Occupation Time and the Lebesgue Measure of the Range for a Levy Process

1988 ◽  
Vol 103 (4) ◽  
pp. 1241
Author(s):  
S. C. Port
Keyword(s):  
Lebesgue Measure ◽  
Lévy Process ◽  
Levy Process ◽  
Occupation Time

For which functions are 𝑓(𝑋_{𝑡})-𝔼𝕗(𝕏_{𝕥}) and 𝕘(𝕏_{𝕥})/𝔼𝕘(𝕏_{𝕥}) martingales?

2021 ◽  
Vol 105 (0) ◽  
pp. 79-91
Author(s):  
F. Kühn ◽  
R. Schilling
Keyword(s):  
Lebesgue Measure ◽  
Lévy Process ◽  
Levy Process ◽  
Content Type ◽  
One Dimensional ◽  
Exponential Moments ◽  
Bounded Functions

Let X = ( X t ) t ≥ 0 X=(X_t)_{t\geq 0} be a one-dimensional Lévy process such that each X t X_t has a C b 1 C^1_b -density w. r. t. Lebesgue measure and certain polynomial or exponential moments. We characterize all polynomially bounded functions f : R → R f\colon \mathbb {R}\to \mathbb {R} , and exponentially bounded functions g : R → ( 0 , ∞ ) g\colon \mathbb {R}\to (0,\infty ) , such that f ( X t ) − E f ( X t ) f(X_t)-\mathbb {E} f(X_t) , resp. g ( X t ) / E g ( X t ) g(X_t)/\mathbb {E} g(X_t) , are martingales.

Occupation times of Lévy processes

2021 ◽  
pp. 2142003
Author(s):  
Lan Wu ◽  
Xiao Zhang
Keyword(s):  
Explicit Formula ◽  
Lévy Process ◽  
Lévy Processes ◽  
Levy Processes ◽  
Levy Process ◽  
Occupation Time ◽  
Applied Probability ◽  
Occupation Times ◽  
Financial Applications ◽  
Pricing Problems

In this paper, we give a complete and succinct proof that an explicit formula for the occupation time holds for all Lévy processes, which is important to the pricing problems of various occupation-time-related derivatives such as step options and corridor options. We construct a sequence of Lévy processes converging to a given Lévy process to obtain our conclusion. Besides financial applications, the mathematical results about occupation times of a Lévy process are of interest in applied probability.

scholarly journals LAN property for a simple Lévy process

2014 ◽  
Vol 352 (10) ◽  
pp. 859-864 ◽  
Author(s):  
Arturo Kohatsu-Higa ◽  
Eulalia Nualart ◽  
Ngoc Khue Tran
Keyword(s):  
Lévy Process ◽  
Levy Process

scholarly journals On the optimal dividend problem for a spectrally negative Lévy process

2007 ◽  
Vol 17 (1) ◽  
pp. 156-180 ◽  
Author(s):  
Florin Avram ◽  
Zbigniew Palmowski ◽  
Martijn R. Pistorius
Keyword(s):  
Lévy Process ◽  
Levy Process ◽  
Spectrally Negative Lévy Process ◽  
Optimal Dividend
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