In this note, we investigate the density of the exponential functional of the fractional Brownian motion. Based on the techniques of Malliavin’s calculus, we provide a log-normal upper bound for the density

Nguyen Thu Hang

Nguyen Tien Dung

Pham Thi Phuong Thuy

Comptes Rendus Mathématique

Tien Dung, Nguyen

Thu Hang, Nguyen

Phuong Thuy, Pham Thi

Numérisation de Documents Anciens Mathématiques

Comptes RendusMathématiqueNguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyDensity estimates for the exponential functionals of fractional BrownianmotionVolume 360 (2022), p. 151-159Published online: 15 February 2022https://doi.org/10.5802/crmath.274This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orge-ISSN : 1778-3569Comptes RendusMathématique2022, 360, p. 151-159https://doi.org/10.5802/crmath.274Probability theory / ProbabilitésDensity estimates for the exponentialfunctionals of fractional Brownian motionNguyen Tien Dung∗, a, Nguyen Thu Hangb and Pham Thi Phuong Thuyca Department of Mathematics, VNU University of Science, Vietnam, NationalUniversity, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnamb Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien,Bac Tu Liem, Hanoi, 084 Vietnamc The faculty of Basic Sciences, Vietnam Air Defence and Air Force Academy, Son Tay,Ha Noi, 084 VietnamE-mails: dung@hus.edu.vn (N. Tien Dung), thuhangmdc@gmail.com (N. Thu Hang),thuypham82.com@gmail.com (P. T. Phuong Thuy)Abstract. In this note, we investigate the density of the exponential functional of the fractional Brownianmotion. Based on the techniques of Malliavin’s calculus, we provide a log-normal upper bound for the density.Mathematical subject classification (2010). 60G22, 60H07.Funding. This research was funded by the Vietnam National University, Hanoi under grant number QG.20.21.Manuscript received 6th March 2021, revised 19th June 2021, accepted 21st September 2021.1. IntroductionLet B H = (B Ht )t ∈ [0,T ] be a fractional Brownian motion (fBm) with Hurst index H ∈ (0,1). We recallthat B H is a centered Gaussian process with covariance functionRH (t , s) := E[B Ht BHs]= 12(t 2H + s2H −|t − s|2H), 0 ≤ s, t ≤ T.We consider the exponential functional of the formF =∫ T0eas+σBHs d s, (1)where T > 0, a ∈ R and σ > 0 are constants. It is known that this functional plays an importantrole in several domains. The special case, where H = 12 , has been well studied and a lot of fruitfulproperties of F can be founded in the literature, see e.g. [3, 4, 10]. However, to the best ourknowledge, the deep properties of F for H 6= 12 are scarce. In a recent paper [8], we have proved theLipschitz continuity of the cumulative distribution function of F with respect to the Hurst index∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/152 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyH . The aim of the present paper is to investigate the density of F. Unlike the case H = 12 , it is noteasy to find the density of F explicitly for H 6= 12 and hence, our work will focus on providing theestimates for the density function. It should be noted that, in the last years, the density estimatesfor random variables related to fBm has been extensively studied, see e.g. [1,2,6,7] and referencestherein.The rest of this article is organized as follows. In Section 2, we briefly recall some of the relevantelements of the Malliavin calculus and two general estimates for densities. Our main results arethen stated and proved in Section 3. Our Theorems 6 and 8 point out that the density of F isbounded from above by log-normal densities.2. PreliminariesIn the whole paper, we assume H > 12 . Under this assumption, fBm admits the Volterra represen-tationB Ht =∫ t0K (t , s)dBs , (2)where (Bt )t∈[0,T ] is a standard Brownian motion and for some normalizing constant cH , the kernelK is given byK (t , s) = cH s1/2−H∫ ts(u − s)H− 32 uH− 12 du, 0 < s ≤ t ≤ T.Let us recall some elements of Malliavin calculus with respect to Brownian motion B , where Bis used to present B Ht as in (2). We suppose that (Bt )t ∈ [0,T ] is defined