Formulas for stopped diffusion processes with stopping times based on drawdowns and drawups

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1 Stochastic Processes and their Applications 119 (009) Formlas for stopped diffsion processes with stopping times based on drawdowns and drawps Libor Pospisil, Jan Vecer, Olympia Hadjiliadis Department of Statistics, Colmbia University, 155 Amsterdam Avene, NY 1007, USA Department of Mathematics, Brooklyn College (C.U.N.Y.), Ingersoll Hall, NY 1109, USA Received 5 Febrary 008; received in revised form 6 Janary 009; accepted 16 Janary 009 Available online 4 Janary 009 Abstract This paper stdies drawdown and drawp processes in a general diffsion model. The main reslt is a formla for the joint distribtion of the rnning minimm and the rnning maimm of the process stopped at the time of the first drop of size a. As a conseqence, we obtain the probabilities that a drawdown of size a precedes a drawp of size b and vice versa. The reslts are applied to several eamples of diffsion processes, sch as drifted Brownian motion, Ornstein Uhlenbeck process, and Co Ingersoll Ross process. c 009 Elsevier B.V. All rights reserved. MSC: primary 91B8; secondary 60G44 Keywords: Diffsion process; Drawdowns and drawps; Stopping time 1. Introdction In this article, we stdy properties of a general diffsion process {X t } stopped at the first time when its drawdown attains a certain vale a. Let s denote this time as T D (a). The drawdown of a process is defined as the crrent drop of the process from its rnning maimm. We present two main reslts here. First, we derive the joint distribtion of the rnning minimm and the rnning maimm stopped at T D (a). Second, we calclate the probability that a drawdown of Corresponding athor at: Department of Statistics, Colmbia University, 155 Amsterdam Avene, NY 1007, USA. Tel.: ; fa: addresses: lp185@colmbia.ed (L. Pospisil), vecer@stat.colmbia.ed (J. Vecer), ohadjiliadis@brooklyn.cny.ed (O. Hadjiliadis) /$ - see front matter c 009 Elsevier B.V. All rights reserved. doi: /j.spa

2 564 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) size a precedes a drawp of size b, where the drawp is defined as the increase of {X t } over the rnning minimm. All formlas are epressed in terms of the drift fnction, the volatility fnction, and the initial vale of {X t }. In addition to the main theorems, this paper contains other reslts that help s to nderstand the behavior of diffsion processes better. For eample, we relate the probability that the drawp process stopped at T D (a) is zero to the epected rnning minimm stopped at T D (a). We apply the reslts to several eamples of diffsion processes: drifted Brownian motion, Ornstein Uhlenbeck process (OU), and Co Ingersoll Ross process (CIR). These eamples play important roles in change point detection and in finance. We also discss how the reslts presented in this paper are related to the problem of qickest detection and identification of twosided changes in the drift of general diffsion processes. Or reslts etend several theorems stated and proved in [1 3]. These reslts inclde the distribtion of a diffsion process stopped at the first time it hits either a lower or an pper barrier, and the distribtion of the rnning maimm of a diffsion process stopped at time T D (a). The formlas for a drifted Brownian motion presented here coincide with the reslts in [4]. The approach sed in [4] is based on a calclation of the epected first passage times of the drawdown and drawp processes to levels a and b. However, while this approach applies to a drifted Brownian motion, it cannot be etended to a general diffsion process. In this paper, we derive the joint distribtion of the rnning maimm and minimm stopped at T D (a), which can be obtained for a general diffsion process. Sbseqently, we se this reslt to calclate the probability that a drawdown precedes a drawp. Properties of drawdown and drawp processes are of interest in change point detection, where the goal is to test whether an abrpt change in a parameter of a dynamical system has occrred. Drawdowns and drawps of the likelihood ratio process serve as test statistics for hypotheses abot the change point. Details can be fond, for eample, in [5 7]. The concept of a drawdown has been also been stdied in applied probability and in finance. The nderlying diffsion process sally represents a stock inde, an echange rate, or an interest rate. Some characteristics of its drawdown, sch as the epected maimm drawdown, can be sed to measre the downside risks of the corresponding market. The distribtion of the maimm drawdown of a drifted Brownian motion was determined in [8]. Cherny and Dpire [9] derived the distribtion of a local martingale and its maimm at the first time when the corresponding range process attains vale a. Salminen and Vallois [10] derived the joint distribtion of the maimm drawdown and the maimm drawp of a Brownian motion p to an independent eponential time. Vecer [11] related the epected maimm drawdown of a market to directional trading. Several athors, sch as Grossman and Zho [1], Cvitanic and Karatzas [13], and Chekhlov et al. [14], discssed the problem of portfolio optimization with drawdown constraints. Meilijson [15] sed stopping time T D (a) to solve an optimal stopping problem based on a drifted Brownian motion and its rnning maimm. Obloj and Yor [16] stdied properties of martingales with representation H(M t, M t ), where M t is a continos local martingale and M t its spremm p to time t. Nikeghbali [17] associated the Skorokhod stopping problem with a class of sbmartingales which incldes drawdown processes of continos local martingales. This paper is strctred in the following way: notation and assmptions are introdced in Section. In Section 3, we derive the joint distribtion of the rnning maimm and the rnning minimm stopped at the first time that the process drops by a certain amont (Theorem 3.1), and in Section 4, we calclate the probability that a drawdown of size a will precede a drawp of size b (Theorems 4.1 and 4.). Special cases, sch as drifted Brownian motion, Ornstein Uhlenbeck process, Co Ingersoll Ross process, are discssed in Section 5. The relevance of the reslt in

