Fractional Lévy Cox-Ingersoll-Ross and Jacobi processes

Transcription

1 Holger Fik ad Georg Schlücherma Fracioal Lévy Cox-Igersoll-Ross ad Jacobi rocesses Workig Paer Number 7, 6 Ceer for Quaiaive Risk Aalysis (CEQURA) Dearme of Saisics Uiversiy of Muich h://

2 FRACTIONAL LÉVY COX-INGERSOLL-ROSS AND JACOBI PROCESSES HOLGER FINK a,, GEORG SCHLÜCHTERMANN a,b, ABSTRACT AUTHORS INFO We rove a geeral Picard-Lidelöf-ye framework for sochasic differeial equaios drive by Madelbro-Va Ness fracioal Lévy rocesses. This allows us o derive he exisece of a fracioal Lévy Cox-Igersoll-Ross ad Jacobi model wih almos surely osiive, resecively bouded, samles ahs. a Faculy of Mahemaics, Iformaics ad Saisics, Ludwig- Maximilias-Uiversiä Müche, Akademiesrasse /I, 8799 Muich, Germay b Dearme of Mechaical, Auomoive ad Aeroauical Egieerig, Uiversiy of Alied Scieces Muich, Dachauer Srasse 98b, 8335 Muich, Germay holger.fik@sa.ui-mueche.de gschluec@hm.edu 6G, 6H, 6H MSC KEYWORDS fracioal Lévy rocess, Cox-Igersoll-Ross rocess, Jacobi rocess, log memory. Iroducio Madelbro-Va Ness fracioal Lévy rocesses (MvN-fLs) have iiially bee iroduced by Marquard (6) while heir codiioal disribuios have bee aalyzed i Fik (6). Addiioally, Fik ad Klüelberg () cosidered MvN-fLdrive sochasic differeial equaios (sdes) ad cosruced exlici soluios based o he geeral idea of Doss (977), Lyos (994), Zähle (998) ad Buchma ad Klüelberg (6). However, as has bee discussed i Secio 5. of Fik ad Klüelberg (), he heory herei oly covers Cox-Igersoll-Ross (CIR) sdes like dx = X d + σ X dl d or dx = X d + σ X dl d which is o suiable for, e.g., volailiy modelig whe aimig for a fracioal versio of he classical Heso seu (cf. Heso (993)). Therefore, we would like o obai a exisece ad uiqueess resul regardig a sricly osiive soluio of he geeral MvN-fL CIR sde give by dx = ()(θ() X )d + σ() X dl d, [,T ], X = x >. () For a fracioal Browia moio (fbm) as a drivig rocess, his has recely bee solved by Schlücherma ad Yag (6). I aricular, due o he fac ha fbm has zero quadraic variaio, oly ()θ() > is eeded o esure osiiviy of a soluio similar o he resul which we will obai below. Addiioally ad of ieres for sochasic correlaio models we shall cosider a fracioal versio of he (quie similar) Jacobi sde as well, i.e. dx = ()(θ() X )d + σ() X X dl d, [,T ], X = x >. () Throughou he aer, we will work o a comlee robabiliy sace (Ω,F,P) wih a give square-iegrable MvN-fL L d = (L d ) R, d (,/), wihou Gaussia ar i he sese of Marquard (6). I aricular, oly he log memory case is icluded as ahs of L d are a.s. Hölder coiuous u uil d (cf. Theorem 4.3 (i) of Marquard (6)). Addiioally, o esure he exisece of iegrals, oly MvN-fLs wih bouded -variaio for <, similar o Fik ad Klüelberg (), are cosidered. Therefore, from ow o, iegraio shall be udersood i he ahwise Riema-Sieljes sese (cf. Youg (936)). Secio shall rovide a geeral framework for MvN-fL drive sdes ossibly allowig for ime-deede coefficie fucios ad herefore exedig he seu of Lyos (994). I Secio 3 we shall use hese resuls o rove ha suiable soluios o () ad () exis a.s. i he ahwise sese. A brief simulaio sudy closes he aer. Preri submied o Elsevier Augus 5, 6

