Subordinated Brownian Motion: Last Time the Process Reaches its Supremum

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1 Sankhyā : The Indian Journal of Statistics 25, Volume 77-A, Part, pp c 24, Indian Statistical Institute Subordinated Brownian Motion: Last Time the Process Reaches its Supremum Stergios B. Fotopoulos, Venkata K. Jandhyala and Yuxing Luo Washington State University, Pullman, USA Abstract The article develops a theory for the last time the subordinated Brownian motion SBM with negative drift reaches its supremum. The study includes obtaining expressions for the Laplace transform of the last time that the SBM reaches its supremum and also for its density. In the process, we establish that the last time that the SBM reaches its supremum is a member of the generalized gamma convolution GGC ) family. The theoretical results for the general case have been explicitly derived for some well-known subordinators. Numerical investigations show close agreement between the theoretical derivations and empirical computations. AMS 2) subject classification. Primary 6J75; Secondary 6J65, 6G5. Keywords and phrases. Subordinated Brownian motion, Brownian motion with negative drift, Wiener-Hopf factorization, generalized gamma convolution. Introduction Laws of random times including the last hitting time of points and intervals for one-dimensional Brownian motion with or without drift have been well studied in the literature. In many cases, explicit expressions can also be found see e.g. Borodin and Salminen, 996 or Mörters and Peres, 2). It turns out that such laws are infinitely divisible ID), and even self-decomposable SD) see, Yano, Yano and Yor, 29). The aim of this article is to investigate the laws of last hitting times to points for one-dimensional subordinated Brownian motion SBM) with negative drift. In particular, it is of interest to study the behavior and law of the last time a SBM reaches its maximum. It is well known in the literature that a Brownian motion with negative drift has continuous paths while the subordinated Brownian motion is a purely discontinuous Lévy process. However, both processes drift to minus infinity and thus they possess a.s. finite suprema. Expressions for the Laplace transform of the first hitting time to its supremum of a general Lévy process can be found in the literature e.g., see Pecherskii and Rogozin, 969). The main contribution of this article is

2 Last time subordinated Brownian motion reaches its supremum 47 to provide more attractive expressions under the special case of SBM processes. Furthermore, these Laplace transforms enable us to invert and obtain explicit forms for the probability density function of the last time that the maximum is attained. To this end, we need to differentiate our current work from the recent study of Baurdoux and van Schaik 22). These authors provide properties of predicting the exact time finding a stopping rule) a Lévy process attains its ultimate supremum. They begin by formulating the prediction problem under a general Lévy process framework, and then study properties of the stopping time that minimizes the L -distance to the time at which a spectrally negative Lévy process attains its ultimate supremum. At a more basic level, literature on the problem of predicting the time of ultimate supremum driven by a linear Brownian motion with a finite horizon is quite extensive see e.g., Graversen, Peskir and Shiryaev, 2; Du Toit and Peskir, 28). In some ways, our study is also motivated by the works of Bondesson 992), James, Roynette and Yor 28a), James, Ljioi and Prünster 28b) and other related works. From applications point of view, it is well known that subordinated Brownian motion processes play an important role in the study of financial log returns see e.g., Barndorff-Nielsen, Mikosch and Resnick, 2). The article is organized as follows: Section 2 introduces some preliminaries from subordinated Brownian motion, fluctuation theory, and generalized gamma convolutions GGC). The main contributions of the article are presented in Section 3. In Section 4, we provide few well known examples together with some illustrations. Section 5 concludes the paper with numerical results related to the examples of Section 4. 2 Preliminaries 2.. Argument Supremum for Lévy Processes. Suppose that Ω, F, F,P) is a filtered probability space with filtration F = {F t : t }. A onedimensional Lévy process on Ω is a strong Markov, F adapted process X = {X t : t } with càdlág paths. In this case, the process X starts from the origin and, for each s t, the increment X t X s is independent of F s and has the same distribution as X t s. Inthissense,itissaid that a Lévy process has stationary independent increments. Consequently, it can be shown that there is a unique continuous function Ψ X : R C, such that Ψ X ) = and ] E [e iθxt = e tψ X θ),t,θ R,

