A Stability Result for Stochastic Differential Equations Driven by Fractional Brownian Motions

Abstract

We study the stability of the solutions of stochastic differential equations driven by fractional Brownian motions with Hurst parameter greater than half. We prove that when the initial conditions, the drift, and the diffusion coefficients as well as the fractional Brownian motions converge in a suitable sense, then the sequence of the solutions of the corresponding equations converge in Hölder norm to the solution of a stochastic differential equation. The limit equation is driven by the limit fractional Brownian motion and its coefficients are the limits of the sequence of the coefficients.

1. Introduction and Main Result

Suppose that 𝐵^𝐻=(𝐵^𝐻_𝑡)_{0≤𝑡≤𝑇} is an 𝑚-dimensional fractional Brownian motion (fBm in short) with Hurst parameter 𝐻 defined on a complete filtered probability space (Ω,ℱ,(ℱ_𝑡)_{0≤𝑡≤𝑇},𝐏). We mean that the components 𝐵^𝐻,𝑗, 𝑗=1,…,𝑚 are independent centered Gaussian processes with the covariance function 𝑅_𝐻()=1𝑠,𝑡2𝑡^2𝐻+𝑠^2𝐻−|𝑡−𝑠|^2𝐻. (1.1) If 𝐻=1/2, then 𝐵^𝐻 is clearly a Brownian motion. Since for any 𝑝≥1, 𝐄|𝐵_𝑡^𝐻,𝑗−𝐵_𝑠^𝐻,𝑗|^𝑝=𝑐_𝑝|𝑡−𝑠|^𝑝𝐻 the processes 𝐵^𝐻,𝑗 have 𝛼-Hölder continuous paths for all 𝛼∈(0,𝐻) (see [1] for further information about fBm).

In this paper we fix 1/2<𝐻<1 and we consider the solution (𝑋_𝑡)_{0≤𝑡≤𝑇} of the following stochastic differential equation (abbreviated by SDE from now on) on 𝐑^𝑑𝑋^𝑖_𝑡=𝑥^𝑖+^𝑚_𝑗=1^𝑡₀𝜎^𝑖,𝑗𝑋_𝑠𝑑𝐵_𝑠^𝐻,𝑗+^𝑡₀𝑏^𝑖𝑋_𝑠𝑑𝑠,0≤𝑡≤𝑇, (1.2)𝑖=1,…,𝑑, 𝑥∈𝐑^𝑑 is the initial value of the process 𝑋.

Under suitable assumptions on 𝜎, the processes 𝜎(𝑋) and 𝐵^𝐻 have trajectories which are Hölder continuous of order strictly larger than 1/2 so we can use the integral introduced by Young in [2]. The stochastic integral in (1.2) is then a path-wise Riemann-Stieltjes integral. A first result on the existence and uniqueness of a solution of such an equation was obtained in [3] using the notion of 𝑝-variation. The theory of rough paths introduced by Lyons in [3] was used by Coutin and Qian in order to prove an existence and uniqueness result for (1.2) (see [4]). The Riemann-Stieltjes integral appearing in (1.2) can be expressed as a Lebesgue integral using a fractional integration by parts formula (see Zähle [5]). Using this formula Nualart and Răşcanu have established in [6] the existence of a unique solution for a class of general differential equations that includes (1.2). Later on, the regularity in the sense of Malliavin calculus and the absolute continuity of the law of the random variables 𝑋_𝑡 have been investigated in [7–10].

In order to obtain moment bounds on the solution of (1.2), we have to estimate the corresponding determinist differential equation very carefully. Indeed, an exponential of the Hölder norm of the fBm may appear and by Fernique’s theorem, it is well known that such exponential moment does not always exist. This fact will be specified in Section 2. Thanks to a technical trick due to Hairer and Pillai in [11] (see also [12]), some estimations that are compatible with exponential moments are now available. This is the starting point of this short communication: first we will estimate the difference between two solutions of SDEs with different coefficients. We will endeavor ourselves to give some bounds that are suitable for stability results.