3 . A geeral Picard-Lidelöf-ye framework I his secio, we wa o rove a geeral Picard-Lidelöf-ye exisece ad uiqueess resul for MvN-fL drive sdes o comac ime ses. I order o do ha, we eed a Baach sace for oeial soluios o live i: For < ad a < b, le ([a,b]) be he se of all coiuous fucios f o [a,b] wih bouded -variaio v (f,[a,b]), where W co v (f,[a,b]) = su f(x i ) f(x i ) wih he su ake over all grids of [a,b]. Alyig Mikowski s iequaliy shows ha by oiwise addiio ad scalar mulilicaio W co ([a,b]) becomes a R-vecor sace ad (v (,[a,b])) is a semiorm o W co ([a,b]) sice, obviously, we have v (f,[a,b]) = for every cosa fucio f. To overcome his roblem ad o obai a ormed vecor sace we could cosider he quoie sace W co ([a,b])/cos([a,b]) where cos([a,b]) is he vecor sace of all cosa fucios o [a,b]. This aroach however causes roblems whe cosiderig iegral equaios soluios would oly be uique u o a.s. cosa shifs. Isead we defie similar o Chisyakov ad Galki (998) a acual orm o W co ([a,b]) by = [a,b] = [a,b] su + (v (,[a,b]) ) where we will suress he [a,b] i he oaio whe he ierval is clear. Proosiio 7. of Chisyakov ad Galki (998) ow imlies ha (W co ([a,b]), ) is a Baach sace. For he cosideraios o come, we shall eed he followig echical lemma.. Lemma. Le [a,b] be a comac ierval, g W co For all x [a,b] we defie φ(x) = x a fdg. The φ Wco ([a,b]) ad f W co q ([a,b]) where q > ad wih + q >. ([a,b]). Moreover we have wih ζ deoig he Riema zea fucio v (φ,[a,b]) ({ + ζ ( + q )} v q (f,[a,b]) q + f su ) v (g,[a,b]). Proof. The firs ar follows from Theorem A.3 of Fik ad Klüelberg (). For he iequaliy, we recall ha for ad x,y R we have x + y ( x + y ) ad herefore calculae for z i [x i,x i ] [a,b] v (φ,[a,b]) = su su fdg x i x i = su fdg f(z i )[g(x i ) g(x i )] + f(z i )[g(x i ) g(x i )] x i ( (f f(z i ))dg + f(z i ) g(x i ) g(x i ) ) x i su Usig (.9) of Youg (936) we obai x i x i x i ( (f f(z i ))dg + f(z i ) g(x i ) g(x i ) ). x i su ( { + ζ ( + q )} v q (f,[x i,x i ]) q v (g,[x i,x i ]) + f su g(x i ) g(x i ) ) su ({ + ζ ( + q )} v q (f,[a,b]) q v (g,[x i,x i ]) + f su g(x i ) g(x i ) ). Fially, wih he iequaliy v (g,[x i,x i ]) v (g,[a,b]), we arrive a su ({ + ζ ( + q )} v q (f,[a,b]) q v (g,[a,b]) + f [a,b] g(x i ) g(x i ) ) which roves he asserio. = ({ + ζ ( + q )} v q (f,[a,b]) q v (g,[a,b]) + f [a,b] v (g,[a,b])) Le Li(A) be he se of all Lischiz coiuous fucios o A R. We ca sae ad rove our mai resul of his secio.

