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Let BH be a fractional Brownian motion with H∈ (0, 1) and g be a deterministic function. We study the asymptotic behaviour of the tail probability as for fixed x and as for fixed T. Our results partially generalise those obtained by Prakasa Rao in.
VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 Review Article Maximal Inequalities for Fractional Brownian Motion with Variable Drift Trinh Nhu Quynh1, Tran Manh Cuong2,* Military Information Technology Institute, 17 Hoang Sam, Cau Giay, Hanoi, Vietnam Department of Mathematics, VNU University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam Received 18 December 2019 Revised 06 March 2020; Accepted 15 June 2020 Abstract: Let BH be a fractional Brownian motion with H∈ (0, 1) and g be a deterministic function We study the asymptotic behaviour of the tail probability as for fixed x and as for fixed T Our results partially generalise those obtained by Prakasa Rao in [1] Keywords: Fractional Brownian motion, Maximal inequalities, Variable drift Introduction Let B H ( BtH )t 0 be a standard fractional Brownian motion (fBm) with Hurst index , i.e BH is a centered Gaussian process with covariance function given by 2H 2H RH t , s : E[ BtH BsH ] (t s H t s ), t , s We refer the readers to the monograph [2] for a short survey of properties of fBm When H , the following limit theorems were proved by Prakasa Rao in [1] k k 1 Theorem 1.1 Let g t ak t ak 1t a1t be a polynomial of degree k with ak > Then, for any T > and k ≥ we have Corresponding author Email address: cuongtm@vnu.edu.vn https//doi.org/ 10.25073/2588-1124/vnumap.4447 T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 log P( sup ( BtH g (t )) x) t[0, T] lim x Theorem 1.2 Let g t ak t ak 1t any x > and k ≥ we have k x k 1 a1t be a polynomial of degree k with ak >0 Then, for log P( sup ( BtH g (t )) x) lim sup t[0, T] T It is known that when H , 2T H T 2k 2 H ak2 BtH reduces to a standard Brownian motion In this case, Prakasa Rao's results reduce to those established previously by Jiao [3] Naturally, one would like to ask the following questions: Q1: Are Theorems 1.1 and 1.2 still true when H < ? Q2: Can we remove the polynomial structure of the drift g(t)? The aim of this paper is to provide an affirmative answer to Q1 and Q2 Our method is different from Prakasa Rao's where he mainly uses the classical Slepian's lemma In the present paper, we employ the techniques of Malliavin calculus which lead us to a shorter proof for more general results The rest of the paper is organized as follows In Section 2, we recall some fundamental concepts of Malliavin calculus The main results of the paper are stated and proved in Section Preliminaries It is well known that BtH admits the so-called Volterra representation (see, e.g [4]) t B K t , s dBs , t 0;T , H t (2.1) where (Bt)t≥0 is a standard Brownian motion, K(t, s) = for s ≥ t and H H 12 t 1 H H t u 2 K (t , s ) CH (t s ) (H ) (u s) du , s t H 1 H s s s where CH H (2 H 1) 1 (2 H )( H ) sin( ( H )) 2 Our proofs will be strongly based on techniques of Malliavin calculus For the reader's convenience, let us recall the definition of Malliavin derivative with respect to Brownian motion B, where B is used to present BtH as in (2.1) H We suppose that ( Bt )t[0,T ] is defined on the complete probability space (, , P) , where T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 ( t )t[0,T ] is a natural filter generated by the Brownian motion B For h L2 0, T we denote by B(h) the Wiener integral T B h = h t dBt Let S denote the dense subset of L2 , , P consisting of smooth random variables of the form F f ( B h1 , , B(hn )), where n , f Cb ( n derivative as the process (2.2) ) , h1, , hn ∈ L [0, T] If F has the form (2.2), we define its Malliavin DF : {Dt F ; t 0;T } given by n Dt F k 1 f ( B h1 , , B(hn ))hk (t ) xk 1,2 We will denote by the space of Malliavin differentiable random variables, it is the closure of S with respect to the norm T 2 : E F E Dt F dt 1,2 0 The next Proposition is a concrete version of Corollary 4.7.4 in [5] F Proposition 2.1 Let F be in 1,2 Assume that T ( D F ) d a.s (2.3) for some Then, for all x > 0, we have x2 P( F - E ( F ) x) exp (2.4) 2 Remark 2.1 The random variable -F also satisfies the conditions of Proposition 2.1 We therefore obtain the same bound for the left tail x2 P( F - E ( F ) x) P( F - E ( F ) x) exp , x 2 (2.5) The main results We firstly establish the following technical result which plays a key role in this paper Proposition 3.1 Suppose that f is a continuously differentiable function on ℝ with bounded derivative H and g is a continuous function on [0, T] Let Bt be a fBm with H 0, 1 , it holds that T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 ( x cT )2 H P sup ( Bt g (t )) x exp 2H t[0, T] 2sup f '( x) T x , x c , T (3.1) and ( x cT )2 H P sup ( Bt g (t )) x exp 2H t[0, T] 2sup f '( x) T x ,0 x c , T (3.2) where cT E sup f ( BtH ) g t t[0,T ] Proof If sup f '( x) , then the estimates (3.1) and (3.2) are trivial Hence, we can and will assume x that sup f '( x) x Consider a countable and dense subset S0 {tn , n 1} of [0, T] Define M n sup{ X t1 , X t2 , , X tn }, where X t : f ( Bt ) +g t Because f is continuous differential with bounded derivative, we know from Proposition 1.2.3 in [4] that X t D1,2 and H D X t f ' BtH D BtH f ' BtH K t , , t It is known from Proposition 2.1.10 in [4] that Mn ∈ D1’2 and Mn converges in L2(Ω) to sup X t In t[0,T ] order to evaluate the Malliavin derivative of Mn, we introduce the following sets: A1 {M n =Xt1 }, Ak {M n Xt1 , , M n X tk 1 , M n =X tk }, k n By the local property of the operator D; on the set Ak the derivatives of the random variables Mn and X tk coincide Hence, we can write n n D M n I Ak f ' BtHk D X tk I Ak f ' BtHk K (tk , ) I Ak k 1 k 1 Consequently, n ( D M n )2 f ' BtHk k 1 K (tk , ) I Ak And hence, T tk ( D M n )2 d ( D M n )2 d sup f '( x) x tn n K k 1 (tk , ) I Ak d Denote by F(t, ) the antiderivative of K (t, ) Since K(t, θ) = for θ ≥ t we can obtain (3.3) a.s T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 tn n K k 1 (tk , ) I Ak d t1 n tn n tn t1 tn1 K (tk , ) I Ak d k 1 K (tk , ) I Ak d k 1 F (t1 , t1 ) F (t1 ,0) I A1 F (tn , tn ) F (tn ,0) I An n K k 1 (tn , ) I An d E BtH1 I A1 E BtHn I An 3.4 t12 H I A1 tn2 H I An T H Combining (3.3) and (3.4) yields T ( D M n )2 d sup f '( x) T H , a.s (3.5) x The inequalitiy (3.5) shows that the random variable Mn satisfies the condition (2.3) of Proposition 2.1 Consequently, we can get x2 P M n E[M n ] x exp 2H 2sup f '( x) T x Then, by Fatou's lemma we deduce , x P sup ( BtH g (t )) cT x lim inf P M n E[M n ] x t[0, T] n ( x cT ) , x 0, exp 2H 2sup f '( x) T x which gives us (3.1) Similarly, we can obtain (3.2) by using the estimate (2.5) So the proof of Proposition is complete Remark 3.1 We state Proposition 3.1 in a general form because it can be useful for the other researches Let us give here an example Consider the fractional stochastic differential equation xt x0 ( xs )dBsH , t [0, T ] t Under suitable assumptions on σ and H, the Doss-Sussmann representation of xt is given by (see, e.g [6, 7]) xt f BtH , where f (x) solves the ordinary differential equation: f ' x f x , f 0 x Thus f (x) will satisfy the condition of Proposition 3.1 if σ(x) is continuous and bounded on ℝ We now are in a position to formulate and prove the first main result which generalises and improves Theorem 1.1 6 T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 H Theorem 3.1 Let Bt be a fBm with H ∈ (0, 1) Fixed T > 0, let g be a continuous function on [0, T] It holds that log P( sup ( BtH g (t )) x) t[0, T] lim x x 2T H Proof Obviously, we have P sup ( BtH g (t )) x P ( BTH g (T )) x P BTH x g (T ) t[0, T] H Since Bt is a normal random variable with mean zero and variance T2H, we can obtain x g (T ) P sup ( BtH g (t )) x P Z , TH t [0, T] (3.7) x g (T ) for sufficiently large x, we can apply TH where Z has a standard normal distribution Since Lemma in [2] to get ( x g (T )) e 2T for sufficiently large x P sup ( BtH g (t )) x t[0, T] 6( x g (T )) TH 2H As a consequence, log P( sup ( BtH g (t )) x) t[0, T] ( x g (T ))2 log(6 x g (T )) log T H , 2T H and hence, log P( sup ( BtH g (t )) x) t[0, T] lim inf x x 2T H (3.8) On the other hand, we obtain from Proposition 3.1 that ( x g (T ))2 P sup ( BtH g (t )) x exp , 2T H t[0, T] which gives us log P( sup ( BtH g (t )) x) t[0, T] x2 Notice ( x cT )2 x 2T H cT E[ sup ( BtH g t )] that is finite because t[0,T ] cT E[ sup BtH ] sup [g t ] E[ sup BtH ]T H sup [g t ] t[0,T ] t[0,T ] t[0,1] t[0,T ] Taking the limit x→∞ we get log P( sup ( BtH g (t )) x) lim sup x t[0, T] x So we can finish the proof by combining (3.8) and (3.9) 2T H □ (3.9) T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 The second main result of this paper is the following theorem H Theorem 3.2 Let Bt be a fBm with H ∈ (0, 1) and g be a continuous function on [0, 1) Assume that there exists a positive constant k > H such that g (T ) k : lim k T T Then, for any x > we have log P( sup ( BtH g (t )) x) t[0, T] lim sup T 2k 2 H T k2 (3.10) and log P( sup ( BtH g (t )) x) t[0, T] lim inf T 2k 2 H T k2 (3.11) Proof It is clear that cT E ( sup BtH g t ) E ( BTH g T ) g (T ) as T→∞ t[0,T ] Hence x < cT for sufficiently large T Once again, we apply Proposition 3.1 to get log P( sup ( BtH g (t )) x) t[0, T] T 2k 2 H ( x cT )2 for sufficiently large T, 2T k which leads us to the following log P( sup ( BtH g (t )) x) lim sup T Since lim T t[0, T] T 2k 2 H c 1 lim Tk T T (3.12) cT g (T ) lim k k This, together with (3.12), yields k T T T log P( sup ( BtH g (t )) x) lim sup t[0, T] T 2k 2 H T k2 Thus the estimate (3.10) was proved The remaining of the proof is to show (3.11) Because αk > and k > H, we have lim T for any x>0 Hence, x g (T ) g (T ) x P Z P Z TH TH for sufficiently large T Recalling (3.7) and using Lemma in [3], we have ( g (T ) x ) e 2T for sufficiently large T P sup ( BtH g (t )) x t[0, T] 6( g (T ) x) TH We therefore obtain 2H x g (T ) TH T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 log P( sup ( BtH g (t )) x) t[0, T] ( g (T ) x)2 log(6 g (T ) x) log T H 2T H and log P( sup ( BtH g (t )) x) t[0, T] lim inf T 2k 2 H T 1 g (T ) lim k k T T 2 The proof of Theorem is complete We end up this paper with a remark Remark 3.2 The method used in the paper can be applied to a larger class of Gaussian processes of form Yt k (t , s)dBs , t [0, T ], t where the Volterra kernel k(t, s) is continuous and satisfies the function t E Yt k t , s ds is t non-decreasing Here we note that the non-decreasing property of E Yt is used to prove the inequality (3.5) T For example, when Yt e (t s ) dBs is an Ornstein-Uhlenbeck process we have log P( sup (Yt g (t )) x) limsup x t[0, T] x T 2 k (T , s)ds e2T Conclusion Thus, we have generalized Rao's studies of fractional Brownian motion with continuous drift, H ∈ (0, 1) And we got the answers to question one and question who are the two issues raised in the introduction In these proofs we also use images of the Malliavin’s calculus, which are quite different from Rao's Acknowledgments This work was partially supported by Vietnam National University, Hanoi (grant no QG.20.26) References [1] B L S Prakasa Rao: Some maximal inequalities for fractional Brownian motion with polynomial drift Stoch Anal Appl 31, no 5, (2013) 785-799 https://doi.org/10.1080/07362994.2013.817240 [2] B L S Prakasa Rao: Statistical Inference for Fractional Diffusion Processes Wiley, Chichester, UK (2010) [3] L Jiao: Some limit results for probabilities estimates of Brownian motion with polynomial drift Indian J Pure Appl Math 41, no 3, (2010) 425-442 https://doi.org/10.1007/s13226-010-0026-9 T.N Quynh, T.M Cuong / VNU Journal of Science: Mathematics – Physics, Vol 36, No (2020) 1-9 [4] D Nualart: The Malliavin calculus and related topics Probability and its Applications Springer¬Verlag, Berlin, second edition (2006) [5] N Privault: Stochastic analysis in discrete and continuous settings with normal martingales Lecture Notes in Mathematics, 1982 Springer-Verlag, Berlin, 2009 [6] E Alịs, J A Ln, D Nualart: Stochastic Stratonovich calculus fBm for fractional Brownian motion with Hurst parameter less than Taiwanese J Math 5, no 3, (2001) 609-632 [7] I Nourdin: A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one Séminaire de probabilities XLI, 181-197, Lecture Notes in Math., 1934, Springer, Berlin (2008) 181-197 https://doi.org/10.1007/978-3-540-77913-1-8 ... Hanoi (grant no QG.20.26) References [1] B L S Prakasa Rao: Some maximal inequalities for fractional Brownian motion with polynomial drift Stoch Anal Appl 31, no 5, (2013) 785-799 https://doi.org/10.1080/07362994.2013.817240... Statistical Inference for Fractional Diffusion Processes Wiley, Chichester, UK (2010) [3] L Jiao: Some limit results for probabilities estimates of Brownian motion with polynomial drift Indian J Pure... settings with normal martingales Lecture Notes in Mathematics, 1982 Springer-Verlag, Berlin, 2009 [6] E Alịs, J A Ln, D Nualart: Stochastic Stratonovich calculus fBm for fractional Brownian motion with