3 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Section 4 to the problem of qickest detection and identification of two-sided alternatives in the drift of general diffsion processes is also presented in Section 5. Finally, Section 6 contains conclding remarks.. Drawdown and drawp processes In this section, we define drawdown and drawp processes in a diffsion model and present the main assmptions. Consider an interval I = (l, r), where l < r. Let (Ω, F, P) be a probability space, {W t } a Brownian motion, and {X t } a niqe strong soltion of the following stochastic differential eqation: dx t = µ(x t )dt + σ (X t )dw t, X 0 = I, (1) where X t I for all t 0. Moreover, we will assme that fnctions µ(.) and σ (.) meet the following conditions: σ (y) > 0, for y I, r l Ψ(, z) z dz =, for all a > 0 sch that a I, (3) z a Ψ(, y)dy Ψ(, z) z+b dz =, for all b > 0 sch that + b I, (4) z Ψ(, y)dy where Ψ(, z) = e z γ (y)dy and γ (y) = µ(y). Drifted Brownian motion, Ornstein Uhlenbeck σ (y) process, and Co Ingersoll Ross process are eamples of diffsion processes satisfying these assmptions. Fnctions µ(.) and σ (.) will be referred to as the drift and the volatility fnctions. Note that a process given by (1) has the strong Markov property. If we need to emphasize that is the starting vale of {X t }, we will write P [. ]. Let s define the rnning maimm, {M t }, and the rnning minimm, {m t }, of process {X t } as: M t = sp X s, m t = inf X s. s [0,t] s [0,t] The drawdown and the drawp of {X t } are defined as: DD t = M t X t, DU t = X t m t. We denote by T D (a) and T U (b) the first passage times of the processes {DD t } and {DU t } to the levels a and b respectively, where a > 0, b > 0, a I, and + b I. We set T D (a) = or T U (b) = if process DD t does not reach a or process DU t does not reach b: T D (a) = inf {t 0; DD t = a}, T U (b) = inf {t 0; DU t = b}. Conditions (3) and (4) ensre that P [T D (a) < ] = lim v r P [M TD (a) v] = 1, P [T U (b) < ] = lim l+ P [m TU (b) > ] = 1. Ths, we assme that T D (a) < and T U (b) < almost srely for any a > 0 and b > 0, sch that a I and + b I. ()

