On the First Hitting Times to Boundary of the Reflected O-U Process with Two-Sided Barriers

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1 Int. J. Contemp. Math. Sciences, Vol. 8, 213, no. 5, HIKARI Ltd, On the First Hitting Times to Boundary of the Reflected O-U Process with Two-Sided Barriers Lidong Zhang 1 School of Management, Tianjin University Tianjin 372, P.R. China 2 School of Science, Tianjin University of Science & Technology Tianjin 3457, P.R. China zhanglidong1979@yahoo.com.cn Ziping Du College of Economics & Management Tianjin University of Science & Technology Tianjin 3222, P.R. China Ximin Rong School of science, Tianjin University Tianjin 372, P.R. China Copyright c 213 Lidong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract. In this paper, we give the Laplace transform of the first hitting times to boundary and obtain its expectation on the reflected Ornstein-Uhlenbeck process with two-sided barriers. Finally,a comparison theorem is derived. Mathematics Subject Classification: 6H1 Keywords: Ornstein-Uhlenbeck process, reflected processes, first hitting times

2 236 Lidong Zhang, Ziping Du and Ximin Rong 1 Introduction Recently the reflected Ornstein-Uhlenbeck (ROU) process has extensively been investigated by many authors. ROU has many applications in different area such as queueing systems, credit risk modeling. In queueing systems, ROU with one barrier can be a limit process of a special queueing systems.ward & Glynn(23) discussed a queueing system with reneging and constructed a ROU process with one barrier via an appropriate Markovian approximation procedure. In their successive article properties of the reflected process was discussed. In passing, Bo(26) consider a fluid model with finite buffer capacity. This leads to the ROU process with two-sided barriers. Zhang(29,21) discuss properties of ROU with two barriers, the Laplace transform of the first hitting times and stationary distribution was obtained separately. Furthermore,ROU with two barriers was also used to model the price dynamics of the regulated goods or services, Bo(211) investigated Conditional default probability in a regulated market. The so-called ROU with two barriers is as follows. Let Z = {Z t,t } be an one-dimensional reflected Ornstein-Uhlenbeck process with barriers and b (b > is a constant), which is defined by { dzt =(μ αz t )dt + σdb t + dl t du t, Z = x [,b, (1.1) where B = {B t,t } is an one-dimensional standard Brownian motion, and μ R, α, σ R +. Here L = {L t,t } and U = {U t,t } are the regulators of point and b, respectively. Further, the processes L and U are uniquely determined by the following properties (see Harrison,M(1986)), Both t L t and t U t are continuous nondecreasing processes with L = U =, L and U increase only when Z = and Z = b, respectively, t I {Z s=}dl s = L t and I {Z s=b}du s = U t, for t. i.e., Lions & Sznitman (1984) guaranteed the existence and uniqueness of the solutions of (1.1).The first passage times or hitting times are important quantities in applications of ROU.Bo (26)and Zhang (21)obtained the Laplace transform of the first passage times of the O-U process with two-sided barriers. In this paper, we focus on the first hitting times to boundary. The paper is organized as follows. In Section 2, we investigate the first hitting times and obtain the Laplace transform of the first hitting times to boundary. Section 3 provide the result on the expectation of of the first hitting times. In the final section, a comparison theorem is obtained.

3 Reflected O-U process with two-sided barriers Laplace transform of the first hitting times In this section we consider the equation (1.1) with assumptions Z = x [,b. Let y [,b, define the first hitting time by T (y) := inf{t : Z t = y}, (2.1) with the usual convention inf Ø =. On the other hand, suppose λ> and f C 2 ([,b), define a linear operator Af(x) := σ2 2 f (x)+(μ αx)f (x) λf(x), for x [,b. For x b,we are going to give the expression of the Laplace transform of T = T () T (b). Theorem 2.1 Suppose that f (λ) is the solution of the following equation then, Af(y) =,y [,b, and f() = f(b), [ E x e λt f (λ) (x) = f (λ) (). (2.2) Proof. Applying Itô formula for h(t, x) := exp( λt)f(x) with f Cb 2([,b), we have h(t, Z t ) = h(,z ) = f(z )+ + = f(z )+ f (b) h t s (s, Z h s)ds + x (s, Z s)dz s 2 h x (s, Z s)d Z, Z 2 s e λs [ λf(z s ) αz s f (Z s )+ σ2 e λs f (Z s )dl s e λs f (Z s )du s + σ e λs A (λ) f(z s )ds + f () e λs du s + σ 2 f (Z s ) e λs dl s ds e λs f (Z s )db s e λs f (Z s )db s, (2.3) since L and U are finite variation (FV) processes. Let T< beastopping time and x [,b. It follows from martingale optional theorem, that [ E x e λt f(z T ) [ T = f(x)+e x e λs A (λ) f(z s )ds [ T [ T +f ()E x e λs dl s f (b)e x e λs du s.