3 48 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo where E denotes expectation with respect to P. According to Lévy- Khintchine formula, the function Ψ X is the logarithm of the characteristic function characteristic exponent) and it is expressed as: Ψ X θ) = iμθ σ2 ) 2 θ2 e ixθ ixθi x < ) Π X dx), R\{} where θ R, μ R, σ. Here, Π X is a measure on R\{}, which satisfies R x 2 ) Π X dx) <, wherex y =min{x, y}. The measure Π X is called the Lévy measure. For the process X = {X t : t } defined on Ω, let X t =sup s t X s denote the supremum by time t, andlet X =sup s X s denote its ultimum supremum. Let τ r, r>, be an exponentially distributed random variable τ r Expr)) independent of the process X. Itcanbeshownthat X and X X are independent. Let J t := sup { } s t : Xt = X s denote the time at which the process reaches its supremum for the last time by time t. Thus, J := sup { } s : X = X s denotes the ultimum argument supremum. From Kyprianou 26, p.74), it is easy to see that the Laplace transform of J τ r is expressed as E [ exp λj τ )] r =exp e λt) e rt t λ, and r>. One can observe that re rt E [ ] E e λj,asr. This in turn yields E [ exp { λj )] =exp ) P X t ) dt, 2.3) [ ] e λj t dt = [ ] e t E e λj t/r dt e λt) P X t ) dt t },λ. 2.4) The main contribution in this investigation is to address the question of characterizing and computing the distribution function of J τ r. It may be seen that the Laplace exponent in 2.3) and 2.4) can be written as: f λ) = e λt) Π J dt),r, 2.5) where Π J dt) = e rt t P X t ) dt. Assuming that P X t ) dt <, it is easy to see that t) Π J dt) <. Consequently, Π J dt) can be considered as a Lévy measure. In what follows, we investigate Π J dt) under the one-dimensional subordinated Brownian motion SBM ), which we discuss in Section 2.2 below.

4 Last time subordinated Brownian motion reaches its supremum The Subordinated Brownian Motion. We say that the process X is a subordinated Brownian motion with negative drift if it can be represented by X t := Y Tt = δt t B Tt,where{B t : t } is a standard one-dimensional Brownian motion, Y t := δt B t is a linear Brownian motion with negative drift δ <), and the process T = {T t : t } is a subordinator independent of {Y t : t }. A subordinator T is a Lévy process taking values on [, ). Following Bertoin 996), the infinitely divisible ID) property of the law of T implies that its Laplace transform can be expressed by E [ e λtt] = e tψ T λ), λ, where ψ T :[, ) [, ) is the so-called Laplace exponent. In this case, the Lévy-Khintchine formula for the subordinator T is expressed by ψ T λ) =a dλ e λx) Π T dx),λ, 2.6), ) where Π T is the Lévy measure of T. In this article, we assume that the killing and drift coefficients a and d are zero. In this case, apart from the fact that, ) x)π T dx) <, the Laplace exponent also satisfies ψ T ) =. Following Sato 999, Th. 3.), the SBM X = {X t : t } is alévy process on R with diffusion coefficient σ X =, drift coefficient a X given by a X = x xp Y s dx)π T ds), and a Lévy measure defined by Π X A) = P Y s A)Π T ds), A B R) \{}), where Π T is the Lévy measure of T. Also, the law of X satisfies P X t A) := P Y s A) P T t ds),a B R). 2.7) Note that the Laplace exponent of any subordinator satisfies 2.6), which in itself agrees with the Laplace exponent for J τ r expressed in 2.5). Hence, it follows that J τ r is a subordinator. We shall now determine the Lévy measure Π J dt) forj τ r, r when X is a SBM. Calling upon 2.7), it follows, for each t fixed), Π J dt) = e rt t P X t ) dt = e rt E [ Φ t )] /2 2qT t dt, t >,r,q >, 2.8) where q = δ / 2,δ <and Φx) = /2 2π x e y2 dy. Utilizing Lemma A. see Appendix A), 2.8) can be expressed as: Π J dt) = t E [ ))] rψ e t q 2 T Z /2,/2 dt, t >,r >,q >, 2.9)

