Hybrid scheme for Brownian semistationary processes


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1 Hybrid scheme for Browia semistatioary processes Mikkel Beedse Asger Lude Mikko S. Pakkae September 8, 28 arxiv:57.34v4 [math.pr] 4 May 27 Abstract We itroduce a simulatio scheme for Browia semistatioary processes, which is based o discretizig the stochastic itegral represetatio of the process i the time domai. We assume that the kerel fuctio of the process is regularly varyig at zero. The ovel feature of the scheme is to approximate the kerel fuctio by a power fuctio ear zero ad by a step fuctio elsewhere. The resultig approximatio of the process is a combiatio of Wieer itegrals of the power fuctio ad a Riema sum, which is why we call this method a hybrid scheme. Our mai theoretical result describes the asymptotics of the mea square error of the hybrid scheme ad we observe that the scheme leads to a substatial improvemet of accuracy compared to the ordiary forward Riemasum scheme, while havig the same computatioal complexity. We exemplify the use of the hybrid scheme by two umerical experimets, where we examie the fiitesample properties of a estimator of the roughess parameter of a Browia semistatioary process ad study Mote Carlo optio pricig i the rough Bergomi model of Bayer et al. [], respectively. Keywords: Stochastic simulatio; discretizatio; Browia semistatioary process; stochastic volatility; regular variatio; estimatio; optio pricig; rough volatility; volatility smile. JEL Classificatio: C22, G3, C3 MSC 2 Classificatio: 6G2, 6G22, 65C2, 9G6, 62M9 Itroductio We study simulatio methods for Browia semistatioary BSS processes, first itroduced by BardorffNielse ad Schmiegel [8, 9], which form a flexible class of stochastic processes that are able to capture some commo features of empirical time series, such as stochastic volatility itermittecy, roughess, statioarity ad strog depedece. By ow these processes have bee Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Ecoomics ad Busiess Ecoomics ad CREATES, Aarhus Uiversity, Fuglesags Allé 4, 82 Aarhus V, Demark. Departmet of Mathematics, Imperial College Lodo, South Kesigto Campus, Lodo SW7 2AZ, UK ad CREATES, Aarhus Uiversity, Demark.
2 applied i various cotexts, most otably i the study of turbulece i physics [7, 6] ad i fiace as models of eergy prices [4, ]. A BSS process X is defied via the itegral represetatio Xt = t gt sσsdw s,. where W is a twosided Browia motio providig the fudametal oise iovatios, the amplitude of which is modulated by a stochastic volatility itermittecy process σ that may deped o W. This drivig oise is the covolved with a determiistic kerel fuctio g that specifies the depedece structure of X. The process X ca also be viewed as a movig average of volatilitymodulated Browia oise ad settig σs =, we see that statioary Browia movig averages are ested i this class of processes. I the applicatios metioed above, the case where X is ot a semimartigale is particularly relevat. This situatio arises whe the kerel fuctio g behaves like a powerlaw ear zero; more specifically, whe for some α 2, 2 \ {}, gx x α for small x >..2 Here we write to idicate proportioality i a iformal sese, aticipatig a rigorous formulatio of this relatioship give i Sectio 2.2 usig the theory of regular variatio [5], which plays a sigificat role i our subsequet argumets. The case α = 6 i.2 is importat i statistical modelig of turbulece [6] as it gives rise to processes that are compatible with Kolmogorov s scalig law for ideal turbulece. Moreover, processes of similar type with α.4 have bee recetly used i the cotext of optio pricig as models of rough volatility [,, 8, 2], see Sectios 2.5 ad 3.3 below. The case α = would roughly speakig lead to a process that is a semimartigale, which is thus excluded. Uder.2, the trajectories of X behave locally like the trajectories of a fractioal Browia motio with Hurst idex H = α + 2, \ { 2 }. While the local behavior ad roughess, measured i terms of Hölder regularity, of X are determied by the parameter α, the