# Hybrid scheme for Brownian semistationary processes

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6 Uder our assumptios, we have the followig characterizatio of the behavior of V X ear zero, which geeralizes a result of Bardorff-Nielse [3, p. 9] ad implies that X has a locally Hölder cotiuous modificatio. Therei, ad i what follows, we write ax bx, x y, to idicate ax that lim x y bx =. The proof of this result is carried out i Sectio 4.. Propositio 2.2 Local behavior ad cotiuity. Suppose that A, A2 ad A3 hold. i The variogram of X satisfies V X h E[σ 2 ] 2α + + y + α y α 2 dy h 2α+ L g h 2, h, which implies that V X is regularly varyig at zero with idex 2α +. ii The process X has a modificatio with locally φ-hölder cotiuous trajectories for ay φ, α + 2. Motivated by Propositio 2.2, we call α the roughess idex of the process X. Igorig the slowly varyig factor L g h 2 i 2.2, we see that the variogram V h behaves like h 2α+ for small values of h, which is remiiscet of the scalig property of the icremets of a fractioal Browia motio fbm with Hurst idex H = α + 2. Thus, the process X behaves locally like such a fbm, at least whe it comes to secod order structure ad roughess. Moreover, the factor 2α+ + y + α y α 2 dy coicides with the ormalizatio coefficiet that appears i the Madelbrot Va Ness represetatio [24, Theorem.3.] of a fbm with H = α + 2. Let us ow look at two examples of a kerel fuctio g that satisfies our assumptios. Example 2.3 The gamma kerel. The so-called gamma kerel gx = x α e λx, x,, with parameters α 2, 2 \ {} ad λ >, has bee used extesively i the literature o BSS processes. It is particularly importat i coectio with statistical modelig of turbulece, see Corcuera et al. [6], but it also provides a way to costruct geeralizatios of Orstei Uhlebeck OU processes with roughess that differs from the usual semimartigale case α =, while mimickig the log-term behavior of a OU process. Moreover, BSS ad LSS processes defied usig the gamma kerel have iterestig probabilistic properties, see [25]. A i-depth study of the gamma kerel ca be foud i [3]. Settig L g x := e λx, which is slowly varyig at sice lim x L g x =, it is evidet that A holds. Sice gx decays expoetially fast to as x, it is clear that also A3 holds. To verify A2, ote that g satisfies g x = α x λ gx, g x = α 2 x λ α x 2 gx, x,, where lim x α x λ2 α x 2 = λ 2 >, so g is ultimately icreasig with g x 2 α + λ 2 gx 2, x [,. Thus, g x 2 dx < sice g is square itegrable. 6

7 Example 2.4 Power-law kerel. Cosider the kerel fuctio gx = x α + x β α, x,, with parameters α 2, 2 \ {} ad β, 2. The behavior of this kerel fuctio ear zero is similar to that of the gamma kerel, but gx decays to zero polyomially as x, so it ca be used to model log memory. I fact, it ca be show that if β, 2, the the autocorrelatio fuctio of X is ot itegrable. Clearly, A holds with L g x := + x β α, which is slowly varyig at sice lim x L g x =. Moreover, ote that we ca write gx = x β K g x, x,, where K g x := + x β α satisfies lim x K g x =. Thus, also A3 holds. We ca check A2 by computig α + βx α + βx 2 g x = gx, g α 2αx βx2 x = + x + x x + x x 2 + x 2 gx, x,, where α 2αx βx 2 whe x as β < 2, so g is ultimately icreasig. Additioally, we ote that g x 2 α + β 2 gx 2, x [,, implyig g x 2 dx < sice g is square itegrable. 2.3 Hybrid scheme Let t R ad cosider discretizig Xt based o its itegral represetatio 2. o the grid G t := {t, t, t 2,...} for N. To derive our discretizatio scheme, let us first ote that if the volatility process σ does ot vary too much, the it is reasoable to use the approximatio Xt = k= t k + t k gt sσsdw s k= σ t k t k + gt sdw s, 2.3 t k that is, we keep σ costat i each discretizatio cell. Here, ad i the sequel, stads for a iformal approximatio used for purely heuristic purposes. If k is small, the due to A we may approximate [ k k gt s t s α L g, t s, k ] \ {}, 2.4 as the slowly varyig fuctio L g varies less tha the power fuctio y y α ear zero, cf If k is large, or at least k 2, the choosig b k [k, k] provides a adequate approximatio [ bk k gt s g, t s, k ], 2.5 7