5 5 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo where Z /2,/2 is a beta random variable with parameters /2, /2). Letting r, one obtains Π J dt) = t E [ )] e tψ q 2 T Z /2,/2 dt, t >,q >. 2.) Moreover, it should be noted that t ) Π J dt) P X t ) dt E [ / ψ T q 2 / Z /2,/2 )],r. Thus, under an SBM with negative drift, the condition t) Π J dt) < is satisfied whenever E [ / ψ T q 2 / )] Z /2,/2 <. Then, one may invoke 2.3) and 2.4) to conclude that J τ r is a non-degenerate random variable. The representations in 2.9) and 2.) lead one to the generalized gamma convolutions GGC ) family ofrandomvariables. TheGGC family of random variables and related results are introduced in the next sub-section The Generalized Gamma Convolutions GGC). The standard gamma process γ = {γ t : t } is a subordinator without drift with Lévy measure Π γ dx) = x e x dx, x> see e.g., James et al., 28a). Thus, the Lévy-Khintchine representation is expressed by E [ e λγt] =exp t e λx ) x e x dx ) =exp tln λ)) = λ) t 2.),λ,t. Note that the above identity is a simple consequence of the Frullani formula. A random variable Γ is said to be of GGC type if it is a weak limit of linear combinations of independent gamma variables with positive coefficients. A detailed study of the GGC family can be found in Bondesson 992) and James et al. 28a). Our main results in Section 3 also benefit from results on GGC subordinators and their stochastic integral representations. For this, we introduce the Wiener-Gamma integral see e.g., James et al., 28a) by: Γ:= Πs) dγ s, where Π : R R is a Borel function such that, ) ln Π t)) dt <. Bondesson 992) and James et al. 28a) have shown results on GGC that are important for our subsequent analysis in Section 3. We summarize some of their results in Appendix A2.

6 Last time subordinated Brownian motion reaches its supremum 5 Using the Frullani formula for a = in Result A see Appendix A2), the Laplace transform of the Wiener-Gamma integral can be expressed as: E [ e λγ] =exp, ) e λt ) ) t, ) e tu Π Γ du) dt =exp, ) e λt ) ) 2.2) ρ t) dt,λ. The last identity shows that the Lévy density of Γ is expressed by ρ t) = t, ) e tu Π Γ du). The measure Π Γ associated with Γ is known as a Thorin measure. Result A3 in Appendix A2 can be stated when the Thorin measure Π Γ has finite total mass. For example, for m>and a Borel function Π:[,m] R, such that m ln Π u)) du <, them-wiener integral of Π is defined by Γ m := m Πs) dγ s. For G> a random variable such that E [ ln G) ] <, wesaythatγism, G)-GGC, if its Laplace transform can be expressed as [ E e λγ] =exp m e λt) t E [ ) e tg] dt., ) In this case, Γ = d Γ m G) throughout the article = d denotes equality in law) and the Thorin measure Π Γ is defined as see also e.g., James et al., 28a) Π Γ dx) =mf G dx), where F G denotes the law of G and the total mass m is just Π Γ, )) = m. Under the restriction E [ ln /G) ] <, and the Frullani formula, we may now say that Γ m G) amemberof m, G)-GGC family whenever its Laplace transform admits the following representation: E [ e λγmg)] =exp m, ) e λt ) t E [ e tg] ) dt =exp me [ ln λ )]) 2.3) G,λ. As presented in Appendix A3, one may also express members of m, G) -GGC in terms of members of m, G) - Dirichlet means see Appendix A3 for definition of m, G) Dirichlet means) through the relation Γ m G) = d γ m D m G), for any positive r.v.g 2.4) It turns out that these random variables are also crucial for identifying the behavior of J τ r. 3 The Main Results Our main goal in this section is to compute convenient and explicit formulae for the Laplace transforms in 2.3) and2.4) when the underlying

