Hölder continuity in the Hurst parameter of functionals of Stochastic Differential Equations driven by fractional Brownian motion
HHölder continuity in the Hurst parameter of functionals ofstochastic differential equations driven by fractional Brownianmotion
Alexandre Richard and Denis Talay
TOSCA team – INRIA Sophia-Antipolis2004 route des Lucioles, F-06902 Sophia-Antipolis Cedex, France
May 31, 2016
Abstract
Sensitivity analysis w.r.t. the long-range / memory noise parameter for probabilitydistributions of functionals of solutions to stochastic differential equations is an im-portant stochastic modeling issue in many applications. However, we have not foundtheoretical results on this topic in the literature.In this paper we consider solutions { X Ht } t ∈ (cid:82) + to stochastic differential equationsdriven by fractional Brownian motions and two sensitivity problems when the Hurstparameter H of the noise tends to the critical Brownian parameter H = . We firstget accurate sensitivity estimates of time marginal probability distributions of X H . Wesecond develop a sensitivity analysis for the Laplace transforms of the first passagetimes of X H at given thresholds.Our technique requires accurate Gaussian estimates on the density of X Ht . TheGaussian estimate we obtain in Section 5 may be of interest by itself. Key words : Fractional Brownian motion, Malliavin calculus, first hitting time.
In many applied situations where continuous-time stochastic differential equations areused, one chooses Markovian dynamics for natural reasons: a huge literature exists ontheir analysis, calibration and simulation; probability distributions of functionals of theirpaths can be obtained by solving partial differential or integro-differential equations, or bymeans of stochastic numerical methods; their semimartingale property allows to use thestochastic calculus theory; asymptotic properties are proved by well developed techniques(homogenization, mean-field limits, convergence to equilibrium analysis, etc.).However, the statistical nature of the driving noise is a difficult issue, and noises havingthe semi-martingale property may often be seen as arbitrary idealizations of the reality.Empirical studies actually tend to show memory effects in biological, financial, physicaldata: see e.g. Rypdal and Rypdal [ ] for a statistical evidence in climatology and Berzinet al. [ ] for the statistics of stochastic differential equations with memory. The Markov1 a r X i v : . [ m a t h . P R ] M a y tructure then becomes questionable and it justifies to propose new models driven by noiseswith long-range memory such as fractional Brownian motions rather than Lévy processes.A natural question then arises. As emphasized by Jolis and Viles [ ] , choosing theHurst parameter H of the noise (for example by means of statistical methods) does not closethe noise modelling problem: since in practice the estimation of H is inevitably crude, oneneeds to check that the model does not exhibit a large sensitivity w.r.t. H . More precisely,one needs to study the sensitivity w.r.t. H of probability distributions of smooth and nonsmooth functionals of the paths of solutions to stochastic differential equations.First passage times at prescribed thresholds is an important class of non smooth func-tionals in many applications. For example, this issue appears in the study of default riskin mathematical finance, ruin probabilities in insurance or spike trains in neuroscience(spike trains are sequences of times at which the membrane potential of neurons reachlimit thresholds and then are reset to a resting value, are essential to describe the neuronalactivity). This issue also appears in complex simulations, e.g. the simulations of stochasticparticle systems which are confined in cells (see e.g. Bernardin et al. [
7, Sec.3 ] ) and inextremely various situations in physical sciences (for nice surveys, see e.g. Metzler et al. [ ] ).Markov properties are crucial to get equations or to construct numerical algorithms forfirst passage time probability distributions: See e.g. Deaconu and Herrmann [ ] and cita-tions therein. On the contrary, the long-range dependence leads to analytical and numericaldifficulties: see e.g. Jeon et al. [ ] and Dalang and Sanz-Solé [ ] .To summarize the preceding discussion, it seems worth developing sensitivity analysesw.r.t. the long-range / memory parameter for solutions to stochastic differential equationsand the probability distributions of their first passage times at given thresholds. Here weconsider the case of stochastic differential equations driven by fractional Brownian motionsand the sensitivity, w.r.t. the Hurst parameter of the noise, of probability distributions ofcertain functionals of the trajectories of the solutions. Our main results.
The fractional Brownian motion { B Ht } t ∈ (cid:82) + with Hurst parameter H ∈ (
0, 1 ) is the only Gaussian process with stationary increments which is self-similar of order H (up to centering of the mean and normalization of the variance). Its covariance reads: R H ( s , t ) = (cid:128) s H + t H − | t − s | H (cid:138) , ∀ s , t ∈ (cid:82) + .Given H ∈ ( , 1 ) , we consider the process { X Ht } t ∈ (cid:82) + solution to the following stochasticdifferential equation driven by { B Ht } t ∈ (cid:82) + : X Ht = x + (cid:90) t b ( X Hs ) d s + (cid:90) t σ ( X Hs ) ◦ d B Hs , (1.1;H)where the last integral is a pathwise Stieltjes integral in the sense of Young [ ] (the notionof solution is explained in Section 2). For H = the process X solves the following SDE inthe classical Stratonovich sense: X t = x + (cid:90) t b ( X s ) d s + (cid:90) t σ ( X s ) ◦ d B s . (1.1; )Our first theorem concerns the sensitivity w.r.t. H around the critical Brownian parameter H = of time marginal probability distributions of { X Ht } t ∈ (cid:82) + .2 heorem 3.1. Let H ∈ ( , 1 ) , and let X H and X be as before. Suppose that b and σ aresmooth enough and σ is strongly elliptic (see Section 3 for a precise condition), and that ϕ isbounded and Hölder continuous of order + β for some β > . Then, for any T > , thereexists C T > such that for any H ∈ [ , 1 ) : sup t ∈ [ T ] (cid:12)(cid:12) (cid:69) ϕ ( X Ht ) − (cid:69) ϕ ( X t ) (cid:12)(cid:12) ≤ C T ( H − ) .Our next theorem concerns the first passage time of X H at threshold 1, assuming x < τ XH : = inf { t ≥ X Ht = } . The probability distribution of the first passage time τ H of afractional Brownian motion is not explictly known. Molchan [ ] obtained the asymptoticbehaviour of its tail distribution function and Decreusefond and Nualart [ ] obtained anupper bound on the Laplace transform of τ HH . The recent work of Delorme and Wiese [ ] proposes an asymptotic expansion (in terms of H − ) of the density of τ H formally obtainedby perturbation analysis techniques. Related works are those of Nourdin and Viens [ ] onthe density of sup t ∈ [ a , b ] B Ht − (cid:69) (cid:128) sup t ∈ [ a , b ] B Ht (cid:138) where 0 < a < b , and Baudoin et al. [ ] onhitting probabilities of multidimensional fractional diffusions. Theorem 4.1.
Suppose that b and σ are smooth enough and σ is strongly elliptic (see Section4 for a precise condition), and let x < . There exist constants λ ≥ , µ ≥ (both dependingon b and σ only) and < η < − x such that: for all ε ∈ ( ) and < η ≤ η , there exist α > and C ε , η > such that ∀ λ ≥ λ , ∀ H ∈ [ , 1 ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:69) (cid:128) e − λτ XH (cid:138) − (cid:69) (cid:18) e − λτ X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε , η ( H − ) − ε e − α S ( − x − η )( (cid:112) λ + µ − µ ) , where S : (cid:82) + → (cid:82) + is an increasing function defined in Section 4. In the pure fBm situation(where b ≡ and σ ≡ ) the result holds with λ = and µ = . To prove the preceding theorem we need accurate estimates on the density of X Ht . OurTheorem 5.1 below, which improves estimates in [
6, 9 ] , may be of interest by itself.A sensitivity analysis of the density of τ XH would certainly be useful for applications.Our estimate on the Laplace transform of τ XH gives information on the robustness of thisdensity around time 0 when H is close to . This seems interesting since the simulationsin [ ] suggest that, when H > , the density of sup t ∈ [ ] B Ht is unbounded near 0. Togo further, we have tried to start from Proposition 4.4 below which provides an expressionfor (cid:69) (cid:128) e − λτ X H (cid:138) − (cid:69) (cid:16) e − λτ X / (cid:17) whose inverse Laplace transform can be computed in principle.However, we have not succeeded to solve technical issues raised by terms whose Malliavinderivatives are highly singular. Organization of the paper.
In Section 2 we review elements of stochastic calculus forfractional Brownian motion and gather a few asymptotic properties of its first passage times.In Section 3 we prove Theorem 3.1. In Section 4 we prove Theorem 4.1. Our techniquerequires a Gaussian-type upper bound on the density of X H (Theorem 5.1 which is provenin Section 5). Various technical lemmas are gathered in Appendix A and B. Notations. If γ ∈ (
0, 1 ) , the Hölder norm of a bounded function f defined on an interval [ a , b ] is denoted by (cid:107) f (cid:107) γ , a , b : = sup a ≤ s < t ≤ b | f ( t ) − f ( s ) | ( t − s ) γ . We use the same notation for γ ∈ ∞ ) \ (cid:78) and adopt the definition of [
23, p.3 ] , that is, we denote by C m + γ ( (cid:82) d ) the space ofbounded functions with bounded derivatives up to order m ≥ f ( m ) is a γ -Höldercontinuous function.The supremum norm is denoted by (cid:107) · (cid:107) ∞ , a , b or simply (cid:107) · (cid:107) ∞ when no confusion canexist.We denote by C any constant which may change from line to line but does not dependon the Hurst parameter H . When necessary we emphasize that a constant depends on thefinal time horizon T : we then denote it by C T . Identically, α will be a constant in theexponential of Theorem 4.1 that might change from line to line.We will consider Brownian and fractional Brownian motion ( B and B H ), Brownian andfractional Brownian diffusions ( X and X H ), which, unless stated otherwise, all start fromthe same initial condition x . W.l.o.g. the first hitting time threshold is fixed at the value 1and therefore, except when explicitly mentioned, we consider x < H ≥ In this section, we briefly review the definitions of Skorokhod, Stieltjes and Stratonovichintegrals w.r.t. fractional Brownian motion (fBm). The material comes from [
6, Section 2 ] , [ ] and Nualart [ ] .Let T >
0. For fixed H ∈ ( , 1 ) , the covariance R H ( s , t ) of the fBm is given by: for any s , t ∈ [ T ] , (cid:168) R H ( s , t ) = (cid:0) s H + t H − | s − t | H (cid:1) = α H (cid:82) s (cid:82) t | u − v | H − d u d v α H = H ( H − ) .The Cameron-Martin space (cid:72) H ( T ) (simply written (cid:72) H if there cannot be any confusion)associated to the covariance R H is defined as the closure of the space of step functions withrespect to the following scalar product: 〈 ϕ , ψ 〉 (cid:72) H = α H (cid:90) T (cid:90) T | u − v | H − ϕ ( u ) ψ ( v ) d u d v .Note that (cid:72) = L [ T ] .A natural subspace of (cid:72) H will be needed in the sequel: |(cid:72) H | is the Banach space ofmeasurable functions ϕ on [ T ] such that (cid:107) ϕ (cid:107) |(cid:72) H | : = α H (cid:90) T (cid:90) T | ϕ s | | ϕ t | | s − t | H − d s d t < ∞ . (2.1) Operators.
For any H ∈ ( , 1 ) define the integral operator K H by its kernel K H ( θ , σ ) : ∀ σ ≥ θ > K H ( θ , σ ) : = ∀ θ > σ > K H ( θ , σ ) : = c H ( H − ) σ − H (cid:90) θσ u H − ( u − σ ) H − d u ,4here c H : = (cid:32) H Γ( / − H )Γ( H + ) Γ( − H ) (cid:33) .Now define the operator K ∗ H as follows: for any ϕ ∈ (cid:72) H , K ∗ H ϕ ( s ) : = (cid:90) Ts ∂ K H ∂ θ ( θ , s ) ϕ ( θ ) d θ = ( H − ) c H (cid:90) Ts (cid:18) θ s (cid:19) H − ( θ − s ) H − ϕ ( θ ) d θ .Notice that 〈 ϕ , ψ 〉 (cid:72) H = 〈 K ∗ H ϕ , K ∗ H ψ 〉 L [ T ] .In addition, one can easily verify that c H tends to 1 as H → and thus ∂∂ θ K H ( θ , σ ) convergesin the distributional sense to the Dirac measure at point σ , which implies that K ∗ H tends tothe identity operator. Representation of fBm as non-anticipating stochastic integrals.
The fBm B H can berepresented as follows: for some standard Brownian motion B ≡ B , ∀ t ≥ B Ht = (cid:90) t K H ( t , u ) d B u ,which results from: R H ( s , t ) = (cid:90) s ∧ t K H ( s , u ) K H ( t , u ) d u . (2.2) Malliavin calculus for fractional Brownian motion.
Similarly to the usual Malliavinderivative D associated to the Brownian motion B , given the fBm B H , the Malliavin deriva-tive D H is defined as on operator on the smooth random variables with values in (cid:72) H . Thedomain of D H in L p (Ω) ( p >
1) is denoted by (cid:68) p and is the closure of the space of smoothrandom variables with respect to the norm: (cid:107) F (cid:107) p p = (cid:69) ( | F | p ) + (cid:69) (cid:128) (cid:107) D H F (cid:107) p (cid:72) H (cid:138) .Equivalently, D H can be defined as D H = ( K ∗ H ) − D (cf [
27, p.288 ] ). In particular, we seethat for any s , t ∈ [ T ] : D Hs B Ht = [ t ] ( s ) and D s B Ht = K ∗ H (cid:128) D H · B Ht (cid:138) ( s ) = K ∗ H ( [ t ] ( · ))( s ) = K H ( t , s ) . (2.3)In case p = (cid:68) is a Hilbert space and is identical to the usual (cid:68) space of Brownianmotion [
27, p.288 ] . The spaces (cid:68) k , p are defined by iterating Malliavin differentiation andthe corresponding norms are denoted by (cid:107) · (cid:107) k , p .Finally, (cid:68) ( |(cid:72) H | ) is defined as the space of |(cid:72) H | -valued random variables such that (cid:69) (cid:107) u (cid:107) |(cid:72) H | + (cid:69) (cid:107) D H u (cid:107) |(cid:72) H | < ∞ ,5here (cid:107) · (cid:107) |(cid:72) H | was defined in (2.1). See Nualart [
27, p.31 and p.288 ] for properties of thisspace. Similarly to (cid:68) k , p , one can define (cid:68) k , p ( |(cid:72) | ) .Given T >
0, the divergence operator δ ( T ) H is defined by the duality relation (cid:69) (cid:128) 〈 u , D H F 〉 (cid:72) H (cid:138) = (cid:69) (cid:128) F δ ( T ) H ( u ) (cid:138) , which holds for any F in (cid:68) and u in the domain of δ ( T ) H , denoted bydom δ ( T ) H ⊂ (cid:72) H ( T ) . In case the context is clear, we will simply write δ H for δ ( T ) H . δ H is also related to the Skorokhod integral w.r.t. the Brownian motion B : for any u suchthat K ∗ H u ∈ dom δ , δ H ( u ) = δ ( K ∗ H u ) .It can be shown that dom δ H = ( K ∗ H ) − ( dom δ ) . A sufficient condition for a process being indom δ H is that it belongs to (cid:68) ( (cid:72) H ) ⊂ dom δ H .In the sequel, the Skorokhod integral δ H ( [ t ] u ) is denoted by (cid:82) t u s d B Hs . Stochastic integrals and Itô’s formula for fractional Brownian motion.
Except when H = , the fBm is not a semimartingale, thus stochastic integrals w.r.t. B H cannot be definedin the Itô sense. However, it is well known from Young [ ] that Stieltjes integrals (cid:82) T f s ◦ d g s can be defined if f is β -Hölder continuous and g is γ -Hölder continuous with β + γ > { u t } t ∈ [ T ] is in C β a.s. with β > , and the Hurst parameter is H > , then the stochastic integral (cid:82) T u s ◦ d B Hs can be defined pathwise (see also [ ] and [ ] ). In addition, if { u t } t ∈ [ T ] belongs to (cid:68) ( |(cid:72) H | ) and satisfies (cid:90) T (cid:90) T | D Hs u t | | s − t | H − d s d t < ∞ , (2.4)and if the map t (cid:55)→ u t is continuous in (cid:68) , then (cid:82) T u s ◦ d B Hs coincides with the Stratonovichintegral w.r.t. B H ( [
27, Chap.5,Rk.2 ] ) and the following equality holds true: (cid:90) T u s ◦ d B Hs = δ H ( u ) + α H (cid:90) T (cid:90) T D Hs u t | s − t | H − d s d t (2.5)(see Alòs and Nualart [ ] and Nualart [
27, Prop.5.2.3 ] ).In the case of Stratonovich integrals and functions F whose first derivative F (cid:48) has aHölder regularity larger than 1 − H , the Itô formula is the classical chain rule: F (cid:130)(cid:90) t u s ◦ d B Hs (cid:140) = F ( ) + (cid:90) t F (cid:48) (cid:130)(cid:90) s u σ ◦ d B H σ (cid:140) u s ◦ d B Hs . (2.6)Observe that if F (cid:48) (cid:16)(cid:82) s u σ ◦ d B H σ (cid:17) u s satisfies condition (2.4) (this is the case in particular if F is a C function and u ∈ (cid:68) ( |(cid:72) H | ) ), Formula (2.5) allows to express the right-hand sideof (2.6) in terms of Skorokhod stochastic integrals. This will be useful in Section 3 whenwe will need to evaluate expectations of terms which are of the type of the left-hand side:the Skorokhod integral has mean zero whereas (cid:69) ( (cid:82) t u s ◦ d B Hs ) cannot be computed directly. Solutions to the stochastic differential equation (1.1;H).