4 . Theorem. Le (L d ) R be a MvN-fL of a.s. bouded ˆ-variaio, ˆ [,), d (, ). Furhermore give < T < < T < le µ(,), σ(,) Li(R) for all [T,T ] ad µ(,z), σ(,z) Li([T,T ]) for all z R, where he Lischiz cosas shall be ideede of ad z, resecively. The for all ˆ < < ad x R he sochasic differeial equaio has a uique ahwise soluio i W co ([T,T ]). dx = µ(,x )d + σ(,x )dl d, [T,T ], X = x Proof. The followig cosideraios will always be i he ahwise sese usig ahs of (L d ) R which are a.s. of bouded ˆ-variaio ad Hölder coiuous. However, for ease of oaio we shall suress he argume ω. Now, cosider some N > such ha vˆ (L d,[t,t ]) < N. Furhermore le K µ, C µ ad K σ, C σ be he Lischiz cosas of µ ad σ wih resec o he firs ad secod argume. Wihou loss of geeraliy, we may assume K µ = C µ ad K σ = C σ. Give suiable small δ (, ) ad r (, ) we cosider he comlee subsace ad defie he oeraor B = {Z W co ([ δ, + δ]) Z x < r} W co ([ δ, + δ]) Ψ B W co ([ δ, + δ]), Ψ(Z) = x + µ(s,z s )ds + σ(s,z s )dl d s The firs iegral o he righ had side is differeiable i is uer boud ad hus of fiie variaio while he secod iegral is of bouded -variaio by Lemma. sice v (σ(,z )) = [su σ( i+,z i+ ) σ( i,z i ) ] [C σ su Z i+ Z i + C σ su [su σ( i+,z i+ ) σ( i+,z i ) + σ( i+,z i ) σ( i,z i ) ] i+ i ] C σ v (Z ) + 4Cσ δ <. Therefore, we ca coclude ha Ψ is well-defied. Now, our firs aim is o show ha Ψ(B) B. By defiiio of B, here exiss some C > such ha su [ µ(,z ) + σ(,z ) ] < C. [ δ, +δ], Z B For [ δ, + δ], ivokig he Hölder coiuiy of L d (cf. Theorem 4.3 (i) of Marquard (6)) ad (3), we obai for ε > such ha + ε <, < d < d ad some s [,] Ψ(Z) x µ(s,z s )ds + σ(s,z s )dl d s δ µ(,z ) su + (σ(s,z s ) σ(s,z s ))dl d s + σ(s,z s )[L d L d ] δ µ(,z ) su + { + ζ ( + + ε )} (v (,σ(,z ))) (v+ε (L d )) +ε + σ(,z ) su C L dδ d C (δ + δ d) + { + ζ ( + + ε )} [C σv (Z ) + 4Cσ δ] C L dδ ε d +ε (v (L d )) +ε where C L d deoes he Hölder cosa of L d ad C (δ + δ d) + C (v (Z )) δ ε d +ε + C3 δ ε d +ε + (3) C = max{ C C L d, C }, C = [ + ζ ( + + ε )] C σc L dn +ε ad C 3 = 4 [ + ζ ( + + ε )] C σc L dn +ε. Sice his uer boudary does o deed o, we ca coclude ha Ψ(Z) x su C (δ + δ d) + C (v (Z )) δ ε d +ε + C3 δ ε d+ +ε. (4) 3

5 Now o he oe had we have v ( µ(s,z s )ds) = (su C (su i i+ µ(s,z s )ds ) µ(,z ) su (su i+ i ) i+ i ) = C δ (5) ad o he oher had usig Lemma. ad (3) we ge for η > such ha ˆ < η v ( σ(s,z s )dl d s) ({ + ζ ( )} v (σ(,z )) + σ(,z ) su) v (L d ) ({ + ζ ( )} [C σ v (Z ) + 4Cσ δ] + C ) CL dδ η d v η (L d ) C 4 v (Z ) δ η d + C5 δ η d +η d + C6 δ (6) where C 4 = + { + ζ ( )} C σ C L dn, C5 = C C L dn ad C 6 = + { + ζ ( )} C σ C L dn. Puig (5) ad (6) ogeher, we obai ad fially v (Ψ(Z) x) C δ + C 4 v (Z ) δ η d + C5 δ η d +η d + C6 δ Ψ(Z) x C (δ + δ d) + C (v (Z )) ε δ d +ε + C3 δ ε d +ε + + C δ + C 4 v (Z ) η δ d + C5 δ η d +η d + C6 δ C (δ + δ d) + C rδ ε d +ε + C3 δ ε d +ε + + C δ + C 4 rδ η d + C5 δ η d +η d + C6 δ < r for δ small eough. Therefore we have show ha Ψ(B) B. To use Baach s fixed oi heorem i remais o rove ha for suiable small δ he oeraor Ψ becomes a coracio, i.e. we eed o show ha for Z, Y B we have Ψ(Z) Ψ(Y ) D Z Y, wih D (,). Therefore, for fixed Z, Y B we sar by observig [µ(s,z s ) µ(s,y s C µ δ Z Y su. (7) )]ds su 4