7 52 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo processisasbm. Mainly, our interest is to characterize them in a way that they are not only easier for interpretation, but can also pave way for inversion. The proofs of various theorems and corollaries of this section are presented in Appendix B. Assuming that the Laplace exponent of the subordinator satisfies the properties stated in Section 2, the main contribution of this study begins with the following theorem. Theorem 3.. Let the subordinated Brownian motion X be defined by X t = δt t B Tt = Y Tt, t, δ<. Assume E [ ln ψ T q 2 / )) ] Z /2,/2 <, where ψ T is as defined in 2.6) with a = d =,andq = δ / 2, r. Then, the random time J τ r m, G) GGCas in 2.3)), withm = 2,and G = r ψ T q 2 / ) Z /2,/2. In the next result, which follows from Theorem 3., we take the Laplace transform of J τ r found in 2.3) and specialize it to the case of SBM. Specifically, we establish the following. Theorem 3.2. Let the subordinated Brownian motion X be defined by X t = δt t B Tt = Y Tt, t, δ<. IfE [ ln ψ T q 2 / Z /2,/2 )) ] <, q= δ / 2, then, the Laplace transform of J τ r, r, can be expressed as: E [ exp λj τ r )] =exp { λ, r. 2 E [ln )]} λ r ψ / ), T q 2 Z /2,/2 Moving further, expressions for the density function of J τ r can also be derived. The following theorem can be seen as a consequence of the duality theorem found in James et al. 28a). Theorem 3.3. Let the subordinated Brownian ) motion X be defined by X t = δt t B Tt, t, δ<. Set G r Z/2,/2 = r ψt q 2 / ) Z /2,/2, r. If E [ ln )) ] / G r Z/2,/2 <, q = δ 2, then, the density function of J τ r, r, can be expressed as follows: 2 f J x) = exp { E [ ln ))]} G πx 3/2 2 r Z/2,/2 [ {Γ/2 ))} /4 )) )] E Gr Z/2,/2 cos 2 xγ /2 Gr Z/2,/2 I x, )) = exp E [ ln ))]) G πx 2 r Z/2,/2 [ ] E e xd /2G rz /2,/2)) I x, )).

8 Last time subordinated Brownian motion reaches its supremum 53 From Theorem 3.2 and Result A of Appendix A2, one may see that J τ r is equal in law with the following stochastic integral: J τ r = d Γ /2 /{ r ψt q 2 / Z /2,/2 )}),r. Applying Theorem 3.2, one may also show that the Laplace transform of the last time the process reaches its supremum for the case of a linear Brownian motion takes a simple and attractive algebraic form. To this end, the following corollary is in order. Corollary 3.. Let the linear Brownian motion Y be defined by Y t = δt B t, t, δ<. Then, for q = δ / 2, the Laplace transform of J τ r, r, is determined by: E [ exp λj τ r )] = q q 2 r q,λ, r. q 2 λ r We shall now take advantage of the convenient form in Corollary 3. and identify the distribution of J τ r in the case of linear Brownian motion. The result is presented in the following corollary. The corollary involves the three parameter Generalized Inverse Gaussian GIG) family of distributions denoted by GIG λ, δ, γ). The parameter space and the form of the density function of GIG family of distributions may be found in Eberlein 2). Corollary 3.2. Let the linear Brownian motion Y be defined by Y t = δt B t, t, δ <. Then, for q = δ / 2, the distribution of J τ r is determined by: J τ r = d GIG )/ 2, γ 2 q 2 r q q 2 r, 2 q 2 r ),r. where γ 2 is a gamma r.v. with parameter 2. Remark. We are not aware of any work in the existing literature that is related to the simple looking expressions in Corollaries 3. and 3.2, under the exponential time τ r, r>. Earlier, Karatzas and Shreve 988), Akahori 995), Buffet 23), Dassios 25), among others, considered the linear Brownian motion and studied properties of J t, for fixed time t. Their results can be seen to be somewhat different from the results we derive for J τ r in Corollaries 3. and 3.2. As demonstrated in 2.4), it

9 54 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo mayalsoberemarkedthatwhenr =, one captures the distributional behavior of J. Clearly, the results in Theorems 3.2, 3.3 and Corollaries 3., 3.2 can be applied directly to obtain the corresponding results for J. Referring back to Theorem 3.2, the density of J τ r for the general SBM may not be very convenient from a numerical point of view. A more convenient expression for the density can be derived for simulation purposes. For this, we define the function G r ) =r ψ T q 2 / ). As in James et al. 28a), we assume that this function remains constant within successive intervals [s i,s i ), i =,, 2,,n on[, /2]. Specifically, we discretize the function G r ) byg r s) = n i= G r s i ) I s [s i,s i )), for any subdivision <s s s n < /2. In this case, it can be seen that J τ r = d Γ /2 r ψt q 2 / Z /2,/2 )) =d /2 = d lim s n i= F rψ Tq 2 /Z /2,/2 ) 2s i) dγ u F rψ Tq 2 /Z /2,/2 ) 2u) γsi γ si ), where s =max j n s j s j. 4 Examples The aim of this section is to illustrate Theorems 3.2 and 3.3 for some familiar subordinators, mainly restricting to the case of J only. Theorems 3.2 and 3.3, however, enable one to extend the analysis more generally to J τ r. One may verify that conditions of Theorems 3.2 and 3.3 are satisfied for all of the examples considered in this section. 4.. Stable Subordinator. We let T t, t, be α-stable, α, ), so that T t S α t /α,, ) see Samorodnitsky and Taqqu, 994, p.9, for notation). Thus, for the SBM defined by X t = δt t B Tt, t, δ<, our goal is to obtain the Laplace transform of the r.v. J. From Theorem 3.2 and the fact the subordinator is α-stable, the Laplace exponent ψ T is given by ψ T x) =x α, x>, α, ). Thus, the Laplace exponent of J is determined as follows: ψ J λ) = 2 E λ ln,λ. q /Z 2α /2,/2 α