We consider pathwise solu-tions to (1.1;H) as defined in Nualart and Rascanu [ ] , based on the generalized Stieltjes6ntegrals defined in Zähle [ ] . We mentioned in the previous paragraph that these inte-grals (generalized Stieltjes, Young, Stratonovich) may coincide if enough regularity on theintegrand is assumed, as this will be the case in this paper.As in [ ] , the solutions to (1.1;H) that appear here are strong solutions, in the sense that X H is an adapted process with Hölder continuous sample paths of order H − ε , ∀ ε > [ ] . In this section, we denote by τ H the first passage time at threshold 1 of the fBm ( B Ht ) t ∈ (cid:82) + starting from x <
1. For H > , Decreusefond and Nualart [ ] get the following boundfor the Laplace transform of τ H : ∀ λ ≥ (cid:69) (cid:128) e − λτ HH (cid:138) ≤ (cid:69) (cid:128) e − λτ / (cid:138) = e − ( − x ) (cid:112) λ .The inverse inequality holds for H < : see Lei and Nualart [ ] .Consider now x = λ → [ ] obtained (cid:80) (cid:128) τ HH > t (cid:138) ∼ t − ( − H ) H as t → ∞ .Hence, it results from a classical Tauberian theorem [
10, p.334 ] that, for any H ∈ ( , 1 ) and some constant C , 1 − (cid:69) (cid:128) exp ( − λτ HH ) (cid:138) ∼ C λ ( − H ) H when λ → H > and λ → ∞ .For H > , one actually has (see [
1, p.90 ] ): (cid:80) ( τ HH ≤ ε ) = (cid:80) sup s ∈ [ ε H ] B Hs ≥ = (cid:80) (cid:130) sup s ∈ [ ] B Hs ≥ ε − (cid:140) ∼ ε → Ψ( ε − ) ,where Ψ is the Gaussian tail distribution function. As Ψ( ε − ) ∼ ε − /
2, de Bruijn’s Taube-rian theorem ( [
10, Thm.4.12.9 ] ) yields: − log (cid:69) (cid:128) exp ( − λτ HH ) (cid:138) ∼ (cid:112) λ as λ → ∞ .Similarly one can get asymptotics on the Laplace transform of τ H (and not of τ HH asabove): (cid:69) (cid:0) exp ( − λτ H ) (cid:1) ∼ − C λ − H as λ → − log (cid:69) (cid:0) exp ( − λτ H ) (cid:1) ∼ (cid:18) + H (cid:19) H / ( H + ) λ H H + as λ → ∞ .7 Sensitivity of time marginal distributions of fBm-driven dif-fusion processes w.r.t. their Hurst parameter
Consider { X Ht } t ∈ [ T ] and { X t } t ∈ [ T ] as in (1.1;H) and (1.1; ). We aim to estimate theconvergence rate of the law of X Ht to the law of X t for every t . We assume the followingconditions:(H1) There exist some γ ∈ (
0, 1 ) such that b , σ ∈ C + γ ( (cid:82) ) ;(H2) The function σ satisfies a strong ellipticity condition: ∃ σ > | σ ( x ) | ≥ σ , ∀ x ∈ (cid:82) .Under (H1) a unique strong solution to (1.1;H) exists for each H ∈ [ , 1 ) (even weakerregularity conditions are sufficient: see [ ] ).The aim of this section is to prove the following theorem. Theorem 3.1.
Let X H and X be the solutions to (1.1;H) and (1.1; ) respectively. Supposethat b and σ satisfy the hypotheses (H1) and (H2), and ϕ is Hölder continuous of order + β for some β > . Then, for any T > , there exists C T > such that ∀ H ∈ [ , 1 ) , sup t ∈ [ T ] (cid:12)(cid:12) (cid:69) ϕ ( X Ht ) − (cid:69) ϕ ( X t ) (cid:12)(cid:12) ≤ C T ( H − ) .We will discuss in Section 6 the hypothesis (H2) as well as possible extensions of Theo-rem 3.1. σ ( x ) ≡ by the Lamperti transform Consider the Lamperti transform F ( x ) : = (cid:82) x σ ( z ) d z . The process Y Ht : = F ( X Ht ) satisfies thefollowing SDE: Y Ht = F ( x ) + B Ht + (cid:90) t ˜ b ( Y Hs ) d s . (3.1)with ˜ b ( Y Hs ) = b ( F − ( Y Hs )) σ ( F − ( Y Hs )) . Hence for Theorem 3.1 to hold true, it suffices for it to be true for Y H , Y and ϕ ◦ F − instead of ϕ . Notice that σ ∈ C + γ implies that ϕ ◦ F − ∈ C + γ ∧ β and ˜ b is in C + γ .Hence without loss of generality we hereafter assume that X H is defined as a solutionto (3.1), or equivalently as a solution to (1.1;H) with σ ≡ We start with proving Hölder norm estimates for X H and its Malliavin derivatives. Lemma 3.2.
Let b satisfy hypothesis (H1). For any α <
H it a.s. holds that (cid:107) X H (cid:107) ∞ , [ T ] ≤ C T ( + | x | + (cid:107) B H (cid:107) ∞ , [ T ] ) , (cid:107) X H (cid:107) α ≤ (cid:107) B H (cid:107) α + C T ( + | x | + (cid:107) B H (cid:107) ∞ ) , (cid:107) D H · X H · (cid:107) ∞ , [ T ] ≤ C T ,sup r ≤ t | D Hr X Ht − | t − r ≤ C T , ∀ t ∈ [ T ] .8 roof. The process X H satisfies (cid:107) X H (cid:107) ∞ , [ T ] ≤ C | x | + (cid:107) B H (cid:107) ∞ , [ T ] + C T , | X Hs − X Ht | ≤ | B Hs − B Ht | + C | t − s | ( + (cid:107) X H (cid:107) ∞ , [ T ] ) .To obtain estimates for the Malliavin derivatives of X H , we use the following represen-tation (see Nualart and Saussereau [ ] ): (cid:40) ∀ r > t , D Hr X Ht = ∀ r ≤ t , D Hr X Ht = + (cid:82) tr D Hr X Hs b (cid:48) ( X Hs ) d s . (3.2)As b (cid:48) is bounded, Gronwall’s lemma leads to: | D Hr X Ht | ≤ { t ≥ r } exp (cid:0) (cid:107) b (cid:48) (cid:107) ∞ ( t − r ) (cid:1) .The last desired inequality also follows from (3.2).Now consider the following parabolic PDE with initial condition ϕ at time t ∈ ( T ] : (cid:40) ∂∂ s u ( s , x ) + b ( x ) ∂∂ x u ( s , x ) + ∂ ∂ x u ( s , x ) = ( s , x ) ∈ [ t ) × (cid:82) , u ( t , x ) = ϕ ( x ) , x ∈ (cid:82) . (3.3) Lemma 3.3.
Let ϕ ∈ C + β ( (cid:82) ) for some < β < . Suppose that b satisfies the hypoth-esis (H1). Let u be the solution to (3.3). Then for any x ∈ (cid:82) , ∂ s u ( · , x ) and ∂ x u ( · , x ) arebounded. For each H > one has (cid:90) T (cid:90) T | r − s | H − (cid:12)(cid:12) D Hr ( ∂ x u ( s , X Hs )) (cid:12)(cid:12) drds < ∞ a . s . In addition, for H = one has (cid:82) T (cid:12)(cid:12) D s ( ∂ x u ( s , X s )) (cid:12)(cid:12) ds < ∞ a . s . Proof.
In view of Lunardi [
23, p.189 ] , there exists C > (cid:107) u (cid:107) C + β ([ T ] × (cid:82) ) ≤ C (cid:107) ϕ (cid:107) C + β ( (cid:82) ) .As D Hr (cid:128) ∂ x u ( s , X Hs ) (cid:138) = D Hr X Hs ∂ x x u ( s , X Hs ) , the result follows from Lemma 3.2.We now are in a position to prove Theorem 3.1. We start with representing (cid:69) ϕ ( X Ht ) − (cid:69) ϕ ( X t ) in integral form by using the solution u of thePDE (3.3). For any s ∈ [ t ] and x ∈ (cid:82) we have u ( s , x ) = (cid:69) x (cid:0) ϕ ( X t − s ) (cid:1) .In particular, u is a C ([ T ] × (cid:82) ) function. Using the integration-by-parts formula forStieltjes integrals we thus get u ( t , X Ht ) = u ( x ) + (cid:90) t (cid:128) ∂ s u ( s , X Hs ) + ∂ x u ( s , X Hs ) b ( X Hs ) (cid:138) d s + (cid:90) t ∂ x u ( s , X Hs ) ◦ d B Hs .9n view of Lemma 3.3 we can use Equality (2.5). Using also Equality (3.2) for r > s , weget: u ( t , X Ht ) = u ( x ) + (cid:90) t (cid:128) ∂ s u ( s , X Hs ) + ∂ x u ( s , X Hs ) b ( X Hs ) (cid:138) d s + δ H (cid:128) [ t ] ∂ x u ( · , X H · ) (cid:138) + α H (cid:90) t (cid:90) s | r − s | H − D Hr (cid:128) ∂ x u ( s , X Hs ) (cid:138) d r d s = u ( x ) + (cid:90) t (cid:128) ∂ s u ( s , X Hs ) + ∂ x u ( s , X Hs ) b ( X Hs ) (cid:138) d s + δ H (cid:128) [ t ] ∂ x u ( · , X H · ) (cid:138) + α H (cid:90) t (cid:90) s | r − s | H − D Hr X Hs ∂ x x u ( s , X Hs ) d r d s .Using the definition of u and the fact that the Skorokhod integral has zero mean we get (cid:69) ϕ ( X Ht ) − (cid:69) x ϕ ( X t ) = (cid:69) u ( t , X Ht ) − u ( x )= − (cid:69) (cid:90) t ∂ x x u ( s , X Hs ) d s + α H (cid:69) (cid:90) t (cid:90) s | r − s | H − ∂ x x u ( s , X Hs ) d r d s + α H (cid:69) (cid:90) t (cid:90) s | r − s | H − ( D Hr X Hs − ) ∂ x x u ( s , X Hs ) d r d s = (cid:69) (cid:90) t ∂ x x u ( s , X Hs ) (cid:128) Hs H − − (cid:138) d s + α H (cid:69) (cid:90) t (cid:90) s | r − s | H − ( D Hr X Hs − ) ∂ x x u ( s , X Hs ) d r d s = : ∆ H + ∆ H .We bound | ∆ H | as follows: | ∆ H | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) t ∂ x x u ( s , X Hs ) (cid:128) Hs H − − (cid:138) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( H − ) (cid:107) ∂ x x u (cid:107) ∞ (cid:90) t ( ∨ s H − ) ( + H | log s | ) d s ≤ C ( H − ) .To bound | ∆ H | we use the last inequality of Lemma 3.2. It comes: | ∆ H | ≤ C α H (cid:90) t (cid:90) s ( s − r ) H − ( s − r ) ∂ x x u ( s , X Hs ) d r d s ≤ C (cid:107) ∂ x x u (cid:107) ∞ α H (cid:90) t (cid:90) s ( s − r ) H − d r d s ≤ C ( H − ) . 10 Sensitivity of first passage time Laplace transform of fBmdriven diffusion process w.r.t. their noise Hurst parameter
The aim of this section is to prove the following theorem which estimates the sensitivity offirst passage time Laplace transform of ( X Ht ) t ∈ (cid:82) + solution to the SDE (1.1;H) and fractionalBrownian motions w.r.t. their Hurst parameter.We slightly reinforce the assumption on the drift and diffusion coefficients and suppose:(H1’) b , σ ∈ C ( (cid:82) ) . Theorem 4.1.
Assume that b and σ satisfy (H1’) and (H2) and let x < . There existconstants λ ≥ , µ ≥ (both depending only on b and σ ) and < η < − x such that: forall ε ∈ ( ) and < η ≤ η , there exist constants α > and C ε , η > such that ∀ λ ≥ λ , ∀ H ∈ [ , 1 ) , (cid:12)(cid:12)(cid:12)(cid:12) (cid:69) (cid:128) e − λτ XH (cid:138) − (cid:69) (cid:18) e − λτ X (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ε , η ( H − ) − ε e − α S ( − x − η )( (cid:112) λ + µ − µ ) , where S : (cid:82) + → (cid:82) + is the function defined by S ( x ) = x ∧ x H . In the pure fBm case, µ = and the result holds true for λ = . In view of Proposition B.1 we have (cid:69) (cid:18) e − λτ X (cid:19) ≤ e − C ( − x ) (cid:16) (cid:112) λ + µ − µ (cid:17) .Part of the difficulties in the proof of Theorem 4.1 comes from the fact that we successfullyobtained a sensitivity estimate which has the same exponential decay at infinity w.r.t λ asthe preceding upper bound for (cid:69) (cid:18) e − λτ X (cid:19) and tends to 0 as ( H − ) − ε when H tends to . Remark 4.2.
The above constant C ε , η tends to infinity either when ε → or η → . Remark 4.3.
When X H is a fractional Brownian motion, we believe that the previous resultcan be as follows: Let ε ∈ ( ) and η > such that − x > η , where x < is the initialposition of X and X H . Then there exist constants α > and C ε , η > such that: for any λ ≥ and any H ∈ [ , 1 ) , (cid:12)(cid:12)(cid:12) (cid:69) (cid:128) e − λτ H (cid:138) − (cid:69) (cid:16) e − λτ (cid:17)(cid:12)(cid:12)(cid:12) ≤ C ε , η ( H − ) − ε e − α S ( − x − η ) T ( λ ) , where T ( λ ) = ( λ ) − H if λ ≤ , and T ( λ ) = (cid:112) λ otherwise. We start with proving Theorem 4.1 in the pure fBm case ( b ( x ) ≡ σ ≡ .1 An error decomposition in the pure fBm case Our Laplace transforms sensititivity analysis is based on a PDE representation of first hittingtime Laplace transforms in the case H = . When b ( x ) ≡ σ ≡ (cid:69) x (cid:16) e − λτ (cid:17) is explicitly known. However, we adopt iteven in this simple case in order to introduce the technique we need in Section 4.7 to treatthe general drift and diffusion coefficient case.For λ > ∀ x ∈ ( −∞ , 1 ] , = (cid:69) x (cid:16) e − λτ (cid:17) = u λ ( x ) ,where the function u λ is the classical solution with bounded continuous first and secondderivatives to u (cid:48)(cid:48) λ ( x ) = λ u λ ( x ) , x < u λ ( ) = x →−∞ u λ ( x ) =
0. (4.1)
Proposition 4.4.
Let x < be fixed. For any λ ≥ , (cid:69) (cid:128) e − λτ H (cid:138) − (cid:69) (cid:16) e − λτ (cid:17) = (cid:69) (cid:150) λ (cid:90) τ H ( Hs H − − ) u λ ( B Hs ) e − λ s ds (cid:153) (4.2) + lim T →∞ (cid:69) (cid:20) δ ( T ) H (cid:128) [ t ] u λ ( B H · ) e − λ · (cid:138)(cid:12)(cid:12)(cid:12) t = τ H ∧ T (cid:21) = : I ( λ ) + I ( λ ) . (4.3) Proof.
