K t dt. N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION19
In addition, denote r ( t ) = exp {− θ R t a ( s ) ds } , ψ ( t ) = f ( t ) exp { θ R t a ( s ) ds } . Apply-ing Fubini theorem several times, we obtain that T H − K T − (cid:16) H − (cid:17) Z T s H − K s ds = T H − Z T l H ( T, t ) f ( t ) G ( t ) dt − (cid:16) H − (cid:17) Z T t H − Z t l H ( t, s ) f ( s ) G ( s ) dsdt = T H − Z T l H ( T, t ) ψ ( t ) Z t r ( s ) dB Hs dt − (cid:16) H − (cid:17) Z T t H − Z t l H ( t, u ) ψ ( u ) Z u r ( s ) dB Hs dudt = Z T r ( s ) Z Ts l H ( T, t ) ψ ( t ) dtdB Hs T H − − (cid:16) H − (cid:17) Z T r ( s ) Z Ts t H − Z ts l H ( t, u ) ψ ( u ) dudtdB Hs = Z T r ( s ) (cid:16) T H − Z Ts l H ( T, t ) ψ ( t ) dt − (cid:16) H − (cid:17) Z Ts t H − Z ts l H ( t, u ) ψ ( u ) dudt (cid:17) dB Hs . Denote F ( T, s ) = T H − Z Ts l H ( T, t ) ψ ( t ) dt − (cid:16) H − (cid:17) Z Ts t H − Z ts l H ( t, u ) ψ ( u ) dudt = Z Ts t − H e θ R t a ( s ) ds f ( t ) (cid:16) T H − ( T − t ) − H − (cid:16) H − (cid:17) Z Tt u H − ( u − t ) − H du (cid:17) dt := F ( T, s ) − F ( T, s ) . and F + ( T, s ) = F ( T, s ) + F ( T, s ) . Function f is bounded, positive for s > s and separated from zero. For the sake oftechnical simplicity, we can put f ( t ) = a ( t ) ≡
1. Besides, we can omit the constantmultiplier c H . Then0 ≤ E ( H (1) T ) = Z T Z T e θs e θt F ( T, s ) F ( T, t ) | s − t | H − dsdt ≤ Z T Z T e θs e θt F + ( T, s ) F + ( T, t ) | s − t | H − dsdt. A , A. MELNIKOV B ∗ AND Y. MISHURA A Consider the terms containing F ( T, s ) F ( T, t ): I = Z T Z T e θs e θt F ( T, s ) F ( T, t ) | s − t | H − dsdt = T H − Z T Z T e θs e θt Z Ts u − H ( T − u ) − H e − θu du × Z Tt v − H ( T − v ) − H e − θv dv | s − t | H − dsdt ≤ T H − Z T Z T ( st ) − H Z Ts ( T − u ) − H e − θ ( u − s ) du × Z Tt ( T − v ) − H e − θ ( v − t ) dv | s − t | H − dsdt. Applying H¨older inequality we conclude that integral R Ts ( T − u ) − H e − θ ( u − s ) du admits the following bound: Z Ts ( T − u ) − H e − θ ( u − s ) du ≤ Z s + T s ( T − u ) − H e − θ ( u − s ) du + Z T s + T ( T − u ) − H e − θ ( u − s ) du ≤ H − ( T − s ) − H Z s + T s e − θ ( u − s ) du + (cid:16) Z T s + T ( T − u ) − H du (cid:17) (cid:16) Z T s + T e − θ ( u − s ) du (cid:17) ≤ C ( H ) (cid:16) ( T − s ) − H + ( T − s ) − H (cid:17) . Therefore I ≤ C ( H ) (cid:16) T H − Z T Z T ( st ) − H (( T − t )( T − s )) − H | s − t | H − dsdt + T Z T Z T ( st ) − H | s − t | H − dsdt (cid:17) ≤ C ( H ) T . Furthermore, function e θs F ( T, s ) admits the following bounds: e θs F ( t, s ) ≤ C ( H ) s − H Z Ts t H − ( T − t ) − H e − θ ( t − s ) dt ≤ C ( H ) T − H s − H Z Ts t H − e − θ ( t − s ) dt. Note that function R Ts t H − e − θ ( t − s ) dt decreases in s since its derivative equals e θs ( R Ts t H − e − θt dt − s H − ) < . Therefore, e θs F ( t, s ) ≤ C ( H ) T − H s − H Z T t H − e − θt dt ≤ C ( H ) T − H s − H . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION21