4 566 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) In the following sections, we derive the joint distribtion of (m TD (a), M TD (a)) (Section 3) and a formla for the probability P [T D (a) < T U (b)] (Section 4). 3. Joint distribtion of the rnning minimm and the rnning maimm stopped at T D (a) The distribtion of random variable M TD (a) was derived in [3]. In or paper, we focs on the joint distribtion of (m TD (a), M TD (a)). Note that the rnning minimm stopped at time T D (a) is bonded by a and : a m TD (a). The joint distribtion of the rnning minimm and maimm stopped at time T D (a) will be denoted as H : H (, v) = P [m TD (a) >, M TD (a) > v], where [ a, ] and v [, ). In the following theorem, we will epress H in terms of fnction Ψ(, z) = e z γ (y)dy, where γ (y) = µ(y) σ (y). Theorem 3.1. Let a > 0 sch that a I. The random variables m TD (a) and M TD (a) have the following joint distribtion: H (, v) = +a e v +a Ψ(+a,z) zz a Ψ(+a,y)dy dz, (5) where [ a, ], v [+a, ), Ψ(, z) = e z γ (y)dy and γ (y) = µ(y). If [ a, ] σ (y) and v [, + a), then: H (, v) = +a. (6) Proof. The process {X t } is given by (1) and X 0 =. First, let s assme that [ a, ] and v [ + a, ). The event { m TD (a) >, M TD (a) > v } occrs if and only if the process {X t } attains + a withot dropping below and then eceeds v before the drawdown reaches a. De to the Markov property of the process {X t }, we can write the probabilities of these events as follows: H (, v) = P [m TD (a) >, M TD (a) > v] = P [X τ(,+a) = + a] P +a [M TD (a) > v] = +a e v +a Ψ(+a,z) zz a Ψ(+a,y)dy dz, (7) where τ(, + a) = inf {t 0; X t = or X t = + a}. The formla for the first probability in (7) follows from [1], page 110. The second probability in (7), representing the srvival fnction of M TD (a), was derived in [3], page 60. Finally, if v < + a, we have { mtd (a) >, M TD (a) > v } = { m TD (a) > } = { X τ(,+a) = + a } becase M TD (a) m TD (a) + a. Ths, H (, v) = +a.

5 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) The distribtion fnction, the srvival fnction, and the density fnction of m TD (a) will be denoted as F, F, and f, respectively: F () = 1 F () = P [m TD (a) > ], f () = df (), d where [ a, ]. We can derive the marginal distribtion of m TD (a) from the reslts in Theorem 3.1. Corollary 3.. Let a > 0 sch that a I. The distribtion fnction, the density fnction, and the epected vale of random variable m TD (a) are: F () = +a, f () = Ψ(, + a) + +a ( ) +a, (9) [ ] E mtd (a) = a +a d, (10) +a where [ a, ], Ψ(, z) = e z γ (y)dy, and γ (y) = µ(y) σ (y). Let s denote the distribtion fnction and the srvival fnction of the rnning maimm stopped at T D (a) as G and G: G (v) = 1 G (v) = P [M TD (a) > v], v < r. Note that the joint distribtion of (m TD (a), M TD (a)) can be represented by the marginal distribtions F and G : H (, v) = F () G +a (v), (11) where [ a, ] and v [ +a, ). Let s calclate the derivative of H (, v) with respect to, which will be sed in the proof of the main theorem. Lemma 3.3. Let a > 0 sch that a I. Let [ a, ] and v + a. Then H v +a (, v) = e Ψ(+a,z) zz a Ψ(+a,y)dy dz +a ( +a where Ψ(, z) = e z γ (y)dy and γ (y) = µ(y) σ (y). Proof. According to (11), H (, v) = F () G +a (v) + F () G +a(v). (8) ), (1) Note that the fnction Ψ has the following property: Ψ(a, b)ψ(b, c) = Ψ(a, c). Therefore,.. Ψ(,z)dz.. Ψ(,z)dz =.. Ψ(C,z)dz.. Ψ(C,z)dz and Ψ(,y). = Ψ(C,y).. Ψ(,z)dz. Ψ(C,z)dz for any constant C. Ths, the first variable