9 A4 For some γ >, N γ+,. 2.4 Asymptotic behavior of mea square error We are ow ready to state our mai theoretical result, which gives a sharp descriptio of the asymptotic behavior of the mea square error MSE of the hybrid scheme as. We defer the proof of this result to Sectio 4.2. Theorem 2.5 Asymptotics of mea square error. Suppose that A, A2, A3 ad A4 hold, so that γ > 2α + 2β +, 2.9 ad that for some δ >, E[ σs σ 2 ] = O s 2α++δ, s. 2. The for all t R, where E[ Xt X t 2 ] Jα, κ, be[σ 2 ] 2α+ L g / 2,, 2. Jα, κ, b := k=κ+ k Remark 2.6. Note that if α 2,, the havig E[ σs σ 2 ] = O s θ, s, y α b α k 2 dy <. 2.2 for all θ,, esures that 2. holds. Take, say, δ := 2 2α+ > ad θ := 2α++δ = α +,. Whe the hybrid scheme is used to simulate the BSS process X o a equidistat grid {,, 2,..., T } for some T > see Sectio 3. o the details of the implemetatio, the followig cosequece of Theorem 2.5 esures that the covariace structure of the simulated process approximates that of the actual process X. Corollary 2.7 Covariace structure. Suppose that the assumptios of Theorem 2.5 hold. The for ay s, t R ad ε >, E[X tx s] E[XtXs] = O α+ 2 +ε,. Proof. Let s, t R. Applyig the Cauchy Schwarz iequality, we get E[X tx s] E[XtXs] E[X t 2 ] /2 E[ Xs X s 2 ] /2 + E[Xs 2 ] /2 E[ Xt X t 2 ] /2. We have sup N E[X t 2 ] /2 < sice E[X t 2 ] E[Xt 2 ] < as, by Theorem 2.5. Moreover, Theorem 2.5 ad the boud 2.2 imply that E[ Xs X s 2 ] /2 = O α+ 2 +ε ad E[ Xt X t 2 ] /2 = O α+ 2 +ε for ay ε >. 9

11 asymptotic RMSE κ = κ = κ = 2 κ = 3 reductio i asymptotic RMSE % κ = κ = κ = 2 κ = α α Figure : Left: The asymptotic RMSE give by Jα, κ, b as a fuctio of α 2, 2 \ {} for κ =,, 2, 3 usig b = b of Propositio 2.8 solid lies ad b = b FWD dashed lies. Right: Reductio i the asymptotic RMSE relative to the forward Riema-sum scheme κ = ad b = b FWD give by the formula 2.3, plotted as a fuctio of α 2, 2 \ {} for κ =,, 2, 3 usig b = b solid lies ad for κ =, 2, 3 usig b = b FWD dashed lies. I all computatios, we have used the approximatios outlied i Remark 2.9 with N =. RMSE approaches % as as α 2. Whe α, 2, the improvemet achieved usig the hybrid scheme is more modest, but still cosiderable. Figure also highlights the importace of usig the optimal sequece b, istead of b FWD, as evaluatio poits i the scheme, i particular whe α, 2. Fially, we observe that icreasig κ beyod 2 does ot appear to lead to a sigificat further reductio. Ideed, i our umerical experimets, reported i Sectio 3.2 ad 3.3 below, we observe that usig κ =, 2 already leads to good results. Remark 2.9. It is o-trivial to evaluate the quatity Jα, κ, b umerically. itegral i 2.2 explicitly, we ca approximate Jα, κ, b by J N α, κ, b := N k=κ+ k 2α+ k 2α+ 2α + 2bα k k α+ k α+ + b 2α k α + Computig the with some large N N. This approximatio is adequate whe α 2,, but its accuracy deteriorates whe α 2. I particular, the sigularity of the fuctio α Jα, κ, b at 2 is difficult to capture usig J N α, κ, b with umerically feasible values of N. To overcome this umerical problem, we itroduce a correctio term i the case α, 2. The correctio term ca be derived iformally as follows. By the mea value theorem, ad sice b k k 2 for large k, we have y α b α k 2 = α 2 ξ 2α 2 y b k 2 α 2 k 2α 2 y k 2, b = b FWD, α 2 k 2α 2 y k + 2 2, b = b,