global behavior of X e.g., whether the process has log or short memory depeds o the behavior of gx as x, which ca be specified idepedetly of α. This should be cotrasted with fractioal Browia motio ad related selfsimilar models, which ecessarily must coform to a restrictive affie relatioship betwee their Hölder regularity local behavior ad roughess ad Hurst idex global behavior, as elucidated by Geitig ad Schlather [2]. Ideed, i the realm of BSS processes, local ad global behavior are coveietly decoupled, which uderlies the flexibility of these processes as a modelig framework. I coectio with practical applicatios, it is importat to be able to simulate the process X. If the volatility process σ is determiistic ad costat i time, the X will be strictly statioary ad Gaussia. This makes X ameable to exact simulatio usig the Cholesky factorizatio or circulat embeddigs, see, e.g., [2, Chapter XI]. However, it seems difficult, if ot impossible, to develop a exact method that is applicable with a stochastic σ, as the process X is the either Markovia or Gaussia. Thus, i the geeral case oe must resort to approximative methods. To this ed, Beth et al. [3] have recetly proposed a Fourierbased method of simulatig BSS 2
3 processes, ad more geeral Lévy semistatioary LSS processes, which relies o approximatig the kerel fuctio g i the frequecy domai. I this paper, we itroduce a ew discretizatio scheme for BSS processes based o approximatig the kerel fuctio g i the time domai. Our startig poit is the Riemasum discretizatio of.. The Riemasum scheme builds o a approximatio of g usig step fuctios, which has the pitfall of failig to capture appropriately the steepess of g ear zero. I particular, this becomes a serious defect uder.2 whe α 2,. I our ew scheme, we mitigate this problem by approximatig g usig a appropriate power fuctio ear zero ad a step fuctio elsewhere. The resultig discretizatio scheme ca be realized as a liear combiatio of Wieer itegrals with respect to the drivig Browia motio W ad a Riema sum, which is why we call it a hybrid scheme. The hybrid scheme is oly slightly more demadig to implemet tha the Riemasum scheme ad the schemes have the same computatioal complexity as the umber of discretizatio cells teds to ifiity. Our mai theoretical result describes the exact asymptotic behavior of the mea square error MSE of the hybrid scheme ad, as a special case, that of the Riemasum scheme. We observe that switchig from the Riemasum scheme to the hybrid scheme reduces the asymptotic root mea square error RMSE substatially. Usig merely the simplest variat the of hybrid scheme, where a power fuctio is used i a sigle discretizatio cell, the reductio is at least 5% for all α, 2 ad at least 8% for all α 2,. The reductio i RMSE is close to % as α approches 2, which idicates that the hybrid scheme ideed resolves the problem of poor precisio that affects the Riemasum scheme. To assess the accuracy of the hybrid scheme i practice, we perform two umerical experimets. Firstly, we examie the fiitesample performace of a estimator of the roughess idex α, itroduced by BardorffNielse et al. [6] ad Corcuera et al. [6]. This experimet eables us to assess how faithfully the hybrid scheme approximates the fie properties of the BSS process X. Secodly, we study Mote Carlo optio pricig i the rough Bergomi stochastic volatility model of Bayer et al. []. We use the hybrid scheme to simulate the volatility process i this model ad we fid that the resultig implied volatility smiles are idistiguishable from those simulated usig a method that ivolves exact simulatio of the volatility process. Thus we are able propose a solutio to the problem of fidig a efficiet simulatio scheme for the rough Bergomi model, left ope i the paper []. The rest of this paper is orgaized as follows. I Sectio 2 we recall the rigorous defiitio of a BSS process ad itroduce our assumptios. We also itroduce the hybrid scheme, state our mai theoretical result cocerig the asymptotics of the mea square error ad discuss a extesio of the scheme to a class of trucated BSS processes. Sectio 3 briefly discusses the implemetatio of the discretizatio scheme ad presets the umerical experimets metioed above. Fially, Sectio 4 cotais the proofs of the theoretical ad techical results give i the paper. 3