10 Last time subordinated Brownian motion reaches its supremum 55 We apply Theorem 3.3 to find the density of J. Setting c =cosπα/2) / q 2α, q = δ / 2, we obtain f J x) = [ E ln cz α )]/ 2 2e /2,/2 πx 3/2 ) /4 E [Γ /2 cz/2,/2 α cos 2 = e E [ ln cz /2,/2 α )]/ 2 x /2 π xγ /2 cz α /2,/2 ) )] [ )] E e xd /2 cz/2,/2 α,x, ). In order to capture the behavior of J following is more convenient: / ) J = d Γ /2 cz α /2,/2 in an empirical manner, the = d /2 dγ u F / 2u) cz /2,/2 α F / 2s i ) cz /2,/2 α = d lim s n j= γsi γ si ), where Z /2,/2 is Beta /2, /2), s = {s,s,,s n } is a partition of the interval, 2 ), i.e., <s s s n < /2 and s =max j n s j s j Gamma Subordinator. We shall now let the subordinator T be the standard gamma process. Then, the corresponding Laplace exponent is given by ψ T x) =lnx), x>. Also, the Laplace transform of J associated with the process X t = δt t B Tt, t, δ<, satisfies [ )] E exp λj =exp { 2 [ )) )]} E ln λ ln q 2 Z /2,/2,λ. Thus, setting L q, Z) =ln q 2 Z/2,/2) andtheninvertingthelaplace transform as indicated in Theorem 3.2, the density of J is expressed as f J x) = { 2exp } 2 E[lnL q,z))] πx 3/2 [ )] E Γ /2 L q, Z)) /4 cos 2 xγ /2 L q,z)) I x, )) { = exp } 2 E[lnL q,z))] πx E [ e xd /2L q,z)) ] I x, )).

11 56 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo Finally, as in the previous case, we have J = d Γ /2 L q, Z)) = d /2 lim = d s n j= F L q,z) 2s i) dγ u F L q,z) 2u) γsi γ si ). Note that, in this case, J is a 2,L q, Z) ) GGC Poisson Subordinator. Here, we assume that T is a Poisson process with parameter θ, θ, In this case, the domain of T can be seen to be the set of natural numbers N, and the Laplace transform of T is E [ e λtt] = exp tθ e λ)) ), λ. Thus, setting L 2 q,z) =θ e q2 /Z /2,/2 and applying Theorems 3.2 and 3.3, we have E [ exp { λj )] =exp )]} [ln 2 E λ, L 2 q,z) { 2exp E [ln L2 q, Z))]} 2 f J x) = πx 3/2 E [Γ /2 L 2 q, Z) ) /4 cos 2 xγ /2 L 2 q, Z) ))] I x, )) = exp { E [ln L2 q, Z))]} [ ] 2 E e xd /2 L 2q,Z)) I x, )) πx and J = d Γ /2 L 2 q, Z)) = d /2 lim = d s n j= F L 2 q,z) 2s i) dγ u F L 2 q,z)) 2u) γsi γ si ). 5 Numerical Results In this section, we illustrate the distribution of J for the three examples of Section 4, where the process T t follows: i) an α-stable subordinator with choices of α =.25,.5,.75, ii) a gamma process with parameter, and iii) a Poisson process with parameter θ =. In addition in case iv), we let the process X t be the simple linear Brownian motion with drift δ =.; in this case T t t. Note that the simple linear Brownian motion case corresponds to α-stable subordinator with α =, with the results for this case appearing in Corollary 3.2. The goal is to compute the theoretical density function of J by numerically inverting the Laplace transform that appears in Theorem