Now we present the kind of results we are interested in and so we need further notations. For a differentiable function 𝜑 from 𝐑^𝑑 to 𝐑^𝑞, we denote (if the following quantities do exist) ‖𝜑‖_∞=max_{𝑗=1,…,𝑞}‖𝜑^𝑗‖_∞ and 𝜑_𝐂1=𝜑_∞+max_{𝑖=1,…,𝑑}max_{𝑗=1,…,𝑞}𝜕𝜑^𝑗𝜕𝑥_𝑖∞. (1.3) The space 𝐂¹ is the space of continuously differentiable functions 𝜑 such that ‖𝜑‖_𝐂¹<∞. The space 𝐂² is defined in a similar way. For 0<𝜆<1 and 0≤𝑎<𝑏≤𝑇, we denote by 𝐂^𝜆(𝑎,𝑏;𝐑^𝑑) the space of 𝜆-Hölder continuous functions 𝑓∶[𝑎,𝑏]→𝐑^𝑑, equipped with the norm 𝑓_𝜆∶=𝑓_{𝑎,𝑏,∞}+𝑓_{𝑎,𝑏,𝜆}, (1.4) where 𝑓_{𝑎,𝑏,∞}=sup_{𝑎≤𝑟≤𝑏}𝑓(𝑟),𝑓_{𝑎,𝑏,𝜆}=sup_{𝑎≤𝑟≤𝑠≤𝑏}𝑓(𝑠)−𝑓(𝑟)|𝑠−𝑟|^𝜆. (1.5) We simply write 𝐂^𝜆(𝑎,𝑏) when 𝑑=1.

The main result of this work is the following theorem.

Theorem 1.1. Let (𝐵^𝐻,𝑛)_𝑛≥1 be a sequence of fractional Brownian motions defined on (Ω,ℱ,(ℱ_𝑡)_{0≤𝑡≤𝑇},𝐏). Let (𝑥_𝑛)_𝑛≥1, (𝑏_𝑛)_𝑛≥1, and (𝜎_𝑛)_𝑛≥1 be some sequences in 𝐑^𝑑, 𝐂¹, and 𝐂². One considers the sequence (𝑋^𝑛)_𝑛≥1 of processes such that for any 𝑛≥1, (𝑋^𝑛_𝑡)_{0≤𝑡≤𝑇} is the unique solution of 𝑋^𝑛_𝑡=𝑥_𝑛+^𝑡₀𝜎_𝑛𝑋^𝑛_𝑠𝑑𝐵_𝑠^𝐻,𝑛+^𝑡₀𝑏_𝑛𝑋^𝑛_𝑠𝑑𝑠,0≤𝑡≤𝑇. (1.6) If there exists(i)𝑥∈𝐑^𝑑, 𝑏∈𝐂¹ and 𝜎∈𝐂^𝟐 such that lim_𝑛→∞𝑥_𝑛−𝑥+𝑏_𝑛−𝑏_∞+𝜎−𝜎_𝑛𝐂1=0; (1.7)(ii)a fractional Brownian motion 𝐵^𝐻 defined on (Ω,ℱ,(ℱ_𝑡)_{0≤𝑡≤𝑇},𝐏) such that for 𝛽∈(1/2,𝐻) and 𝑝≥1 one has lim_𝑛→∞𝐄𝐵^𝐻,𝑛−𝐵^𝐻𝑝_0,𝑇,𝛽=0; (1.8)then lim_𝑛→∞𝐄𝑋^𝑛−𝑋^𝑝_𝛽=0, (1.9) where (𝑋_𝑡)_{0≤𝑡≤𝑇} is the solution of (1.2) with the coefficients 𝑥, 𝑏, and 𝜎 and driven by the fBm 𝐵^𝐻.

As usual in the theory of SDEs driven by fBm, the above theorem will be the counterpart of a deterministic result on ordinary differential equations driven by Hölder continuous functions. More precisely, Theorem 1.1 will be a consequence of an estimation on the Hölder norm of the difference of two solutions of rough differential equations. This result is interesting in itself and it is the subject of Section 2. It is precisely stated in Proposition 2.3 but we present here a brief description of the result we have obtained. We consider for some 𝛽∈(1/2,1) two deterministic rough functions 𝑔 and 𝑔 in 𝐂^𝛽 and two deterministic differential equations 𝑥_𝑡=𝑥₀+^𝑡₀𝑏𝑥_𝑠𝑑𝑠+^𝑡₀𝜎𝑥_𝑠𝑑𝑔_𝑠,𝑥_𝑡=𝑥₀+^𝑡₀𝑏̃𝑥_𝑠𝑑𝑠+^𝑡₀𝜎𝑥_𝑠𝑑𝑔_𝑠, (1.10) where all the coefficients are smooth. Then we will prove that there exists a constant 𝐶 such that ‖𝑥−𝑥‖_𝛽≤𝐶𝑥₀−𝑥₀+𝑏−𝑏̃_∞+‖𝜎−𝜎‖_𝐂¹𝑔_0,𝑇,𝛽+𝑔−𝑔_0,𝑇,𝛽×1+𝑔_0,𝑇,𝛽^2/𝛽exp𝐶𝑔^1/𝛽_0,𝑇,𝛽. (1.11) Since our upper bound is explicit, this estimate can be viewed as a refinement of Theorems 11.3 and 11.6 of [13]. Nevertheless we strength the fact that many results have been obtained in the theory of fractional SDEs thanks to rough paths theory. We may prove the above results by rough paths techniques but we adopt the simplest context of Young’s integral. The stability with respect to the driving noise is a reformulation of the continuity of the Itô map and this is well known. A weaker stability result with respect to the initial condition is proved in [14]. The stability with respect to all the coefficients in (1.2) is new to our knowledge.