4 2 The model ad theoretical results 2. Browia semistatioary process Let Ω, F, {F t } t R, P be a filtered probability space, satisfyig the usual coditios, supportig a twosided stadard Browia motio W = {W t} t R. We cosider a Browia semistatioary process Xt = t gt sσsdw s, t R, 2. where σ = {σt} t R is a {F t } t R predictable process with locally bouded trajectories, which captures the stochastic volatility itermittecy of X, ad g :, [, is a Borel measurable kerel fuctio. To esure that the itegral 2. is welldefied, we assume that the kerel fuctio g is square itegrable, that is, gx 2 dx <. I fact, we will shortly itroduce some more specific assumptios o g that will imply its square itegrability. Throughout the paper, we also assume that the process σ has fiite secod momets, E[σt 2 ] < for all t R, ad that the process is covariace statioary, amely, E[σs] = E[σt], Covσs, σt = Covσ, σ s t, s, t R. These assumptios imply that also X is covariace statioary, that is, E[Xt] =, CovXs, Xt = E[σ 2 ] gxgx + s t dx, s, t R. However, the process X eed ot be strictly statioary as the depedece betwee the volatility process σ ad the drivig Browia motio W may be timevaryig. 2.2 Kerel fuctio As metioed above, we cosider a kerel fuctio that satisfies gx x α for some α 2, 2 \{} whe x > is ear zero. To make this idea rigorous ad to allow for additioal flexibility, we formulate our assumptios o g usig the theory of regular variatio [5] ad, more specifically, slowly varyig fuctios. To this ed, recall that a measurable fuctio L :, ] [, is slowly varyig at if for ay t >, Ltx lim x Lx =. Moreover, a fuctio fx = x β Lx, x, ], where β R ad L is slowly varyig at, is said to be regularly varyig at, with β beig the idex of regular variatio. Remark 2.. Covetioally, slow ad regular variatio are defied at [5, pp. 6, 7 8]. However, L is slowly varyig resp. regularly varyig at if ad oly if x L/x is slowly varyig resp. regularly varyig at. 4
5 A key feature of slowly varyig fuctios, which will be very importat i the sequel, is that they ca be sadwiched betwee polyomial fuctios as follows. If δ > ad L is slowly varyig at ad bouded away from ad o ay iterval u, ], u,, the there exist costats C δ C δ > such that C δ x δ Lx C δ x δ, x, ]. 2.2 The iequalities above are a immediate cosequece of the socalled Potter bouds for slowly varyig fuctios, see [5, Theorem.5.6ii] ad 4. below. Makig δ very small therei, we see that slowly varyig fuctios are asymptotically egligible i compariso with polyomially growig/decayig fuctios. Thus, by multiplyig power fuctios ad slowly varyig fuctios, regular variatio provides a flexible framework to costruct fuctios that behave asymptotically like power fuctios. Our assumptios cocerig the kerel fuctio g are as follows: A For some α 2, 2 \ {}, gx = x α L g x, x, ], where L g :, ] [, is cotiuously differetiable, slowly varyig at ad bouded away from. Moreover, there exists a costat C > such that the derivative L g of L g satisfies L gx C + x, x, ]. A2 The fuctio g is cotiuously differetiable o,, so that its derivative g is ultimately mootoic ad satisfies g x 2 dx <. A3 For some β, 2, gx = Ox β, x. Here, ad i the sequel, we use fx = Ohx, x a, to idicate that lim sup x a fx hx <. Additioally, aalogous otatio is later used for sequeces ad computatioal complexity. I view of the boud 2.2, these assumptios esure that g is square itegrable. It is worth poitig out that A accommodates fuctios L g with lim x L g x =, e.g., L g x = log x. The assumptio A iflueces the shortterm behavior ad roughess of the process X. A simple way to assess the roughess of X is to study the behavior of its variogram also called the secodorder structure fuctio i turbulece literature V X h := E[ Xh X 2 ], h, as h. Note that, by covariace statioarity, V X s t = E[ Xs Xt 2 ], s, t R. 5