The latter implies that the term containing F ( T, s ) F ( T, t ) admits the followingbounds: I = Z T Z T e θs e θt F ( T, s ) F ( T, t ) | s − t | H − dsdt ≤ C ( H ) T − H Z T Z T ( st ) − H | s − t | H − dsdt ≤ C ( H ) T − H . So, E ( H (1) T ) ≍ C ( H ) T − H asymptotically and if we compare this to asymptoticalbehavior of E ( H (2) T ) ≍ C ( H ) T , we can conclude that σ T T ≍ C ( H ) T → ∞ as T → ∞ .(c) Let θ = 0. Then it is easy to verify that E ( H (1) T ) ≍ C ( H ) T and we canrefer to the case θ > (cid:3) Remark . The assumptions of the theorem are fulfilled, for example, if a ( s ) = − b ( s ) = b ∈ R and γ ( s ) = γ >
0. In this case we deal with a standard Ornstein-Uhlenbeck process X with constant coefficients that satisfies the equation dX t = θ ( b − X t ) dt + γdB Ht , t ≥ . This model with constant coefficients was studied in [KlLeBr] where the Laplacetransform Θ T ( λ ) was calculated explicitly and strong consistency of θ (1) T was estab-lished. Therefore, our results generalize the statement of strong consistency to thecase of variable coefficients.II. Consider a simple version of the Ornstein-Uhlenbeck model where a = γ = 1, b = x = 0. The SDE has a form dX t = θX t dt + dB Ht , t ≥ X t = e θt R t e − θs dB Hs . Let us construct an estimate which is a modification of θ (2) T : e θ (2) T = R T e − θs X s dX s R T e − θs X s ds = θ + (cid:16) R T e − θs dB Hs (cid:17) R T (cid:16) R s e − θu dB Hu (cid:17) ds . Theorem 10.
Let θ > . Then estimate e θ (2) T is strongly consistent as T → ∞ .Proof. Applying Remark 5 yields | Z T e − θs dB Hs | ≤ e − θT | B HT | + Z T e − θs | B Hs | ds ≤ ξ (cid:16) e − θT T H + p + Z T e − θs s H + p ds (cid:17) ≤ ζ, where ζ is a random variable independent of T . So, it is sufficient to establish that R ∞ (cid:16) R s e − θu dB Hu (cid:17) ds = 0 to prove the strong consistency of e θ (2) T . Similarly to theproof of Theorem 9, we can consider the moment generation function E exp {− λ Z T (cid:16) Z s e − θu dB Hu (cid:17) ds } ≤ E exp n − λT − (cid:16) Z T Z s e − θu dB Hu ds (cid:17) o = (cid:16) λσ T T + 1 (cid:17) − , A , A. MELNIKOV B ∗ AND Y. MISHURA A where σ T = E (cid:16) Z T Z s e − θu dB Hu ds (cid:17) = Z T Z T Z s Z t e − θu − θv | u − v | H − dudvdsdt = T H +2 Z Z Z s Z t e − T ( θu + θv ) | u − v | H − dudvdsdt ≥ T H +2 Z Z Z s Z t e − T ( θu + θv ) dudvdsdt = T H θ − (cid:16) Z (cid:16) − e − θsT (cid:17) ds (cid:17) ≍ T H θ − , whence the proof follows. (cid:3) Appendix
A.To apply Theorem 2 to the fractional derivative of the fractional Brownian mo-tion and to prove Theorem 3, we need an auxiliary result. In what follows we denoteby C ( H, α ) a constant depending only on H and α and not on other parameters. Lemma 4.
Let z i > for i = 1 , . In addition, let < H < , − H < α < and I = z H + α − + z H + α − + | z − z | H − z H − z H ( z z ) − α . Then I ≤ C ( H, α ) | z − z | H + α − . Proof.
