Occupation time of Lévy processes with jumps rational Laplace transforms

We are interested in occupation times of Lévy processes with jumps rational Laplace transforms. The corresponding boundary value problems via the Feynman-Kac representation are solved to obtain an explicit formula for the joint distribution of the occupation time and the terminal value of the Lévy processes with jumps rational Laplace transforms.


Introduction
The occupation time is the amount of time a stochastic process stays with in a certain range. It is an interesting topic for stochastic processes. Many explicit results on Laplace transforms for occupation times have been obtained for some well known examples of Lévy process. For a standard Brownian motion W = {W t : t ≥ 0}, P. Lévy's arcsine law is a well known result. It states the following, let Γ + (t) be the time W spends above 0 up to time t: Lévy [10] (for more details see Chapter IV of [16]) showed that for each t > 0 the variable Γ + (t)/t follows the arcsine law: This result was then extended to a Brownian motion with drift by Akahori [2] and Takács [14]. After that, the investigation on occupation times of Lévy processes has made much great progress. For recent works in this topic, see [1], [3], [12], [9], [15] and the references therein for more details.
In this paper, we are interested in the joint Laplace transforms of X = (X t ) t≥0 and its occupation times, i.e, where α > 0, β > 0, γ is some suitable constant and e α is an independent (of X) exponential random variable with rate α > 0 and X = (X t ) t≥0 is a Lévy process with jumps * HEC Montréal, CANADA. E-mail: djilali.ait-aoudia@hec.ca rational Laplace transforms proposed by Lewis and Mordecki [11], see also Kuznetsov [8]. And the purpose is deriving formulas for This extends recent results obtained in Ait-Aoudia and Renaud [1], (Theorem 2) on the processes with hyper-exponential jumps. More precisely, to find an explicit formula for the function ψ(x) in Equation (1.2), the corresponding boundary value problem via the Feynman-Kac representation is considered. By direct calculation, the associated ordinary integro-differential equation (OIDE) is transformed into a homogeneous ordinary differential equation (ODE) of higher order, which is then solved in closed form and its solution equals to ψ(x).
Results obtained here can be applied to price occupation time derivatives as in Cai et al. [3], in which the authors have noted that there are several products in the real market with payoffs depending on the occupation times of an interest rate or a spread of swap rates. For other investigations, see, e.g., [15], [17] and [18].
The rest of the paper is organized as follows. In section 2, we introduce the jumpdiffusion process having jumps with rational Laplace transform. Section 3 contains our main results.

The model
where µ ∈ R and σ > 0 represent the drift and volatility of the diffusion part respectively, W = {W t , t ≥ 0} is a (standard) Brownian motion, N = {N t , t ≥ 0} is a homogeneous Poisson process with rate λ and {Y i , i = 1, 2, . . . } are independent and identically distributed random variables supported in R \ {0}; moreover, {W t , t ≥ 0}, {N t , t ≥ 0} and {Y i , i = 1, 2, . . . } are mutually independent; finally, the probability density function (pdf) of Y 1 is given by where, p ij , q ij ≥ 0 and they are such The parameters η j and θ j can in principle take complex values (see [11]) with 0 < η 1 < Re(η 2 ) < · · · < Re(η m ), Another important tool to establish the key result of the article is the infinitesimal generator of X. Note that X is a Markovian process and its infinitesimal generator is given by for any bounded and twice continuously differentiable function h. Throughout the rest of the paper, the law of X such that X 0 = x is denoted by P x and the corresponding expectation by E x ; we write P and E when x = 0. The Lévy exponent ECP 23 (2018), paper 68. of X is given by Accordingly, G is a rational function on C. The equation G(ζ) − α = 0 with α > 0, σ > 0 and µ ∈ R yields S = M + N + 2 zeros with M = m i=1 m i and N = n j=1 m i (see [8] for details). Let us denote the zeros of G(ζ) − α in the half-plane Re(ζ) > 0 {Re(ζ) < 0} as ρ 1,α , ρ 2,α , . . . , ρ M +1,α {ρ 1,α ,ρ 2,α , . . . ,ρ N +1,α }.