on a complete probabilityspace (Ω,F ,F,P ), where F= (Ft )t ∈ [0,T ] is a natural filtration generated by the Brownian motionB. For h ∈ L2[0,T ], we denote by B(h) the Wiener integralB(h) =∫ T0h(t )dBt .Let S denote the dense subset of L2(Ω,F ,P ) consisting of smooth random variables of the formF = f (B(h1), . . . , B(hn)), (3)where n ∈ N, f ∈ C∞b (Rn),h1, . . . , hn ∈ L2[0,T ]. If F has the form (3), we define its Malliavinderivative as the process DF := {D t F, t ∈ [0,T ]} given byD t F =n∑k=1∂ f∂xk(B(h1), . . . , B(hn))hk (t ).More generally, for each k ≥ 1, we can define the iterated derivative operator by settingDkt1, ..., tk F = D t1 . . . D tk F.For any p,k ≥ 1, we shall denote by Dk,p the closure of S with respect to the norm‖F‖pk, p := E |F |p +E[∫ T0∣∣D t1 F ∣∣p d t1]+·· ·+E [∫ T0. . .∫ T0∣∣∣Dkt1, ..., tk F ∣∣∣p d t1 . . . d tk] .A random variable F is said to be Malliavin differentiable if it belongs to D1,2. For any F ∈ D1,2,the Clark–Ocone formula says thatF −E [F ] =∫ T0E[Ds F∣∣Fs]dBs .Moreover, any F,G ∈D1,2, we have the following covariance formulaCov(F,G) = E[∫ T0Ds F E[DsG∣∣Fs]d s] . (4)In order to obtain the density estimates for exponential functionals we need the following generalresults.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 153Proposition 1. Let q,α,β be three positive real numbers such that 1q + 1α+ 1β = 1. Let F be a randomvariable in the spaceD2,α, such that E [‖DF‖−2βH ] <∞. Then the density ρF (x) of F can be estimatedas followsρF (x) ≤ cq,α,β (P (F ≤ x))1/q ×(E[‖DF‖−1H]+∥∥D2F∥∥Lα(Ω;H⊗H) ∥∥‖DF‖−2H ∥∥β) , x ∈R, (5)where cq,α,β is a positive constant and H = L2[0,T ].Proof. This proposition comes from the computations on [5, p. 87]. Proposition 2. Let F ∈D2,4 be such that E [F ] = 0. Define the random variableΦF :=∫ T0Ds F E[Ds F∣∣Fs]d s.Assume that ΦF 6= 0 a.s. and the random variables FΦF and1Φ2F∫ T0 DsΦF E [Ds F |Fs ]d s belong toL2(Ω). Then the law of F has a continuous density given byρF (x) = ρF (0)exp(−∫ x0hF (z)d z)exp(−∫ x0wF (z)d z), x ∈ suppρF , (6)where the functions wF and hF are defined bywF (z) := E[FΦF∣∣∣∣F = z] , hF (z) := E[1Φ2F∫ T0DsΦF E[Ds F∣∣Fs]d s∣∣∣∣∣F = z].Proof. This proposition is Theorem 7 in our recent paper [9]. 3. The main resultsIn this section, we provide explicit estimates for the density ρF (x) of the functional F definedby (1). Our idea is to consider the random variable X := lnF − E [lnF ] and use the relationρF (x) = 1x ρX (ln x −E [lnF ]), x > 0, where ρX denotes the density of X .We need some technical results.Proposition 3. Consider the random variable X := lnF −E [lnF ]. It holds that0 ≤ DθX ≤σK (T,θ) a.s. (7)0 ≤ Dr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s. (8)Proof. By the chain rule for Malliavin derivatives, we have, for 0 ≤ r,θ ≤ T,DθX =σ∫ Tθ K (s,θ)eas+σB Hs d s∫ T0 eas+σB Hs d s(9)andDr DθX =σ2∫ Tθ∨r K (s,θ)K (s,r )eas+σB Hs d s∫ T0 eas+σB Hs d s− σ2∫ Tr K (s,r )eas+σB Hs d s∫ Tθ K (s,θ)eas+σB Hs d s(∫ T0 eas+σB Hs d s)2 .Because the function s 7→ K (s,θ) is non-decreasing for each θ, (7) follows directly from (9). Wealso haveDr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s.To prove the non-negativity of the second order Malliavin derivative, we let U be a randomvariable with the density function defined byf (x) = eax+σB Hx∫ T0 eas+σB Hs d s, 0 ≤ x ≤ T.C. R. Mathématique — 2022, 360, 151-159154 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyDenote by EU the expectation with respect to U . We haveDr DθX = EU [K (U ,θ)K (U ,r )]−EU [K (U ,θ)]EU [K (U ,r )] .Note that the functions s 7→ K (s,θ) and s 7→ K (s,r ) are non-decreasing. Hence, by Chebyshev’sassociation inequality, Dr DθX ≥ 0 a.s. The proof of Proposition 3 is complete. Lemma 4. DefineMr := E [F |Fr ] = E[∫ T0eas+σk f l6BHs d s∣∣∣∣Fr ] , 0 ≤ r ≤ T.Then, for every p ≥ 2, we haveE[(max0≤r ≤TMr)p]≤C <∞,where C is a positive constant depending on p,T, a,σ and H .Proof. The stochastic process M := (Mr )0≤r≤T is a martingale with M0 = E [F ] and MT = F. Hence,by Burkhölder–David–Gundy inequality, we haveE[(max0≤r ≤TMr)p]≤ cp(M p0 +E[〈M〉p/2T])= cp((E [F ])p +E[〈M〉p/2T]), (10)where cp is a positive constant. Using the Clark–Ocone formula we haveMT = E MT +∫ T0E[Dr MT∣∣Fr ]dBr= E [F ]+σ∫ T0E[∫ TrK (s,r )eas+σBHs d s∣∣∣∣Fr ]dBr ,which gives us〈M〉T =∫ T0σ2(E[∫ TrK (s,r )eas+σBHs d s∣∣∣∣Fr ])2 dr≤∫ T0σ2K 2(T,r )M 2r dr a.s.Then, by Hölder inequality, we haveE[〈M〉p/2T]≤σp E[(∫ T0K2p−4p (T,r )K4p (T,r )M 2r dr)p/2]≤σp(∫ T0K 2(T,r )dr) p2 −1 (∫ T0K 2(T,r )E[M pr]dr)≤σp T (p−2)H(∫ T0K 2(T,r )E[F p]dr)=σp T pH E[F p].