6 568 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) of Ψ is redndant in sch fractions and can be omitted dring the calclation of their derivative with respect to. Using formla (9), we have F () = f () = +a The derivative of G +a (v) with respect to is given by: G +a (v) = = 1 Ψ( + a, z)dz e Ψ(, + a) G +a(v). +a +a Combining these reslts yields formla (1). Ψ(, + a) ( ) +a. +a Ψ(+a,z) zz a Ψ(+a,y)dy dz Formla (9) allows s to decompose the density of m TD (a) into two parts: f ()d = P [DU TD (a) > 0, m TD (a) (, + d)] + P [DU TD (a) = 0, m TD (a) (, + d)]. (13) The set { DU TD (a) = 0 } corresponds to the event that the process attained its rnning minimm at time T D (a) : X TD (a) = m TD (a). In the following lemma, we calclate the probabilities introdced in (13). Lemma 3.4. Let a > 0 sch that a I. Then P [DU TD (a) > 0, m TD (a) (, + d)] = +a ( +a ) d, (14) P [DU TD (a) = 0, m TD (a) (, + d)] = Ψ(, + a) ( ) +a d, (15) P [DU TD (a) = 0] = a where Ψ(, z) = e z γ (y)dy and γ (y) = µ(y) σ (y). Ψ(, + a) ( ) +a d, (16) Proof. Let s se the relationship M TD (a) = m TD (a) + DU TD (a) + a to rewrite probability (14) in terms of the fnction H (, v): P [DU TD (a) > 0, m TD (a) (, + d)] = P [M TD (a) > + a, m TD (a) (, + d)] = H (, + a)d for [ a, ]. Ths, reslt (14) follows from Lemma 3.3. Formla (15) can be obtained sing the decomposition of f in (13) and Eqs. (9) and (14). The reslt (15) also implies (16): P [DU TD (a) = 0] = a P [DU TD (a) = 0, m TD (a) (, + d)].

7 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Remark 3.5. If a > 0 sch that a I, P [DU TD (a) = 0] = a E [ ] mtd (a). (17) Proof. Formla (17) can be verified by calclating the derivative of E [m TD (a)], which is given by (10), and comparing the reslt with (16). Let s heristically eplain Remark 3.5 sing the following epression: m TD (a) = (M TD (a) a) I { DU TD (a)=0 } + m TD (a)i { DU TD (a)>0 }. Shifting a by a small nmber h has an impact on m TD (a) only if the process {X t } attained its rnning minimm m at time T D (a): DU TD (a) = X TD (a) m TD (a) = 0. In this case, m TD (a) = M TD (a) a and the change in m TD (a) is h becase the rnning maimm M TD (a) remains the same. On the other hand, if {X t } is greater than m TD (a), that is, DU TD (a) > 0, then small changes in a do not affect m TD (a). As a reslt, m TD (a+h) m TD (a) h I { DU TD (a)=0 } for h small. Applying the epected vale to this relationship and letting h 0 lead to Eq. (17). The knowledge of the joint distribtion H (, v) allows s to determine the distribtion and the epected vale of the range process, R t = M t m t, stopped at time T D (a). Corollary 3.6. Let a > 0 sch that a I. The distribtion of the range process R t = M t m t, stopped at time T D (a) is: P [R TD (a) > r] = P [R TD (a) = a] = a a G +a ( + r) where r > a. The epected vale of R TD (a): where E [R TD (a)] = G +a (v) = e v +a +a ( +a ) d, (18) Ψ(, + a) ( ) +a d, (19) G (v)dv + a Ψ(+a,z) zz a Ψ(+a,y)dy dz, +a d, (0) +a Ψ(, z) = e z γ (y)dy, and γ (y) = µ(y) σ (y).