Let z > z > z > z > I as I = ( z H + α − − z H + α − ) + 2( z z ) H + α − +(( z − z ) H − ( z H − z H ) − z z ) H )( z z ) α − = ( z H + α − − z H + α − ) + ( z − z ) H − ( z H − z H ) ( z z ) − α = I + I . Recall a simple inequality b r − a r ≤ ( b − a ) r for b > a, < r ≤
1. Since 0 1, we can estimate I by ( z − z ) H + α − . Furthermore, I can berewritten as I = ( z − z ) H + α − | z − z | H − ( z H − z H ) ( z z ) − α ( z − z ) H + α − = ( z − z ) H + α − f ( u ) , where u = z z > f ( u ) = ( u − H − ( u H − u − α ( u − H + α − ≥ . Calculate the limit of function f at 1:lim u → f ( u ) = lim u → ( u − H − ( u H − ( u − H + α − . Here lim u → ( u − H ( u − H + α − = lim u → ( u − − α = 0 , and lim u → ( u H − ( u − H + α − = H lim u → ( u − − H − α = 0 , N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION23 since lim u → u H − u − = H. Calculate the limit of the function f at infinity:0 ≤ lim u →∞ f ( u ) = lim u →∞ ( u − H − ( u H − u − β ( u − H + α − ≤ lim u →∞ u H − ( u H − u H + α − = lim u →∞ u H − u H + α − = 0 . This implies that function f is bounded, i.e. there exists C ( H, α ) > I ≤ C ( H, α )( z − z ) H + α − , and the proof follows if we combine the bounds for I and I . (cid:3) We are now ready to check conditions ( D ) and ( D ) for the fractional derivativeof the fractional Brownian motion. Lemma 5. Let X ( t ) = B Ht − B Ht ( t − t ) − α + Z t t B Hu − B Ht ( u − t ) − α du, where ≤ t < t , < H < , − H < α < . Then the following bounds hold:1) for any ≤ t < t (cid:0) E ( X ( t )) (cid:1) ≤ C ( H, α )( t − t ) H + α − ; 2) (a) Let H + α ≤ . Then for any ≤ t < t , ≤ s < s and any < ε < ( H + α − ∧ ( E | X ( t ) − X ( s ) | ) ≤ C ( H, α ) (cid:0) ε − (cid:1) ( | t − s | ∨ | t − s | )) H + α − − ε ( t ∨ s )) ε with C ( H, α ) not depending on X , its arguments and ε .(b) Let H + α > . Then for any ≤ t < t , ≤ s < s ( E | X ( t ) − X ( s ) | ) ≤ C ( H, α )( | t − s | ∨ | t − s | ) ( t ∨ s ) H + α − . Proof. The first statement follows immediately from the Minkowski’s integral in-equality: (cid:0) E ( X ( t )) (cid:1) ≤ E (cid:18) B Ht − B Ht ( t − t ) − α (cid:19) ! + E (cid:18)Z t t B Hu − B Ht ( u − t ) − α du (cid:19) ! ≤ (cid:18) ( t − t ) H ( t − t ) − α ) (cid:19) + Z t t E (cid:18) B Hu − B Ht ( u − t ) − α (cid:19) ! du = ( t − t ) H + α − ++ Z t t (cid:18) ( u − t ) H ( u − t ) − α ) (cid:19) du = α + Hα + H − t − t ) H + α − . In order to prove the second statement, denote X ( t ) = B Ht − B Ht ( t − t ) − α and X ( t ) = R t t B Hu − B Ht ( u − t ) − α du. Evidently,(31) ( E | X ( t ) − X ( s ) | ) ≤ ( E | X ( t ) − X ( s ) | ) + ( E | X ( t ) − X ( s ) | ) . A , A. MELNIKOV B ∗ AND Y. MISHURA A Let t > s , the opposite case can be considered in a similar way. Then(32) ( E | X ( t ) − X ( s ) | ) = E (cid:18) B Ht − B Ht ( t − t ) − α − B Ht − B Hs ( t − s ) − α + B Ht − B Hs ( t − s ) − α − B Hs − B Hs ( s − s ) − α (cid:19) ! ≤ E (cid:18) B Ht − B Ht ( t − t ) − α − B Ht − B Hs ( t − s ) − α (cid:19) ! + E (cid:18) B Ht − B Hs ( t − s ) − α − B Hs − B Hs ( s − s ) − α (cid:19) ! =: I + I . It is more convenient to estimate the squares ( I ) and ( I ) from (32) instead of I and I . As for ( I ) , we can calculate it explicitly and then estimate it with thehelp of Lemma 4; ( I ) can be evaluated similarly.