Let T >
0. It can be verified easily that ∀ t ∈ [ T ] , [ t ] u (cid:48) λ ( B H · ) e − λ · ∈ dom δ ( T ) H .Hence, one can apply Itô’s formula ( [ ] , [
27, p.294 ] ) and, as u λ satisfies (4.1), for any t ≤ T ∧ τ H we get u λ ( B Ht ) e − λ t = u λ ( x ) + δ ( T ) H (cid:128) [ t ] u (cid:48) λ ( B H · ) e − λ · (cid:138) − λ (cid:90) t u λ ( B Hs ) e − λ s d s + H (cid:90) t u (cid:48)(cid:48) λ ( B Hs ) s H − e − λ s d s = u λ ( x ) + δ ( T ) H (cid:128) [ t ] u (cid:48) λ ( B H · ) e − λ · (cid:138) + λ (cid:90) t (cid:128) Hs H − − (cid:138) u λ ( B Hs ) e − λ s d s ,Evaluating the previous equation at T ∧ τ H and then taking expectations leads to (cid:69) (cid:16) u λ ( B HT ∧ τ H ) e − λ ( T ∧ τ H ) (cid:17) − u λ ( x ) = λ (cid:69) (cid:90) T ∧ τ H ( Hs H − − ) u λ ( B Hs ) e − λ s d s + (cid:112) λ (cid:69) (cid:20) δ ( T ) H (cid:128) [ t ] u λ ( B H · ) e − λ · (cid:138)(cid:12)(cid:12)(cid:12) t = T ∧ τ H (cid:21) .By the Dominated Convergence Theorem, the first three terms converge as T → ∞ . Thusthe expectation of the Skorokhod integral also converges.12 .2 Estimate on I ( λ ) (pure fBm case) In view of Fubini’s theorem, we have I ( λ ) = λ (cid:90) ∞ ( Hs H − − ) e − λ s (cid:69) (cid:128) { τ H ≥ s } u λ ( B Hs ) (cid:138) d s .In Subsection 4.3 we will need to work with a smooth extension of u λ on ( + ∞ ) . Inthe present pure fBm setting it suffices to observe that the explicit solution u λ ( x ) = e − ( − x ) (cid:112) λ to (4.1) is well defined and smooth on the whole space (cid:82) . This extension is still denotedby u λ . We start with proving the following elementary lemma which will be used in thissubsection with η = η > Lemma 4.5.
For all η ≥ , there exists C η > such that for any p > s ≥ and λ ≥ , wehave (cid:69) (cid:128) { B Hs ≤ + η } u λ ( B Hs ) p (cid:138) ≤ (cid:168) C η s H e p η (cid:112) λ if s H p (cid:112) λ > − x + η , e − p ( − x − η ) (cid:112) λ if s H p (cid:112) λ ≤ − x + η . In case η = and if s H p (cid:112) λ > − x , we have (cid:69) (cid:128) { B Hs ≤ } u λ ( B Hs ) p (cid:138) ≤ .Proof. Let Φ denote the cumulative distribution function of the standard Gaussian law. Wehave (cid:69) (cid:128) { B Hs ≤ + η } u λ ( B Hs ) p (cid:138) = e − ( − x ) (cid:112) λ p + s H λ p (cid:90) r − H ( + η − x ) −∞ e − ( x − s H (cid:112) λ p ) (cid:112) π d x = e − ( − x ) (cid:112) λ p + s H λ p Φ( s − H ( + η − x ) − s H (cid:112) λ p ) .When s H p (cid:112) λ > − x + η , we then use the inequality: ∀ x > Φ( − x ) ≤ (cid:112) π x e − x ,to get (cid:69) (cid:128) { B Hs ≤ + η } u λ ( B Hs ) p (cid:138) ≤ e p η (cid:112) λ e − ( + η − x ) s H (cid:112) π (cid:128) ps H (cid:112) λ − s − H ( + η − x ) (cid:138) ≤ s H η (cid:112) π e p η (cid:112) λ .When s H p (cid:112) λ ≤ − x + η , we merely bound Φ( x ) by 1. Proposition 4.6.
Let T be the function of λ ∈ (cid:82) + defined by T ( λ ) = ( λ ) − H if λ ≤ andT ( λ ) = (cid:112) λ if λ > . There exists a constant C > such that | I ( λ ) | ≤ C ( H − ) e − S ( − x ) T ( λ ) , where S is the function defined in Theorem 4.1. roof. First, we notice that for H ∈ ( , 1 ) and s ∈ ( ∞ ) , | Hs H − − | ≤ ( H − ) ( ∨ s H − ) | + H log s | .Therefore λ (cid:90) ∞ | Hs H − − | e − λ s d s ≤ C ( H − ) (cid:90) ∞ ( ∨ s H − ) | + H log s | λ e − λ s d s ≤ C ( H − ) . (4.4)We now split I into two parts. We first consider the case s H (cid:112) λ ≤ − x . Lemma 4.5(with η = p =
1) and Inequality (4.4) imply that λ (cid:90) (cid:16) − x (cid:112) λ (cid:17) H | Hs H − − | e − λ s (cid:69) (cid:128) { τ H ≥ s } u λ ( B Hs ) (cid:138) d s ≤ e − ( − x ) (cid:112) λ (cid:90) (cid:16) − x (cid:112) λ (cid:17) H | Hs H − − | e − λ s d s ≤ C ( H − ) e − ( − x ) (cid:112) λ . (4.5)Second, consider the case s H (cid:112) λ > − x . Since (cid:69) (cid:128) { τ H ≥ s } u λ ( B Hs ) (cid:138) ≤ ( ∨ s H − ) | + H log s | e − λ s ≤ C , λ (cid:90) ∞ (cid:16) − x (cid:112) λ (cid:17) H | Hs H − − | e − λ s (cid:69) (cid:128) { τ H ≥ s } u λ ( B Hs ) (cid:138) d s ≤ C ( H − ) (cid:90) ∞ (cid:16) − x (cid:112) λ (cid:17) H λ e − λ s d s ≤ C ( H − ) e − λ (cid:16) − x (cid:112) λ (cid:17) H . (4.6)Thus the two estimates (4.5) and (4.6) lead to: | I ( λ ) | ≤ C ( H − ) max (cid:129) e − ( − x ) (cid:112) λ , e − ( − x ) H ( λ ) − H (cid:139) ,which is the desired result. I ( λ ) (pure fBm case, λ ≥ ) To complete the proof of Theorem 4.1 in the fBm case, we now aim to prove: | I ( λ ) | ≤ C ( H − ) − ε e − α S ( − x − η ) (cid:112) λ . (4.7)First observe that the optional stopping theorem does not hold for Skorokhod integralsof the fBm. However, Proposition 13 of Peccati et al. [ ] shows that: ∀ T > (cid:69) (cid:16) δ ( T ) ( [ t ] ( • ) u λ ( B H • ) e − λ • ) (cid:12)(cid:12) t = T ∧ τ H (cid:17) = I ( λ ) satisfies | I ( λ ) | = (cid:112) λ (cid:12)(cid:12)(cid:12)(cid:12) lim N →∞ (cid:69) (cid:20) δ ( N ) H (cid:128) [ t ] ( • ) u λ ( B H • ) e − λ • (cid:138)(cid:12)(cid:12)(cid:12) t = τ H ∧ N − δ ( N ) (cid:128) [ t ] ( • ) u λ ( B H • ) e − λ • (cid:138)(cid:12)(cid:12)(cid:12) t = τ H ∧ N (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:112) λ (cid:12)(cid:12)(cid:12)(cid:12) lim N →∞ (cid:69) (cid:20) δ ( N ) (cid:128) { K ∗ H − Id } ( [ t ] ( • ) u λ ( B H • ) e − λ • ) (cid:138)(cid:12)(cid:12)(cid:12) t = τ H ∧ N (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ (cid:112) λ lim N →∞ (cid:69) sup t ∈ [ τ H ∧ N ] | δ ( N ) (cid:128) { K ∗ H − Id } ( [ t ] ( • ) u λ ( B H • ) e − λ • ) (cid:138) |≤ (cid:112) λ lim N →∞ (cid:69) sup t ∈ [ N ] (cid:148) { τ H ≥ t } | δ ( N ) (cid:128) { K ∗ H − Id } ( [ t ] ( • ) u λ ( B H • ) e − λ • ) (cid:138) | (cid:151) .14efine the field { U t ( v ) , t ∈ [ N ] , v ≥ } and the process { Υ t , t ∈ [ N ] } by ∀ t ∈ [ N ] , U t ( v ) = { K ∗ H − Id } (cid:128) [ t ] ( • ) u λ ( B H • ) e − λ • (cid:138) ( v ) ,and Υ t = δ ( N ) ( U t ( • )) .For any real-valued function f with f ( ) = { τ H ≥ t } | f ( t ) | ≤ { τ H ≥ t } [ t ] (cid:88) n = sup s ∈ [ n , n + ] { τ H ≥ s } | f ( s ) − f ( n ) | ≤ [ t ] (cid:88) n = sup s ∈ [ n , n + ] { τ H ≥ s } | f ( s ) − f ( n ) | .Therefore | I ( λ ) | ≤ (cid:112) λ lim N →∞ (cid:69) sup t ∈ [ N ] (cid:148) { τ H ≥ t } | Υ t | (cid:151) ≤ (cid:112) λ lim N →∞ N − (cid:88) n = (cid:69) sup t ∈ [ n , n + ] (cid:148) { τ H ≥ t } | Υ t − Υ n | (cid:151) .(4.8)Suppose for a while that we have proven: there exists η ∈ ( − x ) such that for all η ∈ ( η ] and all ε ∈ ( ) , there exist constants C , α > (cid:69) sup t ∈ [ n , n + ] (cid:148) { τ H ≥ t } | Υ t − Υ n | (cid:151) ≤ C ( H − ) − ε e − ( + ε ) λ n e − α S ( − x − η ) (cid:112) λ . (4.9)We would then get: | I ( λ ) | ≤ C (cid:112) λ ∞ (cid:88) n = e − λ n ( + ε ) ( H − ) − ε e − α S ( − x − η ) (cid:112) λ ≤ C ( H − ) − ε e − α S ( − x − η ) (cid:112) λ ,which is the desired result (4.7). It thus remains to prove (4.9). In order to estimate the left-hand side of Inequality (4.9) we aim to apply Garsia-Rodemich-Rumsey’s lemma (see Lemma A.1 in Appendix). However, it seems hard to get the desiredestimate by estimating moments of increments of { τ H ≥ t } | Υ t − Υ n | , in particular because { τ H ≥ t } is not smooth in the Malliavin sense. We thus proceed by localization and constructa process ¯ Υ t which is smooth on the event { τ H ≥ t } and is close to 0 on the complementaryevent. To this end we introduce the following new notations.For some small η > ∀ t ∈ [ N ] , ¯ U t ( v ) = { K ∗ H − Id } (cid:128) [ t ] ( • ) u λ ( B H • ) φ η ( B H • ) e − λ • (cid:138) ( v ) and ¯ Υ t = δ ( N ) (cid:0) ¯ U t (cid:1) ,where φ η is a smooth function taking values in [
0, 1 ] such that φ η ( x ) = ∀ x ≤
1, and φ η ( x ) = ∀ x > + η . 15he crucial property of ¯ Υ t is the following: For all n ∈ (cid:78) and n ≤ r ≤ t < n + { τ H ≥ t } Υ r = { τ H ≥ t } ¯ Υ r a.s. This is a consequence of the local property of δ ( [
27, p.47 ] ).Indeed, it suffices to notice that U t and ¯ U t belong to (cid:68) ( (cid:72) ) and that { τ H ≥ t } (cid:0) U r ( v ) − ¯ U r ( v ) (cid:1) = { τ H ≥ t } { K ∗ H − Id } (cid:128) [ r ] ( • ) u λ ( B H • )( − φ η ( B H • )) e − λ • (cid:138) ( v )= φ η ( B Hs ) = s ∈ [ t ] on the event { t ≤ τ H } .Therefore, for any n ≤ N − (cid:69) (cid:130) sup t ∈ [ n , n + ] { τ H ≥ t } | Υ t − Υ n | (cid:140) = (cid:69) (cid:130) sup t ∈ [ n , n + ] { τ H ≥ t } | ¯ Υ t − ¯ Υ n | (cid:140) ≤ (cid:69) (cid:130) sup t ∈ [ n , n + ] | ¯ Υ t − ¯ Υ n | (cid:140) .(4.10)In order to apply Garsia-Rodemich-Rumsey’s lemma we now need to estimate moments of¯ Υ t − ¯ Υ s . As we will see, one can obtain bounds on the norm (cid:13)(cid:13) ¯ Υ t − ¯ Υ s (cid:13)(cid:13) L (Ω) in terms of ( H − ) . Therefore we notice that (cid:69) (cid:128) | ¯ Υ s − ¯ Υ t | + ε (cid:138) ≤ (cid:13)(cid:13) ¯ Υ t − ¯ Υ s (cid:13)(cid:13) L (Ω) × (cid:69) (cid:128) | ¯ Υ t − ¯ Υ s | + ε (cid:138) and then combine Lemmas 4.7 and 4.8 below to obtain: for every [ n ≤ s ≤ t ≤ n + ] , (cid:69) (cid:128) | ¯ Υ s − ¯ Υ t | + ε (cid:138) ≤ C ( H − ) ( t − s ) − ε e − α S ( − x − η ) (cid:112) λ × ( t − s ) + ε e − λ s e − α S ( − x − η ) (cid:112) λ ≤ C ( H − ) ( t − s ) + ε e − λ s e − α S ( − x − η ) (cid:112) λ .We now are in a position to apply the Garsia-Rodemich-Rumsey lemma with p = + ε and q = + ε/ + ε (recall our convention that α , like C , may change from line to line but does notdepend on H ): (cid:69) (cid:130) sup t ∈ [ n , n + ] { τ H ≥ t } | Υ t − Υ n | (cid:140) ≤ C ( H − ) ( + ε ) e − α + ε S ( − x − η ) (cid:112) λ (cid:32)(cid:90) n + n (cid:90) n + s e − λ s ( t − s ) ε − d t d s (cid:33) + ε ≤ C ( H − ) + ε e − α S ( − x − η ) (cid:112) λ e − ( + ε ) λ n ,from which Inequality (4.9) follows.It now remains to prove the above estimates on (cid:13)(cid:13) ¯ Υ t − ¯ Υ s (cid:13)(cid:13) L (Ω) and (cid:69) (cid:0) | ¯ Υ t − ¯ Υ s | + ε (cid:1) :the estimates are provided by Lemmas 4.7 and 4.8 respectively stated and proven in Sub-sections 4.5 and 4.6 below. Lemma 4.7.
There exists η ∈ ( − x ) such that: for all < η ≤ η , for all H ∈ [ , 1 ) andfor all < ε < , there exist C , α > such that ∀ λ ≥ ∀ ≤ n ≤ s ≤ t ≤ n + ≤ N , (cid:69) (cid:128) | ¯ Υ t − ¯ Υ s | + ε (cid:138) ≤ C ( t − s ) + ε e − λ s e − α S ( − x − η ) (cid:112) λ , where S is the function defined in Theorem 4.1. roof. Setting ¯ u λ : = u λ φ η we have ¯ Υ t − ¯ Υ s = δ ( N ) (cid:128) { K ∗ H − Id } (cid:128) ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • (cid:138)(cid:138) = δ ( N ) (cid:128) K ∗ H ( ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • ) (cid:138) − δ ( N ) (cid:128) ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • (cid:138) .We now use the following result coming from the proof of Theorem 5 of [ ] : for a stochasticprocess u ∈ (cid:76) pH (see the definition of this space in [
27, p.42 ] ), if ˜ pH > ∀ ≤ s ≤ t ≤ N , (cid:69) (cid:20)(cid:12)(cid:12)(cid:12) δ ( N ) (cid:128) K ∗ H ( ( s , t ] u ) (cid:138)(cid:12)(cid:12)(cid:12) ˜ p (cid:21) ≤ C ( t − s ) ˜ pH − (cid:90) ts A H ( r ) d r (4.11)where A H ( r ) : = | (cid:69) ( u r ) | ˜ p + (cid:69) (cid:149)(cid:16)(cid:82) N | D H θ u r | / H d θ (cid:17) ˜ pH (cid:152) . Since ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • ∈ (cid:76) p ∩ (cid:76) pH , for any 0 ≤ s ≤ t ≤ N , Inequality (4.11) with ˜ p = + ε yields (cid:13)(cid:13) ¯ Υ s − ¯ Υ t (cid:13)(cid:13) L ˜ p (Ω) ≤ (cid:13)(cid:13)(cid:13) δ ( N ) (cid:128) K ∗ H ( ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • ) (cid:138)(cid:13)(cid:13)(cid:13) L ˜ p (Ω) + (cid:13)(cid:13)(cid:13) δ ( N ) (cid:128) ( s , t ] ( • ) ¯ u λ ( B H • ) e − λ • (cid:138)(cid:13)(cid:13)(cid:13) L ˜ p (Ω) ≤ (cid:168) C ( t − s ) ( + ε ) H − (cid:90) ts A H ( r ) d r (cid:171) ˜ p + (cid:168) C ( t − s ) ε (cid:90) ts A ( r ) d r (cid:171) p . (4.12)We now proceed in two steps: in the first step , we prove that there exist α > C > ∀ λ ≥ ∀ r ∈ (cid:82) + , A H ( r ) ≤ C e − α ˜ pS ( − x − η ) (cid:112) λ e − λ ˜ pr . (4.13)In the second step , we conclude the proof of Lemma 4.7 by using Inequality (4.13). First step. As u (cid:48) λ ( x ) = (cid:112) λ u λ ( x ) we have (cid:69) (cid:150)(cid:18) (cid:90) N | D H θ ( ¯ u λ ( B Hr ) e − λ r ) | / H d θ (cid:19) ˜ pH (cid:153) = e − λ ˜ pr (cid:69) (cid:32)(cid:90) N { θ ≤ r } (cid:129) ( (cid:112) λφ η ( B Hr ) + φ (cid:48) η ( B Hr )) u λ ( B Hr ) (cid:139) / H d θ (cid:33) ˜ pH ≤ e − ˜ p λ r r ˜ pH ( (cid:112) λ (cid:107) φ η (cid:107) ∞ + (cid:107) φ (cid:48) η (cid:107) ∞ ) ˜ p (cid:69) (cid:148) { B Hr < + η } u λ ( B Hr ) ˜ p (cid:151) .The term e − ˜ p λ r r ˜ pH ( (cid:112) λ (cid:107) φ η (cid:107) ∞ + (cid:107) φ (cid:48) η (cid:107) ∞ ) ˜ p can be bounded uniformly in r and λ ≥ η .When r H ˜ p (cid:112) λ ≤ − x + η , Lemma 4.5 ensures that (cid:69) (cid:148) { B Hr < + η } u λ ( B Hr ) ˜ p (cid:151) ≤ e − ˜ p ( − x − η ) (cid:112) λ .When r H ˜ p (cid:112) λ > − x + η , we now deduce from Lemma 4.5 that e − ˜ p λ r (cid:69) (cid:20) { B Hr < + η } u λ ( B Hr ) ˜ p (cid:21) ≤ C η e − ˜ p λ r + η ˜ p (cid:112) λ ≤ C η exp − ˜ p (cid:112) λ (cid:130) − x + η + H ˜ p (cid:140) H λ − H − η (4.14)17rom which e − ˜ p λ r (cid:69) (cid:148) { B Hr < + η } u λ ( B Hr ) ˜ p (cid:151) ≤ C η e − α ˜ p ( − x − η ) H (cid:112) λ follows for some α > η small enough. Thus, for all r > λ ≥ (cid:69) (cid:32)(cid:90) N | D H θ ( ¯ u λ ( B Hr ) e − λ r ) | / H d θ (cid:33) ˜ pH ≤ C η e − α ˜ pS ( − x − η ) (cid:112) λ e − ˜ p λ r .Similarly, we conclude that there exist C > α > (cid:128) (cid:69) (cid:148) ¯ u λ ( B Hr ) e − λ r (cid:151)(cid:138) ˜ p ≤ e − ˜ p λ r (cid:128) (cid:69) (cid:148) { B Hr < + η } u λ ( B Hr ) (cid:151)(cid:138) ˜ p ≤ C e − α ˜ pS ( − x − η ) (cid:112) λ e − λ ˜ pr .This ends the proof of Inequality (4.13). Second step.