8 570 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Probability of a drawdown preceding a drawp In this section, we derive formlas for the probabilities that a drawdown of size a precedes a drawp of size b and vice versa. The calclation is based on the knowledge of the joint distribtion of (m TD (a), M TD (a)), which appears in Theorem 3.1. Theorem 4.1. Assme that {X t } is a niqe strong soltion of Eq. (1) and that conditions () (4) are satisfied. Let b a > 0 sch that a I and + b I. Then: where P [T D (a) < T U (b)] = 1 a P [T D (a) > T U (b)] = G +a (v) = e v +a a Ψ(+a,z) zz a Ψ(+a,y)dy dz, G +a ( + b) G +a ( + b) +a ( +a +a ( +a Ψ(, z) = e z γ (y)dy, and γ (y) = µ(y) σ (y). Proof. Let b a > 0. First, we will show that ) d, (1) ) d, () {T D (a) < T U (b)} = { 0 < DU TD (a) < b a } { DU TD (a) = 0 }. (3) The range of the process {X t } at t is defined as R t = M t m t, which implies R t = DU t + DD t. One can also prove that { } R t = ma sp DU, sp DD. (4) [0,t] [0,t] The process on the right-hand side of (4) is non-decreasing and eqals zero at time t = 0. Moreover, it increases only if sp [0,t] DU or sp [0,t] DD changes, which occrs when either X t = M t or X t = m t. In this case, the right-hand side is M t m t, which jstifies (4). According to the formla (4), DU TD (a) + a = ma { sp [0,TD (a)] DU, a }. Ths, DU TD (a) = ma { sp [0,TD (a)] DU a, 0 } and { } {T D (a) < T U (b)} = sp DU < b [0,T D (a)] = { 0 < DU TD (a) < b a } { DU TD (a) = 0 }, which proves (3). Frthermore, sing the relationship M TD (a) = m TD (a) + DU TD (a) + a, we have: P [DU TD (a) > b a, m TD (a) (, + d)] = P [M TD (a) > + b, m TD (a) (, + d)] = H (, + b)d.

9 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Now let s calclate the probability of the event {T D (a) < T U (b)} : P [T D (a) < T U (b)] = P [DU TD (a) = 0] + P [0 < DU TD (a) < b a] = 1 P [DU TD (a) > b a] = 1 = 1 a a P [DU TD (a) > b a, m TD (a) (, + d)] P [M TD (a) > + b, m TD (a) (, + d)] { } = 1 H (, + b) d. (5) a { } The derivative of H is calclated in Lemma 3.3. If we replace H (, + b) in (5) with that reslt, we obtain formla (1). Probability () is the complement of (1). If b < a, the formla for P [T D (a) < T U (b)] is a modification of (1). Theorem 4.. Assme that {X t } is a niqe strong soltion of Eq. (1) and that conditions () (4) are satisfied. Let 0 < b < a sch that a I and + b I. Then: where P [T D (a) < T U (b)] = +b P [T D (a) > T U (b)] = 1 v b G v b () = e G v b (v a) v b Ψ(v, z)dz ( v v b Ψ(v, z)dz) dv, (6) +b Ψ(v b,z) z+b z Ψ(v b,y)dy dz, Ψ(, z) = e z γ (y)dy, and γ (y) = µ(y) σ (y). G v b (v a) v b Ψ(v, z)dz ( v v b Ψ(v, z)dz) dv, (7) Proof. The proof is analogos to the proof of Theorem 4.1. The procedre we sed in the proof of Theorem 4.1 allows s to interpret Eqs. (1) and () as follows: P [T D (a) < T U (b)] = 1 = P [T D (a) > T U (b)] = a a a P [M TD (a) > + b, m TD (a) (, + d)] P [M TD (a) + b, m TD (a) (, + d)], P [M TD (a) > + b, m TD (a) (, + d)]. Let s discss this interpretation. If m TD (a) lies in a neighborhood of, the event of {T D (a) > T U (b)} coincides with { M TD (a) > + b }. Probability P [T D (a) > T U (b)] is then the integral of P [M TD (a) > + b, m TD (a) (, + d)] over all possible vales of m TD (a).

