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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On discrete approximation of occupation time of diffusion processes with irregular sampling 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 definedon 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. 0The quantity Γ(A) is the amount of time the diffusion X spends on set A. The problem ofevaluating Γ(A) is very important in many applied domains such as mathematical finance,queueing theory and biology. For example, in mathematical finance, these quantities areof great interest for the pricing of many derivatives, such as Parisian, corridor and Eddokooptions (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 0 nReceived December 25, 2013. Accepted June 26, 2014.Contact Nguyen Thi Lan Huong, e-mail address: nguyenhuong0011@gmail.com 3 Nguyen Thi Lan Huong, Ngo Hoang Long and Tran Quang Vinhand any n > 0. However, in practice for many reasons we can not observe X at regularobservation 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 standardBrownian motion and provide a functional central limit theorem (Theorem 2.2) for theerror process. It should be noted here that assumption A, which is obviously satisfiedfor regular sampling, is the key to construct the limit of the error process for irregularsampling. We then introduce an estimator for Γ(A) for general diffusion process and showthat its error is of order 3/4.2. Main results Throughout out this paper, we suppose that coefficients b and σ satisfy the followingconditions: (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 andmarginal distribution of X (see [11]). We note here that under a more restrictive conditionon 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 definedin [10]. At the nth stage, we suppose that X is observed at times tni , i = 0, 1, 2, ... satisfying0 = tn0 < tn1 < tn2 < ... and there exists a constant k0 > 0 such that ∆n ≤ k0 min ∆ni , ∀n, (2.2) iwhere ∆ni = tni − tni−1 and ∆n = max ∆ni . We assume moreover that limn→∞ ∆n = 0. iWe 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 ofrandom vectors with values in a Polish space (E, E), all defined on the same probabil ...