6 Uder our assumptios, we have the followig characterizatio of the behavior of V X ear zero, which geeralizes a result of BardorffNielse [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 φhölder 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 socalled 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 logterm behavior of a OU process. Moreover, BSS ad LSS processes defied usig the gamma kerel have iterestig probabilistic properties, see [25]. A idepth 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 Powerlaw 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
8 by A2. Applyig 2.4 to the first κ terms, where κ =, 2,..., ad 2.5 to the remaiig terms i the approximatig series i 2.3 yields k= σ t k t k + gt sdw s t k κ k L g k= + k=κ+ g σ t k t k + t s α dw s t k bk σ t k t k + t k dw s, For completeess, we also allow for κ =, i which case we require that b, ] ad iterpret the first sum o the righthad side of 2.6 as zero. To make umerical implemetatio feasible, we trucate the secod sum o the righthad side of 2.6 so that both sums have N κ + terms i total. Thus, we arrive at a discretizatio scheme for Xt, which we call a hybrid scheme, give by where X t := ˇX t + ˆX t, ˇX t := ˆX t := κ k L g k= N k=κ+ g bk 2.6 σ t k t k + t s α dw s, 2.7 t k σ t k W t k + W t k, 2.8 ad b := {b k } k=κ+ is a sequece of real umbers, evaluatio poits, that must satisfy b k [k, k] \ {} for each k κ +, but otherwise ca be chose freely. As it stads, the discretizatio grid G t depeds o the time t, which may seem cumbersome with regard to samplig X t simultaeously for differet times t. However, ote that wheever times t ad t are separated by a multiple of, the correspodig grids G t ad G t will itersect. I fact the hybrid scheme defied by 2.7 ad 2.8 ca be implemeted efficietly, as we shall see i Sectio 3., below. Sice bk g = g t t b k, the degeerate case κ = with b k = k for all k correspods to the usual Riemasum discretizatio scheme of Xt with Itō type forward sums from 2.8. Heceforth, we deote the associated sequece {k} k=κ+ by b FWD, where the subscript FWD alludes to forward sums. However, icludig terms ivolvig Wieer itegrals of a power fuctio give by 2.7, that is havig κ, improves the accuracy of the discretizatio cosiderably, as we shall see. Havig the leeway to select b k withi the iterval [k, k] \ {}, so that the fuctio gt is evaluated at a poit that does ot ecessarily belog to G t, leads additioally to a moderate improvemet. The tructio i the sum 2.8 etails that the stochastic itegral 2. defiig X is trucated at t N I practice, the value of the parameter N should be large eough to mitigate the. effect of trucatio. To esure that the trucatio poit t N asymptotic results, we itroduce the followig assumptio: 8 teds to as i our
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
10 I Theorem 2.5, the asymptotics of the MSE 2. are determied by the behavior of the kerel fuctio g ear zero, as specified i A. The coditio 2.9 esures that error from approximatig g ear zero is asymptotically larger tha the error iduced by the trucatio of the stochastic itegral 2. at t N. I fact, differet kid of asymptotics of the MSE, where trucatio error becomes domiat, could be derived whe 2.9 does ot hold, uder some additioal assumptios, but we do ot pursue this directio i the preset paper. While the rate of covergece i 2. is fully determied by the roughess idex α, which may seem discouragig at first, it turs out that the quatity Jα, κ, b, which we shall call the asymptotic MSE, ca vary a lot, depedig o how we choose κ ad b, ad ca have a substatial impact o the precisio of the approximatio of X. It is immediate from 2.2 that icreasig κ will decrease Jα, κ, b. Moreover, for give α ad κ, it is straightforward to choose b so that Jα, κ, b is miimized, as show i the followig result. Propositio 2.8 Optimal discretizatio. Let α 2, 2 \ {} ad κ. Amog all sequeces b = {b k } k=κ+ with b k [k, k] \ {} for k κ +, the fuctio Jα, κ, b, ad cosequetly the asymptotic MSE iduced by the discretizatio, is miimized by the sequece b give by k b α+ k = k α+ /α, k κ +. α + Proof. Clearly, a