10 57 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Table 1 Fnction Ψ(, z) for eamples of diffsion processes. Process I µ(y) σ (y) Ψ(, z) Brownian motion R µ σ e µ σ (z ) [ κ Ornstein Uhlenbeck process R κ(θ y) σ e σ (z θ) ( θ) ] Co Ingersoll Ross process (0, ) κ(θ y) σ ( y z ) κθ σ κ e σ (z ) When a = b in (1) and (), we have {T D (a) < T U (a)} = { DU TD (a) = 0 } : P [T D (a) < T U (a)] = P [T D (a) > T U (a)] = a a Ψ(, + a) ( ) +a d, +a ( +a ) d. 5. Application of the reslts In this section, we apply the reslts from Corollary 3. and Theorem 4.1 to the following eamples of diffsion processes: Brownian motion, Ornstein Uhlenbeck process and Co Ingersoll Ross process. We also present an application of the reslt in Theorem 4.1 to the problem of qickest detection and identification of two-sided changes in the drift of general diffsion processes Eamples of diffsion processes The formlas for G (v), F (v), H (, v), and P [T D (a) < T U (b)] depend on fnction Ψ(, z). Table 1 shows specific forms of this fnction for several eamples of diffsion processes dx t = µ(x t )dt + σ (X t )dw t, where X t I and X 0 =. We assme that conditions () (4) are satisfied for all these processes. In the cases of a drifted Brownian motion and an Ornstein Uhlenbeck process, the conditions hold tre for any combination of the parameters. In the case of a Co Ingersoll Ross process, we need to make an additional assmption: kθ > σ /. One can derive an analytical epression of the fnction H (, v) and the probability P [T D (a) < T U (b)] for Brownian motion: H (, v) = e µ σ a e µ σ ( ( a)) e µ σ a 1 ep { (v ) where [ a, ] and v [ + a, ), implying { G (v) = P [M TD (a) > v] = ep (v ) µ σ µ σ e µ σ a 1 e µ σ a 1 } },, v [, ),

11 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) F () = P [m TD (a) > ] = e µ σ a e µ [ ] σ E mtd (a) = µ + a, e µ σ a 1 { P [T D (a) < T U (b)] = 1 ep where b a > 0. If a = b, σ ( ( a)) e µ σ a 1 (b a) µ P [T D (a) < T U (a)] = e σ a + µ a 1 σ e µ σ a + e µ σ a. µ σ e µ σ a 1, [ a, ], } e µ σ a µ σ a 1 e µ σ a + e µ σ a Random variable M TD (a) has an eponential distribtion on [, ) and m TD (a) has a trncated eponential distribtion on [ a, ]. Note that the formla for P [T D (a) < T U (b)] is identical with the reslts presented in [4]. When the drift µ eqals zero, the formlas frther redce to: H (, v) = P [m TD (a) >, M TD (a) > v] = e v (+a) a, a where [ a, ] and v [ + a, ), implying G (v) = P [M TD (a) > v] = e v a, v [, ), F () = P [m TD (a) > ] = [ ] a and E mtd (a) = a, P [T D (a) < T U (b)] = 1 1 b a e a, where b a > 0. If a = b, P [T D (a) < T U (a)] = P [T D (a) > T U (a)] = 1. Hence, M TD (a) has an eponential distribtion on [, ) with parameter 1 a and m T D (a) has a niform distribtion on [ a, ]. Calclation of P [T D (a) < T U (b)] for an Ornstein Uhlenbeck process and a Co Ingersoll Ross process involves nmerical integration. In Figs. 1, 3 and 5, we have plotted densities of M TD (a) and m TD (a) for varios diffsion processes. Figs., 4 and 6 captre dependence of the probability P [T D (a) < T U (b)] on the parameters of the processes. Let s discss the interpretation of Fig. 4, which shows the probability P [T D (1) < T U (1)] as a fnction of κ/σ. When κ = 0, the drift term of {X t } vanishes and the probability is 1. Moreover, if the process starts at its long-term mean, = θ, it is symmetric and P [T D (1) < T U (1)] = 1 for any vale of κ/σ. Now let s assme that = θ + 1. As κ/σ increases, the drift term will prevail over the volatility term and the process will be pshed down from to θ. As a reslt, a drawdown of size 1 will tend to occr before a drawp of size 1, which eplains the convergence of P [T D (a) < T U (a)] to 1 as κ/σ. A similar reasoning can be sed to jstify the convergence of P [T D (1) < T U (1)] to 0 if = θ 1.,