12 where ξ = ξy, b k [k, k], for large k. Thus, for large N, we obtai Jα, κ, b J N α, κ, b = k=n+ k y α b α k 2 dy α 2 k=n+ k2α 2 k k y k2 dy, b = b FWD, α 2 k=n+ k2α 2 k k y k dy, b = b, α 2 3 = ζ2 2α, N +, b = b FWD, α 2 2 ζ2 2α, N +, b = b, where ζx, s := k= k+s, x >, s >, is the Hurwitz zeta fuctio, which ca be evaluated x usig accurate umerical algorithms. Remark 2.. Ulike the Fourier-based method of Beth et al. [3], the hybrid scheme does ot require trucatig the sigularity of the kerel fuctio g whe α 2,, which is beeficial to maitaiig the accuracy of the scheme whe α is ear 2. Let us briefly aalyze the effect of trucatig the sigularity of g o the approximatio error, cf. [3, pp ]. Cosider, for ay ε >, the modified BSS process X ε t := t g ε t sσsdw s, t R, defied usig the trucated kerel fuctio gε, x, ε], g ε x := gx, x ε,. Adaptig the proof of Theorem 2.5 i a straightforward maer, it is possible to show that, uder A ad A3, E [ Xt Xε t 2 ] = E[σ 2 ] ε gs gε 2ds 2α + 2 α + + E[σ 2 ]ε 2α+ L g ε 2, ε, }{{} =: Jα for ay t R. While the rate of covergece, as ε, of the MSE that arises from replacig g with g ε is aalogous to the rate of covergece of the hybrid scheme, it is importat to ote that the factor Jα blows up as α 2. I fact, Jα is equal to the first term i the series that defies Jα,, b FWD ad Jα Jα,, b FWD, α 2, which idicates that the effect of trucatig the sigularity, i terms of MSE, is similar to the effect of usig the forward Riema-sum scheme to discretize the process whe α is ear 2. I particular, the trucatio threshold ε would the have to be very small i order to keep the trucatio error i check. 2

13 2.5 Extesio to trucated Browia semistatioary processes It is useful to exted the hybrid scheme to a class of o-statioary processes that are closely related to BSS processes. This extesio is importat i coectio with a applicatio to the so-called rough Bergomi model, which we discuss i Sectio 3.3, below. More precisely, we cosider processes of the form Y t = t gt sσsdw s, t, 2.4 where the kerel fuctio g, volatility process σ ad drivig Browia motio W are as before. We call Y a trucated Browia semistatioary T BSS process, as Y is obtaied from the BSS process X by trucatig the stochastic itegral i 2. at. Of the precedig assumptios, oly A ad A2 are eeded to esure that the stochastic itegral i 2.4 exists i fact, of A2, oly the requiremet that g is differetiable o, comes ito play. The T BSS process Y does ot have covariace statioary icremets, so we defie its timedepedet variogram as V Y h, t := E[ Y t + h Y t 2 ], h, t. Extedig Propositio 2.2, we ca describe the behavior of h V Y h, t ear zero as follows. The existece of a locally Hölder cotiuous modificatio is the a straightforward cosequece. We omit the proof of this result, as it would be straightforward adaptatio of the proof of Propositio 2.2. Propositio 2. Local behavior ad cotiuity. Suppose that A ad A2 hold. i The variogram of Y satisfies for ay t, V Y h, t E[σ 2 ] 2α + +, t y + α y α 2 dy h 2α+ L g h 2, h, which implies that h V Y h, t is regularly varyig at zero with idex 2α +. ii The process Y has a modificatio with locally φ-hölder cotiuous trajectories for ay φ, α + 2. Note that while the icremets of Y are ot covariace statioary, the asymptotic behavior of V Y h, t is the same as that of V X h as h cf. Propositio 2.2 for ay t >. Thus, the icremets of Y apart from icremets startig at time are locally like the icremets of X. We defie the hybrid scheme to discretize Y t, for ay t, as where Y t := ˇY t + Ŷt, 2.5 ˇY t := mi{ t,κ} k= L g k σ t k t k + t s α dw s, t k 3