(11)Here we used the fact that∫ T0 K2(T,r )dr = E |B HT |2 = T 2H . So we obtain the desired conclusion byinserting (11) into (10). Proposition 5. Let X be as in Proposition 3. We defineΦX :=∫ T0 Ds X E [Ds X |Fs ]d s. Then,∣∣ΦX ∣∣−1 ∈ Lp (Ω), ∀ p ≥ 1.We also have (∫ T0∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 155Proof. It follows from (9) thatDθX ≥σTe−2|a|T+σ min0≤ s≤TB Hs −σ max0≤ s≤TB Hs∫ TθK (s,θ)d s a.s. (12)On the other hand, by using the Cauchy–Schwarz inequality, we haveE[DθX∣∣Fθ]≥ σ(E[√∫ Tθ K (s,θ)eas+σB Hs d s∣∣∣∣Fθ])2E[∫ T0 eas+σB Hs d s∣∣∣Fθ] a.s.and √∫ TθK (s,θ)eas+σB Hs d s ≥∫ Tθ K (s,θ)√eas+σB Hs d s√∫ Tθ K (s,θ)d sa.s.We therefore getE[DθX∣∣Fθ]≥ σ(∫ Tθ K (s,θ)E[eas/2+σBHs /2∣∣∣Fθ]d s)2∫ Tθ K (s,θ)d sE[∫ T0 eas+σB Hs d s∣∣∣Fθ] a.s.Furthermore, by Lyapunov’s inequality,E[eas/2+σBHs /2∣∣∣Fθ]≥ eas/2+σE[B Hs ∣∣Fθ]/2 a.s.As a consequence,E[DθX∣∣Fθ]≥ σe−|a|T+σ min0≤θ≤ s≤T Ns,θ ∫ Tθ K (s,θ)d smax0≤θ≤TMθa.s. (13)whereNs,θ := E[B Hs∣∣∣Fθ] and Mθ := E [∫ T0eas+σBHs d s∣∣∣∣Fθ] .Combining (12) and (13) yieldsDθX E[DθX∣∣Fθ]≥ σ2T e−3|a|T+σ min0≤ s≤T B Hs −σ max0≤ s≤T B Hs +σ min0≤θ≤ s≤T Ns,θ(∫ Tθ K (s,θ)d s)2max0≤θ≤TMθa.s.and hence,ΦX ≥ σ2Te−3|a|T+σ min0≤ s≤TB Hs −σ max0≤ s≤TB Hs +σ min0≤θ≤ s≤TNs,θ∫ T0(∫ Tθ K (s,θ)d s)2dθmax0≤θ≤TMθa.s. (14)We observe that ∫ T0(∫ TθK (s,θ)d s)2dθ =∫ T0∫ Tθ∫ TθK (t ,θ)K (s,θ)d sd tdθ=∫ T0∫ T0(∫ s∧t0K (t ,θ)K (s,θ)dθ)d sd t=∫ T0∫ T0E[B Ht BHs]d sd t = T2H+22H +2 .This, together with (14), yields∣∣ΦX ∣∣−1 ≤ 2H +2σ2T 2H+1e3|a|T+2σ max0≤ s≤TB Hs +σ max0≤θ≤ s≤TNs,θmax0≤θ≤TMθ a.s.C. R. Mathématique — 2022, 360, 151-159156 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyWe have (Ns,θ)0≤θ≤ s≤T is a Gaussian field with finite variances because Ns,θ =∫ θ0 KH (s,r )dBr .Hence, by Fernique’s theorem, there exists ε> 0 such thatE[eε max0≤θ≤ s≤T |Ns,θ|2]<∞.Sinceepσ max0≤θ≤ s≤TNs,θ ≤ ep2σ24ε +ε max0≤θ≤ s≤T |Ns,θ|2,this implies thateσ max0≤θ≤ s≤TNs,θ ∈ Lp (Ω) for any p ≥ 1.Similarly, we also havee2σ max0≤ s≤TB Hs ∈ Lp (Ω) for any p ≥ 1.So, recalling Lemma 4, we conclude that |ΦX |−1 ∈ Lp (Ω) for any p ≥ 1.We deduce from (12) that∫ T0∣∣DθX ∣∣2dθ ≥ σ2(2H +2)T 2H e−4|a|T+2σ min0≤ s≤TB Hs −2σ max0≤ s≤TB Hs a.s. (15)Hence, we also have (∫ T0 |DθX |2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1. The proof of Proposition 5 is complete.We now are in a position to bound the density ρF (x) of F. We first use Proposition 1 to estimatethe left tail of the density.Theorem 6. We haveρF (x) ≤ cxexp(− (ln x −E [lnF ])28σ2T 2H), 0 < x ≤ e E [lnF ], (16)where c is a positive constant.Proof. It is known from Proposition 5 that‖D X ‖−2H =(∫ T0∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.In addition, from the estimate (8), we have∥∥D2X ∥∥2L2(Ω ; H⊗H) = ∫ T0∫ T0E∣∣DθDr X ∣∣2dθdr ≤σ4 ∫ T0∫ T0K 2(T,θ)K 2(T,r )dθdr =σ4T 2H <∞.The above estimates allow us to use Proposition 1 with q =β= 4,α= 2 and we obtainρX (x) ≤ cP (X ≤ x)14 , x ∈R, (17)where c is a positive constant.The remaining of the proof is to bound P (X ≤ x) for x ≤ 0. We consider the function ϕ(λ:= E [e−λX ], λ > 0 (this function is well defined because F−1 ∈ Lp (Ω), ,∀ p ≥ 1). By usingrepeatedly the covariance formula (4), we haveC. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 157σ2X :=Var(X ) = E[ΦX]andϕ′(λ) =−E[X e−λX]=λE[e−λXΦX]=λσ2X E[e−λX]+λE[e−λX(ΦX −σ2X)]=λσ2X E[e−λX]−λ2E[e−λX∫ T0Ds X E[DsΦX∣∣Fs]d s]Since Ds X ≥ 0 and Dr Ds X ≥ 0, those imply that∫ T0 Ds X E [DsΦX |Fs ]d s ≥ 0, and hence,ϕ′(λ) ≤λσ2X E[e−λX]=λσ2Xϕ(λ), λ> 0.This, together the fact ϕ(0) = 1, gives usϕ(λ) ≤ eλ2σ2X2 , λ> 0.By Markov’s inequality we have, for all λ> 0,P (X ≤ x) ≤ eλxϕ(λ) ≤ eλx+λ2σ2X2 , x ≤ 0.When x ≤ 0, we can choose λ=− xσ2Xto getP (X ≤ x) ≤ e− x22σ2X , x ≤ 0.From the estimate (7), we have σ2X = E [ΦX ] ≤∫ T0 |Ds X |2d s ≤σ2T 2H . So we deduceP (X ≤ x) ≤ e−x22σ2T 2H , x ≤ 0. (18)Combining (17) and (18) yieldsρX (x) ≤ c e−x28σ2T 2H , x ≤ 0.where c is a positive constant. Recalling X = lnF − E [lnF ], the density of F satisfies ρF (x) =1x ρX (ln x −E [lnF ]). When 0 < x ≤ eE [lnF ], we have y := ln x −E [lnF ] ≤ 0. We thus obtainρF (x) = 1xρX (y) ≤ cxe− y28σ2T 2H = cxe− (ln x−E [lnF ])28σ2T 2H , 0 < x ≤ e E [lnF ].This completes the proof of Theorem 6. Remark 7. Replacing X by F −E [F ] in the proof of Theorem 6, we obtain the following Gaussianbound for the left tailρF (x) ≤ c e− (x−E [F ])28σ2F , x ≤ E [F ],where σ2F := Var(F ) and c is a positive constant.We now use Proposition 2 to estimate the right tail of the density.Theorem 8. We haveρF (x) ≤ cxexp(− (ln x −E [lnF ])22σ2T 2H), x > eE [lnF ], (19)where c is a positive constant.C. R. Mathématique — 2022, 360, 151-159158 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyProof. Let X be as in Proposition 3. Obviously, we have ΦX 6= 0 a.s. Moreover, from the esti-mates (7) and (8) we obtain0 ≤ DsΦX =∫ T0Ds DθX E[DθX∣∣Fθ]dθ+∫ T0DθX E[Ds DθX∣∣Fθ]dθ≤ 4σ3∫ T0K 2(T,θ)K (T, s)dθ = 4σ3K (T, s)T 2Hand0 ≤∫ T0DsΦX E[Ds X∣∣Fs]d s ≤ 4σ4T 2H ∫ T0K 2(T, s)d s = 4σ4T 4HHence, it follows from Proposition 5 that the random variable 1Φ2X∫ T0 DsΦX E [Ds X |Fs ]d s belongto L2(Ω). We also have XΦX ∈ L2(Ω) because−|a|T −σ max0≤ s≤TB Hs + lnT −E [lnF ] ≤ X ≤ |a|T +σ max0≤ s≤TB Hs + lnT −E [lnF ]and hence, X ∈ Lp (Ω) for all p ≥ 2.In view of Proposition 2, the density ρX (x) of X is given byρX (x) = ρX (0)exp(−∫ x0hX (z)d z)exp(−∫ x0wX (z)d z), x ∈ supp ρX , (20)wherewX (z) := E[XΦX∣∣∣∣X = z] and hX (z) := E[1Φ2X∫ T0DsΦX E[Ds X∣∣Fs]d s∣∣∣∣∣X = z].Since hX ≥ 0, this implies thatexp(−∫ x0hX (z)d z)≤ 1, x ≥ 0.From the estimate (7) we have0 ≤ΦX ≤σ2∫ T0K 2(T,θ)dθ =σ2T 2H a.s.and we obtainexp(−∫ x0wF (z)d z)≤ e−x22σ2T 2H , x ∈R.So we can conclude thatρX (x) ≤ ρX (0)e−x22σ2T 2H , x ≥ 0,and (19) follows because ρF (x) = 1x ρX (ln x −E [lnF ]). The proof of Theorem 8 is complete. AcknowledgmentsThe author would like to thank the anonymous referee for their valuable comments for improvingthe paper. A part of this paper was done while the first author was visiting the Vietnam Institutefor Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for financialsupport and hospitality.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 159References[1] M. Besalú, A. Kohatsu-Higa, S. Tindel, “Gaussian-type lower bounds for the density of solutions of SDEs driven byfractional Brownian motions”, Ann. Probab. 44 (2016), no. 1, p. 399-443.[2] J. Liu, C. A. Tudor, “Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solutionand analysis of its density”, Acta Math. Sci. 37 (2017), no. 6, p. 1545-1566.[3] H. Matsumoto, M. Yor, “Exponential functionals of Brownian motion. I. Probability laws at fixed time”, Probab. Surv.2 (2005), p. 312-347.[4] ——— , “Exponential functionals of Brownian motion. II. Some related diffusion processes”, Probab. Surv. 2 (2005),p. 348-384.[5] D. Nualart, The Malliavin calculus and related topics, second ed., Probability and Its Applications, Springer, 2006.[6] N. V. Tan, “Smoothness and Gaussian density estimates for stochastic functional differential equations with frac-tional noise”, Stat. Optim. Inf. Comput. 8 (2020), no. 4, p. 822-833.[7] N. Tien Dung, “The density of solutions to multifractional stochastic Volterra integro-differential equations”, Non-linear Anal., Theory Methods Appl. 130 (2016), p. 176-189.[8] ——— , “Kolmogorov distance between the exponential functionals of fractional Brownian motion”, C. R. Math.Acad. Sci. Paris 357 (2019), no. 7, p. 629-635.[9] ——— , “Gaussian lower bounds for the density via Malliavin calculus”, C. R. Math. Acad. Sci. Paris 358 (2020), no. 1,p. 79-88.[10] M. Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer, 2001.C. R. Mathématique — 2022, 360, 151-159

