Law of the exponential functional of one-sided Levy processes and Asian options

Let ξ = (ξ t ) t≥0 be a real-valued spectrally negative Lévy process with unbounded variation and we denote its law by P y (P = P 0 ) when ξ 0 = y ∈ R. That means that ξ is a process with stationary and independent increments having only negative jumps and its right continuous paths with left-limits are of infinite variation Key words and phrases. Lévy processes, exponential functional, special functions, Asian options 2000 Mathematical Subject Classification: 60G51, 33C15, 91B28.

on every compact time interval a.s. We refer to the excellent monographs [2] and [15] for background. It is well-known that the law of ξ is characterized by its one dimensional distributions and thus by the Laplace exponent ψ : R + → R of the random variable ξ 1 which admits the following Lévy-Khintchine representation

(e ur -1 -urI {|r|<1} )ν(dr), u ≥ 0, where b ∈ R, σ ≥ 0 and the measure ν satisfies the integrability condition 0 -∞ (1 ∧ r 2 ) ν(dr) < +∞. Since we excluded the case when ξ has finite variation, the condition 0 -∞ (1 ∧ r) ν(dr) < +∞ is not allowed. Note, see e.g. [2, VII, Corollary 5], that the property that ξ has unbounded variation is equivalent to the following asymptotic

The aim of this note is to describe the distribution function of the so-called exponential functional Σ eq = eq 0 e ξs ds where e q is a random variable, independent of ξ, which is exponentially distributed with parameter q ≥ 0, where we understand e 0 = ∞. In particular, if q = 0, the strong law of large numbers for Lévy processes gives the following equivalence Σ e0 < ∞ a.s. ⇐⇒ E[ξ 1 ] < 0 and we refer to the paper of Bertoin and Yor [3] for alternative conditions, references on the topic and for motivations for studying the law of Σ eq . We simply mention that this positive random variable appears in various fields such as diffusion processes in random environments, fragmentation and coalescence processes, the classical moment problems, mathematical finance and astrophysics. We also point out that the law of Σ eq was known only for the Brownian motion with drift, see Yor’s monograph [16], and a few other isolated cases, see e.g. Carmona et al. [4], Gjessing and Paulsen [7] and Patie [10]. We are now ready to summarize the conditions which will be in force throughout the remaining part of this note.

H: (1.1) holds and either q > 0 or q = 0 and E[ξ 1 ] < 0.

We mention that in [13] the case when the condition (1.1) does not hold is also considered. The remaining part of this Note is organized as follows. In the next Section, we state the representation of the distribution of Σ eq in terms of a power series. In Section 3, we derive a Geman-Yor type formula for the price of Asian options in a spectrally negative Lévy market. We end up this note by revisiting the Brownian motion with drift case.

Let us start by recalling some basic properties of the Laplace exponent ψ, which can be found in [2]. First, it is plain that lim u→∞ ψ(u) = +∞ and ψ is convex. Note that 0 is always a root of the equation ψ(u) = 0. However, in the case E[ξ 1 ] < 0, this equation admits another positive root, which we denote by θ. Moreover, for any

which is also continuous and increasing. Finally, we write, for any u ≥ 0 and q > 0, ψ(u) = ψ(u) -q, and set the following notation

Recalling that ψ(θ) = 0 and observing that ψ(φ(q)) = 0, the mappings ψ θ , ψ φ(q) are plainly Laplace exponents of conservative Lévy processes. We also point out that ψ ′ θ (0

In order to present our result in a compact form, we write γ = φ(q) if q > 0, θ otherwise, and

Next, set a 0 = 1 and a n (ψ γ ) = ( n k=1 ψ γ (k)) -1 , n = 1, 2, . . . In [12], the author introduced the following power series

and showed by means of classical criteria that the mapping z → I ψ (z) is an entire function. We refer to [12] for interesting analytical properties enjoyed by these power series and also for connections with well known special functions, such as, for instance, the modified Bessel functions, confluent hypergeometric functions and several generalizations of the Mittag-Leffler functions. To simplify the notation, we introduce the following definition, for any z ∈ C,

Next, let G κ be a Gamma random variable independent of ξ, with parameter κ > 0. Its density is given by g(dy) = e -y y κ-1 Γ(κ) dy, y > 0, with Γ the Euler gamma function. Then, in [14], the author suggested the following generalization

∞ 0 e -s s ρ-1 ds, Re(ρ) > 0, and an argument of dominated convergence, we obtain the following power series representation

which is easily seen to be, under the condition (1.1), an entire function in z. Note also, by the principle of analytical continuation, that the mapping ρ → O ψγ (ρ; z) is an entire function for arbitrary z ∈ C. We are now ready to state the main result of this note.

Theorem 2.1. Assume that H holds and write S(t) = P(Σ eq ≥ t), t > 0. Then, there exists a constant

and one has, for any t > 0,

Moreover, the law of Σ eq is absolutely continuous with a density, denoted by s, g

…(Full text truncated)…

This content is AI-processed based on ArXiv data.