sequece b = {b k } k=κ+ miimizes the fuctio Jα, κ, b if ad oly if b k miimizes k yα b α k 2 dy for ay k κ +. By stadard L 2 space theory, c R miimizes the itegral k yα c 2 dy if ad oly if the fuctio y y α c is orthogoal i L 2 to all costat fuctios. This is tatamout to k y α cdy =, ad computig the itegral ad solvig for c yields c = kα+ k α+. α + Settig b k := c/α k, k completes the proof. To uderstad how much icreasig κ ad usig the optimal sequece b from Propositio 2.8 improves the approximatio, we study umerically the asymptotic root mea square error RMSE Jα, κ, b. I particular, we assess how much the asymptotic RMSE decreases relative to RMSE of the forward Riemasum scheme κ = ad b = b FWD usig the quatity Jα, κ, b Jα,, bfwd reductio i asymptotic RMSE = %. 2.3 Jα,, bfwd The results are preseted i Figure. We fid that employig the hybrid scheme with κ leads to a substatial reductio i the asymptotic RMSE relative to the forward Riemasum scheme whe α 2,. Ideed, whe κ, the asymptotic RMSE, as a fuctio of α, does ot blow up as α 2, while with κ = it does. This explais why the reductio i the asymptotic
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 Riemasum 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 otrivial 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 Fourierbased 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 Riemasum 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 ostatioary processes that are closely related to BSS processes. This extesio is importat i coectio with a applicatio to the socalled 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 φhölder 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
15 we ca compute 3. via the formula i κ k X = L g σi k W N b i k,k + g k σi k W i k. 3.4 k= k=κ+ }{{}}{{} = ˇX i = ˆX i I order to simulate 3.2 ad 3.3, it is istrumetal to ote that the κ + dimesioal radom vectors W i := W i, W i,,..., W i,κ, i = N, N +,..., T, are i.i.d. accordig to a multivariate Gaussia distributio with mea zero ad covariace matrix Σ give by Σ, =, Σ,j = Σ j, = j α+ j 2 α+ α + α+, Σ j,j = j 2α+ j 2 2α+ 2α + 2α+, for j = 2,..., κ +, ad Σ j,k = j α + 2α+ α+ k α2f α,, α + 2, j k j 2 α+ k 2 α 2F α,, α + 2, j 2, 3.5 k 2 for j, k = 2,..., κ + such that j < k, where 2 F stads for the Gauss hypergeometric fuctio, see, e.g., [7, p. 56] for the defiitio. Whe k < j, set Σ j,k = Σ k,j. For the coveiece of the reader, we provide a proof of 3.5 i Sectio 4.3. Thus, {Wi } T i= N ca be geerated by takig idepedet draws from the multivariate Gaussia distributio N κ+, Σ. If the volatility process σ is idepedet of W, the {σi } T i= N ca be geerated separately, possibly usig exact methods. Exact methods are available, e.g., for Gaussia processes, as metioed i the itroductio, ad diffusios, see [4]. I the case where σ depeds o W, simulatig {Wi } T i= N ad {σi } T i= N is less straightforward. That said, if σ is drive by a stadard Browia motio Z, correlated with W, say, oe could rely o a factor decompositio Zt := ρw t + ρ 2 W t, t R, 3.6 where ρ [, ] is the correlatio parameter ad {W t} t [,T ] is a stadard Browia motio idepedet of W. The oe would first geerate {Wi } T i= N, use 3.6 to geerate {Z i+ Z i } T i= N } T i= N thereafter. ad employ some appropriate approximate method to produce {σi This approach has, however, the caveat that it iduces a additioal approximatio error, ot quatified i Theorem 2.5. Remark 3.. I the case of the T BSS process Y, itroduced i Sectio 2.5, the observatios Y i, i =,,..., T, give by the hybrid scheme 2.5 ca be computed via i Y = mi{i,κ} k= L g k σ i k W i k,k + i k=κ+ 5 b g k σi k W i k, 3.7