6 Addiioally, for θ > such ha θ < < + θ < ad > we esimae usig he Lischiz roery of σ +θ v +θ (σ(,z ) σ(,y )) +θ = [su [su σ( i+,z i+ ) σ( i,z i ) σ( i+,y i+ ) + σ( i,y i ) +θ ] θ Cσ Z θ Y θ su σ( i+,z i+ ) σ( i,z i ) σ( i+,y i+ ) + σ( i,y i ) ] θ θ +θ C +θ σ +θ θ +θ C +θ σ +θ θ +θ Z Y su [su σ( i+,z i+ ) σ( i,z i ) σ( i+,y i+ ) + σ( i Y,i ) ] θ +θ Z Y su [su σ( i+,z i+ ) σ( i+,z i ) σ( i+,y i+ ) + σ( i+,y i ) θ + su +θ +θ σ( i,z i ) σ( i+,z i ) σ( i,y i ) + σ( i+,y i ) ] +θ +θ Cσ Z Y su [v (Z ) + v (Y ) + δ ] +θ +θ +θ +θ+ +θ C σ Z Y su (8) where, i he las lie, we used ha Z Y su <, δ < ad v (Z ) + v (Y ) < r + r <. The laer follows by defiiio sice v (Z ) = v (Z x) Z x < r. Similar o he σ-ar of (3) ad (4) we obai via (8) wih θ as above [σ(s,z s ) σ(s,y s )]dl d s su C 7 δ θ d +θ v+θ (σ(,z ) σ(,y )) +θ + C8 δ d Z Y su C 7 δ θ d +θ Z Y su + C 8 δ d Z Y su (9) where C 7 = { + ζ ( + θ )} N +θ, C7 = C 7 +θ+ +θ C σ ad C 8 = C σ C L d. Havig he su-ar of he -orm covered, we coiue wih he -variaio by observig ha due o Lischiz coiuiy Havig (8) i mid ad usig Lemma. agai, we calculae v ( [µ(s,z s ) µ(s,y s )]ds) Cµ δ Z Y su. () v ( [σ(s,z s ) σ(s,y s )]dl d s) +θ ({ + ζ ( + θ + )} +θ v +θ (σ(,z ) σ(,y )) +θ +θ { + ζ ( + θ + +θ )} +θ+ C 9 δ η d Z Y su (+θ) C + σ(,z ) σ(,y ) su) v (L d ) σ Z Y su + C σ Z Y su C L dδ η d v η (L d ) () wih C 9 = +θ { + ζ ( + θ + )} +θ +θ+ (+θ) C σ + C σ C L dn. 5

7 Combiig (7), (9) () ad () we obai Ψ(Z) Ψ(Y ) C µ δ Z Y su + C 7 δ θ d +θ Z Y su + C 8 δ d Z Y su + C µ δ Z Y su + C 9 δ η d Z Y su [3C µ δ + C 7 δ θ d +θ + C8 δ d + C 9 δ η d ] Z Y. Choosig δ small eough we ca coclude ha Ψ is a coracio ad accordig o Baach s fixed oi heorem here exiss a uique X B wih X = x + µ(s,x s )ds + σ(s,x s )dl d s, [ δ, + δ]. As δ did o exlicily deed o we ca ierae his rocedure o obai X o he whole ierval [T,T ]. 3. Fracioal Lévy Cox-Igersoll-Ross ad Jacobi rocesses Obviously, a volailiy rocess give by σ(,z) = {z } σ() z for some σ() >, [,T ] ad z R, does o fulfill he global Lischiz assumios of Theorem 3.. However, followig he geeral idea of, e.g., Gikhma (), we ca sill use Theorem 3. i a ahwise sese ad coclude he exisece of a global, osiive, soluio o (). 3. Theorem. Le (L d ) R be a MvN-fL of a.s. bouded ˆ-variaio, ˆ [,), d (, ) ad T >. If (), θ(), σ() Li([,T ]) are sricly osiive, he he sde dx = ()(θ() X )d + σ() X dl d, [,T ], X = x > has a uique osiive soluio i W co ([,T ]) for all ˆ < <. Proof. For some (oeially ah-deede) δ >, he ahwise roof of Theorem 3. esures he exisece ad uiqueess of a soluio i [,δ]. The remaiig cosideraios shall all be resriced o some measurable se A Ω which cosiss of he L d -ahs for which he heorem does o hold. Now ake ε > ad defie a radom variable via τ ε = mi{ [,T ] X ε}. Furhermore, we se f(z) = z /. Due o he a.s. sric osiiviy of (X τε ) [,T ] i fac, (X τε ) [,T ] is bouded below by ε we ca ivoke a chai rule ad a desiy formula similar o A. ad A.3 of Fik ad Klüelberg (), o obai f(x τε ) = f(x τε ) + τ ε = X / f (X s τε )dx s τε = f(x ) + (s)θ(s)x (+/) f (X s )dx s ds + (s)x / ds Furhermore, sice he execaios below exis er defiiio, we ca deduce ha for ε small eough E [X / ] = X / E [ X / + (s)θ(s)x (+/) ds] + E [ (s)e [X / ] ds X / + ad obai by Growall s iegral iequaliy for o-decreasig fucios (s)x / ds] E [ (s)e [X / ] ds σ(s)x dl d s. σ(s)x dl d s] E [(X τε ) / ] X / ex { (s)ds}. () Fially, we ge for [,T ] via Chebyshev s iequaliy for bouded radom variables ad () P(τ ε ) = P(X τε ε) = P (X / ε / ) = P (X / ε / ) ε / X / ex { (s)ds} as ε, which allows as o coclude ha P(A) = ad X says osiive a.s. (i.e. ahwise) o [,T ]. Now, a similar heorem ca be rove for he fracioal versio of he Jacobi sde (). 3. Theorem. Le (L d ) R be a MvN-fL of a.s. bouded ˆ-variaio, ˆ [,), d (, ) ad T >. If (), θ(), σ() Li([,T ]) are sricly osiive, wih θ() (,), he he sde dx = ()(θ() X )d + σ() X X dl d, [,T ], X = x > 6