12 Last time subordinated Brownian motion reaches its supremum 57 Figure : Theoretical density continuous curve) and empirical density histogram) of J :forα-stable subordinator in case i) with a) c) representing α =.25,.5,.75, respectively, and d) representing case iv) for the linear Brownian motion with α =,e) representing gamma process in case ii), and f) representing the Poisson process in case iii). 3.2 for all three examples. We also compute the corresponding empirical densities based upon the expressions provided within each example. The empirical computations are all based upon 5, simulations. In carrying out these computations, we consider the drift coefficient to be δ =.. The graphs of the density for J are represented in Fig. a)-c) for case i), Fig. d) for case iv), Fig. e) for case ii) and Fig. f) for case iii). It may be noted that there is a close match between the theoretical and empirical densities in all cases. Acknowledgement. The authors thank two anonymous reviewers as well as the Co-Editor for their constructive comments and suggestions that have led to substantial improvements in both content and readability of the article. References akahori, j. 995). Some formulae for a new type of path-dependent option. Ann. Appl. Probab., 5, barndorff-nielsen, o.e., mikosch, t. and resnick, s.i. 2). Lévy processes: theory and applications. Birkhäuser, New York. baurdoux, e. and van schaik, k. 22) Predicting the time at which a Lévy process attains its supremum. Acta Applicandae Mathematicae in press). arxiv:

13 58 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo bertoin, j. 996). Lévy processes. Cambridge University Press, Cambridge. bondesson, l. 992). Generalized gamma convolutions and related classes of distributions and densities, lectures notes in statistics. Springer-Verlag, New York. borodin, a.n. and salminen, p. 996). Handbook of brownian motion-facts and formulae. Birkhäuser Verlag, Boston Berlin. buffet, e. 23). On the maximum of Brownian motion with drift. J. Appl. Math. Stoch. Anal., 6, cifarelli, d.m. and melilli, e. 2). Some new results for Dirichlet priors. Ann. Stat., 28, cifarelli, d.m. and regazzini, e. 99). Distribution functions of means of a Dirichlet process. Ann. Stat., 8, dassios, a. 25). On the quantiles of Brownian motion and their hitting times. Bernoulli,, du toit, j. and peskir, g. 28) Predicting the Time of the Ultimate Maximum for Brownian Motion with Drift. In Proceedings of the Mathematical Control Theory and Finance. Springer, pp eberlein, e. 2) Application of Hyperbolic Lévy motions to finance. In Lévy Processes: Theory and Applications O.E.Barndorff-Nielsen,T.MikoschandS.I.Resnick,eds.). Birkhäuser, pp gradshteyn, i.s. and ryzhik, i.m. 2). Tables of Integrals, Series and Products, 5th edn. Academic Press, New York. graversen, s.e., peskir, g. and shiryaev, a.n. 2). Stopping Brownian motion without anticipation as close as possible to its ultimate maximum. Theory Prob. Appl., 45, james, l.f., roynette, b. and yor, m. 28a). Generalized gamma convolutions, Dirichlet means, Thorin measures, with explicit examples. Probab. Surv., 5, james, l.f., ljioi, a. and prünster, i. 28b). Distributions of linear functionals of two parameter Poisson-Dirichlet random measures. Ann. Probab., 8, karatzas, i. and shreve, s.e 988). Brownian motion and stochastic calculus. Springer- Verlag, New York. kyprianou, a.e. 26) Introductory Lectures on Fluctuation of Lévy Processes with Applications. Springer-Verlag. lijoi, a. and prünster, i. 29). Distributional properties of means of random probability measures. Stat. Surv., 3, mörters, p. and peres, y. 2). Brownian motion. Cambridge University Press, Cambridge. pecherskii, e.a. and rogozin, b.a. 969). On joint distributions of random variables associated with fluctuations of a process with independent increments. Theory Probab. Appl., 4, samorodnitsky, g. and taqqu, m.s. 994). Stable Non-gaussian random processes. Chapman & Hall, New York, London. sato, k.-i. 999). Lévy processes and infinitely divisible distributions. University Press, Cambridge. yano, k., yano, y. and yor, m. 29) On the laws of first hitting times of points for one-dimensional symmetric stable Levy processes. Seminaire de Probabilites XLII, Lecture Notes in Math. Springer, Berlin, pp