The paper is organized as follows. In Section 2 we present the case of deterministic differential equations driven by Hölder continuous function and we state an estimation on the difference between the solutions of such equations (see Proposition 2.3). Theorem 1.1 will be a straightforward consequence of this work on deterministic differential equations. Finally some proofs are gathered in Section 3 and in the appendix.

2. Deterministic Differential Equation Driven by Rough Functions

This section deals with deterministic differential equations driven by Hölder’s continuous functions. These equations are the one satisfied by the trajectories of the solution of (1.2). Our aim is to prove an estimate for the difference of two solutions of deterministic differential equations driven by two different Hölder continuous functions. In [8, Theorem 3.3], such estimates are proved but are unfortunately unusable in our context (see the discussion below). Proposition 2.3 hereafter will strengthen the result of [8].

Suppose that 𝑓∈𝐂^𝜆(𝑎,𝑏) and 𝑔∈𝐂^𝜇(𝑎,𝑏) with 𝜆+𝜇>1. From [2], the Riemann-Stieltjes integral ∫^𝑏_𝑎𝑓𝑑𝑔 exists. In [5], the author provides an explicit expression for the integral ∫^𝑏_𝑎𝑓𝑑𝑔 in terms of fractional derivatives. In order to give some precisions, we consider 𝛼>0 being such that 𝛽>1−𝛼. Supposing that the following limit exists and is finite, we denote 𝑔_𝑏−(𝑡)=𝑔(𝑡)−lim_𝜀↓0𝑔(𝑏−𝜀). Then the Riemann-Stieltjes integral can be expressed as ^𝑏_𝑎𝑓_𝑡𝑑𝑔_𝑡=(−1)^𝛼𝑏_𝑎𝐷^𝛼_𝑎+𝑓(𝑡)𝐷^1−𝛼_𝑏−𝑔_𝑏−(𝑡)𝑑𝑡, (2.1) where 𝐷^𝛼_𝑎+𝑓(𝑡)=1Γ()𝑓(𝑡)1−𝛼(𝑡−𝑎)^𝛼+𝛼^𝑡_𝑎𝑓(𝑡)−𝑓(𝑠)(𝑡−𝑠)^𝛼+1,𝐷𝑑𝑠_𝛼𝑏−𝑔_𝑏−(𝑡)=(−1)^𝛼Γ()𝑔(𝑡)(𝑏)1−𝛼−𝑔(𝑏−𝑡)^𝛼+𝛼^𝑏_𝑡𝑔(𝑡)−𝑔(𝑠)(𝑠−𝑡)^𝛼+1𝑑𝑠. (2.2) We refer to [15] for further details on fractional operators. The following useful lemma is now classical. Its proof is postponed in the appendix.

Lemma 2.1. Let 𝑓 and 𝑔 in 𝐂^𝛽(𝑠,𝑡) with 1/2<𝛽<1 and 0≤𝑠<𝑡≤𝑇, then there exists a universal constant 𝜅 such that ^𝑡_𝑠𝑓_𝑟𝑑𝑔_𝑟≤𝜅𝑔𝛽−1/2^0,𝑇,𝛽𝑓_{𝑠,𝑡,∞}(𝑡−𝑠)^𝛽+𝑓_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽. (2.3)

Set 1/2<𝛽<1 and let 𝑔,𝑔∈𝐂^𝛽(0,𝑇;𝐑^𝑚). We will work with the following deterministic differential equations on 𝐑^𝑑: 𝑥^𝑖_𝑡=𝑥^𝑖₀+^𝑡₀𝑏^𝑖𝑥_𝑠𝑑𝑠+^𝑚_𝑗=1^𝑡₀𝜎^𝑖,𝑗𝑥_𝑠𝑑𝑔^𝑗_𝑠,𝑥^𝑖_𝑡=𝑥^𝑖₀+^𝑡₀𝑏̃^𝑖𝑥_𝑠𝑑𝑠+^𝑚_𝑗=1^𝑡₀𝜎^𝑖,𝑗𝑥_𝑠𝑑𝑔^𝑗_𝑠, (2.4) for 0≤𝑡≤𝑇,𝑖=1,…,𝑑 and 𝑥₀,𝑥₀∈𝐑^𝑑.