16 usig the radom vectors {Wi } T i= ad radom variables {σi } T i=. I the hybrid scheme, it typically suffices to take κ to be at most 3. Thus, i 3.4, the first sum ˇX i requires oly a egligible computatioal effort. By cotrast, the umber of terms i the secod sum ˆX i icreases as. It is the useful to ote that i N ˆX = Γ k Ξ i k = Γ Ξ i, k= where, k =,..., κ, Γ k := g b k, k = κ +, κ + 2,..., N, Ξ k := σ k W k, k = N, N +,..., T. ad Γ Ξ stads for the discrete covolutio of the sequeces Γ ad Ξ. It is wellkow that the discrete covolutio ca be evaluated efficietly usig a fast Fourier trasform FFT. The computatioal complexity of simultaeously evaluatig Γ Ξ i for all i =,,..., T usig a FFT is ON log N, see [23, pp. 79 8], which uder A4 traslates to O γ+ log. The computatioal complexity of the etire hybrid scheme is the O γ+ log, provided that {σi } T i= N is geerated usig a scheme with complexity ot exceedig O γ+ log. As a compariso, we metio that the complexity of a exact simulatio of a statioary Gaussia process usig circulat embeddigs is O log [2, p. 36], whereas the complexity of the Cholesky factorizatio is O 3 [2, p. 32]. Remark 3.2. With T BSS processes, the computatioal complexity of the hybrid scheme via 3.7 is O log. Figure 2 presets examples of trajectories of the BSS process X usig the hybrid scheme with κ =, 2 ad b = b. We choose the kerel fuctio g to be the gamma kerel Example 2.3 with λ =. We also discretize X usig the Riemasum scheme, κ = with b {b FWD, b } that is, the forward Riemasum scheme ad its couterpart with optimally chose evaluatio poits. We ca make two observatios: Firstly, we see how the roughess parameter α cotrols the regularity properties of the trajectories of X as we decrease α, the trajectories of X become icreasigly rough. Secodly, ad more importatly, we see how the simulated trajectories comig from the Riemasum ad hybrid schemes ca be rather differet, eve though we use the same iovatios for the drivig Browia motio. I fact, the two variats of the hybrid scheme κ =, 2 yield almost idetical trajectories, while the Riemasum scheme κ = produces trajectories that are comparatively smoother, this differece becomig more apparet as α approaches 2. Ideed, i the extreme case with α =.499, both variats of the Riemasum scheme break dow ad yield aomalous trajectories with very little variatio, while the hybrid scheme cotiues to produce accurate results. The fact that the hybrid scheme is able to reproduce the fie properties of rough BSS processes, eve for values of α very close to 2, is backed up by a further experimet reported i the followig sectio. 6
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 Riemasum 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
18 3.2 Estimatio of the roughess parameter Suppose that we have observatios X i m, i =,,..., m, of the BSS process X, give by 2., for some m N. BardorffNielse et al. [6] ad Corcuera et al. [6] discuss how the roughess idex α ca be estimated cosistetly as m. The method is based o the chageoffrequecy COF statistics COFX, m = m k=5 m k=3 X k m 2X k 2 m + X k 4 2 m X k m 2X k m + X k 2 2, m 5, m which compare the realized quadratic variatios of X, usig secodorder icremets, with two differet lag legths. Corcuera et al. [6] have show that uder some assumptios o the process X, which are similar to A, A2 ad A3 albeit slightly more restrictive, it holds that ˆαX, m := log COFX, m P α, m log 2 2 A idepth study of the fiite sample performace of this COF estimator ca be foud i [2]. To examie how well the hybrid scheme reproduces the fie properties of the BSS process i terms of regularity/roughess, we apply the COF estimator to discretized trajectories of X, where the kerel fuctio g is agai the gamma kerel Example 2.3 with λ =, geerated usig the hybrid scheme with κ =, 2, 3 ad b = b. We cosider the case where the volatility process satisfies σt =, that is, the process X is Gaussia. This allows us to quatify ad cotrol for the itrisic bias ad oisiess, measured i terms of stadard deviatio, of the estimatio method itself, by iitially applyig the estimator to trajectories that have bee simulated usig a exact method based o the Cholesky factorizatio. We the study the behavior of the estimator whe applied to a discretized trajectory, while decreasig the step size of the discretizatio scheme. More precisely, we simulate ˆαX, m, where m = 5 ad X is the hybrid scheme for X with = ms ad s {, 2, 5}. This meas that we compute ˆαX, m usig m observatios obtaied by subsamplig every