14 Last time subordinated Brownian motion reaches its supremum 59 Appendix A: Some Requisite Results Appendix A. The lemma below requires results about the beta random variable. For a, b >, we write Z a,b for a beta variable with parameters a, b): f Za,b x) dx = P Z a,b dx) = B a, b) xa x) b dx, x, ), where B a, b) = Γa)Γb)/Γa b). For a standard Brownian motion {B t : t }, the occupation time A = P B s > ) ds is also related to the beta random variable. In particular, A has density f Z/2,/2, i.e., P A x) = 2 π sin x, x [, ], which is the well-known Lévy s arcsine law. Lemma A.. For q = δ / 2, the following identity holds Φ ) 2qu /2 = ] [e 2 E uq2 /Z /2,/2 = 2 e uq2 /x f Z/2,/2 x) dx, where f Z/2,/2 x)is the density of a beta random variable with parameters /2, /2) and Φx) = /2 2π x e y2 dy. Proof of Lemma A.. To evaluate the survival normal distribution, we express it in terms of the complement of the error function. Specifically, it can be seen that Φ 2qu /2) = erfc qu /2)/ 2, where erfc ) isthecomplementary error function, see e.g., Gradshteyn and Ryzhik 2, eq ). Also, it can be seen from Gradshteyn and Ryzhik 2, eq ) that erfc qu /2) = 2q π e t 2 q 2 )u dt. Upon substituting t/q = s, it follows that t 2 q 2 Φ ) 2qu /2 = e uq2 s 2 ) ds π s 2 = 2 e uq2s M ds), A.) where M dt) = yields, πt t I t, )) dt. Further, transformation on A.) Φ ) 2qu /2 = e uq2y dy π y y = e uq2 /y f Z/2,/2 y) dy A.2) where f Z/2,/2 y) = π y /2 y) /2 is again the density of a beta r.v. with parameters /2, /2).

15 6 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo Appendix A2. Results A-A3 stated below are known in the literature see e.g., James et al., 28a). Result A. A random variable Γ is of GGC type if and only if there exists a non-negative measure Π Γ on, ) with,] ln t)π Γ dt) <, and, ] t Π Γ dt) < such that [ E e λγ] =exp aλ ln λ ) ) Π Γ dt),a,λ., ) t Result A2. Suppose that Γ is of GGC type with a = and b := Π Γ, )) <. Then, Γ may be represented as Γ = d γ b G, for some positive r.v. G independent of γ b. The total mass of the Thorin measure is then given by { b =sup p : lim x } f Γ x) =, xp where f Γ is the density of Γ given by f Γ x) = Γb) xb E [ G exp x/g) ]. Result A3. The class of positive GGC variables coincides with the class of Wiener-Gamma integrals. Thus i IfΓ = Πs) dγ s,thene [ e λγ] =exp, ) ln ) λ x) ΠΓ dx), where Π Γ denotes the image of Lebesgue s measure on R under the application t Πt).Inotherwords: e x/πs) ds = e xs Π Γ ds),x>. ii Let Γ be a GGC r.v. with Thorin measure Π. Define F Π x) :=,x] Πdy), x>, and let FΠ denote its right continuous inverse. Then, we have Π s) = F s). Π

16 Last time subordinated Brownian motion reaches its supremum 6 Appendix A3. The Wiener-gamma representation Γ := Πs) dγ s presented in 2.) can also be expressed in terms of a jump process. To see this, we let J m) J m) 2 denote a sequence of lengths of jumps of the process {γ t : t m} ranked in decreasing order. Note that the intensity measure of the Poisson point process {s, e s ):s } is ds e x x dx. Then, the jump times U m),u m) 2, constitute a sequence of independently and identically distributed random variables with uniform law on [,m], which is independent of the sequence J m),j m) 2,. In this case, the Wiener-gamma integral 2.) and Result A3 yield the following representations: Γ= m F G u/m)dγ u = k F G U m) k / ) J m) k = m k G k J m) k, A.3) where G,G 2, are independent and identically distributed random variables with a common distribution function F G. The above representation allows us to express the GGC family in terms of the Dirichlet means. For this, we introduce the m-dirichlet process over [,m], m>, by D m) = { } { } D u m) γu : u [,m] = : u [,m]. A.4) γ m Hence, from 2.), A.3) and A.4), we have γ m m Πs) dγ s = m Πs) dd m) s = J m) k. G k γ m k A.5) Note that k γ m J m) k =. Then, one may introduce the Dirichlet measure P /G),m dx) as P /G),m dx) := k J m) k γ m δ /Gk dx), A.6)