8 has a uique soluio i W co ([,T ]) which lives i [,] for all ˆ < <. Proof. Similar o he roof of 3. we shall oly work o a measurable se A Ω which cosiss agai of he L d -ahs for which Theorem 3. fails. Showig ha X says always osiive works aalogously o he CIR roof. For he secod boudary, ake ε (,), defie τ ε = mi{ [,T ] X ε} ad se f(z) = ( z). Now, similarly o he roof of 3., we obai f(x τε ) = f(x ) + ( X s τε ) [(s)( θ(s))]ds + σ(s)( X s τε ) X s τε Xs τε dl d s. (s)( X s τε ) ds Sice we work o A ad θ() (,), for ε close o ad large eough, he firs iegral s absolue execaio domiaes he σ-iegral ad we ge E [f(x τε )] f(x ) + Now, he roof ca be cocluded as for Theorem 3.. (s)e [f(x s τε )] ds Figure shows simulaed samles ahs of a soluio o () ad () wih cosa coefficie fucios obaied via a classical Euler-Maruyama scheme. As ca be see, osiiviy for he CIR rocess oly holds for θ > ad θ > is addiioally ecessary for he Jacobi sde o have soluios i (,). These visualizaio are ealy ilie wih he resuls of Schlücherma ad Yag (6) regardig he ahwise fracioal Browia CIR rocess. Figure : To: Samle ahs of a soluio o he Cox Igersoll Ross model () wih X =.5 (lef: =., θ =.5, σ =, righ: =, θ =, σ =.). Boom: Samle ahs of a soluio o he Jacobi model () wih X =.5 (lef: =., θ =.5, σ =, righ: =, θ =, σ =.). All simulaios are based o a fracioal Poisso rocess (cf. Secio 5 of Fik ad Klüelberg ()) wih iesiy λ = 4 ad fracioal iegraio arameer d =.5 Refereces Buchma, B., Klüelberg, C., 6. Fracioal iegral equaios ad sae sace rasforms. Beroulli (3), Chisyakov, V. V., Galki, O. E., 998. O mas Of bouded -variaio wih >. Posiiviy (). Doss, H., 977. Lies ere equaios differeielles sochasiques e ordiaires. Aals de Isiue Heri Poicare 3,

9 Fik, H., 6. Codiioal disribuios of Madelbro-va Ness fracioal Lévy rocesses ad coiuous-ime ARMA-GARCH-ye models wih log memory. Joural of Time Series Aalysis 37 (), Fik, H., Klüelberg, C.,. Fracioal Lévy drive Orsei-Uhlebeck rocesses ad sochasic differeial equaios. Beroulli 7 (), Gikhma, I. I.,. A shor remark o Feller s square roo codiio, available olie: h://aers.ssr.com/sol3/aers. cfm?absrac_id= Heso, S., 993. A closed-form soluio for oios wih sochasic volailiy wih alicaios o bod ad currecy oios. Review of Fiacial Sudies 6, Lyos, T., 994. Differeial equaios drive by rough sigals (I): a exesio of a equaliy of L. C. Youg. Mah. Research Leers, Marquard, T., 6. Fracioal Lévy rocesses wih a alicaio o log memory movig average rocesses. Beroulli (6), 9 6. Schlücherma, G., Yag, Y., 6. Noe o fracioal CLKS-ye sochasic differeial equaio ah-wise ad i he Wick sese, available olie: hs:// sochasic_differeial_equaio_-ah-wise_ad_i_he_wick_sese. Youg, L., 936. A iequaliy of he Hölder ye, coeced wih Sieljes iegraio. Aca Mahemaica 67, 5 8. Zähle, M., 998. Iegraio wih resec o fracal fucios ad sochasic calculus. I. Probabiliy Theory ad Relaed Fields,