14 Ŷ t := t k=κ+ g bk σ t k W t k + W t k. I effect, we simply drop the summads i 2.7 ad 2.8 that correspod to itegrals ad icremets o the egative real lie. We make remarks o the implemetatio of this scheme i Sectio 3., below. The MSE of hybrid scheme for the T BSS process Y has the followig asymptotic behavior as, which is, i fact, idetical to the asymptotic behavior of the MSE of the hybrid scheme for BSS processes. We omit the proof of this result, which would be a simple modificatio of the proof of Theorem 2.5. Theorem 2.2 Asymptotics of mea square error. Suppose that A ad A2 hold, ad that for some δ >, E[ σs σ 2 ] = O s 2α++δ, s. The for all t >, E[ Y t Y t 2 ] Jα, κ, be[σ 2 ] 2α+ L g / 2,, where Jα, κ, b is as i Theorem 2.5 Remark 2.3. Uder the assumptios of Theorem 2.2, the coclusio of Corollary 2.7 holds mutatis mutadis. I particular, the covariace structure of the discretized T BSS process approaches that of Y whe. 3 Implemetatio ad umerical experimets 3. Practical implemetatio Simulatig the BSS process X o the equidistat grid {,, 2,..., T } for some T > usig the hybrid scheme etails geeratig i X, i =,,..., T. 3. Provided that we ca simulate the radom variables i+ Wi,j i + j := i W i := σ i i+ i i := σ α s dw s, i = N, N +,..., T, j =,..., κ, 3.2 dw s, i = N, N +,..., T, 3.3, i = N, N +,..., T, 4

17 3 α=.5 2 Xt κ = κ = κ = t 3 α=.45 2 Xt t 3 α= Xt t Figure 2: Discretized trajectories of a BSS process, where g is the gamma kerel Example 2.3, λ = ad σt = for all t R. Trajectories cosistig of = 5 observatios o [, ] were geerated with the hybrid scheme κ =, 2 ad b = b ad Riema-sum scheme κ = ad b = b solid lies, b = b FWD dashed lies, usig the same iovatios for the drivig Browia motio i all cases ad N = 5.5 = 353. The simulated processes were ormalized to have uit statioary variace. 7

19 Bias.4.2 s = exact κ =.2 κ = κ = 2 κ = α.2.4 s =2.4 Stadard deviatio s = exact κ = κ = κ = 2 κ = α.2.4 s =2 Bias.2.2 Stadard deviatio α.2.4 s = α.2.4 s =5 Bias.2.2 Stadard deviatio α α Figure 3: Bias ad stadard deviatio of the COF estimator 3.8 of the roughess idex α, whe applied to discretized trajectories of a BSS process with the gamma kerel Example 2.3, λ = ad σt = for all t R. Trajectories were geerated usig a exact method based o the Cholesky factorizatio, the hybrid scheme κ =, 2, 3 ad b = b ad Riema-sum scheme κ = ad b = b solid lies, b = b FWD dashed lies. I the experimet, = ms observatios were geerated, where m = 5 ad s {, 2, 5}, o [, ] usig N =.5. Every s-th observatio was the subsampled, resultig i m = 5 observatios that were used to compute the estimate ˆαX, m of the roughess idex α. Number of Mote Carlo replicatios:. 9

