Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time
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Some Explicit Results on First Exit Times for a Jump Diffusion Process Involving Semimartingale Local Time Shiyu Song1 Received: 3 May 2020 / Revised: 28 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract In this paper, we consider the one-sided and the two-sided first exit problem for a jump diffusion process with semimartingale local time. Denote this process by X = / (l, u)} {X t , t ≥ 0} and set τl = inf{t ≥ 0, X t ≤ l} and τl,u = inf{t ≥ 0, X t ∈ with l < u. We first establish the existence and uniqueness of strong solutions of the stochastic differential equation (SDE) satisfied by X . Then we investigate the Laplace transforms associated with τl and τl,u . It turns out that the explicit expressions for those Laplace transforms can be expressed in terms of exponential functions. Keywords First exit time · Jump diffusion process · Semimartingale local time · Laplace transform Mathematics Subject Classification (2010) 60E05 · 60G55 · 60H30 · 60J55
1 Introduction We consider an SDE of the form: dX t = μdt + σ dBt + d
N t
Vi
+ (2α − 1)d L t (β),
X 0 ∈ R.
(1.1)
i=1
Here B = {Bt , t ≥ 0} is a standard Brownian motion, N = {Nt , t ≥ 0} is a Poisson process with rate λ, the constants μ and σ > 0 are the drift and volatility of the diffusion part, respectively, and the parameter α takes value in (0, 1). The jump sizes {Vi } form a sequence of independent identically distributed random variables with
This work was partially supported by the National Natural Science Foundation of China (No. 11801407).
B 1
Shiyu Song [email protected] School of Mathematics, Tianjin University, Tianjin 300354, People’s Republic of China
123
Journal of Theoretical Probability
double exponential density f V (x) =
pδ1 e−δ1 x , qδ2 eδ2 x ,
x ≥ 0, x < 0,
where p, q ≥ 0, p + q = 1 and δ1 , δ2 > 0. We denote by F(dx) the corresponding L(β) = probability distribution of Vi such that F(dx) = f V (x)dx. The process { L t (β), t ≥ 0} represents the symmetric semimartingale local time of X at the level β in the sense of Protter [14] (see “Appendix”). Moreover, we assume that B, N and {Vi } are independent. Two cases of (1.1) are of special interest. When the jump rate λ = 0, the solution of (1.1) becomes a well-known process named the skew Brownian motion which has been extensively studied in the literature (see, e.g., [3,6,7,12,16]). When α = 1/2, it is a simple but useful Lévy process prevailing as an asset price dynamics model in the field of financial engineering (see, e.g., [9–11]). In this paper, we are concerned with the distributions of the first times at which X crosses a given level or exits from a finite interval. More precisely, we derive closedform expressions for Laplace transforms associated with τl = inf{t ≥ 0, X t ≤ l} / (l, u)} with l < u. As is known, with the jump part, the and τl,u = inf{t ≥ 0, X t ∈ “overshoot” or “undershoot” may be incurred when the process passes through a given level from below or above, which makes the problem much more complicated tha
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