From Inequality (4.13), we now have (cid:90) ts A H ( r ) d r ≤ C e − α ˜ pS ( − x − η ) (cid:112) λ (cid:90) ts e − λ ˜ pr d r ≤ C ( t − s ) e − λ ˜ ps e − α ˜ pS ( − x − η ) (cid:112) λ .Then, since t − s ∈ [
0, 1 ] and H > , ( t − s ) ( + ε ) H − (cid:82) ts A H ( r ) d r and ( t − s ) ε (cid:82) ts A ( r ) d r are both bounded by C ( t − s ) + ε e − λ ˜ ps e − α ˜ pS ( − x − η ) (cid:112) λ .Therefore, in view of Inequality (4.12), we obtain (cid:69) (cid:128) | ¯ Υ t − ¯ Υ s | + ε (cid:138) = (cid:107) ¯ Υ t − ¯ Υ s (cid:107) ˜ p L ˜ p (Ω) ≤ C ( t − s ) + ε e − ˜ p λ s e − α ˜ p S ( − x − η ) (cid:112) λ ,which ends the proof since ˜ p = + ε > . Lemma 4.8.
There exists η ∈ ( − x ) such that: for all < η ≤ η and < ε < , thereexist C , α > such that for any n ∈ [ N ] , we have ∀ H ∈ [ , 1 ) , ∀ n ≤ s ≤ t ≤ n + ∀ λ ≥ (cid:13)(cid:13) ¯ Υ t − ¯ Υ s (cid:13)(cid:13) L (Ω) ≤ C ( H − ) ( t − s ) − ε e − α S ( − x − η ) (cid:112) λ . Proof.
Recall that (cid:72) = L [ N ] . We apply Meyer’s inequality ( [
27, p.80 ] ) to (cid:107) ¯ Υ t − ¯ Υ s (cid:107) L (Ω) to obtain: (cid:13)(cid:13) ¯ Υ t − ¯ Υ s (cid:13)(cid:13) L (Ω) ≤ C (cid:16) (cid:69) (cid:107) ¯ U t − ¯ U s (cid:107) L [ N ] (cid:17) + C (cid:16) (cid:69) (cid:107) D · ( ¯ U t − ¯ U s )( · ) (cid:107) L [ N ] (cid:17) . (4.15)We only estimate the second term in the right-hand side, the first term being easier toestimate by using similar arguments.Let s ≤ t ∈ [ n , n + ] , then D r ( ¯ U t ( v ) − ¯ U s ( v )) = D r (cid:32)(cid:90) Nv ( s , t ] ( θ ) ∂ K H ∂ θ ( θ , v ) ¯ u λ ( B H θ ) e − λθ d θ − ( s , t ] ( v ) ¯ u λ ( B Hv ) e − λ v (cid:33) .In view of (2.3) we have D r B H θ = K H ( θ , r ) . Set D r ( ¯ u λ ( B H θ )) = K H ( θ , r ) u λ ( B H θ ) (cid:129)(cid:112) λφ η ( B H θ ) + φ (cid:48) η ( B H θ ) (cid:139) = : K H ( θ , r ) u λ ( B H θ ) Q ( B H θ ) Q in λ and η ). We have D r ( ¯ U t ( v ) − ¯ U s ( v )) = (cid:90) tv ∨ s ∂ K H ∂ θ ( θ , v ) K H ( θ , r ) u λ ( B H θ ) Q ( B H θ ) e − λθ d θ − ( s , t ] ( v ) K H ( v , r ) u λ ( B Hv ) Q ( B Hv ) e − λ v .The norm of this expression is splitted as follows: (cid:69) (cid:107) D · ( ¯ U t − ¯ U s )( · ) (cid:107) L [ N ] = (cid:90) N (cid:90) s (cid:69) (cid:104)(cid:0) D r ( ¯ U s ( v ) − ¯ U t ( v )) (cid:1) (cid:105) d v d r + (cid:90) N (cid:90) Ns (cid:69) (cid:104)(cid:0) D r ( ¯ U s ( v ) − ¯ U t ( v )) (cid:1) (cid:105) d v d r = : I s > v + I s ≤ v . First step: estimation of I s > v . I s > v = (cid:90) N (cid:90) s (cid:90) ts (cid:90) ts ∂ K H ∂ θ ( θ , v ) ∂ K H ∂ θ ( θ , v ) K H ( θ , r ) K H ( θ , r ) e − λ ( θ + θ ) × (cid:69) (cid:104) Q ( B H θ ) Q ( B H θ ) u λ ( B H θ ) u λ ( B H θ ) (cid:105) d θ d θ d v d r . (4.16)In view of Equality (2.2) and since K H ( θ , v ) = v ≥ θ , we have (cid:82) N K H ( θ , r ) K H ( θ , r ) d r = R H ( θ , θ ) for any θ , θ ∈ [ N ] . Thus, by Fubini’s theorem, I s > v = (cid:90) ts (cid:90) ts (cid:130)(cid:90) s ∂ K H ∂ θ ( θ , v ) ∂ K H ∂ θ ( θ , v ) d v (cid:140) R H ( θ , θ ) e − λ ( θ + θ ) × e − λ ( θ + θ ) (cid:69) (cid:104) Q ( B H θ ) Q ( B H θ ) u λ ( B H θ ) u λ ( B H θ ) (cid:105) d θ d θ .Notice that: ∃ C > ∀ θ , θ ∈ [ N ] , ∀ λ ≥ R H ( θ , θ ) e − λ ( θ + θ ) ≤ C . Further-more, Lemma A.3 implies that I s > v ≤ C e − α S ( − x − η ) (cid:112) λ (cid:90) ts (cid:90) ts (cid:130)(cid:90) s ∂ K H ∂ θ ( θ , v ) ∂ K H ∂ θ ( θ , v ) d v (cid:140) d θ d θ ,from which, in view of Lemma A.2, I s > v ≤ C ( H − ) e − α S ( − x − η ) (cid:112) λ (cid:90) ts (cid:90) ts ( θ ∨ θ − s ) H − ( θ ∧ θ − s ) H − d θ d θ ≤ C ( H − ) e − α S ( − x − η ) (cid:112) λ H + (cid:32) H − ( t − s ) H + ( t − s ) H + (cid:33) .We therefore conclude that, since 0 ≤ t − s ≤ I s > v ≤ C ( H − ) e − α S ( − x − η ) (cid:112) λ ( t − s ) H . (4.17) Second step: estimation of I s ≤ v . 19et G H ( θ , r ) = K H ( θ , r ) u λ ( B H θ ) Q ( B H θ ) e − λθ and recall that D r ¯ u λ ( B H θ ) e − λθ = G H ( θ , r ) .Thus, for s ≤ v , D r ( ¯ U t ( v ) − ¯ U s ( v )) = (cid:90) tv ∂ K H ∂ θ ( θ , v ) G H ( θ , r ) d θ − G H ( v , r )= (cid:90) tv ∂ K H ∂ θ ( θ , v ) ( G H ( θ , r ) − G H ( v , r )) d θ + G H ( v , r )( K H ( t , v ) − ) .Therefore, I s ≤ v ≤ (cid:69) (cid:90) N (cid:90) Ns (cid:130)(cid:90) tv ∂ K H ∂ θ ( θ , v ) ( G H ( θ , r ) − G H ( v , r )) d θ (cid:140) + (cid:0) G H ( v , r )( K H ( t , v ) − ) (cid:1) d v d r = : 2 I ( ) s ≤ v + I ( ) s ≤ v We have: I ( ) s ≤ v = (cid:90) N (cid:90) Ns (cid:90) tv (cid:90) tv (cid:69) (cid:89) i = ∂ K H ∂ θ ( θ i , v )( G H ( θ i , r ) − G H ( v , r )) d θ d θ d v d r ≤ (cid:90) N (cid:90) Ns (cid:90) tv (cid:90) tv (cid:89) i = ∂ K H ∂ θ ( θ i , v ) (cid:69) (cid:148) ( G H ( θ i , r ) − G H ( v , r )) (cid:151) d θ d θ d v d r . (4.18)Intuitively, each term in the above expectations should have the same regularity as the frac-tional Brownian motion. We thus integrate w.r.t “ r ” and use the Cauchy-Schwartz inequalityto get: (cid:89) i = (cid:32)(cid:90) N (cid:69) (cid:148) ( G H ( θ i , r ) − G H ( v , r )) (cid:151) d r (cid:33) .Now, (cid:69) (cid:148) ( G H ( θ , r ) − G H ( v , r )) (cid:151) ≤ (cid:69) (cid:148) ( K H ( θ , r ) − K H ( v , r )) ( e − λθ u λ ( B H θ ) Q ( B H θ )) (cid:151) (4.19) + (cid:69) (cid:104) K H ( v , r ) (cid:128) e − λθ u λ ( B H θ ) Q ( B H θ ) − e − λ v u λ ( B Hv ) Q ( B Hv ) (cid:138) (cid:105) .(4.20)For θ ≥ v , (cid:82) N (cid:0) K H ( θ , r ) − K H ( v , r ) (cid:1) d r = ( θ − v ) H . Therefore, (cid:90) N (cid:69) (cid:148) ( G H ( θ , r ) − G H ( v , r )) (cid:151) d r ≤ ( θ − v ) H (cid:69) (cid:128) e − λθ u λ ( B H θ ) Q ( B H θ ) (cid:138) + v H (cid:69) (cid:148) ( e − λθ u λ ( B H θ ) Q ( B H θ ) − e − λ v u λ ( B Hv ) Q ( B Hv )) (cid:151) ≤ ( θ − v ) H (cid:69) (cid:128) e − λθ u λ ( B H θ ) Q ( B H θ ) (cid:138) + v H λ ( θ − v ) e − λ v (cid:69) (cid:148) u λ ( B Hv ) Q ( B Hv ) (cid:151) + v H e − λθ (cid:69) (cid:148) ( u λ ( B H θ ) Q ( B H θ ) − u λ ( B Hv ) Q ( B Hv )) (cid:151) .20n the above inequality, the first two terms are bounded by using Lemma A.3. The last oneis bounded by using Lemma A.4. Eventually we obtain that for η small enough (see againEq. (4.14)), there exist C and α > (cid:90) N (cid:69) (cid:148) ( G H ( θ , r ) − G H ( v , r )) (cid:151) d r ≤ C ( θ − v ) e − α S ( − x − η ) (cid:112) λ e − λ v . (4.21)Inequality (4.18) thus becomes: I ( ) s ≤ v ≤ C e − α S ( − x − η ) (cid:112) λ (cid:90) Ns (cid:90) tv (cid:90) tv e − λ v (cid:89) i = ∂ K H ∂ θ ( θ i , v ) ( θ i − v ) d θ d θ d v .As s ≤ v , we have: (cid:90) Ns (cid:90) [ v , t ] e − λ v (cid:89) i = ∂ K H ∂ θ ( θ i , v ) ( θ i − v ) d θ i d v = c H ( H − ) (cid:90) ts d v (cid:90) [ s , t ] e − λ v (cid:89) i = { θ i ≥ v } (cid:18) θ i v (cid:19) H − ( θ i − v ) H − d θ i = c H ( H − ) (cid:90) [ s , t ] (cid:90) θ ∧ θ s e − λ v (cid:89) i = (cid:18) θ i v (cid:19) H − ( θ i − v ) H − d v d θ i ≤ C ( H − ) ( t − s ) H | log ( t − s ) | where the last inequality results from Lemma A.5. We thus have obtained: I ( ) s ≤ v ≤ C e − α S ( − x − η ) (cid:112) λ ( H − ) ( t − s ) H | log ( t − s ) | .We now bound I ( ) s ≤ v . Using Lemma A.3 again, I ( ) s ≤ v = (cid:90) ts v H e − λ v ( K H ( t , v ) − ) (cid:69) (cid:128) Q ( B Hv ) u λ ( B Hv ) (cid:138) d v ≤ C e − α S ( − x − η ) (cid:112) λ (cid:90) ts e − λ v v H ( K H ( t , v ) − ) d v ≤ C e − α S ( − x − η ) (cid:112) λ ( H − ) ( t − s ) | log ( t − s ) | ,where the last inequality results from Lemma A.6.Therefore, we obtained that I s ≤ v = I ( ) s ≤ v + I ( ) s ≤ v ≤ C e − α S ( − x − η ) (cid:112) λ ( H − ) ( t − s ) | log ( t − s ) | .since t − s ∈ [
0, 1 ] and H ≥ . This and Inequality (4.17) on I s > v yields: (cid:69) (cid:107) D · ( ¯ U t − ¯ U s )( · ) (cid:107) L [ N ] = I s > v + I s ≤ v ≤ C e − α S ( − x − η ) (cid:112) λ ( H − ) ( t − s ) | log ( t − s ) | .In view of Inequality (4.15), this concludes the proof of this lemma.21 .7 Proof of Theorem 4.1 We now prove Theorem 4.1 in the general case of Equation (1.1;H).We start as in Subsection 3.1 and use the Lamperti transform. In this paragraph only,we write τ XH ( x ) for the hitting time of 1 by X H started from x <
1. It is easily seen that τ XH ( x ) = τ YH ( F ( x )) , where τ YH ( F ( x )) is the hitting time of F ( ) by Y H started from F ( x ) .Thus (cid:69) (cid:128) e − λτ XH ( x ) (cid:138) − (cid:69) (cid:18) e − λτ X ( x ) (cid:19) = (cid:69) (cid:128) e − λτ YH ( F ( x )) (cid:138) − (cid:69) (cid:18) e − λτ Y ( F ( x )) (cid:19) .We therefore bound the right-hand side of this equality, using the following notations: y = F ( x ) and Θ = F ( ) .Let us first extend the error decomposition (4.2). The second-order ODE satisfied by w λ ( y ) : = (cid:69) (cid:16) e − λτ Y / ( y ) (cid:17) is ˜ b ( y ) w (cid:48) λ ( y ) + w (cid:48)(cid:48) λ ( y ) = λ w λ ( y ) , y < Θ , w λ (Θ) = y →−∞ w λ ( y ) = < t ≤ τ YH ∧ T , e − λ t w λ ( Y Ht ) − w λ ( y ) = (cid:90) t e − λ s (cid:128) w (cid:48) λ ( Y Hs ) ˜ b ( Y Hs ) − λ w λ ( Y Hs ) (cid:138) d s + (cid:90) t e − λ s w (cid:48) λ ( Y Hs ) ◦ d B Hs = (cid:90) t e − λ s (cid:128) w (cid:48) λ ( Y Hs ) ˜ b ( Y Hs ) − λ w λ ( Y Hs ) (cid:138) d s + (cid:90) t e − λ s w (cid:48) λ ( Y Hs ) d B Hs + α H (cid:90) t (cid:90) t D Hv (cid:128) e − λ s w (cid:48) λ ( Y Hs ) (cid:138) | s − v | H − d v d s .Using D Hv Y Hs = [ s ] ( v ) (cid:16) + (cid:82) s ˜ b (cid:48) ( Y H θ ) D Hv Y H θ d θ (cid:17) and the ODE (4.22) satisfied by w λ , weget e − λ t w λ ( Y Ht ) − w λ ( y ) = − (cid:90) t e − λ s w (cid:48)(cid:48) λ ( Y Hs ) d s + α H (cid:90) t (cid:90) s e − λ s w (cid:48)(cid:48) λ ( Y Hs ) | s − v | H − d v d s + (cid:90) t e − λ s w (cid:48) λ ( Y Hs ) d B Hs + α H (cid:90) t (cid:90) s e − λ s w (cid:48)(cid:48) λ ( Y Hs ) I ( v , s ) | s − v | H − d v d s ,where I ( v , s ) = { v ≤ s } (cid:82) sv ˜ b (cid:48) ( Y H θ ) D Hv Y H θ d θ . To simplify the notations, we now denote τ YH ( F ( y )) by τ YH . We apply the previous equality at time t = τ YH ∧ T , take the expecta-22ion and take the limit T → + ∞ as in Proposition 4.4 to obtain (cid:69) (cid:128) e − λτ YH (cid:138) − w λ ( y ) = (cid:69) (cid:90) τ YH (cid:130) α H (cid:90) s | s − v | H − d v − (cid:140) e − λ s w (cid:48)(cid:48) λ ( Y Hs ) d s (D) + lim T → + ∞ (cid:69) (cid:20) δ H (cid:128) [ t ] e − λ · w (cid:48) λ ( Y H · ) (cid:138)(cid:12)(cid:12)(cid:12) t = τ YH ∧ T (cid:21) (Sk) + (cid:69) α H (cid:90) τ YH (cid:90) s e − λ s w (cid:48)(cid:48) λ ( Y Hs ) I ( v , s ) | s − v | H − d v d s , (R)The three terms (D), (Sk) and (R) are treated below as follows: • the difference (D) is treated below similarly to I ( λ ) in Subsection 4.2. However, wehave to bound (cid:69) (cid:128) { τ YH ≥ s } w (cid:48)(cid:48) λ ( Y Hs ) (cid:138) which requires to estimate the density of Y Ht : seeSection 5 (Theorem 5.1). We also need estimates on w (cid:48)(cid:48) λ ( y ) which are obtained inAppendix B (Proposition B.1); • for (Sk), we will explain the differences with δ H (cid:128) ( s , t ] e − λ · u (cid:48) λ ( B H · ) (cid:138) ; • a bound for the remainder (R) will appear when estimating (Sk).In the pure fBm case we had to extend the function u λ to the whole space. Similarly,for any λ ≥
0, we can extend w λ to a (cid:67) function on [ + ∞ ) , still denoted by w λ , whichsatisfies ∀ λ ≥
0, sup x ∈ [ ] (cid:128) | w λ ( x ) | , | w (cid:48) λ ( x ) | , | w (cid:48)(cid:48) λ ( x ) | (cid:138) ≤ C ( + λ ) ,where C is a constant independent of λ . In particular, the bounds of Proposition B.1 holdalso true on [ + ∞ ) . One actually can set w λ ( x ) = a x + b x + c for any x ≥ Θ with a = w (cid:48)(cid:48) λ ( ) b = w (cid:48) λ ( ) − w (cid:48)(cid:48) λ ( ) c = − w (cid:48) λ ( ) + w (cid:48)(cid:48) λ ( ) . The main ingredient is the following extension of Lemma 4.5 and Lemma 4.7:
Lemma 4.5’.