We introduce the following assumptions on the coefficients of the above equations. For a function 𝜑 from 𝐑^𝑑 to 𝐑^𝑞, ∇𝜑 denotes the matrix of first order derivatives and ∇²𝜑 denotes its Hessian.(H1) There exists some positive constants 𝑏₀, 𝑏₁, 𝑏̃₀, 𝑏̃₁ such that ‖𝑏‖_∞≤𝑏₀, ‖𝑏̃‖_∞≤𝑏̃₀, ‖∇𝑏‖_∞≤𝑏₁, and ‖∇𝑏̃‖_∞≤𝑏̃₁. (H2) There exists some positive constants 𝑐₀, 𝑐₁, 𝑐₂, 𝑐₀̃, 𝑐₁̃, and 𝑐₂̃ such that ‖𝜎‖_∞≤𝑐₀, ‖∇𝜎‖_∞≤𝑐₁, ‖∇²𝜎‖_∞≤𝑐₂, ‖𝜎‖_∞≤𝑐₀̃, ‖∇𝜎‖_∞≤𝑐₁̃, and ‖∇²𝜎‖_∞≤𝑐₂̃. It is proved in [6, Theorem 5.1] that if 1−𝛽<𝛼<1/2, each of the above equations has a unique (1−𝛼)-Hölder continuous solution. The estimates on the solution (𝑥_𝑡)_{0≤𝑡≤𝑇} obtained in [6] were improved in [8]. Let us recall the following observations concerning Theorem 3 in Hu and Nualart [8] (see also [11] for similar comments). It has been proved in [8] that if 𝑏=0 and 𝜎 is twice continuously differentiable with bounded second order derivatives, then there exists a constant 𝑘 that depends on 𝑇, 𝛽, and 𝜎 such that ‖𝑥−𝑥‖_0,𝑇,∞≤𝑘exp𝑘𝑔^1/𝛽_0,𝑇,𝛽1+‖𝑥‖_0,𝑇,𝛽+‖𝑥‖_0,𝑇,𝛽^1/𝛽1+‖𝑥‖_0,𝑇,𝛽𝑔−𝑔_0,𝑇,𝛽. (2.5) Replacing 𝑔 and 𝑔 by the trajectories of the fractional Brownian motions 𝐵^𝐻 and 𝐵^𝐻, we obtain estimation in the supremum norm for the difference of the processes 𝑋 and 𝑋 satisfying 𝑋_𝑡=𝑥₀+^𝑡₀𝜎𝑋_𝑠𝑑𝐵^𝐻_𝑠,𝑋_𝑡=𝑥₀+^𝑡₀𝜎𝑋_𝑠𝑑𝐵^𝐻_𝑠. (2.6) Thus the estimation (2.5) holds almost-surely and one have to take expectation. For this purpose we use the following Fernique’s type result on the exponential moments of the Hölder norm of the fBm.

Lemma 2.2. Let 𝑇>0, 1/2<𝛽<𝐻<1. Then for any 𝛼<1/(128(2𝑇)^{2(𝐻−𝛽)})𝐄exp𝛼𝐵^𝐻2_0,𝑇,𝛽≤1−128𝛼(2𝑇)^{2(𝐻−𝛽)−1/2}. (2.7)

One refers to [16] for a proof of this lemma. Lemma 2.2 implies the following integrability property for 𝑋 (or 𝑋): for any 𝜆>0 and 𝛾<2𝐻, 𝐄exp𝜆‖𝑋‖^𝛾_0,𝑇,∞<∞. (2.8) Inequality (2.8) together with (2.5) will unfortunately be useless since the quantities ‖𝑋‖^1/𝛽_0,𝑇,∞ and ‖𝐵‖^1/𝛽_0,𝑇,∞ appear in a multiplicative way in the exponent in the right hand side of (2.5). Indeed, Young’s inequality yields ‖𝑋‖^1/𝛽_0,𝑇,∞‖𝐵‖^1/𝛽_0,𝑇,∞≤𝑐‖𝑋‖^𝑝/𝛽_0,𝑇,∞+𝑐‖𝐵‖^𝑞/𝛽_0,𝑇,∞ and if one imposes 𝑝/𝛽<2 then necessarily 𝑞/𝛽>2 and the finiteness of the expression such as 𝐄exp𝑘‖𝐵‖^1/𝛽_0,𝑇,∞‖𝑋‖^1/𝛽_0,𝑇,∞ (2.9) cannot be deduced from Lemma 2.2 and (2.8) (in fact we do not even know if this expectation is finite). Hence we need a suitable estimate to obtain moment bounds on the quantities we are interested in. Such investigations have been carried out in [11, 12] but we need some nontrivial modifications to handle the difference 𝑥−𝑥 when 𝑥 and 𝑥 are the solutions of (2.4).