17 62 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo where δ /Gj ), j, denote Dirac delta functions. In light of A.3)-A.6), it follows that D m G) := xp /G) m,m dx) = u/m)ddm) F u. A.7) G Relation A.7) constitutes the well-known Dirichlet means. Then, one may express members of the m, G) GGC in terms of members of the m, G) Dirichlet means through the relation Γ m G) = d γ m D m G), for any positive r.v. G. A.8) For a survey of the main properties of m, G) Dirichlet means, see e.g., James et al. 28a), Cifarelli and Regazzini 99), and Lijoi and Prünster 29). Appendix B: Proofs of theorems and corollaries Proof of Theorem 3.. Since Y t and T t, t, are independent, it follows from 2.7) that for q = δ / 2, we have P X t ) = = [, ) [, ) P Y s > ) P T t ds) Φ q ) 2s P T t ds) =E [ Φ q )] 2T /2 t. Next, using Lemma A. and the fact that T t is a subordinator, a straight forward computation shows that P X t ) = 2 E [ e q2 T t/z /2,/2 ] = 2 E [e tψ T q 2 /Z /2,/2) ], B.) where ψ T is the Laplace exponent of T. Substituting B.) into 2.3), the Laplace transform of J τ r is then expressed as E [ exp λj τ )] r =exp { e λt) } Π J dt),λ>, B.2) with Π J dt) = e rt t 2 [e ] E tψ T q 2 /Z /2,/2) dt. The representation of the Lévy measure Π J ) in B.2) enables one to conclude that J τ r 2,r ψ T q 2 / )) Z /2,/2 GGC.

18 Last time subordinated Brownian motion reaches its supremum 63 Proof of Theorem 3.3. First note that J /2 x) = J ν denotes the Bessel function with index ν: J ν z) = j= 2 πx cos x, where ) j z ) ν2j, z < and Arg z <π. j!γ j ν ) 2 The theorem can then be established by utilizing the duality theorem in James et al. 28a, Theorem 2.). Proof of Corollary 3.. Note that the Lévy measure for the linear Brownian motion with negative drift is now expressed as Π J dt) = e rt ] t E [e tq2 /Z /2,/2 dt. Thus, the Laplace transform in 2.3), Lemma A. and Frullani formula together yield E [ exp λj τ r )] { =exp =exp e λt ) e rt t E [ { 2 E ln ] } [e q2 t/z /2,/2 dt )]} λz /2,/2,λ,r >. rz /2,/2 q 2 B.3) To proceed, we only utilize the Laplace exponent in B.3). In particular, the Identity 3.3) and Theorem 3. yield that [ 2 E ln λz )] /2,/2 rz /2,/2 q 2 = [ln 2 E λ r) Z )] /2,/2 q 2 2 [ E ln rz )] /2,/2 q 2 2 B.4) =ln { E [ λ r)γ /2 q 2 / Z /2,/2 )]} ln { E [ rγ/2 q 2 / Z /2,/2 )]}. Thus, applying the Cifarelli and Melilli 2) identity, i.e., E [ / )] ) λγ t Z/2,/2 = 2 2t, the result follows by just letting t = /2. λ Proof of Corollary 3.2. Note that the right hand side expression in the Laplace transform in Corollary 3. is expressed q q 2 r q q 2 λ r = ),λ, r. q 2 r λ q 2 r q q 2 r

19 64 S. B. Fotopoulos, V. K. Jandhyala and Y. Luo Thus, it is not hard to see that q q 2 r q q 2 λ r { )} = exp u u q 2 r q λ q 2 r q 2 r du, λ, r. Moreover, it is known that { λ )} y exp y = 2π e λ 2 t t 3/2 e y t)2 /2t dt. B.5) B.6) In light of B.6), B.5) becomes q q 2 r q q 2 λ r q = 2 r 2π q ) ue u e λ 2q 2 r) t t 3/2 q 2 r exp u ) 2 / q 2 r q q 2 r t 2t dtdu [ λ = E exp 2q 2 r) GIG ))] 2, γ 2 q 2 r q q 2 r,. The last equation follows upon applying the probability density function of a random variable following GIG λ, δ, γ) see Eberlein, 2). This completes the proof of the corollary. Stergios B. Fotopoulos and Yuxing Luo Department of Finance & Management Science Washington State University Pullman, USA fotopo@wsu.edu Venkata K. Jandhyala Department of Mathematics Washington State University Pullman, WA USA jandhyala@wsu.edu Paper received: 2 January 23; revised: 6 May 24.