Density estimates for the exponential functionals of fractional Brownian motion

Comptes RendusMathématiqueNguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyDensity estimates for the exponential functionals of fractional BrownianmotionVolume 360 (2022), p. 151-159<https://doi.org/10.5802/crmath.274>© Académie des sciences, Paris and the authors, 2022.Some rights reserved.This article is licensed under theCreative Commons Attribution 4.0 International License.http://creativecommons.org/licenses/by/4.0/Les Comptes Rendus. Mathématique sont membres duCentre Mersenne pour l’édition scientifique ouvertewww.centre-mersenne.orgComptes RendusMathématique2022, 360, p. 151-159https://doi.org/10.5802/crmath.274Probability theory / ProbabilitésDensity estimates for the exponentialfunctionals of fractional Brownian motionNguyen Tien Dung∗, a, Nguyen Thu Hangb and Pham Thi Phuong Thuyca Department of Mathematics, VNU University of Science, Vietnam, NationalUniversity, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnamb Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien,Bac Tu Liem, Hanoi, 084 Vietnamc The faculty of Basic Sciences, Vietnam Air Defence and Air Force Academy, Son Tay,Ha Noi, 084 VietnamE-mails: dung@hus.edu.vn (N. Tien Dung), thuhangmdc@gmail.com (N. Thu Hang),thuypham82.com@gmail.com (P. T. Phuong Thuy)Abstract. In this note, we investigate the density of the exponential functional of the fractional Brownianmotion. Based on the techniques of Malliavin’s calculus, we provide a log-normal upper bound for the density.Mathematical subject classification (2010). 60G22, 60H07.Funding. This research was funded by the Vietnam National University, Hanoi under grant number QG.20.21.Manuscript received 6th March 2021, revised 19th June 2021, accepted 21st September 2021.1. IntroductionLet B H = (B Ht )t ∈ [0,T ] be a fractional Brownian motion (fBm) with Hurst index H ∈ (0,1). We recallthat B H is a centered Gaussian process with covariance functionRH (t , s) := E[B Ht BHs]= 12(t 2H + s2H −|t − s|2H), 0 ≤ s, t ≤ T.We consider the exponential functional of the formF =∫ T0eas+σBHs d s, (1)where T > 0, a ∈ R and σ > 0 are constants. It is known that this functional plays an importantrole in several domains. The special case, where H = 12 , has been well studied and a lot of fruitfulproperties of F can be founded in the literature, see e.g. [3, 4, 10]. However, to the best ourknowledge, the deep properties of F for H 6= 12 are scarce. In a recent paper [8], we have proved theLipschitz continuity of the cumulative distribution function of F with respect to the Hurst index∗Corresponding author.ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/152 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyH . The aim of the present paper is to investigate the density of F. Unlike the case H = 12 , it is noteasy to find the density of F explicitly for H 6= 12 and hence, our work will focus on providing theestimates for the density function. It should be noted that, in the last years, the density estimatesfor random variables related to fBm has been extensively studied, see e.g. [1,2,6,7] and referencestherein.The rest of this article is organized as follows. In Section 2, we briefly recall some of the relevantelements of the Malliavin calculus and two general estimates for densities. Our main results arethen stated and proved in Section 3. Our Theorems 6 and 8 point out that the density of F isbounded from above by log-normal densities.2. PreliminariesIn the whole paper, we assume H > 12 . Under this assumption, fBm admits the Volterra represen-tationB Ht =∫ t0K (t , s)dBs , (2)where (Bt )t∈[0,T ] is a standard Brownian motion and for some normalizing constant cH , the kernelK is given byK (t , s) = cH s1/2−H∫ ts(u − s)H− 32 uH− 12 du, 0 < s ≤ t ≤ T.Let us recall some elements of Malliavin calculus with respect to Brownian motion B , where Bis used to present B Ht as in (2). We suppose that (Bt )t ∈ [0,T ] is defined on a complete probabilityspace (Ω,F ,F,P ), where F= (Ft )t ∈ [0,T ] is a natural filtration generated by the Brownian motionB. For h ∈ L2[0,T ], we denote by