20 the bias i the case α 2, is positive, idicatig too smooth discretized trajectories, which is coected with the failure of the Riema-sum scheme with α ear 2, illustrated i Figure 2. With s = 2 ad s = 5, the results improve with both schemes. Notably, i the case s = 5, the performace of the hybrid scheme eve with κ = is o a par with the exact method. However, the improvemets with the Riema-sum scheme are more meager, as cosiderable bias persists whe α is ear Optio pricig uder rough volatility As aother experimet, we study Mote Carlo optio pricig i the rough Bergomi rbergomi model of Bayer et al. []. I the rbergomi model, the logarithmic spot variace of the price of the uderlyig is modelled by a rough Gaussia process, which is a special case of 2.4. By virtue of the rough volatility process, the model fits well to observed implied volatility smiles [, pp ]. More precisely, the price of the uderlyig i the rbergomi model with time horizo T > is defied, uder a equivalet martigale measure idetified with P, as t St := S exp usig the spot variace process vt := ξ t exp η 2α + vsdzs 2 t t t s α dw s }{{} =:Y t vsds, t [, T ], η2 2 t2α+, t [, T ]. Above, S >, η > ad α 2, are determiistic parameters, ad Z is a stadard Browia motio give by Zt := ρw t + ρ 2 W t, t [, T ], 3.9 where ρ, is the correlatio parameter ad {W t} t [,T ] is a stadard Browia motio idepedet of W. The process {ξ t} t [,T ] is the so-called forward variace curve [, p. 89], which we assume here to be flat, ξ t = ξ > for all t [, T ]. We aim to compute usig Mote Carlo simulatio the price of a Europea call optio struck at K > with maturity T, which is give by CS, K, T := E[ST K + ]. 3. The approach suggested by Bayer et al. [] ivolves samplig the Gaussia processes Z ad Y o a discrete time grid usig exact simulatio ad the approximatig S ad v usig Euler discretizatio. We modify this approach by usig the hybrid scheme to simulate Y, istead of the computatioally more costly exact simulatio. As the hybrid scheme ivolves simulatig icremets of the Browia motio W drivig Y, we ca coveietly simulate the icremets of Z, eeded for the Euler discretizatio of S, usig the represetatio

22 4 Proofs Throughout the proofs below, we rely o two useful iequalities. The first oe is the Potter boud for slow variatio at, which follows immediately from the correspodig result for slow variatio at [5, Theorem.5.6]. Namely, if L :, ], is slowly varyig at ad bouded away from ad o ay iterval u, ], u,, the for ay δ > there exists a costat C δ > such that { Lx x Ly C δ max y δ, x y The secod oe is the elemetary iequality } δ, x, y, ]. 4. x α y α α mi{x, y} α x y, x, y,, α,, 4.2 which ca be easily show usig the mea value theorem. Additioally, we use the followig variat of Karamata s theorem for regular variatio at. Its proof is similar to the oe of the usual Karamata s theorem for regular variatio at [5, Propositio.5.]. Lemma 4. Karamata s theorem. If α, ad L :, ] [, is slowly varyig at, the y x α Lxdx α + yα+ Ly, y. 4. Proof of Propositio 2.2 Proof of Propositio 2.2. i By the covariace statioarity of the volatility process σ, we may express the variogram V h for ay h as V h = E[ Xh X 2 ] = h gh u g u, u 2 E[σu 2 ]du h = E[σ 2 ] gx 2 dx + gx + h gx 2 dx. 4.3 Ivokig A ad Lemma 4., we fid that h gx 2 dx 2α + h2α+ L g h 2, h. 4.4 We may clearly assume that h <, which allows us to work with the decompositio where gx + h gx 2 dx = A h + A h, A h := h gx + h gx 2 dx, A h := gx + h gx 2 dx. h 22

