Analytical properties of scale functions for spectrally negative Lévy processes
|
|
- Meryl Morrison
- 6 years ago
- Views:
Transcription
1 Analytical properties of scale functions for spectrally negative Lévy processes Andreas E. Kyprianou 1 Department of Mathematical Sciences, University of Bath 1 Joint work with Terence Chan and Mladen Savov 1/ 11
2 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. 2/ 11
3 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. Use P x to denote the law of X when X 0 = x > 0 with associated expectation operator E x. 2/ 11
4 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. Use P x to denote the law of X when X 0 = x > 0 with associated expectation operator E x. Let us define the Laplace exponent ψ on [0, ) by E 0(e βxt ) = e ψ(β)t. 2/ 11
5 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. Use P x to denote the law of X when X 0 = x > 0 with associated expectation operator E x. Let us define the Laplace exponent ψ on [0, ) by E 0(e βxt ) = e ψ(β)t. Then for q 0 we may define the q-scale function W (q) : R [0, ) by W (q) (x) = 0 for x < 0 and on (0, ) it is the unique right continuous function such that for β > Φ(q) Z 0 e βx W (q) (x)dx = 1 ψ(β) q. 2/ 11
6 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. Use P x to denote the law of X when X 0 = x > 0 with associated expectation operator E x. Let us define the Laplace exponent ψ on [0, ) by E 0(e βxt ) = e ψ(β)t. Then for q 0 we may define the q-scale function W (q) : R [0, ) by W (q) (x) = 0 for x < 0 and on (0, ) it is the unique right continuous function such that for β > Φ(q) Z Notation! W := W (q). 0 e βx W (q) (x)dx = 1 ψ(β) q. 2/ 11
7 Spectrally negative Lévy processes and scale functions Suppose that X = {X t : t 0} is a general spectrally negative Lévy process. Use P x to denote the law of X when X 0 = x > 0 with associated expectation operator E x. Let us define the Laplace exponent ψ on [0, ) by E 0(e βxt ) = e ψ(β)t. Then for q 0 we may define the q-scale function W (q) : R [0, ) by W (q) (x) = 0 for x < 0 and on (0, ) it is the unique right continuous function such that for β > Φ(q) Z Notation! W := W (q). 0 e βx W (q) (x)dx = 1 ψ(β) q. Note, formally we need to prove that scale functions exist - they do! 2/ 11
8 Why are scale functions important? They appear in virtually all fluctuation identities for spectrally negative Lévy processes. 3/ 11
9 Why are scale functions important? They appear in virtually all fluctuation identities for spectrally negative Lévy processes. For example, resolvent in a strip: for any a > 0, x, y [0, a], q 0 Z 0 where e qt P x (X t dy, t < τ a + τ 0 )dt j W (q) (x)w (q) (a y) = W (q) (a) ff W (q) (x y) dy. τ + a = inf{t > 0 : X t > a} and τ 0 = inf{t > 0 : X t < 0}. 3/ 11
10 Why are scale functions important? They appear in virtually all fluctuation identities for spectrally negative Lévy processes. For example, resolvent in a strip: for any a > 0, x, y [0, a], q 0 Z 0 where e qt P x (X t dy, t < τ a + τ 0 )dt j W (q) (x)w (q) (a y) = W (q) (a) ff W (q) (x y) dy. τ + a = inf{t > 0 : X t > a} and τ 0 = inf{t > 0 : X t < 0}. Or another example is: for a > 0, x [0, a], E x (e qτ 0 1 {τ 0 <τ + a } ) = Z (q) (x) W (q) (x) Z (q) (a) W (q) (a) where Z x Z (q) (x) = 1 + q W (q) (y)dy. 0 3/ 11
11 A basis of fundamental solutions to the generator equation 4/ 11
12 A basis of fundamental solutions to the generator equation It is not difficult to check that for t 0 and a (0, ] e q(t τ 0 τ + a ) W (q) (X t τ 0 τ + a are martingales. ) and e q(t τ0 τ a + ) Z (q) (X t τ 0 τ a + ) 4/ 11
13 A basis of fundamental solutions to the generator equation It is not difficult to check that for t 0 and a (0, ] e q(t τ 0 τ + a ) W (q) (X t τ 0 τ + a are martingales. ) and e q(t τ0 τ a + ) Z (q) (X t τ 0 τ a + ) Suppose that Γ is the infinitessimal generator of X. Then another way of expressing these martingale properties is by writing (in a loose sense) on (0, a) (Γ q)w (q) (x) = 0 and (Γ q)z (q) (x) = 0 4/ 11