B(h) the Wiener integralB(h) =∫ T0h(t )dBt .Let S denote the dense subset of L2(Ω,F ,P ) consisting of smooth random variables of the formF = f (B(h1), . . . , B(hn)), (3)where n ∈ N, f ∈ C∞b (Rn),h1, . . . , hn ∈ L2[0,T ]. If F has the form (3), we define its Malliavinderivative as the process DF := {D t F, t ∈ [0,T ]} given byD t F =n∑k=1∂ f∂xk(B(h1), . . . , B(hn))hk (t ).More generally, for each k ≥ 1, we can define the iterated derivative operator by settingDkt1, ..., tk F = D t1 . . . D tk F.For any p,k ≥ 1, we shall denote by Dk,p the closure of S with respect to the norm‖F‖pk, p := E |F |p +E[∫ T0∣∣D t1 F ∣∣p d t1]+·· ·+E [∫ T0. . .∫ T0∣∣∣Dkt1, ..., tk F ∣∣∣p d t1 . . . d tk] .A random variable F is said to be Malliavin differentiable if it belongs to D1,2. For any F ∈ D1,2,the Clark–Ocone formula says thatF −E [F ] =∫ T0E[Ds F∣∣Fs]dBs .Moreover, any F,G ∈D1,2, we have the following covariance formulaCov(F,G) = E[∫ T0Ds F E[DsG∣∣Fs]d s] . (4)In order to obtain the density estimates for exponential functionals we need the following generalresults.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 153Proposition 1. Let q,α,β be three positive real numbers such that 1q + 1α+ 1β = 1. Let F be a randomvariable in the spaceD2,α, such that E [‖DF‖−2βH ] <∞. Then the density ρF (x) of F can be estimatedas followsρF (x) ≤ cq,α,β (P (F ≤ x))1/q ×(E[‖DF‖−1H]+∥∥D2F∥∥Lα(Ω;H⊗H) ∥∥‖DF‖−2H ∥∥β) , x ∈R, (5)where cq,α,β is a positive constant and H = L2[0,T ].Proof. This proposition comes from the computations on [5, p. 87]. Proposition 2. Let F ∈D2,4 be such that E [F ] = 0. Define the random variableΦF :=∫ T0Ds F E[Ds F∣∣Fs]d s.Assume that ΦF 6= 0 a.s. and the random variables FΦF and1Φ2F∫ T0 DsΦF E [Ds F |Fs ]d s belong toL2(Ω). Then the law of F has a continuous density given byρF (x) = ρF (0)exp(−∫ x0hF (z)d z)exp(−∫ x0wF (z)d z), x ∈ suppρF , (6)where the functions wF and hF are defined bywF (z) := E[FΦF∣∣∣∣F = z] , hF (z) := E[1Φ2F∫ T0DsΦF E[Ds F∣∣Fs]d s∣∣∣∣∣F = z].Proof. This proposition is Theorem 7 in our recent paper [9]. 3. The main resultsIn this section, we provide explicit estimates for the density ρF (x) of the functional F definedby (1). Our idea is to consider the random variable X := lnF − E [lnF ] and use the relationρF (x) = 1x ρX (ln x −E [lnF ]), x > 0, where ρX denotes the density of X .We need some technical results.Proposition 3. Consider the random variable X := lnF −E [lnF ]. It holds that0 ≤ DθX ≤σK (T,θ) a.s. (7)0 ≤ Dr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s. (8)Proof. By the chain rule for Malliavin derivatives, we have, for 0 ≤ r,θ ≤ T,DθX =σ∫ Tθ K (s,θ)eas+σB Hs d s∫ T0 eas+σB Hs d s(9)andDr DθX =σ2∫ Tθ∨r K (s,θ)K (s,r )eas+σB Hs d s∫ T0 eas+σB Hs d s− σ2∫ Tr K (s,r )eas+σB Hs d s∫ Tθ K (s,θ)eas+σB Hs d s(∫ T0 eas+σB Hs d s)2 .Because the function s 7→ K (s,θ) is non-decreasing for each θ, (7) follows directly from (9). Wealso haveDr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s.To prove the non-negativity of the second order Malliavin derivative, we let U be a randomvariable with the density function defined byf (x) = eax+σB Hx∫ T0 eas+σB Hs d s, 0 ≤ x ≤ T.C. R. Mathématique — 2022, 360, 151-159154 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyDenote by EU the expectation with respect to U . We haveDr DθX = EU [K (U ,θ)K (U ,r )]−EU [K (U ,θ)]EU [K (U ,r )] .Note that the functions s 7→ K (s,θ) and s 7→ K (s,r ) are non-decreasing. Hence, by Chebyshev’sassociation inequality, Dr DθX ≥ 0 a.s. The proof of Proposition 3 is complete. Lemma 4. DefineMr := E [F |Fr ] = E[∫ T0eas+σk f l6BHs d s∣∣∣∣Fr ] , 0 ≤ r ≤ T.Then, for every p ≥ 2, we haveE[(max0≤r ≤TMr)p]≤C <∞,where C is a positive constant depending on p,T, a,σ and H .Proof. The stochastic process M := (Mr )0≤r≤T is a martingale with M0 = E [F ] and MT = F. Hence,by Burkhölder–David–Gundy inequality, we haveE[(max0≤r ≤TMr)p]≤ cp(M p0 +E[〈M〉p/2T])= cp((E [F ])p +E[〈M〉p/2T]), (10)where cp is a positive constant. Using the Clark–Ocone formula we haveMT = E MT +∫ T0E[Dr MT∣∣Fr ]dBr= E [F ]+σ∫ T0E[∫ TrK (s,r )eas+σBHs d s∣∣∣∣Fr ]dBr ,which gives us〈M〉T =∫ T0σ2(E[∫ TrK (s,r )eas+σBHs d s∣∣∣∣Fr ])2 dr≤∫ T0σ2K 2(T,r )M 2r dr a.s.Then, by Hölder inequality, we haveE[〈M〉p/2T]≤σp E[(∫ T0K2p−4p (T,r )K4p (T,r )M 2r dr)p/2]≤σp(∫ T0K 2(T,r )dr) p2 −1 (∫ T0K 2(T,r )E[M pr]dr)≤σp T (p−2)H(∫ T0K 2(T,r )E[F p]dr)=σp T pH E[F p].