(33) ( I ) = ( t − t ) H + α − + ( t − s ) H + α − − E ( B Ht − B Ht )( B Ht − B Hs )( t − t ) − α ( t − s ) − α = ( t − t ) H + α − + ( t − s ) H + α − − t − t ) − α ( t − s ) − α × [ t H − (cid:0) t H + t H − ( t − t ) H (cid:1) − (cid:0) t H + s H − ( t − s ) H (cid:1) + 12 (cid:0) t H + s H − | t − s | H (cid:1) ] = ( t − t ) H + α − + ( t − s ) H + α − + | t − s | H − ( t − t ) H − ( t − s ) H ( t − t ) − α ( t − s ) − α ≤ C ( H, α ) | t − s | H + α − . We derive from (33) that(34) I ≤ C ( H, α ) | t − s | H + α − , and similarly,(35) I ≤ C ( H, α ) | t − s | H + α − . It follows immediately from (34) and (35) that(36) ( E | X ( t ) − X ( s ) | ) ≤ C ( H, α ) ( | t − s | ∨ | t − s | ) H + α − . Now estimate F ( t , s ) = ( E | X ( t ) − X ( s ) | ) = E (cid:18)Z t t B Hu − B Ht ( u − t ) − α du − Z s s B Hu − B Hs ( u − s ) − α du (cid:19) ! . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION25 Let, for instance, 0 ≤ t < s < s < t (other types of relation between thesepoints can be handled similarly). Then(37) F ( t , s ) ≤ E (cid:18)Z s t B Hu − B Ht ( u − t ) − α du (cid:19) ! + E (cid:18)Z s s (cid:18) B Hu − B Ht ( u − t ) − α − B Hu − B Hs ( u − s ) − α (cid:19) du (cid:19) ! + E (cid:18)Z t s B Hu − B Ht ( u − t ) − α du (cid:19) ! =: I + I + I . Using the Minkowski’s integral inequality we immediately obtain(38) I ≤ Z s t E (cid:18) B Hu − B Ht ( u − t ) − α (cid:19) ! du = Z s t ( u − t ) H + α − du = 1 H + α − s − t ) H + α − . Similarly,(39) I ≤ H + α − t − s ) H + α − . Again, using the Minkowski’s integral inequality and Lemma 4 we conclude that A , A. MELNIKOV B ∗ AND Y. MISHURA A (40) I ≤ Z s s E (cid:18) B Hu − B Ht ( u − t ) − α − B Hu − B Hs ( u − s ) − α (cid:19) ! du = Z s s " ( u − t ) H + α − + ( u − s ) H + α − + ( s − t ) H − ( u − t ) H − ( u − s ) H ( u − t ) − α ( u − s ) − α du = Z s s ( u − s ) − ( u − t ) − " ( u − t ) H + α − ( u − s )( u − t )+( u − s ) H + α − ( u − s )( u − t )+ ( s − t ) H − ( u − t ) H − ( u − s ) H ( u − t ) − α ( u − s ) − α du ≤ Z s s ( u − s ) − ( u − t ) − " ( u − t ) H + α − + ( u − s ) H + α − +( u − s ) H + α − ( s − t ) + ( t − s ) H − ( u − t ) H − ( u − s ) H ( u − t ) − α ( u − s ) − α du ≤ C ( H, α ) Z s s ( u − s ) − ( u − t ) − ( s − t ) H + α − du + C ( H, α ) Z s s ( u − s ) H + α − ( u − t ) − ( s − t ) du =: I + I . Evidently, I = ( s − t ) H + α − Z s s ( u − s ) − ( u − t ) − du = ( s − t ) H + α − I up to the constant multiplier and for any 0 < ε < integral I can be rewrittenas I = Z s s ( u − s ) − ( u − t ) − du = Z s − s s − t ( y + 1) − y − dy ≤ (cid:18) s − s s − t (cid:19) ε Z s − s s − t ( y + 1) − y − − ε dy ≤ (cid:18) s − s s − t (cid:19) ε Z ∞ ( y + 1) − y − − ε dy ≤ C (cid:0) ε − (cid:1) (cid:18) s − s s − t (cid:19) ε . Therefore, for any 0 < ε < ( H + α − ∧ (41) I ≤ C ( H, α ) (cid:0) ε − (cid:1) ( s − t ) H + α − − ε ( s − s ) ε . N DRIFT PARAMETER ESTIMATION IN MODELS WITH FRACTIONAL BROWNIAN MOTION27 Furthermore, I = ( s − t ) Z s s ( u − s ) H + α − ( u − t ) − du = ( s − t ) I up to a constant multiplier. In the case when H + α < the integral I can berewritten as I = Z s s ( u − s ) H + α − ( u − t ) − du = Z s − s s − t y H + α − (1 + y ) − ( s − t ) H + α − du ≤ ( s − t ) H + α − Z ∞ y H + α − (1 + y ) − du ≤ C ( H, α )( s − t ) H + α − . In case when H + α > integral I admits an obvious bound I ≤ Z s s ( u − s ) H + α − ( u − s ) − du ≤ C ( H, α )( s − s ) H + α − . Finally, for H + α = integral I admits the same bound as I . Therefore,(42) I ≤ C ( H, α )( s − t ) H + α − for H + α < ,(43) I ≤ C ( H, α )( s − t ) ( s − s ) H + α − for H + α > , and(44) I ≤ C ( H, α )( s − t ) − ε ( s − s ) ε for H + α = . This implies that(45) F ( t , s ) ≤ C ( H, α ) (cid:0) ε − (cid:1) ( | t − s | ∨ | t − s | ) H + α − − ε ( s ∨ t ) ε for H + α ≤ . In case H + α > we can put ε = H + α − ∈ (0 , ) in (41) andconclude that(46) F ( t , s ) ≤ C ( H, α )( | t − s | ∨ | t − s | ) ( s ∨ t ) H + α − . The proof follows immediately from (31) and (36)-(46). (cid:3) Proof of Theorem 3: First of all we should verify conditions ( D ) − ( D ). Condi-tion ( D ) is evident, since X is continuous in both variables. According to the 2ndstatement of Theorem 5, condition ( D ) holds with β = ε , 0 < ε < ( H + α − ∧ and γ = H + α − − ε in case when α + H ≤ , and with β = H + α − and γ = in case when α + H > . According to the first statement of Theorem 5, condition( D ) holds with δ = H + α − A ( t ) = ( t H + α − | log t | p ) ∨ p > t > b l = e l , l ≥ 0. Then δ l = ( e l ( H + α − l p ) ∨ A ( b l ) = e l ( H + α − . Therefore, inthis case series S ( δ ) converges since S ( δ ) = e H + α − + ∞ X l =1 e ( l +1)( H + α − e l ( H + α − l p = e H + α − (1 + ∞ X l =1 l − p ) < ∞ . A , A. MELNIKOV B ∗ AND Y. MISHURA A Moreover, it is easy to check that 1 + βγ − δγ = 0 for any values of α + H , hence κ = 0. This implies that all conditions of Theorem 2 hold true and we can applythe theorem with A ( t ) = ( t H + α − | log t | p ) ∨ (cid:3) Remark . Instead of the fractional derivative, we can consider the fractional Brow-nian motion B Ht itself and apply the same reasoning to it. This case is much simplerand we immediately obtain that sup ≤ s ≤ t | B Hs | ≤ (( t H (log( t )) p ) ∨ ξ ( p ) for any p > Acknowledgments This paper was partially supported by NSERC grant 261855.We are thankful to Ivan Smirnov for the assistance in the preparation of the man-uscript. References [BTT] Bertin K., Torres S., Tudor C.: Drift parameter estimation in fractional diffusions drivenby perturbed random walks. Statistics & Probability Letters, , 243249 (2011)[Bish08] Bishwal, J. P. N.: Parameter estimation in stochastic differential equations Springer,Lecture Notes Math., (2008)[BulKoz] Buldygin, V.V.; Kozachenko, Yu.V.: Metric characterization of random variables andrandom processes. Translations of Mathematical Monographs. . Providence, RI: AMS,American Mathematical Society. 257 p. (2000)[GuNu] Guerra, J., Nualart, D.: Stochastic differential equations driven by fractional Brownianmotion and standard Brownian motion. Stochastic Anal. Appl. 26, No. 5, 1053-1075(2008)[HuNu] Hu, Y., Nualart, D.: Parameter estimation for fractional Ornstein-Uhlenbeck processes.Statistics & Probability Letters, , 1030-1038 (2010)[HuXZ] Hu, Y., Xiao, W., Zhang W.: Exact maximum likelihood estimators for drift fractionalBrownian motions. Arxiv preprint arXiv:0904.4186, 2009 - arxiv.org[KlLeBr] Kleptsyna, M.L., Le Breton, A.: Statistical analysis of the fractional Ornstein-Uhlenbecktype process, Statistical Inference for Stochastic Processes, , 229-248 (2002)[LipSh] Liptser, R., Shiryaev A.: Statistics of Random Processes: II. Applications. Springer(1978)[MN88] Melnikov, A., Novikov, A.: Sequential inferences with prescribed accuracy for semimartin-gales. Theory Probab. Appl., , 446–459 (1988)[MMV] Memin, J., Mishura, Y., Valkeila, E.: Inequalities for the moments of Wiener integralswith respect to a fractional Brownian motion, Statistics & Probability Letters, , 197-206 (2001)[Mish08] Mishura, Y.: Stochastic calculus for fractional Brownian motion and related processes.Springer, Lecture Notes Math., (2008)[MiSh] Mishura, Yu., Shevchenko, G.: Stochastic differential equation involving Wiener processand fractional Brownian motion with Hurst index H > /