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On discrete approximation of occupation time of diffusion processes with irregular sampling

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Thus, in this paper, we will construct an estimation scheme for Γ(A) based on an irregular sample {Xti, i = 0, 1, . . .} of X and study its asymptotic behavior. In particular, we first introduce an unbiased estimator for when X is a standard Brownian motion and provide a functional central limit theorem (Theorem 2.2) for the error process.

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Nội dung Text: On discrete approximation of occupation time of diffusion processes with irregular sampling

  1. JOURNAL OF SCIENCE OF HNUE Interdisciplinary Science, 2014, Vol. 59, No. 5, pp. 3-16 This paper is available online at http://stdb.hnue.edu.vn ON DISCRETE APPROXIMATION OF OCCUPATION TIME OF DIFFUSION PROCESSES WITH IRREGULAR SAMPLING Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh Faculty of Mathematics and Informatics, Hanoi National University of Education Abstract. Let X be a diffusion processes and A be some Borel subsetR of R. In this t paper, we introduce an estimator for the occupation time Γ(A)t = 0 I{Xs ∈A} ds based on an irregular sample of X and study its asymptotic behavior. Keywords: Occupation time, diffusion processes, irregular sample. 1. Introduction Let X be a solution to the following stochastic differential equation dXt = b(Xt )dt + σ(Xt )dWt , X0 = x0 ∈ R, (1.1) where b and σ are measurable functions and Wt is a standard Brownian motion defined on a filtered probability space (Ω, F , (Ft)t>0 , P). For each set A ∈ B(R) the occupation time of X in A is defined by Z t Γ(A)t = I{Xs ∈A} ds. 0 The quantity Γ(A) is the amount of time the diffusion X spends on set A. The problem of evaluating Γ(A) is very important in many applied domains such as mathematical finance, queueing theory and biology. For example, in mathematical finance, these quantities are of great interest for the pricing of many derivatives, such as Parisian, corridor and Eddoko options (see [1, 2, 9]). In practice, one cannot observe the whole trajectory of X during a fixed interval. In other words, we can only collect the values of X at some discrete times, say 0 = t1 < t2 < . . . Recently, Ngo and Ogawa [10] and Kohatsu-Higa et al. [7] have introduced an estimate for Γ(A) by using a Riemann sum and they studied the rate of convergence of this i approximation when X is observed at regular points, i.e. {ti = , i 6 [nt]} for all i > 0 n Received December 25, 2013. Accepted June 26, 2014. Contact Nguyen Thi Lan Huong, e-mail address: nguyenhuong0011@gmail.com 3
  2. Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh and any n > 0. However, in practice for many reasons we can not observe X at regular observation points. Thus, in this paper, we will construct an estimation scheme for Γ(A) based on an irregular sample {Xti , i = 0, 1, . . .} of X and study its asymptotic behavior. In particular, we first introduce an unbiased estimator for Γ(A) when X is a standard Brownian motion and provide a functional central limit theorem (Theorem 2.2) for the error process. It should be noted here that assumption A, which is obviously satisfied for regular sampling, is the key to construct the limit of the error process for irregular sampling. We then introduce an estimator for Γ(A) for general diffusion process and show that its error is of order 3/4. 2. Main results Throughout out this paper, we suppose that coefficients b and σ satisfy the following conditions: (i) σ is continuously differentiable and σ(x) ≥ σ0 > 0 for all x ∈ R, (2.1) (ii) |b(x) − b(y)| + |σ(x) − σ(y)| ≤ C|x − y| for some constant C > 0. The above conditions on b and σ guarantee the continuity of sample path and marginal distribution of X (see [11]). We note here that under a more restrictive condition on the smoothness and boundedness of b, σ and their derivatives, Kohatsu-Higa et al. [7] have studied the strong rate of approximation of Γ(A) via a Riemann sum as one defined in [10]. At the nth stage, we suppose that X is observed at times tni , i = 0, 1, 2, ... satisfying 0 = tn0 < tn1 < tn2 < ... and there exists a constant k0 > 0 such that ∆n ≤ k0 min ∆ni , ∀n, (2.2) i where ∆ni = tni − tni−1 and ∆n = max ∆ni . We assume moreover that limn→∞ ∆n = 0. i We denote ηn (s) = tni if tni ≤ s < tni+1 . 