23 Accordig to A2, there exists M > such that x g x is o-icreasig o [M,. Thus, usig the mea value theorem, we deduce that gx + h gx = g sup y h,m] g y h, x h, M, ξ h g x h, x [M,. where ξ = ξx, h [x, x + h]. It follows the that lim sup h which i tur implies that A h M sup g y 2 + h2 y [,M] g x 2 dx <, A h = Oh2, h. 4.5 where Makig a substitutio y = x h, we obtai A h = h G h y := gx + h gx 2 dx = h /h ghy + ghy 2dy = h 2α+ L g h 2 G h ydy, y + α L ghy + y α L 2 ghy L g h L g h,/h y, y,. By the defiitio of slow variatio at, lim G hy = y + α y α 2, h y,. We shall show below that the fuctios G h, h,, have a itegrable domiat. Thus, by the domiated covergece theorem, A h h 2α+ L g h 2 y + α y α 2 dy, h. 4.6 Sice α < 2, we have lim h h gx 2 dx + A h h 2α+ L gh 2 = by 2.2 ad 4.5, so we get from 4.4 ad 4.6 gx + h gx 2 dx 2α + + y + α y α 2 dy h 2α+ L g h 2, h, which, together with 4.3, implies the assertio. It remais to justify the use of the domiated covergece theorem to deduce 4.6. For ay y, ], we have by the Potter boud 4. ad the elemetary iequality u + v 2 2u 2 + 2v 2, G h y 2y + 2α Lg hy + L g h 2C 2 δ y + 2α+δ + y 2α δ, 2 + 2y 2α Lg hy L g h 23 2

24 where we choose δ, α+ 2 to esure that 2α δ >. Cosider the y [,. By addig ad substractig the term y + α L ghy L gh ad usig agai the iequality u + v 2 2u 2 + 2v 2, we get G h y = y + α L ghy + y + α L ghy L g h L g h + y + α L ghy L g h L 2 ghy yα L g h,/h y 2y + 2α Lg hy + L g hy 2 L g h,/h y + 2 y + α y α 2 Lg hy 2 L g h,/h y. We recall that L g := if x,] L g x > by A, so L g hy + L g hy L g h,/h y L g hy + L g hy L,/h y. g Usig the mea value theorem ad the boud for the derivative of L g from A, we observe that L g hy + L g hy = L gξ hy + hy hc + C h +, ξ y where ξ = ξy, h [hy, hy + ]. Notig that the costrait y < h we obtai further L g hy + L g hy L g h,/h y C h+ L g y is equivalet to h < y+,,/h y C L g y + + C 3 y L g y +, as y, which we the use to deduce that 2y + 2α Lg hy + L g hy 2 L g h,/h y 8C2 y + 2α. Additioally, we observe that, by 4. ad 4.2, 2 y + α y α 2 Lg hy 2 L g h,/h y 2Cδ 2 2 α 2 y 2α +δ2, where we choose δ 2, 2 α, esurig that 2α +δ 2 <. We may fially defie a fuctio 2C 2 δ Gy := y + 2α+δ + y 2α δ, y, ], 8C 2 y + 2α + 2C L 2 δ 2 g 2 α 2 y 2α +δ2, y,, which satisfies G h y Gy for ay y, ad h,, ad is itegrable o, with the aforemetioed choices of δ ad δ 2. ii To show existece of the modificatio, we eed a localizatio procedure that ivolves a acillary process F t := g s 2 σt s 2 ds, t R. 24 L 2 g

25 We check first that F is locally bouded uder A ad A2, which is essetial for localizatio. To this ed, let T,, ad write for ay t [ T, T ], where F t = F t + F t, F t := M+2T g s 2 σt s 2 ds, F t := M+2T g s 2 σt s 2 ds, ad M > is such that x g x is o-icreasig o [M,, as i the proof of i. Sice g is cotiuous o, ad σ locally bouded, we have for ay t [ T, T ], F t M + 2T sup g y 2 y [,M+2T ] Further, whe t [ T, T ], F t = t M+2T g t u 2 σu 2 du, sup u [ M 3T,T ] where g t u 2 g T u 2 sice the argumets satisfy Thus, t u T u T t M + 2T M. F t M+T g T u 2 σu 2 du σu 2 <. g s 2 σ T s 2 ds < for ay t [ T, T ] almost surely, as we have [ ] E g s 2 σ T s 2 ds = g s 2 E[σ T s 2 ]ds = E[σ 2 ] g s 2 ds <, where we chage the order of expectatio ad itegratio relyig o Toelli s theorem ad where the fial equality follows from the covariace statioarity of σ. So we ca coclude that F is ideed locally bouded. Let ow m N ad, for localizatio, defie a sequece of stoppig times τ m, := if{t [ m, : F t > or σt > }, N, that satisfies τ m, almost surely as sice both F ad σ are locally bouded. We follow the usual covetio that if =. Cosider ow the modified BSS process X m,t := t gt sσmi{s, τ m, }dw s, t [ m,, that coicides with X o the stochastic iterval m, τ m,. The process X m, satisfies the assumptios of [5, Lemma ], so for ay p > there exists a costat Ĉp > such that E[ X m,s X m,t p ] ĈpV s t p/2, s, t [ m,