Therefore the next result is a strengthening of Theorem  3.3 in [8] and is based on the method used in [11, Lemma 3.2]. It may also be viewed as a refinement of Theorems 11.3 and 11.6 of [13].

Proposition 2.3. Let 𝑇 be fixed and let 𝑔 and 𝑔 be Hölder continuous of order 1/2<𝛽<1. Under (H1) and (H2), there exists a constant 𝐶 that depends only on 𝑇, 𝛽, 𝑏₀, 𝑏̃₀, 𝑐₀, 𝑐₀̃, 𝑐₁, 𝑐₁̃, 𝑐₂, and 𝑐₂̃ such that ‖𝑥−𝑥‖_𝛽≤𝐶𝑥₀−𝑥₀+𝑏−𝑏̃_∞+‖𝜎−𝜎‖_𝐂¹𝑔_0,𝑇,𝛽+𝑔−𝑔_0,𝑇,𝛽×1+𝑔_0,𝑇,𝛽^2/𝛽exp𝐶𝑔^1/𝛽_0,𝑇,𝛽. (2.10)

It is worth to notice that a careful reading of the proof shows that 𝐶 depends continuously on its parameters. This is important when we apply this proposition to stability properties of stochastic differential equations.

3. Proofs

The subject of this section is the proof of Proposition 2.3 and Theorem 1.1. We follow the arguments developed in the proof of [8, Theorem 3.2], [11, Lemma 3.2] and we give some precisions. We restrict ourselves to the case 𝑑=𝑚=1 for simplicity. Thus for a function 𝜑, we denote 𝜑^′ its derivative and 𝜑^′′ its second order derivative. Moreover 𝑐_𝛽∶=𝜅/(𝛽−1/2) designates the constant in (2.3).

3.1. A Preliminary Lemma

We will need the following lemma whose proof is borrowed from [11]. For the sake of completeness and to give some information on the constants that are involved in the statement, we briefly recall the arguments that are used in the proof.

Lemma 3.1. There exists an explicit constant 𝑀₁ that depends on 𝑇, 𝑏₀, 𝑏₁, 𝑐₀, 𝑐₁, and 𝑐_𝛽 such that ‖𝑥‖_0,𝑇,∞≤𝑇1+𝑥₀𝑒^{2𝑀1𝑏1𝑇}2𝑀₁1+𝑔_0,𝑇,𝛽^1/𝛽. (3.1)

Proof. Let 0≤𝑠≤𝑡≤𝑇. Since |𝑏(𝑥_𝑟)|≤|𝑏(𝑥_𝑟)−𝑏(𝑥_𝑠)|+|𝑏(𝑥_𝑠)| and |𝑏(𝑥_𝑠)|≤𝑏₀+𝑏₁|𝑥_𝑠|, we have ^𝑡_𝑠𝑏𝑥_𝑟𝑑𝑟≤𝑏₀(𝑡−𝑠)+𝑏₁𝑥_𝑠(𝑡−𝑠)+𝑏₁‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^1+𝛽. (3.2) Clearly ‖𝜎(𝑥_⋅)‖_{𝑠,𝑡,𝛽}≤𝑐₁‖𝑥‖_{𝑠,𝑡,𝛽}, thus Inequality (2.3) yields ‖𝑥‖_{𝑠,𝑡,𝛽}≤𝑏₀(𝑡−𝑠)^1−𝛽+𝑏₁𝑥_𝑠(𝑡−𝑠)^1−𝛽+𝑏₁‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)+𝑐_𝛽𝑔_0,𝑇,𝛽𝑐₀+𝑐₁‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽≤𝑀₁1+𝑔_0,𝑇,𝛽+𝑏₁𝑥_𝑠(𝑡−𝑠)^1−𝛽+1+𝑔_0,𝑇,𝛽‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽 (3.3) with 𝑀₁=max(1,𝑇^1−𝛽‖𝑏‖_𝐂¹,𝑐_𝛽‖𝜎‖_𝐂¹). We denote Δ₁=2𝑀₁1+𝑔_0,𝑇,𝛽^−1/𝛽 (3.4) and when (𝑡−𝑠)≤Δ₁ we may write ‖𝑥‖_{𝑠,𝑡,𝛽}≤2𝑀₁1+𝑔_0,𝑇,𝛽+𝑏₁𝑥_𝑠Δ₁^1−𝛽. (3.5) Since ‖𝑥‖_{𝑠,𝑡,∞}≤|𝑥_𝑠|+‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽, we obtain for (𝑡−𝑠)≤Δ₁ that ‖𝑥‖_{𝑠,𝑡,∞}≤𝑥_𝑠1+2𝑀₁𝑏₁Δ₁+1. (3.6) By induction with 𝑁=𝑇/Δ₁ it follows that ‖𝑥‖_0,𝑇,∞≤𝑥₀1+2𝑀₁𝑏₁Δ₁^𝑁+^𝑁_𝑘=11+2𝑀₁𝑏₁Δ₁^𝑘≤𝑇Δ₁1+𝑥₀1+2𝑀₁𝑏₁Δ₁^𝑇/Δ1 (3.7) and since for any 𝑎>0, (1+𝑥/𝑎)^𝑎≤𝑒^𝑥, we have (1+2𝑀₁𝑏₁Δ₁)^𝑇/Δ1≤𝑒^{2𝑀1𝑏1𝑇} and we finally deduce (3.1).
As noticed in [11], if 𝑏 is the null function, then 𝑏₁=0 and the last above argument fails. In this case it is impossible to obtain a bound without any exponential of the quantity ‖𝑔‖_0,𝑇,𝛽.