12 574 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Fig. 1. Densities of M TD (a) and m TD (a), where a = 1, for a drifted Brownian motion: X t = µt + σ W t. The densities depend on the parameters throgh ratio µ/σ. M TD (a) has an eponential distribtion on [0, ), while m TD (a) has a niform distribtion on [ 1, 0] if µ = 0 and a trncated eponential distribtion on [ 1, 0] otherwise. Fig.. Probability P [T D (a) < T U (b)] as a fnction of µ/σ for different vales of a and b. We assme that {X t } is a drifted Brownian motion, X t = µt + σ W t. Note that P [T D (1) < T U (1)] = 0.5 for µ = The problem of qickest detection and identification In this eample, we present the problem of qickest detection and identification of twosided changes in the drift of a general diffsion process. More specifically, we give precise

13 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Fig. 3. Densities of M TD (a) and m TD (a), where a = 1, assming that {X t } is an Ornstein Uhlenbeck process: dx t = κ(θ X t )dt + σ dw t, X 0 =. The densities depend on parameters κ and σ throgh ratio κ/σ. Note that if = θ, process {X t } is symmetric and conseqently, m TD (a) has a symmetric distribtion. Fig. 4. Probability P [T D (a) < T U (b)], where a = b = 1, as a fnction of κ/σ. Process {X t } is an Ornstein Uhlenbeck process: dx t = κ(θ X t )dt + σ dw t, X 0 =. If = θ, then the process is symmetric and the probability is 0.5 for any vale of κ/σ. calclations of the probability of misidentification of two-sided alternatives. In particlar, let {X t } be a diffsion process with the initial vale X 0 = and the following dynamics p to a

14 576 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) Fig. 5. Densities of M TD (a) and m TD (a), where a = 1. {X t } is a Co Ingersoll Ross process: dx t = κ(θ X t )dt + σ X t dw t, X 0 =. We se the same vales of parameters as in Fig. 3. Fig. 6. Probability P [T D (a) < T U (b)], where a = b = 1, as a fnction of κ/σ, where κθ > σ /. Process {X t } is a Co Ingersoll Ross process: dx t = κ(θ X t )dt + σ X t dw t, X 0 =. We se the same vales of parameters as in Fig. 4. deterministic time τ: dx t = σ (X t )dw t, t τ. (8)

15 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) For t > τ, the process evolves according to one of the following stochastic differential eqations: dx t = µ(x t )dt + σ (X t )dw t t > τ, (9) dx t = µ(x t )dt + σ (X t )dw t t > τ. (30) with initial condition y = X τ. We assme that the fnctions µ(.) and σ (.) are known and the stochastic differential eqations (8) (30) satisfy conditions () (4) stated in Section 1. The time of the regime change, τ, is deterministic bt nknown. We observe the process {X t } seqentially and or goal is to identify which regime is in effect after τ. In this contet sppose that the first passage time of the drawp process to a threshold a, T U (a), can be sed as a means of detecting the change of dynamics of {X t } from (8) to (9). Similarly, sppose that the first passage time of the drawdown process to a threshold b, T D (b) may be sed as a means of detecting the change of dynamics of {X t } from (8) to (30) (see [7, 5]). The simplest eample is when µ(x t ) = µ. The probability measres P τ,(1) and P τ,() are the measres generated on the space of continos fnctions C[0, ) by the process {X t }, if the regime changes at time τ from (8) to (9) and from (8) to (30), respectively. The stopping rle proposed and sed widely in the literatre for detecting sch a change is known as the two-sided CUSUM (Cmlative Sm) test, T (a) = min {T D (a), T U (a)}. This rle was proposed in 1959 by Barnard []. Its properties have been widely stdied by many athors [18 1,7] and a version of this rle was also proven asymptotically optimal in [6]. It is ths the rle that has been established in the literatre for detecting two-sided changes in the set-p described above. Theorem 4.1 can be sed to compte the probability of a false identification of the change. More specifically, P 0,(1) [T (a) = T D (a)] = P 0,(1) [T D (a) T U (a)] Ψ(, + a) = ( a ) +a d, (31) with Ψ(, z) = e z γ (y)dy and γ (y) = µ(y), epresses the probability that an alarm indicating σ (y) that the regime switched to (30) will occr before an alarm indicating that the regime switched to (9) given that in fact (9) is the tre regime. Ths (31) can be seen as the probability of a false regime identification. Moreover, in the case that the density of the random variable X τ admits a closed-form representation, we can also compte P τ,(1) y [T (a) = T D (a)] f Xτ (y )dy = P τ,(1) y [T D (a) T U (a)] f Xτ (y )dy, which can be seen as the aggregate probability (or nconditional probability) of a false identification for any given change point τ. 6. Conclsion In this paper, we discssed properties of a diffsion process stopped at time T D (a), the first time when the process drops by amont a from its rnning maimm. We derived the joint distribtion of the rnning minimm and the rnning maimm stopped at time T D (a). This allowed s to obtain a formla for the probability that a drawdown of size a precedes a drawp of size b.