4 238 Lidong Zhang, Ziping Du and Ximin Rong In particular, take T =T for y [,b, and note that [ T E x e λs du s =, and [ T E x e λs dl s =. Then, E x [ e λt f(z T ) = f(x)+e x [ T e λs Af(Z s )ds (2.4) Replace f by f (λ) in (2.4), we immediately get (2.2) by Z T =or b, f (λ) () = f (λ) (b). Thus the proof of the theorem is completed. Remark 2.1 For the Laplace transform of T can be given by f which satisfies Af(x) =and f() = f(b), we must find the solution of second order equation Af(x) =with initial condition f() = f(b). Next, we are concerned with the second order equation Af(x) = with initial condition f() = f(b). Notice that Af(x) := σ2 2 f (x)+(μ αx)f (x) λf(x), so Af(x) = with initial condition f() = f(b) equals to σ 2 2 f (x)+(μ αx)f (x) λf(x) =,f() = f(b) (2.5) Using the variable transformation z = αx μ and the notation g(z) =f( z α + μ α ), (2.5) can be reduced to the following equation where g (z) azg (z) bg(z) =,g( μ) =g(αb μ), (2.6) a = 2,b= 2λ α 2 σ 2. According to the theory of second order linear ODE, the general solution of (2.6) can be given by the following lemma.

5 Reflected O-U process with two-sided barriers 239 { Q Lemma 2.1 g(z) =C AΣ n 1 Q i= (2ai+b) n= z 2n +Σ n 1 } i= (2ai+a+b) (2n)! n= z 2n+1 is the (2n+1)! general solution of g (z) azg (z) bg(z) =,g( μ) =g(αb μ), where A = Σ n= Q n 1 i= (2ai+a+b) Σ n=1 Q n 1 i= (2ai+b) (2n)! (μ 2n (μ αb) 2n ) (2n+1)! (μ 2n+1 (μ αb) 2n+1 ) and C is the constant. convention 1 (λ +2jα)= 1 (λ +(2j +1)α) =1. j= j= we adopt the Proof.g (z) azg (z) bg(z) = is a second order linear ODE, so we must find two linearly independent solutions and the general solution will be a linear combination of them. We exploits the method of Taylor s series to obtain two linearly independent solutions. Indeed two linearly independent solutions can be expressed by Taylor s series, g 1 (z) =Σ n= n 1 i= (2ai + b) z 2n, g 2 (z) =Σ n= (2n)! n 1 i= (2ai + a + b) z 2n+1. (2n + 1)! So the linear combination of g 1 and g 2 is the general solution of g (z) azg (z) bg(z) =. Noting that g( μ) = g(αb μ), the general solution can be obtained easily. Theorem 2.2 is the general solution of σ 2 where A = Σ n= f (λ) (x) =C{Ag 1 (αx μ)+g 2 (αx μ)} 2 f (x)+(μ αx)f (x) λf(x) =,f() = f(b), (2.7) Q n 1 i= (2ai+a+b) (μ (2n+1)! 2n+1 (μ αb) 2n+1 ) Q n 1 Σ i= (2ai+b) n=1 (μ (2n)! 2n (μ αb) 2n ) and C is the constant. Proof. Noting that variable transformation z = αx μ and the notation g(z) =f 1 ( z + μ ), the accuracy of this theorem can be proved. α α 3 Expectation of the first hitting time In this section we give the analytic expression of the expectation of the first hitting times. In convenience, define a operator Af(x) := σ2 2 f (x)+(μ αx)f (x), for x [,b. With it,we have the following theorem.

