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Exponential Functionals of Brownian Motion, II: Some Related Diffusion Processes

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Exponential Functionals of Brownian Motion, II: Some Related Diffusion Processes Probability Surveys Vol. 2 (2005) 348–384 ISSN: 1549-5787 DOI: 10.1214/154957805100000168 Exponential functionals of Brownian motion, II: Some related diffusion processes∗ Hiroyuki Matsumoto Graduate School of Information Science, Nagoya University, Chikusa-ku, Nagoya 464-8601, Japan e-mail: [email protected] Marc Yor Laboratoire de Probabilit´es and Institut universitaire de France, Universit´ePierre et Marie Curie, 175 rue du Chevaleret, F-75013 Paris, France e-mail: [email protected] Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of L´evy’s and Pitman’s theorems are discussed. AMS 2000 subject classifications: Primary 60J65; secondary 60J60, 60H30. Keywords and phrases: Brownian motion, hyperbolic space, heat kernel, random environment, L´evy’s theorem, Pitman’s theorem. Received September 2005. 1. Introduction Let B = Bt,t ≧ 0 be a one-dimensional Brownian motion starting from 0 and { } (µ) (µ) defined on a probability space (Ω, , P ). Denoting by B = Bt = Bt + µt the corresponding Brownian motionF with constant drift µ R{, we consider the} (µ) (µ) ∈ arXiv:math/0511519v2 [math.PR] 26 Apr 2007 exponential functional A = A defined by { t } t (µ) (µ) At = exp(2Bs )ds, t ≧ 0. (1.1) Z0 (µ) In Part I [49] of our survey, we have discussed about the probability law of At for fixed t and about several related topics. ∗This is an original survey paper. 348 H. Matsumoto, M. Yor/Exponential functionals of BM, II 349 Among the results, we have shown some explicit (integral) representations (µ) for the density of At . In particular, we have proven the following formula originally obtained in Yor [63]: 2x (µ) (µ) µx µ2t/2 1+ e x dudx P (A du,B dx)= e − exp θ(e /u,t) , (1.2) t ∈ t ∈ − 2u u where, for r> 0 and t> 0, r π2/2t ∞ ξ2/2t r cosh(ξ) πξ θ(r, t)= e e− e− sinh(ξ) sin dξ. (1.3) (2π3t)1/2 t Z0 The function θ(r, t) appears in the representation for the (unnormalized) density of the so-called Hartman-Watson distribution and satisfies ∞ α2t/2 e− θ(r, t)dt = Iα(r), α> 0, (1.4) Z0 where Iα is the usual modified Bessel function. For details, see Part I and the references cited therein. Another important fact, which has been used in several domains and also discussed in Part I, is the following identity in law due to Dufresne [18]. Let µ> 0. Then one has ( µ) ∞ ( µ) (law) 1 A − exp(2Bs− )ds = , (1.5) ∞ ≡ 2γ Z0 µ where γµ is a gamma random variable with parameter µ, that is, 1 µ 1 x P (γ dx)= x − e− dx, x ≧ 0. µ ∈ Γ(µ) The purpose of this second part of our surveys is to present some results ob- tained by applying the formulae and identities mentioned in Part I to Brownian motion and some related stochastic processes. In Section 2 we discuss about the pricing formula for the average option, so called, Asian option in the Black-Scholes model. In Section 3 we present some formulae for the heat kernels of the semigroups generated by the Laplacians on hyperbolic spaces. By reasoning in probabilistic terms, we obtain not only the classical formulae but also new expressions. In Section 4 we apply the results on exponential functionals to a question pertaining to a class of diffusion processes in random environments. In Section 5 we show Dufresne’s recursion relation for the probability density (µ) of At with respect to µ which, as we have seen in Part I, plays an important role in several studies on exponential functionals. Dufresne’s relation is important in studying extensions or analogues of L´evy’s (µ) (µ) (µ) (µ) and Pitman’s theorems about, respectively, Mt Bt and 2Mt Bt , (µ) (µ) { − } { − } where Mt = max0≦s≦t Bs , by means of exponential functionals. These topics H. Matsumoto, M. Yor/Exponential functionals of BM, II 350 are finally discussed in Section 6 of this survey, where we consider the stochastic process defined by 1 t M (µ),λ = log exp(λB(µ))ds , t> 0. t λ s Z0 (µ),λ By the Laplace method, we easily see that, as λ , Mt converges to (µ) (µ),λ (µ) →(µ) ∞,λ (µ) Mt , and we prove that Mt Bt and 2Mt Bt are diffusion processes for any λ R. Hence,{ the− classical} L´evy{ and Pitman− } theorems may be seen as limiting results∈ of those as λ . → ∞ 2. Asian options In this section we consider the Asian or average call option in the framework of the Black-Scholes model and present some identities for the pricing formula. By the Black-Scholes model, we mean a market model which consists of a riskless bond b = bt with a constant interest rate and a risky asset S = St with a constant appreciation{ } rate and volatility. That is, letting r > 0, µ { R} and σ > 0 be constants, we let b and S be given by the stochastic differential∈ equation dbt dSt = rdt, = µdt + σdBt, bt St where B = Bt is a one-dimensional Brownian motion with B0 = 0 defined on a complete{ probability} space (Ω, , P ). F For simplicity we normalize them by setting b0 = 1. Then we have b = exp(rt) and S = S exp(σB + (µ σ2/2)t). t t 0 t − Following the standard procedure, we consider the discounted stock price S = S given by { t} rt 2 e e St = e− St = S0 exp(σBt + (µ r σ /2)t). − − Then, by Girsanov’se theorem, there exists a unique probability measure Q which is absolutely continuous with respect to P and under which S is a mar- tingale. Q is called the martingale measure for S and we have e dQ µ r (µ r)2 = exp − B e − T , dP − σ T − 2σ2 T F 1 where T = σ Bs,s ≦ T . B = Bt = Bt + σ− (µ r)t is a Brownian motion under QF. { } { − } Let us consider the Europeane e and the Asian call options with fixed strike price k> 0 and maturity T . The payoffs are given by (S k) and ( (T ) k) , T − + A − + H. Matsumoto, M. Yor/Exponential functionals of BM, II 351 respectively, where x = max x, 0 and + { } 1 t (t)= S du, 0 <t ≦ T. A t u Z0 By the Black-Scholes formula or by the non-arbitrage argument, we can show that the theoretical price CE (k,T ) and CA(k,T ) of these call options at time t = 0 are given by rT Q C (k,T )= e− E [(S k) ] E T − + and rT Q C (k,T )= e− E [( (T ) k) ], A A − + where EQ denotes the expectation with respect to the martingale measure Q. Proposition 2.1. If r ≧ 0, one has CA(k,T ) ≦ CE(k,T ) for every k > 0 and T > 0. Proof. We have T rT Q 1 C (k,T )= e− E S dt k . A T t − Z0 + Using Jensen’s inequality, we get T rT 1 Q C (k,T ) ≦ e− E [(S k) ]dt. A T t − + Z0 2 Since r ≧ 0, St = S0 exp(σBt + (r σ /2)t) is a submartingale under Q. Therefore, using{ Jensen’s inequality again,− we see} that (S k) is also a { t − +} submartingale. Hence we obtaine T rT 1 Q C (k,T ) ≦ e− E [(S k) ]dt A T T − + Z0 rT Q = e− E [(S k) ]= C (k,T ). T − + E For more discussions on CA(k,T ), see Geman-Yor [24], Rogers-Shi [55] and the references cited therein. (µ) By using explicit expressions for the density of At discussed in Part I, we obtain several integral representations for CA(k,T ). However, they are compli- cated. Hence we omit this approach and consider instead the Laplace transform of CA(k,T ) in T . In the following we set σ = 2 and consider t (µ) (µ) At = exp(2Bs )ds Z0 H. Matsumoto, M. Yor/Exponential functionals of BM, II 352 under the original probability measure P to follow the same convention as in Part I and in other parts of the present article. Let Tλ be an exponential random variable with parameter λ> 0 independent of B. Yor [62] (see also Part I) has shown the identity in law (law) Z A(µ) = 1,a , Tλ 2γb 2 where a = (ν + µ)/2,b = (ν µ)/2, ν = 2λ + µ , Z1,a is a beta variable with parameters (1,a), γ is a gamma− variable with parameter b and Z and γ are b p 1,a b independent. From this identity we deduce the following result. Theorem 2.2. For all µ R,λ> max 2(1 + µ), 0 and k> 0, we have ∈ { } ∞ λt (µ) λ e− E[(A k) ]dt t − + Z0 1/2k 1 t b 2 a+1 = e− t − (1 2kt) dt. (λ 2(1 + µ))Γ(b 1) − − − Z0 The same formula has been proven in [24] with the help of some properties of Bessel processes. We next present another proof of Theorem 2.2, following Donati-Martin, Ghomrasni and Yor [16], who have used, as an auxiliary tool, the stochastic process Y (µ)(x)= Y (µ)(x) given by { t } t Y (µ)(x) = exp(2B(µ)) x + exp( 2B(µ))ds .
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