14 A basis of fundamental solutions to the generator equation It is not difficult to check that for t 0 and a (0, ] e q(t τ 0 τ + a ) W (q) (X t τ 0 τ + a are martingales. ) and e q(t τ0 τ a + ) Z (q) (X t τ 0 τ a + ) Suppose that Γ is the infinitessimal generator of X. Then another way of expressing these martingale properties is by writing (in a loose sense) on (0, a) (Γ q)w (q) (x) = 0 and (Γ q)z (q) (x) = 0 As many known fluctuation identites turn out to be linear combinations of W (q) and Z (q), this suggest that potentially a small theory would be possible in which these functions play the role of a fundamental basis of solutions to the equation (Γ q)u(x) = 0 on (0, a). 4/ 11
15 A basis of fundamental solutions to the generator equation It is not difficult to check that for t 0 and a (0, ] e q(t τ 0 τ + a ) W (q) (X t τ 0 τ + a are martingales. ) and e q(t τ0 τ a + ) Z (q) (X t τ 0 τ a + ) Suppose that Γ is the infinitessimal generator of X. Then another way of expressing these martingale properties is by writing (in a loose sense) on (0, a) (Γ q)w (q) (x) = 0 and (Γ q)z (q) (x) = 0 As many known fluctuation identites turn out to be linear combinations of W (q) and Z (q), this suggest that potentially a small theory would be possible in which these functions play the role of a fundamental basis of solutions to the equation (Γ q)u(x) = 0 on (0, a). Before even pursuing that objective, one needs to ask whether (Γ q)w (q) (x) makes sense mathematically. 4/ 11
16 How smooth is W (q)? Where to start looking? 5/ 11
17 How smooth is W (q)? Where to start looking? Enough to answer the question for q = 0 and E(X 1) = ψ (0+) 0 as all other cases can be reduced to this case via the relation W (q) (x) = e Φ(q)x W Φ(q) (x) where W Φ(q) plays the role of W after an exponential change of measure under which X drifts to +. 5/ 11
18 How smooth is W (q)? Where to start looking? Enough to answer the question for q = 0 and E(X 1) = ψ (0+) 0 as all other cases can be reduced to this case via the relation W (q) (x) = e Φ(q)x W Φ(q) (x) where W Φ(q) plays the role of W after an exponential change of measure under which X drifts to +. Two key theories that will help analyse the issue of smoothness: 5/ 11
19 How smooth is W (q)? Where to start looking? Enough to answer the question for q = 0 and E(X 1) = ψ (0+) 0 as all other cases can be reduced to this case via the relation W (q) (x) = e Φ(q)x W Φ(q) (x) where W Φ(q) plays the role of W after an exponential change of measure under which X drifts to +. Two key theories that will help analyse the issue of smoothness: Excursion theory 5/ 11
20 How smooth is W (q)? Where to start looking? Enough to answer the question for q = 0 and E(X 1) = ψ (0+) 0 as all other cases can be reduced to this case via the relation W (q) (x) = e Φ(q)x W Φ(q) (x) where W Φ(q) plays the role of W after an exponential change of measure under which X drifts to +. Two key theories that will help analyse the issue of smoothness: Excursion theory Renewal theory 5/ 11
21 The connection with excursion theory 6/ 11
22 The connection with excursion theory It is known that j Z a ff W (x) = W (a) exp ν(ɛ > t)dt x where ν( ) is the intensity measure associated with the Poisson point process of excursions and ɛ is the canonical excursion and ɛ is its maximum value. 6/ 11
23 The connection with excursion theory It is known that j Z a ff W (x) = W (a) exp ν(ɛ > t)dt x where ν( ) is the intensity measure associated with the Poisson point process of excursions and ɛ is the canonical excursion and ɛ is its maximum value. Hence W (x+) W (x ) = ν(ɛ = x) (which immediately implies that for unbounded variation processes W C 1 (0, )). 6/ 11
24 The connection with excursion theory 7/ 11