6 24 Lidong Zhang, Ziping Du and Ximin Rong Theorem 3.1 For x [,b, E x (T ) = 4π x ( ) ) (αv μ) 2 μα σ2 exp P ( N(, 2α ) v (3.1) 4π x ( ) (αv μ) 2 D exp (3.2) where D = b exp ( (αv μ) 2 ( N( μ α, σ2 2α ) v ) d ) P ( ) b (3.3) and N( μ, σ2 ) denote the normal random variable with mean μ α 2α α σ 2 2α. and variance Proof. Applying Itô formula for twice differentiable function h(x) and recalling the notation of the operatora, we have h(z t ) = h(z )+ = h(z )+ + = h(z )+ h x (Z s)dz s h 2 x (Z s)d Z, Z 2 s [(μ αz s )h (Z s )+ σ2 2 h (Z s ) ds h (Z s )dl s h (Z s )du s + σ h (Z s )db s Ah(Z s )ds + h ()L s h (b)u s + σ h (Z s )db s The last equation holds, for L and U increase only when Z = and Z = b respectively. Let T < be a stopping time and Z = x [,b. It follows from martingale optional theorem, that E x (h(z T )) = h(x)+e x [ T In particular, take T =T for y [,b, and note that Ah(Z s )ds + h ()E x (L T ) h (b)e x (U T ). E x (U T )=,E x (L T )=. Then, E x (h(z T )) = h(x)+e x [ T Ah(Z s )ds (3.4)

7 Reflected O-U process with two-sided barriers 241 Denoting by where 4π h 1 (x) = x h (x) =h 1 (x) Dh 2 (x), ( ) ) (αv μ) 2 μα σ2 exp P ( N(, 2α ) v and 4π x ( ) (αv μ) 2 h 2 (x) = exp, we immediately obtain Ah (x) = 1,h () = h (b) =. Replacing h by h in(3.4) and noting Z T =or b, Theorem 3.1 can be proved easily. Thus the proof of the theorem is completed. 4 Comparison of the first hitting times This section mainly focus on a comparison theorem, namely the relation between T () and T (b). Theorem 4.1 For x [,b,we have P (T () <T(b)) = ( ) b x ( ) b (4.1) Proof. Denoting by u(x) = ( ) x, and applying Itô formula for u(x), we have u(z t ) = u(z )+ = u(z )+ + = u(z )+ u x (Z s)dz s u 2 x (Z s)d Z, Z 2 s [(μ αz s )u (Z s )+ σ2 2 u (Z s ) ds u (Z s )dl s u (Z s )du s + σ u (Z s )db s Au(Z s )ds + u ()L s u (b)u s + σ u (Z s )db s The last equation holds, for L and U increase only when Z = and Z = b respectively. For the stopping time T and Z = x [,b. It follows from martingale optional theorem, that [ T E x (u(z T )) = u(x)+e x Au(Z s )ds + u ()E x (L T ) u (b)e x (U T ).

8 242 Lidong Zhang, Ziping Du and Ximin Rong In particular, note that E x (u(z T )) = u() P (T () <T(b)) + u(b) P (T () >T(b)) and Au(x) =,u() =,E x (L T )=,E x (U T )=, we immediately obtain ( ) b x ασ P (T () <T(b)) = 2 ( ) b. Thus the proof of the theorem is completed. Acknowledgements:This paper is supported by the national natural science foundation of China(Grant No , ), The research projects of the social science and humanity on Young Fund of the ministry of Education (Grant No.11YJC917)and the Research Foundation of Tianjin university of Science and technology (Grant No ,21211) References [1 L.J.Bo, D.Tang, Y.J.Wang and X.W.Yang, On the regulated market: a structural approach, Quantitative Finance, 11 (211), [2 L.J.Bo, L.D.Zhang and Y.J.Wang, On the first passange times of reflected O-U processes with Two-sided barriers.queueing Systems 54 (26), [3 M.Harrison,Brownian Motion and Stochastic Flow Systems. John Wiley Sons, New York,1986. [4 P.Lions and A.Sznitman, Stochastic differential equations with reflecting boundary conditions. Comm. Pure Appl. Math.37 (1984), [5 A.Ward and P.A.Glynn, A diffusion approximation for Markovian queue with reneging. Queueing Systems 43 (23), [6 A.Ward and P.A.Glynn, Properties of the reflected Ornstein-Uhlenbeck process. Queueing Systems 44 (23), [7 L.D.Zhang and Z.P.Du,Properties of the first passage times of the reflected O-U process with two sided barriers. Queueing Systems 65 (21), [8 L.D.Zhang and C.M.Jiang,Stationary distribution of reflected O-U process with two-sided barriers. Statist.Probab.Lett,79 (29), Received: December, 212