Main results
Throughout this paper X = {X t , t ≥ 0} will be a Lévy process of the type described before, that is with jumps rational Laplace transforms. The time spent by X between the lower barrier h and the upper barrier H, from time 0 to time T , is given by Our main objective is to obtain the joint distribution of eα 0 1 {h<Xt<H} dt and X eα , where e α is an independent (of X) exponential random variable with rate α > 0. In order to do so, we will compute the following joint Laplace-Carson transform with respect to T : for where β ≥ 0, α > 0 and we assume that 0 < γ < min(η 1 , θ 1 ) and G(γ) < α. Clearly, we Now, our goal is to solve the boundary problem (3.3) and find explicit formulae for ψ(x). We first show that ψ satisfies an integro-differential equation and then derive an ordinary differential equation for ψ. Based on the ODE, we show ψ can be written as a linear combination of known exponential functions.
is a polynomial whose zero coincide with those of G(ζ) − α. Also, denote by D α the differential operator such that its characteristic polynomial is P α (ζ).
The following Lemma will be needed for our proof of Proposition 3.2.
Lemma 3.1. Let d (k) indicate the k-th derivative with respect to x of any differentiable function. Let φ be a bounded and continuous function on R and for δ > 0, we define two functions F + and F − such that Then for all i ≥ 1, Proof. We need only to prove first part of the Lemma, the proof of the second part is similar. We proceed by induction on i. For i = 1, we have Moreover, for all i ≥ 2, It follows inductively that for all i ≥ 2, which is the desired result.
We may now state.
is infinitely differentiable and satisfies the ODE x ≥ H, (3.8) for some constants Q L k , Q 0 k , Q 1 k and Q U k .
To complete the proof,D must be shown to coincide with Dα. To show that the characteristic polynomials of Dα andD will suffice. WriteP(ζ) as the characteristic polynomial ofD. Then, by (3.11),P is given bŷ This equation reveals that the characteristic polynomial Pα(ζ) of Dα equals that,P(ζ), ofD. Therefore, any solution to (3.3) can be expressed as Furthermore, we can argue that the coefficients Q L 0,k and Q U 0,k should be zero. In fact, we know that Thus, lim x→±∞ ψ(x)/e γx < +∞, which implies Q L 0,k and Q U 0,k must be zero and the proof is complete.   Here V is an 2S = 2(M + N + 2)-dimensional vector,   Proof. We suppose that ψ is a bounded solution to the boundary value problem (3.3) and and for x > H, (3.20) Now, observe that G(ρ k,α ) − α = 0 for all k and with Γ (i, u) is the incomplete gamma function (see [7], p. 342). Consequentely, substituting w 1 (x), w 2 (x) and w 3 (x) into (3.19) and (3.20) yields that for any x < h ECP 23 (2018), paper 68.
and for j = 1, . . . , n, i = 1, . . . , n j , In adition, we can also obtain another four equations from the fact that ψ(x) is continuously differentiable at x = h and x = H: Consequently, since all of these equations are linear with respect to the undetermined parameters, it follows that the constant vector Q = Q L i , Q 0 i , Q 1 j , Q U j , i = 1, . . . , M + 1, j = 1, . . . , N + 1 satisfies a linear system (3.14) which completes the proof.  Proof. Using the same idea as in Cai and Kou [4] (Theorem 4.1). Applying Ito's formula to the process {ψ(X t )e −αt−β t 0 1 {h<Xs <H} ds , t ≥ 0} we obtain that the process Since G(γ) < α, it follows from Fubini's theorem that So, using Lebesgue's dominated convergence theorem, we have that {M t , t ≥ 0} is actually a positive martingale. In particular which ends the proof.    Because the e ρ k , are distinct, the Vandermonde matrix in equation (3.24) is invertible. Consequently C = 0 and A is invertible.