16 578 L. Pospisil et al. / Stochastic Processes and their Applications 119 (009) A possible etension of or work is the calclation of the probability that a drawp precedes a drawdown in a finite time horizon. This wold reqire a combination of or reslts with the distribtions of times T D (a) and T U (b). We do not epect that this wold lead to a closed-form soltion. References [1] I.I. Gihman, A.V. Skorokhod, Stochastic Differential Eqations, Springer-Verlag, New York, 197. [] H.M. Taylor, A stopped Brownian motion formla, Annals of Probability 3 () (1975) [3] J.P. Lehoczky, Formlas for stopped diffsion processes with stopping times based on the maimm, Annals of Probability 5 (4) (1977) [4] O. Hadjiliadis, J. Vecer, Drawdowns preceding rallies in a Brownian motion process, Qantitative Finance 6 (5) (006) [5] H.V. Poor, O. Hadjiliadis, Qickest Detection, Cambridge University Press, Cambridge, UK, 008. [6] O. Hadjiliadis, G.V. Mostakides, Optimal and asymptotically optimal CUSUM rles for change point detection in the Brownian motion model with mltiple alternatives, Theory of Probability and its Applications 50 (1) (006) [7] R.A. Khan, Distribtional properties of CUSUM stopping times, Seqential Analysis 7 (4) (008) [8] M. Magdon-Ismail, A. Atiya, A. Pratap, Y. Ab-Mostafa, On the maimm drawdown of a Brownian motion, Jornal of Applied Probability 41 (1) (004). [9] A. Cherny, B. Dpire, On certain distribtions associated with the range of martingales, 007, Preprint. [10] P. Salminen, P. Vallois, On maimm increase and decrease of Brownian motion, Annales de l Institt Henri Poincare (B) Probability and Statistics 43 (6) (007). [11] J. Vecer, Maimm drawdown and directional trading, Risk 19 (1) (006) [1] S.J. Grossman, Z. Zho, Optimal investment strategies for controlling drawdowns, Mathematical Finance 3 (3) (1993) [13] J. Cvitanic, I. Karatzas, On portfolio optimization nder drawdown constraints, IMA Lectre Notes in Mathematics & Applications 65 (1995) [14] A. Chekhlov, S. Uryasev, M. Zabarankin, Drawdown measre in portfolio optimization, International Jornal of Theoretical and Applied Finance 8 (1) (005) [15] I. Meilijson, The time to a given drawdown in Brownian motion, Seminaire de Probabilités XXXVII (003) [16] J. Obloj, M. Yor, On local martingale and its spremm: Harmonic fnctions and beyond, in: From Stochastic Calcls to Mathematical Finance, Springer, 006, pp [17] A. Nikeghbali, A class of remarkable sbmartingales, Stochastic Processes and their Applications 116 (6) (006) [18] K. Kemp, The average rn-length of the cmlative sm chart when a V-mask is sed, Jornal of the Royal Statistical Society B 3 (1961) [19] D.S. van Dobben De Bryn, Cmlative Sm Tests: Theory and Practice, Hafner, New York, [0] A. Bissell, Csm techniqes for qality control, Applied Statistics 18 (1969) [1] W.H. Woodall, On the Markov chain approach to the two-sided CUSUM procedre, Technometrics 6 (1984) [] G. Barnard, Control charts and stochastic processes, Jornal of the Royal Statistical Society: B 1 (1959)