There exists η ∈ ( Θ − y ) such that: for any < η ≤ η , there exist α > λ , C > such that for all p ≥ s ≥ λ ≥ λ ,e − λ ps (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ (cid:40) C e − α p (Θ − y + η ) H R ( λ ) if s H p R ( λ ) > Θ − y + η , C e − p (Θ − y − η ) R ( λ ) if s H p R ( λ ) ≤ Θ − y + η , where R ( λ ) = (cid:112) λ + µ − µ for some constant µ ≥ which only depends on b and σ .In case η = and if s H p R ( λ ) > Θ − y , we have (cid:69) (cid:128) { Y Hs ≤ } w λ ( Y Hs ) p (cid:138) ≤ .
23n particular, this lemma implies that for any s ≥ λ ≥ λ , e − λ ps (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ e − α pS (Θ − y − η ) R ( λ ) ,where S is the function defined in Theorem 4.1. Proof.
We denote by C > Y Ht , as obtained from Theorem 5.1. Let v λ ( x ) be the Laplace transform of τ X ( x ) . Notethat v λ ( x ) = w λ ( F ( x )) . Thus the bounds on v λ and its derivatives provided by PropositionB.1 in the appendix apply to w λ . These bounds and the estimate of Theorem 5.1 on thedensity of Y Ht yield (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ (cid:90) Θ+ η −∞ e − p (Θ − y ) R ( λ ) e C s e − ( y − y ) s H (cid:112) π s H d y ≤ e C s − p (Θ − y ) R ( λ )+ ( ps H R ( λ )) Φ (cid:18) Θ + η − y s H − ps H R ( λ ) (cid:19) . (4.23)As in Lemma 4.5, we obtain: (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ (cid:168) C η s H e p η R ( λ )+ C s if s H p R ( λ ) > Θ − y + η , e − p R ( λ )(Θ − y − η )+ C s if s H p R ( λ ) ≤ Θ − y + η .Indeed, the case s H p R ( λ ) ≤ Θ − y + η is very similar to Lemma 4.5 and the proof inthat case follows easily after choosing λ appropriately (which we explain below).We thus consider the case s H p R ( λ ) > Θ − y + η in the rest of this proof.In Eq. (4.23), the classical Gaussian bound on Φ( − x ) , x >
0, yields e − p (Θ − y ) R ( λ )+ ( ps H R ( λ )) Φ (cid:18) Θ + η − y s H − ps H R ( λ ) (cid:19) = e p η R ( λ ) e − (Θ − y + η ) s H (cid:112) π (cid:0) ps H R ( λ ) − s − H (Θ − y + η ) (cid:1) ≤ s H η e p η R ( λ ) .Let λ > C p . For any λ ≥ λ and s ≥
0, we have: s H e − λ ps + C s ≤ C for some constant C >
0. Thus we obtain that for any s ≥ λ ≥ λ e − λ ps (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ C e p η R ( λ ) .Using s > (cid:16) Θ − y + η p R ( λ ) (cid:17) H , we get e − λ ps (cid:69) (cid:0) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:1) ≤ C e − λ ps e p η R ( λ ) ≤ exp (cid:40) − p R ( λ )(Θ − y + η ) H (cid:32) λ p − H R ( λ ) + H − η (Θ − y + η ) H (cid:33)(cid:41) .Since λ R ( λ ) + H is bounded from below, one can choose η small enough so that: thereexists α > λ ≥ λ , for all s ≥ e − λ ps (cid:69) (cid:128) { Y Hs ≤ Θ+ η } w λ ( Y Hs ) p (cid:138) ≤ e − α p (Θ − y + η ) H R ( λ ) . 24e are now in a position to study (D). Recall that C > Y Ht . We will thus prove the following generalization ofProposition 4.6: Proposition 4.6’.
Let ˜ T be the function of λ ∈ (cid:82) + defined by ˜ T ( λ ) = λ R ( λ ) − H ∧ R ( λ ) .There exist constants α , C > such that for any λ > C : (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ e − λ s (cid:130) α H (cid:90) s ( s − v ) H − dv − (cid:140) (cid:69) (cid:128) { τ YH ≥ s } w (cid:48)(cid:48) λ ( Y Hs ) (cid:138) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C H − λ − C e − α S (Θ − y ) ˜ T ( λ ) , where S is the function defined in Theorem 4.1.Proof. From the inequality (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) α H (cid:90) s ( s − v ) H − d v − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Hs H − − (cid:12)(cid:12)(cid:12) ≤ ( H − ) ρ ( H , s ) where ρ ( H , s ) = | + s | (cid:0) ∨ s H − (cid:1) , we deduce that | ( D ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) ∞ e − λ s (cid:130) α H (cid:90) s ( s − v ) H − d v − (cid:140) (cid:69) (cid:128) { τ Y H ≥ s } w (cid:48)(cid:48) λ ( Y Hs ) (cid:138) d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ( H − ) (cid:90) ∞ e − λ s ρ ( H , s ) (cid:69) (cid:128) { τ YH ≥ s } | w (cid:48)(cid:48) λ ( Y Hs ) | (cid:138) d s .In view of Proposition B.1, | w (cid:48)(cid:48) λ ( y ) | ≤ C ( + λ ) w λ ( y ) . Thus we can apply Lemma 4.5’ with p = η =
0, which yields for s H R ( λ ) ≤ Θ − y : (cid:69) (cid:128) { τ YH ≥ s } | w (cid:48)(cid:48) λ ( Y Hs ) | (cid:138) ≤ C ( + λ ) (cid:69) (cid:128) { Y Hs ≤ T } w λ ( Y Hs ) (cid:138) ≤ C ( + λ ) e − (Θ − y ) R ( λ )+ C s .When s H R ( λ ) > Θ − y , we have: (cid:69) (cid:128) { τ YH ≥ s } | w (cid:48)(cid:48) λ ( Y Hs ) | (cid:138) ≤ C ( + λ ) (cid:69) (cid:128) { Y Hs ≤ Θ } | w λ ( Y Hs ) | (cid:138) ≤ C ( + λ ) , since w λ ( y ) ≤ y ≤ Θ . Therefore, | ( D ) | ≤ ( H − ) C ( + λ ) e − (Θ − y ) R ( λ ) (cid:90) (cid:16) Θ − y R ( λ ) (cid:17) H e − λ s e C s ρ ( H , s ) d s + ( H − ) C ( + λ ) (cid:90) ∞ (cid:16) Θ − y R ( λ ) (cid:17) H e − λ s ρ ( H , s ) d s ,from which the result follows when λ > C . We proceed as for the bound of I ( λ ) in Subsection 4.3. Similarly to I ( λ ) , (Sk) satisfiesthe following inequality: | ( Sk ) | = lim N → + ∞ (cid:12)(cid:12)(cid:12)(cid:12) (cid:69) (cid:20) δ H (cid:128) [ t ] w (cid:48) λ ( Y H · ) e − λ · (cid:138)(cid:12)(cid:12)(cid:12) t = τ Y H ∧ N (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ lim N → + ∞ (cid:69) (cid:150) sup t ∈ [ N ] { τ YH ≥ t } (cid:12)(cid:12)(cid:12) δ ( N ) (cid:128) { K ∗ H − Id } ( [ t ] w (cid:48) λ ( Y H · ) e − λ · ) (cid:138)(cid:12)(cid:12)(cid:12)(cid:153) .25e introduce similar notations as in Subsection 4.3 and define, for each N ∈ (cid:78) : the pro-cesses { U Yt ( v ) , t ∈ [ N ] , v ≥ } and { Υ Yt , t ∈ [ N ] } and for η > { ¯ U Yt ( v ) , t ∈ [ N ] , v ≥ } and { ¯ Υ Yt , t ∈ [ N ] } . We also keep all other notations from Subsection 4.4. We then get | ( Sk ) | ≤ lim N →∞ N − (cid:88) n = (cid:69) (cid:150) sup t ∈ [ n , n + ] | ¯ Υ Yt − ¯ Υ Yn | (cid:153) .Lemmas 4.7 and 4.8 are extended as follows. Lemma 4.7’.
There exists η ∈ ( Θ − y ) such that: for all < η ≤ η there exist C , α , λ > such that: for any λ ≥ λ , for all r ∈ (cid:82) + and for all H ∈ [ , 1 ) , (cid:69) (cid:32)(cid:90) N | D H θ (cid:128) w (cid:48) λ ( Y Hr ) φ η ( Y Hr ) e − λ r (cid:138) | / H d θ (cid:33) ˜ pH ≤ C e − α ˜ pS (Θ − y − η ) R ( λ ) e − λ ˜ pr , where S and R are the functions defined respectively in Theorem 4.1 and Lemma 4.5’.Proof. Proceed as in the proof of Lemma 4.7. Use Proposition B.1 to bound w (cid:48) λ and w (cid:48)(cid:48) λ anduse Lemma 4.5’ instead of Lemma 4.5. Lemma 4.8’.
There exists η ∈ ( − x ) such that: for all < η ≤ η and for all < ε < ,there exist C , α > such that for any n ∈ [ N ] , we have ∀ s ≤ t ∈ [ n , n + ] , ∀ λ ≥ λ , ∀ H ∈ [ , 1 ) , (cid:13)(cid:13) ¯ Υ Ys − ¯ Υ Yt (cid:13)(cid:13) L (Ω) ≤ C ( H − ) ( t − s ) − ε e − α S (Θ − y − η ) R ( λ ) . Proof.
One can essentially proceed as in the proof of Lemma 4.8 and easily adapt its proof.Thus we only explain the beginning of the new proof and exhibit the few additional termsto estimate.Let ¯ w (cid:48) λ : = w (cid:48) λ φ η . Observe that: D r ¯ w (cid:48) λ ( Y H θ ) = D r Y H θ (cid:16) φ η ( Y H θ ) w (cid:48)(cid:48) λ ( Y H θ ) + φ (cid:48) η ( Y H θ ) w (cid:48) λ ( Y H θ ) (cid:17) = : D r Y H θ Q Y ( Y H θ ) ,and D r Y H θ = K H ( θ , r ) + (cid:82) θ r ˜ b (cid:48) ( Y H σ ) D r Y H σ d σ . Apply Meyer’s inequality to (cid:13)(cid:13) ¯ Υ Ys − ¯ Υ Yt (cid:13)(cid:13) L (Ω) .We thus have to estimate (cid:13)(cid:13) D · ( ¯ U Ys ( · ) − ¯ U Yt ( · )) (cid:13)(cid:13) L (Ω) , (cid:72) ⊗ . For s ≤ t ∈ [ n , n + ] , we have D r ( ¯ U Yt ( v ) − ¯ U Ys ( v )) = (cid:90) tv ∨ s ∂ K H ∂ θ ( θ , v ) D r Y H θ Q Y ( Y H θ ) e − λθ d θ − ( s , t ] ( v ) D r Y Hv Q Y ( Y Hv ) e − λ v .Therefore, the main differences with Subsection 4.6 are: • the term D r Y H θ replaces K H ( θ , r ) in several places: – in Eq. (4.16), (cid:82) N K H ( θ , r ) K H ( θ , r ) d r appeared and is now replaced by (cid:82) N D r Y H θ D r Y H θ d r .Lemma 3.2 implies that this new integral is bounded by e C ( θ ∧ θ ) , which is even-tually controlled by e − λ ( θ + θ ) . 26 Similarly to Eq. (4.19), we now have to estimate (cid:82) N ( D r Y H θ − D r Y Hv ) d r . UsingEq. (2.3) and (3.2), we can compute D r Y Ht : D r Y Ht = K ∗ H (cid:128) D H · Y Ht (cid:138) ( r ) = K ∗ H (cid:130) [ t ] ( · ) + (cid:90) t D H · Y H σ b (cid:48) ( Y H σ ) d σ (cid:140) ( r )= K H ( t , r ) + (cid:90) t D r Y H σ b (cid:48) ( Y H σ ) d σ .For θ ≥ v , we thus have (cid:90) N ( D r Y H θ − D r Y Hv ) d r ≤ (cid:90) N ( K H ( θ , r ) − K H ( v , r )) d r + (cid:90) θ (cid:32)(cid:90) θ v ∨ r b (cid:48) ( Y H σ ) D r Y H σ d σ (cid:33) d r ≤ ( θ − v ) H + e C θ ( θ − v ) .Choosing λ > C , we deduce that for all λ ≥ λ and for all v ≤ θ ∈ [ n , n + ] , e − λθ (cid:82) N ( D r Y H θ − D r Y Hv ) d r ≤ C ( θ − v ) H , which is what we need. – The term (cid:82) N K H ( θ , r ) d r from (4.20) now becomes (cid:82) N (cid:128) D r Y H θ (cid:138) d r . It is boundedby using the previous argument. • Equalities (4.16) and (4.19) show that we need bounds on e − λθ (cid:69) (cid:148) Q Y ( Y H θ ) (cid:151) and e − λ ( θ + θ ) (cid:69) (cid:104)(cid:81) i = Q Y ( Y H θ i ) (cid:105) . In view of Proposition B.1, Q Y ( Y H θ ) is bounded by C ( + λ ) { Y H θ < Θ+ η } w λ ( Y H θ ) . Lemma 4.5’ provides the appropriate bound; • the term (cid:69) (cid:149)(cid:16) Q Y ( Y H θ ) − Q Y ( Y H θ ) (cid:17) (cid:152) , which appears as in Eq. (4.20) is bounded byusing Lemma A.4’ from the appendix.Keeping in mind the above observations, one can proceed as in Subsection 4.6.In view of Lemmas 4.7’ and 4.8’, we obtain that there exists η > C , α and λ ≥ n ∈ (cid:78) , s ≤ t ∈ [ n , n + ] and any λ ≥ λ , (cid:69) (cid:148) ( ¯ Υ Ys − ¯ Υ Yt ) + ε (cid:151) ≤ C ( H − ) ( t − s ) + ε e − ηλ s e − α S (Θ − y − η ) R ( λ ) .As in Subsections 4.3-4.4, this inequality can then be combined with Garsia-Rodemich-Rumsey’s lemma (see Subsection 4.4) to obtain: for any λ ≥ λ , | ( Sk ) | ≤ C ( H − ) − ε e − α S (Θ − y − η ) R ( λ ) . (4.24)Coming back to the equation (cid:69) (cid:128) e − λτ YH (cid:138) − w λ ( y ) = ( D ) + ( Sk ) + ( R ) , we are now in aposition to end the proof of Theorem 4.1. Indeed, Proposition 4.6’ and Inequality (4.24)give the desired bound, while ( R ) is bounded similarly by performing the same analysis.27 Gaussian-type upper bound for the density of X Ht Consider { X Ht } t ≥ the solution to (1.1;H) with initial condition x . Assume that b and σ satisfy the hypotheses (H1’) and (H2). Let H ∈ ( , 1 ) . Under these assumptions, the densityof X Ht is known to satisfy: • There exists C > ∀ t ∈ (
0, 1 ] , ∀ x ∈ (cid:82) , p t ( x ) ≤ C (cid:112) π t H exp (cid:130) − C ( x − x ) t H (cid:140) ,(see Besalú et al. [ ] ). • There exist functions of t , C ( ) t and C ( ) t , and constants C , C such that: p t ( x ) ≤ C ( ) t exp (cid:32) − ( | x | − C ( ) t ) C e C t t H (cid:33) ,where C ( ) t diverges as t → ∞ and as H → (see Baudoin et al. [ ] ).To improve on the two above estimates, we obtain in the following theorem a similar boundwhich is accurate for all t > H ∈ [ , 1 ) . Theorem 5.1.