(11)Here we used the fact that∫ T0 K2(T,r )dr = E |B HT |2 = T 2H . So we obtain the desired conclusion byinserting (11) into (10). Proposition 5. Let X be as in Proposition 3. We defineΦX :=∫ T0 Ds X E [Ds X |Fs ]d s. Then,∣∣ΦX ∣∣−1 ∈ Lp (Ω), ∀ p ≥ 1.We also have (∫ T0∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 155Proof. It follows from (9) thatDθX ≥σTe−2|a|T+σ min0≤ s≤TB Hs −σ max0≤ s≤TB Hs∫ TθK (s,θ)d s a.s. (12)On the other hand, by using the Cauchy–Schwarz inequality, we haveE[DθX∣∣Fθ]≥ σ(E[√∫ Tθ K (s,θ)eas+σB Hs d s∣∣∣∣Fθ])2E[∫ T0 eas+σB Hs d s∣∣∣Fθ] a.s.and √∫ TθK (s,θ)eas+σB Hs d s ≥∫ Tθ K (s,θ)√eas+σB Hs d s√∫ Tθ K (s,θ)d sa.s.We therefore getE[DθX∣∣Fθ]≥ σ(∫ Tθ K (s,θ)E[eas/2+σBHs /2∣∣∣Fθ]d s)2∫ Tθ K (s,θ)d sE[∫ T0 eas+σB Hs d s∣∣∣Fθ] a.s.Furthermore, by Lyapunov’s inequality,E[eas/2+σBHs /2∣∣∣Fθ]≥ eas/2+σE[B Hs ∣∣Fθ]/2 a.s.As a consequence,E[DθX∣∣Fθ]≥ σe−|a|T+σ min0≤θ≤ s≤T Ns,θ ∫ Tθ K (s,θ)d smax0≤θ≤TMθa.s. (13)whereNs,θ := E[B Hs∣∣∣Fθ] and Mθ := E [∫ T0eas+σBHs d s∣∣∣∣Fθ] .Combining (12) and (13) yieldsDθX E[DθX∣∣Fθ]≥ σ2T e−3|a|T+σ min0≤ s≤T B Hs −σ max0≤ s≤T B Hs +σ min0≤θ≤ s≤T Ns,θ(∫ Tθ K (s,θ)d s)2max0≤θ≤TMθa.s.and hence,ΦX ≥ σ2Te−3|a|T+σ min0≤ s≤TB Hs −σ max0≤ s≤TB Hs +σ min0≤θ≤ s≤TNs,θ∫ T0(∫ Tθ K (s,θ)d s)2dθmax0≤θ≤TMθa.s. (14)We observe that ∫ T0(∫ TθK (s,θ)d s)2dθ =∫ T0∫ Tθ∫ TθK (t ,θ)K (s,θ)d sd tdθ=∫ T0∫ T0(∫ s∧t0K (t ,θ)K (s,θ)dθ)d sd t=∫ T0∫ T0E[B Ht BHs]d sd t = T2H+22H +2 .This, together with (14), yields∣∣ΦX ∣∣−1 ≤ 2H +2σ2T 2H+1e3|a|T+2σ max0≤ s≤TB Hs +σ max0≤θ≤ s≤TNs,θmax0≤θ≤TMθ a.s.C. R. Mathématique — 2022, 360, 151-159156 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyWe have (Ns,θ)0≤θ≤ s≤T is a Gaussian field with finite variances because Ns,θ =∫ θ0 KH (s,r )dBr .Hence, by Fernique’s theorem, there exists ε> 0 such thatE[eε max0≤θ≤ s≤T |Ns,θ|2]<∞.Sinceepσ max0≤θ≤ s≤TNs,θ ≤ ep2σ24ε +ε max0≤θ≤ s≤T |Ns,θ|2,this implies thateσ max0≤θ≤ s≤TNs,θ ∈ Lp (Ω) for any p ≥ 1.Similarly, we also havee2σ max0≤ s≤TB Hs ∈ Lp (Ω) for any p ≥ 1.So, recalling Lemma 4, we conclude that |ΦX |−1 ∈ Lp (Ω) for any p ≥ 1.We deduce from (12) that∫ T0∣∣DθX ∣∣2dθ ≥ σ2(2H +2)T 2H e−4|a|T+2σ min0≤ s≤TB Hs −2σ max0≤ s≤TB Hs a.s. (15)Hence, we also have (∫ T0 |DθX |2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1. The proof of Proposition 5 is complete.We now are in a position to bound the density ρF (x) of F. We first use Proposition 1 to estimatethe left tail of the density.Theorem 6. We haveρF (x) ≤ cxexp(− (ln x −E [lnF ])28σ2T 2H), 0 < x ≤ e E [lnF ], (16)where c is a positive constant.Proof. It is known from Proposition 5 that‖D X ‖−2H =(∫ T0∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.In addition, from the estimate (8), we have∥∥D2X ∥∥2L2(Ω ; H⊗H) = ∫ T0∫ T0E∣∣DθDr X ∣∣2dθdr ≤σ4 ∫ T0∫ T0K 2(T,θ)K 2(T,r )dθdr =σ4T 2H <∞.The above estimates allow us to use Proposition 1 with q =β= 4,α= 2 and we obtainρX (x) ≤ cP (X ≤ x)14 , x ∈R, (17)where c is a positive constant.The remaining of the proof is to bound P (X ≤ x) for x ≤ 0. We consider the function ϕ(λ:= E [e−λX ], λ > 0 (this function is well defined because F−1 ∈ Lp (Ω), ,∀ p ≥ 1). By usingrepeatedly the covariance formula (4), we haveC. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 157σ2X :=Var(X ) = E[ΦX]andϕ′(λ) =−E[X e−λX]=λE[e−λXΦX]=λσ2X E[e−λX]+λE[e−λX(ΦX −σ2X)]=λσ2X E[e−λX]−λ2E[e−λX∫ T0Ds X E[DsΦX∣∣Fs]d s]Since Ds X ≥ 0 and Dr Ds X ≥ 0, those imply that∫ T0 Ds X E [DsΦX |Fs ]d s ≥ 0, and hence,ϕ′(λ) ≤λσ2X E[e−λX]=λσ2Xϕ(λ), λ> 0.This, together the fact ϕ(0) = 1, gives usϕ(λ) ≤ eλ2σ2X2 , λ> 0.By Markov’s inequality we have, for all λ> 0,P (X ≤ x) ≤ eλxϕ(λ) ≤ eλx+λ2σ2X2 , x ≤ 0.When x ≤ 0, we can choose λ=− xσ2Xto getP (X ≤ x) ≤ e− x22σ2X , x ≤ 0.From the estimate (7), we have σ2X = E [ΦX ] ≤∫ T0 |Ds X |2d s ≤σ2T 2H . So we deduceP (X ≤ x) ≤ e−x22σ2T 2H , x ≤ 0. (18)Combining (17) and (18) yieldsρX (x) ≤ c e−x28σ2T 2H , x ≤ 0.where