2.1. Occupation time of Brownian Motions We first recall the concept of stable convergence: Let (Xn )n≥0 be a sequence of random vectors with values in a Polish space (E, E), all defined on the same probability space (Ω, F , (Ft )t≥0 , P) and let G be a sub-σ-algebra of F . We say that Xn converges G−st G-stably in law to X, denote Xn → X, if X is an E−valued random vector defined on an extension (Ω′ , F ′, P′ ) of the original probability space and limn→∞ E(g(Xn )Z) = E′ (g(X)Z), for every bounded continuous functions g : E → R and all bounded st G−measurable random variables Z (see [4, 5, 8]). When G = F we write Xn → X G−st instead of Xn → X. We denote by Lt (a) the local time of a standard Brownian motion B at a, up to and including t given by Z t Lt (a) = |Bt − a| − |a| − sign(Bs − a)dBs . 0 4
  3. On discrete approximation of occupation time of diffusion processes with irregular sampling For each Borel function g defined on R and γ > 0, we set Z Z γ βγ (g) = |x| |g(x)|dx, λ(g) = g(x)dx. In order to study the asymptotic behavior of the estimation error, we need the following assumption. Assumption A: There exists a non-decreasing function Ft (x) such that 1 P → Ft (x), ∀ t > 0, x ∈ R. (2.3) X 3/2 (tni − tni−1 )3/2 E(Ltni (x) − Ltni−1 (x)|Ftni−1 ) − (∆n ) tn 6t i Theorem 2.1 (The approximation of local time). Suppose that g satisfies the following conditions: g(x) = o(x) as x → ∞, β1 (g) < ∞, and λ(|g|) < ∞. (2.4) Then for all x ∈ R it holds  Btn − x  P Xp tni − tni−1 g p i−1 n n → λ(g)Lt (x). − tn ti − ti−1 i 6t Now, we proceed to state the functional central limit theorem for the error process. First, let us recall the definition of F -progressive conditional martingale (see [5] for more details). We call extension of B another stochastic basis B˜ = (Ω, ˜ constructed ˜ F˜ , (F˜t), P) as follows: We have an auxiliary filtered space (Ω , F , (Ft )t≥0 ) such that each σ-field ′ ′ ′ Ft− is separable, and a transition probability Qω (dω ) from (Ω, F ) into (Ω , F ), and we ′ ′ ′ ′ set ˜ = Ω × Ω′ , F˜ = F˜ ⊗ F ′ , F˜ = ∩s>t Fs ⊗ F ′ , Ω s ˜ P(dω, ′ ′ dω ) = P(dω)Qω (dω ). A process X on the extension B˜ is called an F -progressive conditional martingale if it is adapted to F˜ and if for P-almost all ω in the process X(ω, .) is a martingale on the basis ′ ′ ′ Bω = (Ω , F , (Ft )t≥0 , Qω ). Theorem 2.2. Suppose that B is a standard Brownian motion defined on a filtered space ˜ For each n > 1, t > 0 and K ∈ R we set B = (Ω, F , Ft , P). Z t  ˜ n Bη (s) − K  Γ(K)t = Φ pn ds, 0 s − ηn (s) where Φ is the standard normal distribution function.Then, ˜ (i) Γ(K) n t is an unbiased estimator for the occupation time Γ([K, ∞))t ; 5
  4. Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh ˜ n P (ii) Γ(K) t − → Γ([K, ∞))t ; (iii) Moreover, suppose that A holds then there exists a good extension B˜ of B and a continuous B-biased F -progressive conditional martingale with independent increment X ′ on this extension with 9 hX ′ , X ′ it = √ Ft (K), hX ′ , Bi = 0. 20 2π 1 ˜ n  st Γ(K) t − Γ([K, ∞)) t → X ′. − (∆n )3/4 Remark 2.1. Assumption A make sense. It is is obviously satisfied for regular sampling. The following condition is sufficient to have (2.3), lim min ∆ni (∆n )−1 = 1. (2.5) n→∞ i Moreover, we have Ft (x) = Lt (x). min(∆ni ) Indeed, we set γn = i and max(∆ni ) i 1 X Ftn (x) = (tn − tni−1 )3/2 E(Ltni (x) − Ltni−1 (x)|Ftni ). (∆n )3/2 tn 6t i i Hence Ftn (x) = E(Ltni (x) − Ltni−1 (x)|Ftni ) − Stn (x), where P tn i 6t X (∆ni )3/2  0 6 Stn (x) = 1− E(Ltni (x) − Ltni−1 (x)|Ftni ) tn 6t (∆n )3/2 i P X 6 (1 − γn ) E(Ltni (x) − Ltni−1 (x)|Ftni ) − → 0, tn i 6t P P P since → Lt (x). Thus, Sn (t) − → 0, and Ftn (x) − → Lt (x) as P E(Ltni (x) − Ltni−1 (x)|Ftni ) − tn i 6t n → ∞. The condition (2.5) can be localized a little as follows: Suppose that there exists a sequence of fixed times S1 < S2 < ..., which does not depend on n such that in each interval (Si , Si+1 ) the condition (2.5) is satisfied. Then the condition (2.3) also holds. 