26 Usig the upper boud i 2.2, we ca deduce from i that for ay δ > there are costats C δ > ad h δ > such that V h C δ h 2α+ δ, h, h δ. 4.8 Applyig 4.8 to 4.7, we get E[ X m,s X m,t p ] Ĉp C p/2 δ s t +pα+ 2 δ 2 p, s, t [ m,, s t < h δ. We may ote that pα + 2 δ 2 p > for small eough δ ad large eough p ad, i particular, pα + 2 δ 2 p α + p 2, as δ ad p. Thus it follows from the Kolmogorov Chetsov theorem [22, Theorem 3.22] that X m, has a modificatio with locally φ-hölder cotiuous trajectories for ay φ, α + 2. Moreover, a modificatio of X o R, havig locally φ-hölder cotiuous trajectories for ay φ, α + 2, ca the by costructed from these modificatios of X m,, m N, N, by lettig first ad the m. 4.2 Proof of Theorem 2.5 As a preparatio, we shall first establish a auxiliary result that deals with the asymptotic behavior of certai itegrals of regularly varyig fuctios. Lemma 4.2. Suppose that L :, ] [, is bouded away from ad o ay set of the form u, ], u,, ad slowly varyig at. Moreover, let α 2, ad k. If b [k, k] \ {}, the i lim k x α Lx/ Lb/ bα L/ L/ 2 dx = x α b α 2 dx <, k Lx/ ii lim x 2α k L/ Lb/ 2 dx =. L/ Proof. We oly prove i as ii ca be show similarly. By the defiitio of slow variatio at, the fuctio f x := x α Lx/ Lb/ 2 bα, x [k, k] \ {}, L/ L/ satisfies lim f x = x α b α 2 for ay x [k, k]\{}. I view of the domiated covergece theorem, it suffices to fid a itegrable domiat for the fuctios f, N. The costructio of the domiat is quite similar to the oe see i the proof of Propositio 2.2, but we provide the details for the coveiece of the reader. Usig the Potter boud 4. ad the iequality u + v 2 2u 2 + 2v 2, we fid that for ay x [k, k] \ {}, Lx/ f x 2x 2α L/ 2C 2 δ 2 Lb/ 2 + 2b 2α L/ x 2α max { x δ, x δ} 2 + b 2α max { b δ, b δ} 2 =: fx, 26

27 where we choose δ, α + 2. Whe k 2, we have x ad b, so fx = 2Cδ 2 x 2α+δ + b 2α+δ is a bouded fuctio of x o [k, k]. Whe k =, we have x ad b, implyig that fx = 2C 2 δ x 2α δ + b 2α δ, where 2α δ > with our choice of δ, so f is a itegrable fuctio o, ]. Proof of Theorem 2.5. Let t R be fixed. It will be coveiet to write X t as X t = κ t k k= t k N t k + k=κ+ t k t s α L g k g bk σ t k dw s σ t k dw s. Moreover, we itroduce a acillary approximatio of Xt, amely, N X t = k= t k t k By Mikowski s iequality, we have gt sσ t k t N dw s + gt sσsdw s. E [ X t Xt 2] 2 E [ X t X t 2] 2 E [ X t Xt 2] 2, E [ X t Xt 2] 2 E [ X t X t 2] 2 + E [ X t Xt 2] 2, which together, after takig squares, imply that E E 2 + E E [ X t Xt 2] E E E, 4.9 E E E E where E := E [ X t X t 2], E := E [ Xt X t 2]. Usig the Itō isometry, ad recallig that σ is covariace statioary, we obtai N E = k= t k t k [ gt s 2 E σ t k 2 ] σs ds sup E [ σu σ 2] u, ] gs 2 ds 27

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