Assume that b and σ satisfy the conditions (H1’) and (H2). Then for everyH ∈ [ , 1 ) , X H has an absolutely continuous density and there exists C ( b , σ ) ≡ C > suchthat, for all t ∈ (cid:82) + and H ∈ [ , 1 ) , ∀ x ∈ (cid:82) , p t ( x ) ≤ e C t (cid:112) π t H exp (cid:130) − ( x − x ) (cid:107) σ (cid:107) ∞ t H (cid:140) . (5.1) Proof.
As in [ ] , the proof uses the Girsanov and Doss-Sussman transforms. We add twoimprovements: we get a better bound on the Girsanov exponential martingale and ourestimates depend explicitely on H and t . Note that if t ∈ (
0, 1 ] , our result coincides withthe one in [ ] . Thus we fix t >
1. As above, we denote by C constants which do not dependon t , H or x .We recall notations from [ ] . The process X H is defined on some probability space (Ω , (cid:70) , (cid:80) ) . Let ( ˜ Ω , ˜ (cid:70) , ˜ (cid:80) ) be a second probability space, and ˜ B H be a fractional Brownianmotion on this space. Let Y H be the solution to Y Ht = x + (cid:82) t σ ( Y Hs ) ◦ d ˜ B Hs . Then by theDoss-Sussman transform, Y Ht = ϕ ( ˜ B Ht , x ) for any t ≥
0, where ϕ is the solution to: ∂ ϕ∂ x ( x , y ) = σ (cid:0) ϕ ( x , y ) (cid:1) and ϕ ( y ) = y .Let us denote by K H the integral operator on L ([ T ]) with kernel K H ( · , · ) , and K − H itsinverse (see [ ] for a definition using fractional operators). Under assumptions (H1’) and(H2), Theorem 2 of [ ] ensures that the Radon-Nikodym derivative ξ t = d (cid:80) XHt d˜ (cid:80) Y Ht exists andis given by ξ t = exp (cid:130)(cid:90) t (cid:77) s d ˜ W s − (cid:90) t (cid:77) s d s (cid:140) ,28here (cid:77) s = K − H (cid:129)(cid:82) · b ( Y Hv ) σ ( Y Hv ) d v (cid:139) ( s ) (the explicit expression of (cid:77) s is given in equation (5.4)below).Let p t ( x ) be written as: p t ( x ) = p ( ) t ( x ) − p ( ) t ( x ) , where p ( ) t ( x ) and p ( ) t ( x ) are givenby formula (43) in [ ] : p ( ) t ( x ) = ˜ (cid:69) (cid:130) { ϕ ( ˜ B Ht , x ) ≥ x } ξ t ˜ B Ht t H ∂ x ϕ ( ˜ B Ht , x ) (cid:140) , p ( ) t ( x ) = ˜ (cid:69) (cid:32) { ϕ ( ˜ B Ht , x ) ≥ x } 〈 D · (cid:128) ξ t ∂ x ϕ ( ˜ B Ht , x ) − (cid:138) , K H ( t , · ) 〉 L [ t ] t H (cid:33) .We now prove that p ( ) t and p ( ) t satisfy the inequality (5.1). First part: estimate for p ( ) t ( x ) .Let r > [ ] on ˜ (cid:80) ( ϕ ( ˜ B Ht , x ) ≥ x ) and hypothesis (H2) on σ yield: | p ( ) t ( x ) | ≤ Ct H exp (cid:130) − ( x − x ) (cid:107) σ (cid:107) ∞ t H (cid:140) (cid:128) ˜ (cid:69) ξ + rt (cid:138) + r . (5.2)Consider a new measure ˆ (cid:80) under which ˆ B H : = ˜ B H − r ( + r ) (cid:82) · b ( Y Hu ) σ ( Y Hu ) d u is a fBm. Equation(46) of [ ] yields ˜ (cid:69) (cid:128) ξ + rt (cid:138) = ˆ (cid:69) (cid:150) exp (cid:130) r ( + r ) (cid:90) t (cid:77) s d s (cid:140)(cid:153) .Now we use our bound on (cid:77) s from Lemma 5.2 below instead of the one from [
9, Lemma4.2 ] . Choose 0 < γ < . For A > (cid:90) tA − (cid:77) s d s ≤ C (cid:90) tA − ( s − H − A H − ) d s + CA − + ( H − γ ) (cid:90) tA − (cid:107) ˜ B H (cid:107) γ , s − A − , s d s .Since ( s − H − A H − ) ≤ A H − and (cid:107) ˜ B H (cid:107) γ , s − A − , s ≤ (cid:107) ˆ B H (cid:107) γ , s − A − , s + r ( + r ) (cid:13)(cid:13) (cid:90) · b ( Y Hu ) σ ( Y H u ) d u (cid:13)(cid:13) γ , s − A − , s ≤ (cid:107) ˆ B H (cid:107) γ , s − A − , s + r ( + r ) CA γ − ,we deduce that (cid:90) tA − (cid:77) s d s ≤ CA H − t + CA − + ( H − γ ) (cid:90) tA − (cid:16) (cid:107) ˆ B H (cid:107) γ , s − A − , s + A γ − (cid:17) d s .For s ≤ A − , the following estimate from [
9, Lemma 4.2 ] is enough for our purpose: (cid:90) A − (cid:77) s d s ≤ C (cid:90) A − (cid:16) s − H + (cid:107) ˜ B H (cid:107) γ ,0, A − (cid:17) d s .29hen, using again the above bound on (cid:107) ˜ B H (cid:107) γ ,0, A − , (cid:90) A − (cid:77) s d s ≤ C (cid:130) A H − − H + A − (cid:107) ˆ B H (cid:107) γ ,0, A − + A γ − (cid:140) .Combining our estimates on (cid:82) A − (cid:77) s d s and (cid:82) tA − (cid:77) s d s we get ˆ (cid:69) (cid:150) exp (cid:130) r ( + r ) (cid:90) t (cid:77) s d s (cid:140)(cid:153) ≤ e C t × ˆ (cid:69) (cid:150) exp (cid:130) CA − + ( H − γ ) (cid:90) tA − (cid:107) ˆ B H (cid:107) γ , s − A − , s d s (cid:140)(cid:153) ≤ e C t t − A − (cid:90) tA − ˆ (cid:69) (cid:104) exp (cid:16) CA − + ( H − γ ) (cid:107) ˆ B H (cid:107) γ , s − A − , s (cid:17)(cid:105) d s .We now need estimates on the tail probabilities of (cid:107) B H (cid:107) γ , a , b , which are used without proofin [
9, 18 ] . In these papers, the dependence of the constants in the Hurst parameter is notmade explicit and we could not find it in the literature. We thus use our Lemma 5.3 belowwith ε = ( H − γ ) and get ˆ (cid:69) (cid:104) exp (cid:16) CA − + ( H − γ ) (cid:107) ˆ B H (cid:107) γ , s − A − , s (cid:17)(cid:105) = CA − ( H − γ ) (cid:90) (cid:82) x e CA − + ( H − γ ) x ˆ (cid:80) (cid:128) (cid:107) ˆ B H (cid:107) γ , s − A − , s > x (cid:138) d x ≤ CA − ( H − γ ) (cid:90) (cid:82) ( + (cid:112) A − ) x e CA − + ( H − γ ) x − K ( γ ) A H − γ x d x ,where K ( γ ) : = K ( γ , H − γ ) is the constant in Lemma 5.3. Hence for A large enough (whichdepends on b and σ ), the integral on the right-hand side is convergent. Since γ < , theabove integral is finite for any H ∈ [ , 1 ) . This yields:˜ (cid:69) (cid:128) ξ + rt (cid:138) ≤ e C t . (5.3)In view of Inequality (5.2), we conclude that p ( ) t satisfies the inequality (5.1). Second part:
Estimate for p ( ) t . Set Z t = ξ t ∂ x ϕ ( ˜ B Ht , x ) − . Applying first the Cauchy-Schwarz inequality and then the Hölder inequality, we have | p ( ) t ( x ) | ≤ t H (cid:69) (cid:128) { ϕ ( ˜ B Ht , x ) ≥ x } (cid:107) D · Z t (cid:107) L [ t ] (cid:107) K H ( t , · ) (cid:107) L [ t ] (cid:138) ≤ t H (cid:69) (cid:128) { Y Ht ≥ x } (cid:107) D · Z t (cid:107) L [ t ] (cid:138) ≤ t H ˜ (cid:80) ( Y Ht ≥ x ) / p ˜ (cid:69) (cid:16) (cid:107) DZ t (cid:107) qL [ t ] (cid:17) / q with p − + q − =
1, where D u Z t = D u ξ t ∂ x ϕ ( ˜ B Ht , x ) − K H ( t , u ) ξ t ∂ x x ϕ ( ˜ B Ht , x ) | ∂ x ϕ ( ˜ B Ht , x ) | .The derivatives of ϕ are controlled by (cid:107) σ (cid:107) ∞ , σ min and (cid:107) σ (cid:48) (cid:107) ∞ , thus (cid:16) ˜ (cid:69) (cid:107) DZ t (cid:107) qL [ t ] (cid:17) / q ≤ C (cid:16) ˜ (cid:69) (cid:107) D ξ t (cid:107) qL [ t ] (cid:17) / q + C t H (cid:128) ˜ (cid:69) ξ qt (cid:138) / q .30et us bound the norm of D u ξ t , which reads: D u ξ t = ξ t (cid:130) (cid:77) u + (cid:90) t D u (cid:77) s d ˜ W s − (cid:90) t (cid:77) s D u (cid:77) s d s (cid:140) = : (cid:78) u ( t ) .Now choose q ∈ (
1, 2 ) , (cid:16) ˜ (cid:69) (cid:107) D ξ t (cid:107) qL [ t ] (cid:17) / q = (cid:16) ˜ (cid:69) ξ qt (cid:107)(cid:78) · ( t ) (cid:107) qL [ t ] (cid:17) / q ≤ (cid:128) ˜ (cid:69) ξ q / ( − q ) t (cid:138) ( − q ) / ( q ) (cid:16) ˜ (cid:69) (cid:107)(cid:78) · ( t ) (cid:107) L [ t ] (cid:17) / Each of the three summands in the definition of (cid:78) u ( t ) can be bounded by using similararguments. We only detail the calculations for the second one.˜ (cid:69) (cid:130)(cid:13)(cid:13) (cid:90) t D · (cid:77) s d ˜ W s (cid:13)(cid:13) L [ t ] (cid:140) = ˜ (cid:69) (cid:90) t (cid:130)(cid:90) t D u (cid:77) s d ˜ W s (cid:140) d u = (cid:90) t ˜ (cid:69) (cid:130)(cid:90) t ( D u (cid:77) s ) d s (cid:140) d u .Since Y Ht = ϕ ( ˜ B Ht , x ) , we have D u Y Ht = K H ( t , u ) ∂ x ϕ ( ˜ B Ht , x ) = K H ( t , u ) σ ( Y Ht ) , and there-fore: D u (cid:77) s = K − H (cid:130)(cid:90) · K H ( v , u ) σ ( Y Hv ) ∂ x (cid:18) b σ (cid:19) ( Y Hv ) d v (cid:140) ( s ) .We then bound | D u (cid:77) s | by using the same argument as in the proof of Lemma 5.2. Hence,˜ (cid:69) (cid:130)(cid:13)(cid:13) (cid:90) t D · (cid:77) s d ˜ W s (cid:13)(cid:13) L [ t ] (cid:140) ≤ C (cid:90) t (cid:90) t K H ( s , u ) ( s − H − A H − + ) + ˜ (cid:69) (cid:16) (cid:107) ˜ B H (cid:107) γ , s − A − , s (cid:17) A − ( H − γ ) d s d u ≤ C (cid:90) t (cid:90) t K H ( s , u ) (cid:110) ( s − H − A H − + ) + (cid:111) d s d u ≤ C t .We deduce that ˜ (cid:69) (cid:16) (cid:107)(cid:78) · ( t ) (cid:107) L [ t ] (cid:17) is bounded by C t . Furthermore (cid:16) ˜ (cid:69) ξ q / ( − q ) t (cid:17) ( − q ) / ( q ) isbounded as in (5.3). We conclude that p ( ) t satisfies Inequality (5.1). Lemma 5.2.
Let γ <
H. With the notations of the previous proposition, we have that for anyA > , H ∈ [ , 1 ) and s > A − , almost surely: |(cid:77) s | ≤ C (cid:16) A H − − s − H + A − + H − γ (cid:107) ˜ B H (cid:107) γ , s − A , s (cid:17) .31 roof. (cid:77) s is given by the following formula (see [
9, Eq.(39) ] ): (cid:77) s = s H − Γ( − H ) s − H b ( Y Hs ) σ ( Y Hs ) + ( H − ) (cid:90) s s − H b ( Y Hs ) σ ( Y Hs ) − u − H b ( Y Hu ) σ ( Y Hu ) ( s − u ) H + d u (5.4) = s − H b ( Y Hs ) σ ( Y Hs ) Γ( − H ) + H − Γ( − H ) (cid:90) s − ( s / u ) H − ( s − u ) H + b ( Y Hu ) σ ( Y Hu ) d u + H − Γ( − H ) (cid:90) s b ( Y Hs ) σ ( Y Hs ) − b ( Y Hu ) σ ( Y Hu ) ( s − u ) H + d u = : (cid:77) s + (cid:77) s + (cid:77) s .As in [ ] , we observe that (cid:77) s and (cid:77) s are bounded by Cs H − . We bound (cid:77) s as follows: |(cid:77) s | = H − Γ( / − H ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:90) s b ( Y Hs ) σ ( Y Hs ) − b ( Y Hu ) σ ( Y Hu ) ( s − u ) H + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d u ≤ H − Γ( / − H ) (cid:168) C (cid:90) s − A − ( s − u ) − ( H + ) d u + (cid:90) ss − A − (cid:12)(cid:12)(cid:12) b σ ◦ ϕ ( ˜ B Hs , x ) − b σ ◦ ϕ ( ˜ B Hu , x ) (cid:12)(cid:12)(cid:12) ( s − u ) H + d u (cid:171) ≤ C Γ( / − H ) (cid:40) A H − − s − H + ( H − ) (cid:90) ss − A − (cid:12)(cid:12) ˜ B Hs − ˜ B Hu (cid:12)(cid:12) ( s − u ) H + d u (cid:41) ,where the last term follows from the Lipschitz continuity of b σ and ϕ . We also notice that H (cid:55)→ Γ( / − H ) is bounded away from 0 for H ∈ [ , 1 ) . Hence the desired result followsfrom: (cid:90) ss − A − (cid:12)(cid:12) ˜ B Hs − ˜ B Hu (cid:12)(cid:12) ( s − u ) H + d u ≤ (cid:107) ˜ B H (cid:107) γ , s − A , s (cid:90) ss − A ( s − u ) γ − H − d u ≤ γ − H + (cid:107) ˜ B H (cid:107) γ , s − A , s A − + H − γ . Lemma 5.3.