sth observatio i the sequece X i, i =,,...,. As a compariso, we repeat these simulatios substitutig the hybrid scheme with the Riemasum scheme, usig κ = with b {b FWD, b }. The results are preseted i Figure 3. We observe that the itrisic bias of the estimator with m = 5 observatios is egligible ad hece the bias of the estimates computed from discretized trajectories is the attributable to approximatio error arisig from the respective discretizatio scheme, where positive resp. egative bias idicates that the simulated trajectories are smoother resp. rougher tha those of the process X. Cocetratig first o the baselie case s =, we ote that the hybrid scheme produces essetially ubiased results whe α 2,, while there is moderate bias whe α, 2, which disappears whe passig from κ = to κ = 3, eve for values of α very close to 2. The largest value of α cosidered i our simulatios is α =.49; oe would expect the performace to weake as α approaches 2, cf. Figure, but this rage of parameter values seems to be of limited practical iterest. The stadard deviatios exhibit a similar patter. The correspodig results for the Riemasum scheme are clearly iferior, exhibitig sigificat bias, while usig optimal evaluatio poits b = b improves the situatio slightly. I particular, 8
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 Riemasum 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 sth 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 Riemasum 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 Riemasum 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 socalled 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
21 Table : Parameter values used i the rbergomi model. S ξ η α ρ T =.4 exact κ = κ = κ = T = exact κ = κ = κ = 2 IVk,T.4 IVk,T k.5.5 k Figure 4: Implied volatility smiles correspodig to the optio price 3., computed usig Mote Carlo simulatio 5 time steps, replicatios, with two maturities: T =.4 left ad T = right. The spot variace process v was simulated usig a exact method, the hybrid scheme κ =, 2 ad b = b ad Riemasum scheme κ = ad b = b solid lies, b = b FWD dashed lies. The parameter values used i the rbergomi model are give i Table. We map the optio price CS, K, T to the correspodig Black Scholes implied volatility IVS, K, T, see, e.g., [9]. Reparameterizig the implied volatility usig the logstrike k := logk/s allows us to drop the depedece o the iitial price, so we will abuse otatio slightly ad write IVk, T for the correspodig implied volatility. Figure 4 displays implied volatility smiles obtaied from the rbergomi model usig the hybrid ad Riemasum schemes to simulate Y, as discussed above, ad compares these to the smiles obtaied usig a exact simulatio of Y via Cholesky factorizatio. The parameter values are give i Table. They have bee adopted from Bayer et al. [], who demostrate that they result i realistic volatility smiles. We cosider two differet maturities: short, T =.4, ad log, T =. We observe that the Riemasum scheme κ =, b {b FWD, b } is able capture the shape of the implied volatility smile, but ot its level. Alas, the method eve breaks dow with more extreme logstrikes the prices are so low that the rootfidig algorithm used to compute the implied volatility would retur zero. I cotrast, the hybrid scheme with κ =, 2 ad b = b yields implied volatility smiles that are idistiguishable from the bechmark smiles obtaied usig exact simulatio. Further, there is o discerible differece betwee the smiles obtaied usig κ = ad κ = 2. As i the previous sectio, we observe that the hybrid scheme is ideed capable of producig very accurate trajectories of T BSS processes, i particular i the case α 2,, eve whe κ =. 2
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 oicreasig 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 oicreasig 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 φhölder cotiuous trajectories for ay φ, α + 2. Moreover, a modificatio of X o R, havig locally φhölder 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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