Remark 3.2. When (𝑡−𝑠)≤Δ₁, we substitute (3.1) into (3.5) and we deduce the estimate ‖𝑥‖_{𝑠,𝑡,𝛽}≤𝑀₂1+𝑔_0,𝑇,𝛽, (3.8) where 𝑀₂=2𝑀₁1+𝑏₁𝑇1+𝑥₀𝑒^{2𝑀1𝑏1𝑇}. (3.9)

In the sequel, we naturally denote 𝑀₁, Δ₁, and 𝑀₂ the corresponding quantities that are related to (7).

3.2. Proof of Proposition 2.3

Proof. Let 0≤𝑠≤𝑡≤𝑇. We write 𝑥_𝑡−𝑥_𝑡−𝑥_𝑠−𝑥_𝑠=𝐼₁+𝐼₂+𝐼₃+𝐼₄+𝐼₅, (3.10) where 𝐼₁=^𝑡_𝑠𝑏𝑥_𝑟−𝑏𝑥_𝑟𝐼𝑑𝑟,₂=^𝑡_𝑠𝑏𝑥_𝑟−𝑏̃𝑥_𝑟𝐼𝑑𝑟,₃=^𝑡_𝑠𝜎𝑥_𝑟−𝜎𝑥_𝑟𝑑𝑔_𝑟,𝐼₄=^𝑡_𝑠𝜎𝑥_𝑟−𝜎𝑥_𝑟𝑑𝑔_𝑟,𝐼₅=^𝑡_𝑠𝜎𝑥_𝑟𝑑𝑔_𝑟−𝑔_𝑟. (3.11) Since for any 𝑠≤𝑟≤𝑡, |𝑥_𝑟−𝑥_𝑟|≤‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}+|𝑥_𝑠−𝑥_𝑠|, we may write 𝐼₁≤𝑏₁‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^1+𝛽+𝑥_𝑠−𝑥_𝑠(𝑡−𝑠) (3.12) and clearly 𝐼₂≤𝑏−𝑏̃_∞(𝑡−𝑠). (3.13) We use (2.3) to obtain 𝐼₄≤𝑐_𝛽𝑔_0,𝑇,𝛽‖𝜎−𝜎‖_∞(𝑡−𝑠)^𝛽+𝜎^′−𝜎^′_∞‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽,𝐼₅≤𝑐_𝛽𝑔−𝑔_0,𝑇,𝛽‖𝜎‖_∞(𝑡−𝑠)^𝛽+𝜎^′_∞‖𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽. (3.14) Then by (3.8) we have for (𝑡−𝑠)≤Δ₁: 𝐼₄≤𝑀₃𝑔_0,𝑇,𝛽‖𝜎−𝜎‖_𝐂¹(𝑡−𝑠)^𝛽,𝐼₅≤𝑀₃𝑔−𝑔_0,𝑇,𝛽‖𝜎‖_𝐂¹(𝑡−𝑠)^𝛽, (3.15) where 𝑀₃=𝑐_𝛽(1+𝑏̃₁𝑇(1+|𝑥₀|)𝑒²𝑀₁𝑏̃_1𝑇).
The term 𝐼₃ is a little bit more elaborate. First we have 𝜎𝑥_𝑟−𝜎𝑥_𝑟−𝜎𝑥_𝑟′+𝜎𝑥_𝑟′≤𝜎^′_∞𝑥_𝑟−𝑥_𝑟−𝑥_𝑟′+𝑥_𝑟′+𝜎^′′_∞𝑥_𝑟−𝑥_𝑟′𝑥_𝑟−𝑥_𝑟+𝑥_𝑟′−𝑥_𝑟′≤𝑐₁‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽+2𝑐₂‖𝑥‖_{𝑠,𝑡,𝛽}‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽+𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^𝛽 (3.16) and thus 𝜎𝑥_⋅−𝜎𝑥_{⋅𝑠,𝑡,𝛽}≤𝑐₁‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}+2𝑐₂‖𝑥‖_{𝑠,𝑡,𝛽}‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽+𝑥_𝑠−𝑥_𝑠. (3.17) Since 𝜎𝑥_⋅−𝜎𝑥_{⋅𝑠,𝑡,∞}≤𝜎^′_∞‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽+𝜎^′_∞𝑥_𝑠−𝑥_𝑠. (3.18) Inequality (2.3) yields 𝐼₃≤𝑐_𝛽𝑔_0,𝑇,𝛽𝑐₁‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽+𝑐₁𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^𝛽+𝑐₁‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}+2𝑐₂‖𝑥‖_{𝑠,𝑡,𝛽}‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^𝛽+𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^2𝛽 (3.19) and by (3.8) we deduce that for (𝑡−𝑠)≤Δ₁: 𝐼₃≤𝑀₄𝑔_0,𝑇,𝛽‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽+𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^𝛽, (3.20) with 𝑀₄=2𝑐_𝛽‖𝜎‖_𝐂¹(1+𝑏₁𝑇(1+|𝑥₀|)𝑒^{2𝑀1𝑏1𝑇}). We use (3.12), (3.13), (3.20), and (3.15), in (3.10) and we may write that for (𝑡−𝑠)≤Δ₂∶=min(Δ₁,Δ₁)‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}≤𝑏₁𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^1−𝛽+𝑀₄𝑔_0,𝑇,𝛽𝑥_𝑠−𝑥_𝑠+𝑏−𝑏̃_∞(𝑡−𝑠)^1−𝛽+𝑀₃‖𝜎−𝜎‖_𝐂¹𝑔_0,𝑇,𝛽+𝑀₃‖𝜎‖_𝐂¹𝑔−𝑔_0,𝑇,𝛽+‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}𝑏₁(𝑡−𝑠)+𝑀₄𝑔_0,𝑇,𝛽(𝑡−𝑠)^𝛽≤𝑏₁𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^1−𝛽+𝑀₄𝑔_0,𝑇,𝛽𝑥_𝑠−𝑥_𝑠+𝐸+𝑀₅‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}1+𝑔_0,𝑇,𝛽(𝑡−𝑠)^𝛽, (3.21) where 𝐸=𝑏−𝑏̃_∞(𝑡−𝑠)^1−𝛽+𝑀₃‖𝜎−𝜎‖_𝐂¹𝑔_0,𝑇,𝛽+𝑀₃‖𝜎‖_𝐂¹𝑔−𝑔_0,𝑇,𝛽 (3.22) and 𝑀₅=max(𝑏₁𝑇^1−𝛽,𝑀₄). With Δ₃=(2𝑀₅(1+‖𝑔‖_0,𝑇,𝛽))^−1/𝛽 we may write when (𝑡−𝑠)≤Δ∶=min(Δ₂,Δ₃) that ‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}≤2𝑏₁𝑥_𝑠−𝑥_𝑠(𝑡−𝑠)^1−𝛽+2𝑀₄𝑔_0,𝑇,𝛽𝑥_𝑠−𝑥_𝑠+2𝐸. (3.23) The final arguments are the same as in the proof of (3.1). For 𝑡−𝑠≤Δ we have ‖𝑥−𝑥‖_{𝑠,𝑡,∞}≤𝑥_𝑠−𝑥_𝑠1+2𝑏₁Δ+2𝑀₄𝑔_0,𝑇,𝛽Δ^𝛽+2𝐸 (3.24) that implies by induction if we denote 𝑁=𝑇/Δ‖𝑥−𝑥‖_0,𝑇,∞≤𝑥₀−𝑥₀+2𝑁𝐸1+2𝑏₁Δ+2𝑀₄𝑔_0,𝑇,𝛽Δ^𝛽𝑁≤𝑁𝑥₀−𝑥₀+2𝐸1+2𝑏₁𝑇+2𝑀₄𝑇𝑔_0,𝑇,𝛽Δ^𝛽−1𝑇/Δ^𝑇/Δ≤𝑇Δ𝑥₀−𝑥₀+2𝐸exp2𝑏₁𝑇+2𝑀₄𝑇𝑔_0,𝑇,𝛽Δ^𝛽−1. (3.25) Now we see that we may find a constant 𝑀 depending only on 𝑇, 𝑐_𝛽 and all the coefficients of (2.4) (but not on 𝑔 and 𝑔) such that ‖𝑥−𝑥‖_0,𝑇,∞≤𝐶𝑥₀−𝑥₀+𝐸1+𝑔_0,𝑇,𝛽^1/𝛽exp𝐶𝑔^1/𝛽_0,𝑇,𝛽. (3.26) Substituting the bound (3.26) in (3.23) yields that for (𝑡−𝑠)≤Δ: ‖𝑥−𝑥‖_{𝑠,𝑡,𝛽}≤𝐶𝑥₀−𝑥₀+𝐸1+𝑔_0,𝑇,𝛽^1+1/𝛽exp𝐶𝑔^1/𝛽_0,𝑇,𝛽. (3.27) Thus by Lemma A.2 from [11] we finally obtain ‖𝑥−𝑥‖_0,𝑇,𝛽≤𝐶𝑥₀−𝑥₀+𝐸1+𝑔_0,𝑇,𝛽^2/𝛽exp𝐶𝑔^1/𝛽_0,𝑇,𝛽. (3.28) Proposition 2.3 is now a consequence of (3.26) and (3.28).