Let H ∈ (
0, 1 ) , γ ∈ ( H ) and ε ∈ ( H − γ ) . Let B H be a fBm and fix some ≤ a < b < ∞ . Then, letting K ( γ , ε ) = ε ( ( γ + ε )) − : ∀ x ∈ (cid:82) + , (cid:80) (cid:128) (cid:107) B H (cid:107) γ , a , b > x (cid:138) ≤ ( + (cid:112) ( b − a ) ) exp (cid:130) − K ( γ , ε ) x ( b − a ) ( H − γ − ε ) (cid:140) . Proof.
Let ε ∈ ( H ) . Observe that for any u , v ∈ [ a , b ] , (cid:69) (cid:130) exp (cid:168) η | B Hu − B Hv | | u − v | ( H − ε ) (cid:171)(cid:140) = (cid:112) π (cid:90) (cid:82) exp (cid:166) − ( − η | u − v | ε ) x (cid:169) d x η < ( b − a ) − ε . For γ >
0, let ζ γ be the following modulus: ζ γ ( x ) = (cid:82) x u γ − (cid:112) log ( + u − ) d u . This modulus satisfies: ζ γ + ε ( x ) ≤ C ε x γ (5.5)for ε > C ε ∈ [ ε − / ε − ] . For γ ∈ ( H ) and ε < H − γ . Set η : = η ( b − a ) ( γ + ε − H ) . A corollary of Garsia-Rodemich-Rumsey’s lemma (see [
17, p.576 ] )leads to (cid:69) (cid:130) exp (cid:168) η K sup a ≤ u < v ≤ b | B Hu − B Hv | ζ γ + ε ( v − u ) (cid:171)(cid:140) ≤ (cid:69) (cid:32) ∨ (cid:90) ba (cid:90) ba exp (cid:168) η | B Hu − B Hv | | u − v | ( γ + ε ) (cid:171) d u d v (cid:33) where K : = ( γ + ε ) (see [
17, p.577 ] ). Now use Inequality (5.5) and choose η = ( C ε K ) − to get (cid:69) (cid:16) exp (cid:110) η (cid:107) B H (cid:107) γ , a , b (cid:111)(cid:17) ≤ (cid:69) (cid:32) ∨ (cid:90) ba (cid:90) ba exp (cid:168) η C ε K | B Hu − B Hv | | u − v | ( γ + ε ) (cid:171) d u d v (cid:33) ≤ + (cid:90) ba (cid:90) ba (cid:69) (cid:130) exp (cid:168) η C ε K | B Hu − B Hv | | u − v | ( γ + ε ) (cid:171)(cid:140) d u d v ≤ + (cid:90) ba (cid:90) ba (cid:90) (cid:82) (cid:112) π exp (cid:40) − (cid:32) − η C ε K (cid:18) | u − v | b − a (cid:19) ( H − γ − ε ) (cid:33) x (cid:41) d x d u d v ≤ + (cid:112) ( b − a ) .It then remains to use Tchebychev’s inequality. In this paper we have developed a sensitivity analysis w.r.t. the Hurst parameter of thedriving noise for the probability distribution of functionals of solutions to stochastic differ-ential equations, including the probability distribution of first hitting times, when the Hurstparameter is close to 1 /
2, that is, when the noise is close to the pure Brownian case. Ourestimates seem accurate. However many open questions deserve future works.One first open question concerns the extension of our results to a Gaussian processwith general kernel K and estimate the sensitivity of first hitting time Laplace transforms interms of the L distance between K and K .Another important open question concerns the sensitivity of the first passage time den-sity. Unfortunately, this cannot be easily derived from theorems 4.1 because of the singular-ity of the inverse Laplace transform, so that specific methodologies need to be developed.A last open question concerns the extension of our analysis to the case H < .We now comment Theorem 3.1. In [ ] the convergence of (cid:69) ϕ (cid:128) { X Ht } t ∈ [ T ] (cid:138) when H → H ≥ is obtained for symmetric Russo-Vallois integrals X Ht = (cid:82) t u s ◦ d B Hs rather thanfor solutions to Equations (1.1;H)). Choosing u s = σ ( X Hs ) does not allow us to obtain thisresult since we above needed that σ is elliptic. However, we do not know how to satisfy33he strong integrability conditions on u uniformly in H imposed in [ ] without assumingthe strong ellipticity of σ : sup ˜ H ∈ [ , H ] (cid:90) T sup t ∈ [ T ] (cid:69) | D Hr σ ( X Ht ) | p d r < ∞ .In addition, our technique to obtain accurate convergence rate estimates is less heavy thanthe one developed in [ ] ; where the convergence is proven for H → H where H is notnecessarily ).This ellipticity condition ( H ) may seem restrictive but it is natural in our context for thefollowing reasons. First, we need estimates on derivatives of the solution to the parabolicpartial differential equation associated to the diffusion process with coefficients b and σ and initial condition ϕ . Second, the estimates on the supremum of Y H on [ T ] obtainedin [ ] do not require the ellipticity of σ but depend on the Hölder norm (cid:107) B H (cid:107) α ,0, T , where α ∈ ( , H ) so that, as we already mentioned it, the constants tend to infinity when H → .To avoid the Lamperti transform and relax hypothesis (H2) a natural attempt is asfollows. When H = the SDE (1.1;H) can be written in the following Itô’s form: X t = x + (cid:90) t (cid:128) b ( X s ) + σ ( X s ) σ (cid:48) ( X s ) (cid:138) d s + (cid:90) t σ ( X s ) d W s .Denote the new drift by ˜ b and by ˜ L the generator of ( X t ) . Using parabolic PDEs driven by˜ L and , chain rules to u ( t , X Ht ) lead to (cid:69) (cid:128) u ( t , X Ht ) (cid:138) − u ( x ) = − (cid:69) (cid:90) t ( σσ (cid:48) )( X Hs ) ∂ x u ( s , X Hs ) d s + α H (cid:69) (cid:90) t (cid:90) T | r − s | H − σ ( X Hr ) σ (cid:48) ( X Hs ) ∂ x u ( s , X Hs ) d r d s − (cid:69) (cid:90) t σ ( X Hs ) ∂ x x u ( s , X Hs ) d s + α H (cid:69) (cid:90) t (cid:90) T | r − s | H − σ ( X Hr ) σ ( X Hs ) ∂ x x u ( s , X Hs ) d r d s + α H (cid:69) (cid:90) t (cid:90) T | r − s | H − ( D r X Hs − σ ( X Hr )) (cid:128) ∂ x x u ( s , X Hs ) σ ( X Hs ) + ∂ x u ( s , X Hs ) σ (cid:48) ( X Hs ) (cid:138) d r d s .However so far we have not succeeded to obtain accurate enough bounds on the sup andHölder norm of D r X Hs and on the density of X Hs to deduce relevant sensitivity estimatesw.r.t H .In our calculations we often used our assumption λ ≥
1. However, as already noticedin Remark 4.3, Theorem 4.1 should extend to 0 < λ ≤
1. One of the main issues consists ingetting accurate enough bounds on the term I ( λ ) in Equation (4.8) and more precisely onthe quantity (cid:112) λ (cid:69) (cid:130) sup t ∈ [ a λ , b λ ] { τ H ≥ t } | δ H (cid:128) [ t ] ( • ) u λ ( B H • ) e − λ • (cid:138) | (cid:140) ,where a λ = λ − H and b λ = − log (cid:112) λλ with λ <
1. It is possible to get a sharp estimate onthe previous quantity in terms of the probability distribution of ( ϑ Ht , S Ht ) , where S Ht is thelaw of the running supremum of the fBm up to time t and ϑ Ht is the time at which this34upremum is attained. In the Brownian motion case, the joint law of ( ϑ t , S t ) is known (see [
21, p.96–102 ] ). In particular, for p ∈ (
2, 3 ) one has ∀ t ≥ (cid:69) (cid:149) { S / t ≤ + η } ϑ p t (cid:152) ≤ C . (6.1)Numerical simulations and formal calculations inspired by [ ] and [ ] let us think thatInequality (6.1) holds also true for H > , which should be enough to treat the case 0 <λ ≤ A Technical lemmas
Lemma A.1 (Garsia-Rodemich-Rumsey) . Let { X t , t ∈ [ a , b ] } be an (cid:82) -valued continuousstochastic process. Then, for p ≥ and q > such that pq > , (cid:69) (cid:130) sup t ∈ [ a , b ] | X t − X a | (cid:140) ≤ C pqpq − ( b − a ) q − p (cid:69) (cid:32)(cid:90) ba (cid:90) ba | X s − X t | p | t − s | pq ds dt (cid:33) p ≤ C pqpq − ( b − a ) q − p (cid:32)(cid:90) ba (cid:90) ba (cid:69) (cid:0) | X s − X t | p (cid:1) | t − s | pq ds dt (cid:33) p provided the right-hand side in each line is finite.Proof. With the notations of [
27, p.353-354 ] , apply the general GRR lemma with ψ ( x ) = x p and p ( x ) = x q to obtain the first line. The second line is Hölder’s inequality. Lemma A.2.
For all θ , θ > and all s < θ ∧ θ , one has (cid:90) s ∂ K H ∂ θ ( θ , v ) ∂ K H ∂ θ ( θ , v ) dv ≤ c H ( H − ) − H ( θ ∨ θ − s ) H − ( θ ∧ θ − s ) H − . Proof.
Without loss of generality, we assume θ ≥ θ . We recall from Section 2 that (cid:90) s ∂ K H ∂ θ ( θ , v ) ∂ K H ∂ θ ( θ , v ) d v = c H ( H − ) (cid:90) s (cid:18) θ θ v (cid:19) H − (cid:0) ( θ − v )( θ − v ) (cid:1) H − d v .By successively using the changes of variables z = θ − v θ − v and x = θ θ z we have (cid:90) s (cid:18) θ θ v (cid:19) H − (cid:0) ( θ − v )( θ − v ) (cid:1) H − d v = ( θ θ ) H − ( θ − θ ) H − × (cid:90) θ − s θ − s θ /θ z H − / ( θ z − θ ) − H d z = ( θ − θ ) H − (cid:90) θ ( θ − s ) θ ( θ − s ) x H − / ( − x ) − H d x .35ince θ ( θ − s ) θ ( θ − s ) ≤ (cid:90) s (cid:18) θ θ v (cid:19) H − (cid:0) ( θ − v )( θ − v ) (cid:1) H − d v ≤ ( θ − θ ) H − (cid:18) θ ( θ − s ) θ ( θ − s ) (cid:19) H − (cid:90) θ ( θ − s ) θ ( θ − s ) ( − x ) − H d x ≤ ( θ − θ ) H − − H (cid:18) θ ( θ − s ) θ ( θ − s ) (cid:19) H − (cid:18) s ( θ − θ ) θ ( θ − s ) (cid:19) − H ≤ s − H θ H − θ H − − H ( θ − s ) H − ( θ − s ) H − ,and since s < θ ≤ θ , this gives the desired result.The following lemma extends Lemma 4.5 and Lemma 4.7. We keep the notation ofSubsection 4.6. Lemma A.3.
Assume that η ∈ ( − x ) is small enough. Then there exist constants C , α > such that for all H ∈ [ , 1 ) , λ ≥ and θ , θ ∈ (cid:82) + , (cid:69) (cid:32) (cid:89) i = e − λθ i Q ( B H θ i ) u λ ( B H θ i ) (cid:33) ≤ C e − α S ( − x − η ) (cid:112) λ . Proof.
By using Cauchy-Schwartz inequality we get (cid:69) (cid:16) e − λθ Q ( B H θ ) u λ ( B H θ ) (cid:17) ≤ ( (cid:112) λ (cid:107) φ η (cid:107) ∞ + (cid:107) φ (cid:48) η (cid:107) ∞ ) e − λθ (cid:69) (cid:16) { B H θ < + η } u λ ( B H θ ) (cid:17) .Use Lemma 4.5 and proceed as in the second step of the proof of Lemma 4.7. Note that theconstant C depends on η but not λ and H . The constant α appearing here is strictly smallerthan 1 (see Equation (4.14)). Lemma A.4.
Assume that η ∈ ( − x ) is small enough. There exist positive constants C and α such thate − λθ (cid:69) (cid:128) ( u λ ( B H θ ) Q ( B H θ ) − u λ ( B Hv ) Q ( B Hv )) (cid:138) ≤ C ( θ − v ) e − α S ( − x − η ) (cid:112) λ , for all λ ≥ and θ ≥ v ≥ .Proof. By using Itô’s formula for fractional Brownian motion we have (cid:69) (cid:148) ( u λ ( B H θ ) Q ( B H θ ) − u λ ( B Hv ) Q ( B Hv )) (cid:151) = (cid:69) (cid:32) δ H ( ( v , θ ] ( u λ Q ) (cid:48) ( B H · )) + H (cid:90) θ v ( u λ Q ) (cid:48)(cid:48) ( B Hs ) s H − d s (cid:33) ≤ (cid:69) (cid:104) δ H (cid:128) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:138) (cid:105) + H (cid:69) (cid:32)(cid:90) θ v ( u λ Q ) (cid:48)(cid:48) ( B Hs ) s H − d s (cid:33) . (A.1)36sing now Meyer’s inequality we obtain (cid:69) (cid:16) δ H (cid:128) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:138) (cid:17) ≤ C (cid:107) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:107) L (Ω , (cid:72) H ) + C (cid:13)(cid:13)(cid:13) D H · (cid:128) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:138)(cid:13)(cid:13)(cid:13) L (Ω , (cid:72) ⊗ H ) .(A.2)Note that ( u λ Q ) (cid:48) ( x ) = u λ ( x ) (cid:129) Q (cid:48) ( x ) + (cid:112) λ Q ( x ) (cid:139) , (A.3) ( u λ Q ) (cid:48)(cid:48) ( x ) = u λ ( x ) (cid:129) Q (cid:48)(cid:48) ( x ) + (cid:112) λ Q (cid:48) ( x ) + λ Q ( x ) (cid:139) . (A.4)Hence (cid:13)(cid:13)(cid:13) D H · (cid:128) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:138)(cid:13)(cid:13)(cid:13) L (Ω , (cid:72) ⊗ H ) , is bounded by (cid:69) (cid:16)(cid:13)(cid:13) ( v , θ ] ( s ) [ s ] ( r )( u λ Q ) (cid:48)(cid:48) ( B Hr ) (cid:13)(cid:13) (cid:72) ⊗ H (cid:17) ≤ α H (cid:32) θ H − H − (cid:33) (cid:69) (cid:16)(cid:13)(cid:13) ( v , θ ] ( s ) [ s ] ( r )( u λ Q ) (cid:48)(cid:48) ( B Hr ) (cid:13)(cid:13) L ([ T ] ) (cid:17) ,where we used the inequality between the (cid:72) ⊗ H and L ([ T ] ) norms given for examplein [
27, p.281 ] . Thus the previous inequality, Equality (A.4) and α H = H ( H − ) imply that (cid:13)(cid:13)(cid:13) D H · (cid:128) ( v , θ ] ( u λ Q ) (cid:48) ( B H · ) (cid:138)(cid:13)(cid:13)(cid:13) L (Ω , (cid:72) ⊗ H ) ≤ α H (cid:32) θ H − H − (cid:33) (cid:90) θ v d s (cid:90) s (cid:69) (cid:128) ( u λ Q ) (cid:48)(cid:48) ( B Hr ) (cid:138) d r ≤ C α H (cid:32) θ H − H − (cid:33) λ (cid:107) φ η (cid:107) ∞ (cid:90) θ v d s (cid:90) s (cid:69) (cid:128) { B Hr < + η } u λ ( B Hr ) (cid:138) d r ≤ C η H θ H − λ (cid:90) θ v d s (cid:90) s (cid:69) (cid:128) { B Hr < + η } u λ ( B Hr ) (cid:138) d r .In view of Lemma A.3 we see that e − λ r (cid:69) (cid:128) { B Hr < + η } u λ ( B Hr ) (cid:138) ≤ C e − α S ( − x − η ) (cid:112) λ . There-fore for all v and θ we get e − λθ θ H − λ (cid:90) θ v d s (cid:90) s (cid:69) (cid:128) { B Hr < + η } u λ ( B Hr ) (cid:138) d r ≤ C ( θ − v ) e − α S ( − x − η ) (cid:112) λ .We proceed similarly to estimate the first term in the right-hand side of (A.2) and thesecond term in the right-hand side of (A.1) (for this last term we also use Lemma A.3). Inparticular, the expectation of (A.1) is bounded by ( θ − v ) . Lemma A.4’.