c is a positive constant. Recalling X = lnF − E [lnF ], the density of F satisfies ρF (x) =1x ρX (ln x −E [lnF ]). When 0 < x ≤ eE [lnF ], we have y := ln x −E [lnF ] ≤ 0. We thus obtainρF (x) = 1xρX (y) ≤ cxe− y28σ2T 2H = cxe− (ln x−E [lnF ])28σ2T 2H , 0 < x ≤ e E [lnF ].This completes the proof of Theorem 6. Remark 7. Replacing X by F −E [F ] in the proof of Theorem 6, we obtain the following Gaussianbound for the left tailρF (x) ≤ c e− (x−E [F ])28σ2F , x ≤ E [F ],where σ2F := Var(F ) and c is a positive constant.We now use Proposition 2 to estimate the right tail of the density.Theorem 8. We haveρF (x) ≤ cxexp(− (ln x −E [lnF ])22σ2T 2H), x > eE [lnF ], (19)where c is a positive constant.C. R. Mathématique — 2022, 360, 151-159158 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong ThuyProof. Let X be as in Proposition 3. Obviously, we have ΦX 6= 0 a.s. Moreover, from the esti-mates (7) and (8) we obtain0 ≤ DsΦX =∫ T0Ds DθX E[DθX∣∣Fθ]dθ+∫ T0DθX E[Ds DθX∣∣Fθ]dθ≤ 4σ3∫ T0K 2(T,θ)K (T, s)dθ = 4σ3K (T, s)T 2Hand0 ≤∫ T0DsΦX E[Ds X∣∣Fs]d s ≤ 4σ4T 2H ∫ T0K 2(T, s)d s = 4σ4T 4HHence, it follows from Proposition 5 that the random variable 1Φ2X∫ T0 DsΦX E [Ds X |Fs ]d s belongto L2(Ω). We also have XΦX ∈ L2(Ω) because−|a|T −σ max0≤ s≤TB Hs + lnT −E [lnF ] ≤ X ≤ |a|T +σ max0≤ s≤TB Hs + lnT −E [lnF ]and hence, X ∈ Lp (Ω) for all p ≥ 2.In view of Proposition 2, the density ρX (x) of X is given byρX (x) = ρX (0)exp(−∫ x0hX (z)d z)exp(−∫ x0wX (z)d z), x ∈ supp ρX , (20)wherewX (z) := E[XΦX∣∣∣∣X = z] and hX (z) := E[1Φ2X∫ T0DsΦX E[Ds X∣∣Fs]d s∣∣∣∣∣X = z].Since hX ≥ 0, this implies thatexp(−∫ x0hX (z)d z)≤ 1, x ≥ 0.From the estimate (7) we have0 ≤ΦX ≤σ2∫ T0K 2(T,θ)dθ =σ2T 2H a.s.and we obtainexp(−∫ x0wF (z)d z)≤ e−x22σ2T 2H , x ∈R.So we can conclude thatρX (x) ≤ ρX (0)e−x22σ2T 2H , x ≥ 0,and (19) follows because ρF (x) = 1x ρX (ln x −E [lnF ]). The proof of Theorem 8 is complete. AcknowledgmentsThe author would like to thank the anonymous referee for their valuable comments for improvingthe paper. A part of this paper was done while the first author was visiting the Vietnam Institutefor Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for financialsupport and hospitality.C. R. Mathématique — 2022, 360, 151-159Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 159References[1] M. Besalú, A. Kohatsu-Higa, S. Tindel, “Gaussian-type lower bounds for the density of solutions of SDEs driven byfractional Brownian motions”, Ann. Probab. 44 (2016), no. 1, p. 399-443.[2] J. Liu, C. A. Tudor, “Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solutionand analysis of its density”, Acta Math. Sci. 37 (2017), no. 6, p. 1545-1566.[3] H. Matsumoto, M. Yor, “Exponential functionals of Brownian motion. I. Probability laws at fixed time”, Probab. Surv.2 (2005), p. 312-347.[4] ——— , “Exponential functionals of Brownian motion. II. Some related diffusion processes”, Probab. Surv. 2 (2005),p. 348-384.[5] D. Nualart, The Malliavin calculus and related topics, second ed., Probability and Its Applications, Springer, 2006.[6] N. V. Tan, “Smoothness and Gaussian density estimates for stochastic functional differential equations with frac-tional noise”, Stat. Optim. Inf. Comput. 8 (2020), no. 4, p. 822-833.[7] N. Tien Dung, “The density of solutions to multifractional stochastic Volterra integro-differential equations”, Non-linear Anal., Theory Methods Appl. 130 (2016), p. 176-189.[8] ——— , “Kolmogorov distance between the exponential functionals of fractional Brownian motion”, C. R. Math.Acad. Sci. Paris 357 (2019), no. 7, p. 629-635.[9] ——— , “Gaussian lower bounds for the density via Malliavin calculus”, C. R. Math. Acad. Sci. Paris 358 (2020), no. 1,p. 79-88.[10] M. Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer, 2001.C. R. Mathématique — 2022, 360, 151-159

Directory of Open Access Journals

http://www.numdam.org/item/10.5802/crmath.274.pdf

Density estimates for the exponential functionals of fractional Brownian motion

Abstract

Similar works

Full text

Available Versions

Numérisation de Documents Anciens Mathématiques

Comptes Rendus Mathématique

Directory of Open Access Journals