3.3. Proof of Theorem 1.1

Proof. The proof is now very simple. We use (2.10) from Proposition 2.3 with 𝑔=𝐵^𝐻 and 𝑔=𝐵^𝐻,𝑛. Moreover for any 𝑘≥1 we have the following estimate of the moment of the Hölder norm of the fBm (see [16, Lemma 8] for instance) 𝔼𝐵^𝐻2𝑘_0,𝑇,𝛽≤32^𝑘(2𝑇)^{2𝑘(𝐻−𝛽)}(2𝑘)!.𝑘! (3.29) Then the estimate on the moments of ‖𝑋^𝑛−𝑋‖_𝛽 are deduced from easy algebra using Lemma 2.2, (3.29), Hölder’s inequality and Young’s inequality. The convergence (1.9) follows from the stability assumptions on the coefficients.

Appendix

Proof of Lemma 2.1

Proof. With 1−𝛽<𝛼<1/2, we use (2.1) and we obtain for all 0≤𝑠,𝑡≤𝑇 and all 𝑓∈𝐂^𝛽(𝑠,𝑡): ^𝑡_𝑠𝑓_𝑟𝑑𝑔_𝑟≤^𝑡_𝑠𝐷^𝛼_𝑠+𝑓_𝑟𝐷^1−𝛼_𝑡−𝑔_𝑡−(𝑟)𝑑𝑟. (A.1) We have 𝐷^1−𝛼_𝑡−𝑔_𝑡−(𝑟≤𝛽)Γ(𝛼)𝑔𝛼+𝛽−1^0,𝑇,𝛽|𝑡−𝑟|^{𝛼+𝛽−1},𝐷^𝛼_𝑠+𝑓_𝑟≤1Γ()𝑓1−𝛼^{𝑠,𝑟,∞}(𝑟−𝑠)^−𝛼+𝛼Γ()𝑓𝛽−𝛼1−𝛼^{𝑠,𝑟,𝛽}(𝑟−𝑠)^𝛽−𝛼. (A.2) It follows that ^𝑡_𝑠𝑓_𝑟𝑑𝑔_𝑟≤𝛽Γ(𝛼)Γ()𝑔𝛼+𝛽−11−𝛼^0,𝑇,𝛽×𝑓_{𝑠,𝑡,∞}^𝑡_𝑠(𝑟−𝑠)^−𝛼(𝑡−𝑟)^{𝛼+𝛽−1}𝑑𝑟+𝛼𝑓_{𝑠,𝑡,𝛽}^𝑡_𝑠(𝑟−𝑠)^𝛽−𝛼(𝑡−𝑟)^{𝛼+𝛽−1}𝑑𝑟. (A.3) We use the change of variables 𝑟=(𝑡−𝑠)𝜉+𝑠 and we recall that the Beta function is defined by 𝐵(𝑎,𝑏)=∫¹₀(1−𝜉)^𝑎−1𝜉^𝑏−1𝑑𝜉=Γ(𝑎)Γ(𝑏)/Γ(𝑎+𝑏). Then we get ^𝑡_𝑠𝑓_𝑟𝑑𝑔_𝑟≤𝑘_𝛼,𝛽𝑔_0,𝑇,𝛽𝑓_{𝑠,𝑡,∞}(𝑡−𝑠)^𝛽+𝑓_{𝑠,𝑡,𝛽}(𝑡−𝑠)^2𝛽𝑘with_𝛼,𝛽=𝛽𝐵𝛼+𝛽,1−𝛼Γ(𝛼)Γ()+𝛼+𝛽−11−𝛼𝛼𝛽𝐵𝛼+𝛽,1+𝛽−𝛼Γ(𝛼)Γ().𝛼+𝛽−1𝛽−𝛼1−𝛼 (A.4) It is proved in [12] that in fact 𝑘_𝛼,𝛽≤𝜅/(𝛽−1/2) where 𝜅 is a universal constant.