Let the process Y H be as in Subsection 4.7. Recall that Q Y ( Y H θ ) = (cid:0) φ η ( Y H θ ) w (cid:48)(cid:48) λ ( Y H θ )+ φ (cid:48) η ( Y H θ ) w (cid:48) λ ( Y H θ ) (cid:1) which slightly differs from the definition of Q in the previous lemma. All othernotations come from Subsection 4.7. Let η ∈ ( Θ − y ) be small enough. There exist constantsC , α > and λ ≥ , such thate − λθ (cid:69) (cid:148) ( Q Y ( Y H θ ) − Q Y ( Y Hv )) (cid:151) ≤ C ( θ − v ) e − α S (Θ − y − η ) R ( λ ) for all λ ≥ λ and θ ≥ v ≥ . roof. Apply Itô’s formula to get (cid:69) (cid:148) ( Q Y ( Y H θ ) − Q Y ( Y Hv )) (cid:151) = (cid:69) (cid:150)(cid:18) (cid:90) θ v ( Q Y ) (cid:48) ( Y Hs ) ˜ b ( Y Hs ) d s + (cid:90) θ v ( Q Y ) (cid:48) ( Y Hs ) d B Hs + α H (cid:90) θ v (cid:90) θ v D Hr (cid:128) ( Q Y ) (cid:48) ( Y Hs ) (cid:138) | s − r | H − d r d s (cid:19) (cid:153) .We then proceed as in the proof of Lemma A.4 and use Lemma 4.5’ and Lemma 3.2 tobound D H · Y H · . The bounds on w (cid:48) λ and w (cid:48)(cid:48) λ come from Proposition B.1. Lemma A.5.
There exists a constant C such that for any ≤ t − s ≤ and any λ ≥ , (cid:90) [ s , t ] (cid:90) θ ∧ θ s e − λ v (cid:89) i = (cid:18) θ i v (cid:19) H − ( θ i − v ) H − dv d θ i ≤ C ( t − s ) H ( + | log ( t − s ) | ) . Proof.
Consider the integral w.r.t. the v variable and assume that θ > θ . The exponentialterm is bounded by e − λ s . Use the same changes of variables as in Lemma A.2: z = ( θ − v ) / ( θ − v ) and then x = z − θ /θ to obtain (cid:90) θ ∧ θ s (cid:18) θ θ v (cid:19) H − ( θ − v ) H − ( θ − v ) H − d v = ( θ − θ ) H − (cid:112) θ θ (cid:90) θ ( θ − s ) θ ( θ − s ) ( − x ) − H (cid:0) θ − θ x (cid:1) − x H − d x .Since θ ( θ − s ) θ ( θ − s ) < (cid:90) θ ( θ − s ) θ ( θ − s ) ( − x ) − H (cid:0) θ − θ x (cid:1) − x H − d x ≤ (cid:90) x H − (cid:128) ( − x ) − H − (cid:138) (cid:0) θ − θ x (cid:1) − d x + (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x .The first integral in the right-hand side is bounded by ( H − ) (cid:90) x H − | log ( − x ) | ( − x ) − H ( θ − θ x ) − d x ≤ C ( H − )( θ − θ ) − ,so that e − λ s (cid:90) ts (cid:90) t θ (cid:112) θ θ ( θ − θ ) H − (cid:90) x H − (cid:128) ( − x ) − H − (cid:138) (cid:0) θ − θ x (cid:1) − d x d θ d θ ≤ C ( t − s ) H .38e also have (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x = (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x + (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x ≤ − H H (cid:128) θ − θ (cid:138) − + − H (cid:90) (cid:0) θ − θ x (cid:1) − d x ≤ − H H θ − + − H θ − log (cid:32) θ − θ θ − θ (cid:33) ,so that (cid:90) ts (cid:90) t θ (cid:112) θ θ ( θ − θ ) H − (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x d θ d θ ≤ − H H (cid:90) ts (cid:90) t θ (cid:200) θ θ ( θ − θ ) H − d θ d θ + − H (cid:90) ts (cid:90) t θ (cid:114) θ θ ( θ − θ ) H − log (cid:32) θ − θ θ − θ (cid:33) d θ d θ .Since (cid:82) t θ ( θ − θ ) H − log (cid:16) θ − θ (cid:17) d θ ≤ (cid:82) t − s x H − log (cid:128) x (cid:138) d x = H ( t − s ) H (cid:16) H + log (cid:16) t − s (cid:17)(cid:17) ,we finally obtain e − λ s (cid:90) ts (cid:90) t θ (cid:112) θ θ ( θ − θ ) H − (cid:90) x H − (cid:0) θ − θ x (cid:1) − d x d θ d θ ≤ C (cid:90) ts (cid:32)(cid:90) t θ ( θ − θ ) H − d θ (cid:33) d θ + C ( t − s ) H ( + | log ( t − s ) | ) e − λ s (cid:90) ts θ − d θ which is the desired result. Lemma A.6.
There exists a constant C > such that for all ≤ t − s ≤ and H ∈ [ , 1 ) , (cid:90) ts e − λ v v H ( K H ( t , v ) − ) dv ≤ C ( H − ) | log ( t − s ) | ( t − s ) . Proof.
Set ˜ c H = c H ( H − ) . Observe that v H ( K H ( t , v ) − ) = ˜ c H v (cid:130)(cid:90) tv u H − ( u − v ) H − / d u (cid:140) + v H − c H v H + (cid:90) tv u H − ( u − v ) H − / d u = (cid:168) ˜ c H v (cid:90) tv u H − ( u − v ) H − / d u (cid:130) ˜ c H (cid:90) tv u H − ( u − v ) H − / d u − v H − (cid:140)(cid:171) + (cid:168) v H − ˜ c H v H + (cid:90) tv u H − ( u − v ) H − / d u (cid:171) = : A + A . 39otice that A = ˜ c H v (cid:90) tv u H − ( u − v ) H − / d u ˜ c H (cid:90) tv ( u H − − v H − )( u − v ) H − / d u + ˜ c H v (cid:90) tv u H − ( u − v ) H − / d u v H − ˜ c H ( t − v ) H − H − − : = A + A .Using u H − − v H − ≤ ( H − ) ( u − v ) v H − / , we see that A can be bounded from aboveby C ˜ c H ( v t ) H − ( t − v ) H , for some constant C >
0. The singularity in A is cancelled by A which can be written as A = v H − ˜ c H v H + (cid:90) tv u H − ( u − v ) H − / d u = − ˜ c H v H + (cid:90) tv ( u H − − v H − )( u − v ) H − / d u + v H − ˜ c H v H ( t − v ) H − H − .The absolute value of the first term in the right-hand side can be bounded by C ( H − ) ˜ c H v H − ( t − v ) H + , for some positive constant C . It then remains to estimate R : = (cid:18) c H ( t − v ) H − − (cid:19) × (cid:130) − v H + ˜ c H v H + (cid:90) tv u H − ( u − v ) H − / d u (cid:140) = v H (cid:18) c H ( t − v ) H − − (cid:19) + ˜ c H v H + (cid:18) c H ( t − v ) H − − (cid:19) (cid:90) tv ( u H − − v H − )( u − v ) H − / d u .Observe that (cid:90) ts e − λ v v H ( c H ( t − v ) H − − ) d v ≤ c ( H − ) | log ( t − s ) | ( t − s ) .The other term in the definition of R can be bounded similarly. That ends the proof. B Bound on v λ and its derivatives As already noticed, in the Brownian motion case, Laplace transforms of first hitting timesare explicit, which allowed us to easily get suitable bounds on u (cid:48) λ . We aim to get similarbounds on the two first derivatives of w λ ( x ) : = (cid:69) x (cid:128) e − λτ Y (cid:138) , where Y is the solution toEquation (3.1) with H = and τ Y is the first time Y hits Θ = F ( ) . Proposition B.1.
Under assumptions (H1’) and (H2) on b and σ , there exist constants C > µ > such that, for any x ≤ λ ≥ ,w λ ( x ) ≤ e − C ( − x ) (cid:16) (cid:112) λ + µ − µ (cid:17) w (cid:48) λ ( x ) ≤ C ( + λ ) w λ ( x ) | w (cid:48)(cid:48) λ ( x ) | ≤ C ( + λ ) w λ ( x ) , where C , µ depend only on the uniform norms of b, σ , σ (cid:48) and σ (the constant in (H2)). roof. Let Y + be defined for the same Brownian motion W as: Y + t = F ( x ) + W t + µ t ,where µ : = sup x ∈ (cid:82) | ˜ b ( x ) | (which is finite by hypothesis). F ( x ) ≤ F ( ) . Denote by τ + thefirst time Y + hits F ( ) . Since F is increasing, the comparison principle for SDEs impliesthat a.s., Y t ≤ Y + t , ∀ t ≥
0, thus τ + ≤ τ Y a.s. and thus (cid:69) (cid:128) e − λτ Y (cid:138) ≤ (cid:69) (cid:128) e − λτ + (cid:138) = e µ ( F ( ) − F ( x )) − ( F ( ) − F ( x )) (cid:112) λ + µ .where the last equality can be found in [ ] for example.Observe that F ( ) − F ( x ) ≥ ( − x ) σ , by (H2). Hence, (cid:69) (cid:128) e − λτ Y (cid:138) ≤ e − µ σ ( − x ) (cid:16) (cid:112) λµ − + − (cid:17) ,which is the desired inequality for w λ .Let us now prove the estimate on w (cid:48) λ . We use a trick provided to us by P-E. Jabin. Inview of (4.22) we have w (cid:48) λ ( x ) − w (cid:48) λ ( y ) = − (cid:90) xy ˜ b ( z ) w (cid:48) λ ( z ) d z + λ (cid:90) xy w λ ( z ) d z , ∀ x , y < Θ .Integrate w.r.t. y between x − x to obtain w (cid:48) λ ( x ) = w λ ( x ) − w λ ( x − ) + (cid:90) xx − (cid:32) − (cid:90) xy ˜ b ( z ) w (cid:48) λ ( z ) d z + λ (cid:90) xy w λ ( z ) d z (cid:33) d y .Since w λ is positive and increasing,0 ≤ w (cid:48) λ ( x ) ≤ w λ ( x ) + (cid:107) ˜ b (cid:107) ∞ (cid:90) xx − (cid:90) xy w (cid:48) λ ( z ) d z + λ (cid:90) xx − (cid:90) xy w λ ( z ) d z d y ≤ C ( + λ ) w λ ( x ) ,which is the desired result.The last desired inequality follows from the above estimate and the equation (4.22). References [ ] R.J. Adler and J.E. Taylor.
Random Fields and Geometry . Springer Verlag, 2007. [ ] E. Alòs, O. Mazet, and D. Nualart. Stochastic calculus with respect to Gaussian pro-cesses.
Ann. Probab. , 29(2):766–801, 2001. [ ] E. Alòs and D. Nualart. Stochastic integration with respect to the fractional Brownianmotion.
Stochastics and Stochastic Reports , 75(3):129–152, 2003. [ ] F. Aurzada. On the one-sided exit problem for fractional Brownian motion.
Electron.Commun. Probab [ ] F. Baudoin, E. Nualart, C. Ouyang, and S. Tindel. On probability laws of solutions todifferential systems driven by a fractional Brownian motion. 2014. arXiv:1401.3583. [ ] F. Baudoin, C. Ouyang, and S. Tindel. Upper bounds for the density of solutions tostochastic differential equations driven by fractional Brownian motions.
Ann. Inst.Henri Poincaré Probab. Stat. , 50(1):111–135, 2014. [ ] F. Bernardin, M. Bossy, C. Chauvin, J.-F. Jabir, and A. Rousseau. Stochastic Lagrangianmethod for downscaling problems in meteorology.
M2AN Math. Model. Numer. Anal. ,44(5):885–920, 2010. 41 ] C. Berzin, A. Latour, and J.R. León.
Inference on the Hurst Parameter and the Varianceof Diffusions Driven by Fractional Brownian Motion . Springer, 2014. [ ] M. Besalú, A. Kohatsu-Higa, and S. Tindel. Gaussian type lower bounds for the densityof solutions of SDEs driven by fractional Brownian motions.
Ann. Probab. , 44(1):399–443, 2016. [ ] N. H. Bingham, C. M. Goldie, and J. L. Teugels.
Regular Variation , volume 27. Cam-bridge University Press, 1989. [ ] A. N. Borodin and P. Salminen.
Handbook of Brownian Motion-Facts and Formulae .Birkhäuser, 2012. [ ] L. Coutin and Z. Qian. Stochastic analysis, rough path analysis and fractional Brown-ian motions.
Probab. Theory Related Fields , 122:108–140, 2002. [ ] R. Dalang and M. Sanz-Solé. Hitting probabilities for nonlinear systems of stochasticwaves.
Mem. Am. Math. Soc. , 237 (1120), 2015. [ ] M. Deaconu and S. Herrmann. Simulation of hitting times for Bessel processes withnon integer dimension. arXiv preprint arXiv:1401.4843 , 2014. [ ] L. Decreusefond and D. Nualart. Hitting times for Gaussian processes.
Ann. Probab. ,36(1):319–330, 2008. [ ] M. Delorme and K. J. Wiese. Maximum of a fractional Brownian motion: analyticresults from perturbation theory.
Phys. Rev. Lett. , 115:210601, Nov 2015. [ ] P. K. Friz and N. B. Victoir.
Multidimensional Stochastic Processes as Rough Paths:Theory and Applications , volume 120. Cambridge University Press, 2010. [ ] Y. Hu and D. Nualart. Differential equations driven by Hölder continuous functionsof order greater than 1 / Abel Symposium , pages 399–413, 2006. [ ] J.-H. Jeon, A. V. Chechkin, and R. Metzler. First passage behaviour of multi-dimensional fractional Brownian motion and application to reaction phenomena. InR. Metzler, S. Redner, and G. Oshani, editors,
First-Passage Phenomena and their Ap-plications . World Scientific. [ ] M. Jolis and N. Viles. Continuity in the Hurst parameter of the law of the symmetricintegral with respect to the fractional Brownian motion.
Stochastic Process. Appl. , 120(9):1651–1679, 2010. [ ] I. Karatzas and S. Shreve.
Brownian Motion and Stochastic Calculus . Springer, 1988. [ ] P. Lei and D. Nualart. Stochastic calculus for Gaussian processes and application tohitting times.
Communications in stochastic analysis , 6(3):379–402, 2012. [ ] A. Lunardi.
Analytic Semigroups and Optimal Regularity in Parabolic Problems .Birkhäuser, 2012. [ ] R. Metzler, S. Redner, and G. Oshani, editors.
First-Passage Phenomena and theirApplications . World Scientific. [ ] G. M. Molchan. Maximum of a fractional Brownian motion: probabilities of smallvalues.
Communications in mathematical physics , 205(1):97–111, 1999. [ ] I. Nourdin and F.G. Viens. Density formula and concentration inequalities with Malli-avin calculus.
Electron. J. Probab. , 14(78):2287–2309, 2009. [ ] D. Nualart.
The Malliavin Calculus and Related Topics . Springer, 2006. [ ] D. Nualart and Y. Ouknine. Regularization of differential equations by fractional noise.
Stochastic Process. Appl. , 102(1):103–116, 2002. [ ] D. Nualart and A. Rascanu. Differential equations driven by fractional Brownianmotion.
Collectanea Mathematica , 53(1):55–81, 2002. [ ] D. Nualart and B. Saussereau. Malliavin calculus for stochastic differential equations42riven by a fractional Brownian motion.
Stochastic Process. Appl. , 119(2):391–409,2009. [ ] G. Peccati, M. Thieullen, and C. Tudor. Martingale structure of Skorohod integralprocesses.
Ann. Probab. , 34(3):1217–1239, 2006. [ ] F. Russo and P. Vallois. Forward, backward and symmetric stochastic integration.
Probab. Theory Related Fields , 97(3):403–421, 1993. [ ] M. Rypdal and K. Rypdal. Testing hypotheses about sun-climate complexity linking.
Phys. Rev. Lett. , 104(12):128501, 2010. [ ] L. C. Young. An inequality of the Hölder type, connected with Stieltjes integration.
Acta Math. , 67(1):251–282, 1936. [ ] M. Zähle. Integration with respect to fractal functions and stochastic calculus. I.