2.2. Occupation time of general diffusion In order to study the rate of convergence, we recall the definition of C-tightness. First, we denote by D(R) the Polish space of all càdlàg function: R+ → R with Skorokhod topology. A sequence of D(R) -valued random vector (Xn ) defined on (Ω, F , (F )t , P) is tight if inf sup P(Xn ∈ / K) = 0, where the infimum is taken over all K n 6
  5. On discrete approximation of occupation time of diffusion processes with irregular sampling compact sets K in D(R). The sequence (Xn ) of processes is called C-tightness if it is tight, and if all limit points of the sequence {L(Xn )} are laws of continuous processes (see [5]). Rx 1 m Denote S(x) = and For each set S x0 du Y t = S(X t ). A ∈ B(R), A = [a2i , a2i+1 ) σ(u) i=0 where −∞ 6 a0 < a1 < · · · < a2m+1 6 +∞ we introduce the following estimate for Γ(A)t : m Z t  ˜ n X S(a2j+1 ) − S(Xηn (s) )   S(a ) − S(X 2j ηn (s) )  Γ(A)t = Φ p −Φ p ds. j=0 0 s − ηn (s) s − ηn (s) In particular, Rt if A = [K, +∞) then the biased and consistent estimator for the occupation time 0 I{Xs >K}ds is defined by Z t  ˜ n S(Xηn (s) ) − S(K)  Γ([K, ∞))t = Φ p ds. 0 s − ηn (s) n Theorem 2.3. For each set A ∈ B(R), A = [a2i , a2i+1 ) where −∞ 6 a0 < a1 < S i=0 · · · < a2n+1 6 +∞ the sequence of stochastic processes  1 Z t  ˜ Γ(A) n − I {Xs ∈A} ds) t (∆n )3/4 0 t≥0 is C-tight. 3. Proofs √ We denote (Pt )t>0 a Brownian semigroup given by Pt k(x) = R k(x + y t)ρ(y)dy, where ρ(y) = √12π e−y /2 and k is a Lebesgue integrable function. 2 3.1. Some preliminary estimates Throughout this section we denote by K a constant which may change from line to line. If K depends on an additional parameter γ, we write Kγ . We first recall some estimates on the semigroup (Pt ). Lemma 3.1 (Jacod [4]). Let k : R → R be an integrable function. If t > s > 0 and γ > 0 we have: λ(|k|) |Pt k(x)| 6 K √ , (3.1) t
  6. λ(k)
  7. Kγ  β1 (k) β (k)  √ + 1+γ γ , (3.2) 2
  8. Pt k(x) − √ e−x /2t
  9. 6
  10. 2πt t 1 + |x/ t|γ 1 + |x|
  11. λ(k) −x2 /2t
  12. K
  13. Pt k(x) − √ e
  14. 6 (β2 (k) + β1 (k)|x|). (3.3)
  15. 2πt t3/2 7
  16. Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinh We will need the following estimate. Lemma 3.2. Let k : R → R be an integrable function. Suppose that the sequence {tni } satisfies (2.2). Denote X x + Btn  Xp x + Btn  γ1 (k, x)nt = E (∆ni )2 k( p ni−1 ) , γ2 (k, x)nt = E ∆ni k( p ni−1 ) . tn 6t ∆i tn 6t ∆i i i i>2 i>2 Then √ (i) |γ1 (k, x)nt | 6 Kλ(|k|)(∆n )3/2 t, (3.4) √ (ii) |γ2(k, x)nt | 6 Kλ(|k|) t. (3.5) Moreover, if λ(k) = 0 then tk0 |γ1 (k, x)nt | 6 Kβ1 (k)(∆n )2 k0 (1 + log+ ( )), (3.6) ∆n |γ1 (k, x)nt | 6 K(∆n )2 (β2 (k) + β1 (k)|x|), (3.7) and tk0 (3.8) p |γ2 (k, x)nt | 6 Kβ1 (k) ∆n k0 (1 + log+ ( )), ∆n (3.9) p p n |γ2 (k, x)t | 6 K k0 ∆n (β2 (k) + β1 (k)|x|). Proof. From (3.1) and estimates (4.1), (4.2), we obtain (3.4) and (3.6). Furthermore, from (3.3) in the Lemma 3.1 we get X K |γ1 (k, x)nt | 6 (∆ni )2 tn 3 (β2 (k) + β1 (k)|x|)) tn i 6t, i>2 n )2 ( ∆i−1 i Z t 5/2 6 Kk0 (∆n ) (β2 (k) + β1 (k)|x|)) x−3/2 dx ∆n /k0 2 6 K(∆n ) (β2 (k) + β1 (k)|x|). By using analogous arguments as above, we obtain (3.5), (3.8) and (3.9). Lemma 3.3. Assume that λ(g) = 0 and g satisfies (2.4), then  !2 1 X x + Btni−1 n→∞ (i) E  (tni − tni−1 )2 g p  −−−→ 0. (∆n )3 tn tni − tni−1 i 6t  !2 Xp x + Btn n→∞ (ii) E  tni − tni−1 g p n i−1 n  − −−→ 0. tn 6t ti − ti−1 i 8
  17. On discrete approximation of occupation time of diffusion processes with irregular sampling Proof. We first note that condition (2.4) implies that λ(g 2 ) < ∞. We write  !2 1 X x + Btni−1 E  (tni − tni−1 )2 g p  (∆n )3 n n n t 6t ti − ti−1 i 1 X x + Btni−1 2  = E((tni − tni−1 )4 g( p n )) + (∆n )3 tn ∆i ) i 6t 2  X  x + Btni−1 x + Btnj−1  (3.10) X n 2 n 2 + E (∆i ) g( )( (∆j ) g( p n )) . (∆n )3 i:tn
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