25 The connection with excursion theory For processes with bounded variation paths (then necessarily X t = δt S t where δ > 0 and S is a driftless subordinator): ν(ɛ > t) = 1 δ Π(, t) + 1 Z «W (t + z) Π(dz) 1 δ W (t) [ t,0] showing W C 1 (0, ) Π has no atoms. 7/ 11
26 The connection with excursion theory For processes with bounded variation paths (then necessarily X t = δt S t where δ > 0 and S is a driftless subordinator): ν(ɛ > t) = 1 δ Π(, t) + 1 Z «W (t + z) Π(dz) 1 δ W (t) [ t,0] showing W C 1 (0, ) Π has no atoms. In the presence of a Gaussian component (σ 0): «ν(ɛ passes x continuously) = σ2 W (x) 2 2 W (x) W (x) showing that W C 2 (0, ). 7/ 11
27 The connection with renewal theory 8/ 11
28 The connection with renewal theory An important observation which helps us understand what kind of answer we might expect comes from the Wiener-Hopf factorization: ψ(β) = βφ(β) where φ is the Laplace exponent of the descending ladder height process and hence an integration by parts gives Z e βx W (dx) = 1 φ(β). [0, ) 8/ 11
29 The connection with renewal theory An important observation which helps us understand what kind of answer we might expect comes from the Wiener-Hopf factorization: ψ(β) = βφ(β) where φ is the Laplace exponent of the descending ladder height process and hence an integration by parts gives Z e βx W (dx) = 1 φ(β). [0, ) This means that W can be seen as the renewal function of a (killed) subordinator, say H, which has Laplace exponent φ, from which it can be calculated that its jump measure is given by Π(, x)dx, its drift is given σ 2 /2 and its killing rate E(X 1) > 0. Here, renewal function means W (dx) = Z 0 P(H t dx)dt. 8/ 11
30 The connection with renewal theory 9/ 11
31 The connection with renewal theory Then appealing to the formula 1 = P(H τ + x = x) + P(H τ + x > x) using Kesten s classical result for the probability of continuous crossing for a subordinator and Kesten-Horowitz-Bertoin formula for overshoots of subordinators one derrives that 1 = σ2 2 W (x) + Z x where Π(x) = R Π(, y)dy. x 0 W (x y){π(y) + E(X 1)}dy 9/ 11
32 The connection with renewal theory Then appealing to the formula 1 = P(H τ + x = x) + P(H τ + x > x) using Kesten s classical result for the probability of continuous crossing for a subordinator and Kesten-Horowitz-Bertoin formula for overshoots of subordinators one derrives that 1 = σ2 2 W (x) + Z x 0 W (x y){π(y) + E(X 1)}dy where Π(x) = R x Π(, y)dy. This can be seen as a renewal equation of the form f = 1 + f g for appropriate f and g. But only when σ 0, otherwise it takes the form f = f g. 9/ 11
33 The connection with renewal theory Then appealing to the formula 1 = P(H τ + x = x) + P(H τ + x > x) using Kesten s classical result for the probability of continuous crossing for a subordinator and Kesten-Horowitz-Bertoin formula for overshoots of subordinators one derrives that 1 = σ2 2 W (x) + Z x 0 W (x y){π(y) + E(X 1)}dy where Π(x) = R Π(, y)dy. x This can be seen as a renewal equation of the form f = 1 + f g for appropriate f and g. But only when σ 0, otherwise it takes the form f = f g. When X has bounded variation paths (X t = δt S t) it a direct inverse of the Laplace transform for W also gives us δw (x) = 1 + Z x 0 W (x y)π(, y)dy which is again a renewal function of the form f = 1 + f g. 9/ 11
34 Some more results 10/ 11
35 Some more results Suppose that σ 0 and Z inf{β 0 : x β Π(dx) < } [0, 2). x <1 Then for k = 0, 1, 2, W C k+3 (0, ) if and only if Π C k (0, ). 10/ 11
36 Some more results Suppose that σ 0 and Z inf{β 0 : x <1 x β Π(dx) < } [0, 2). Then for k = 0, 1, 2, W C k+3 (0, ) if and only if Π C k (0, ). Suppose that X has paths of bounded variation and Π has a derivative π such that π(x) Cx 1 α in the neighbourhood of the origin for some α < 1 and C > 0. Then for k = 0, 1, 2, W C k+3 (0, ) if and only if Π C k (0, ). 10/ 11
37 Doney s conjecture for scale functions For k = 0, 1, 2, 11/ 11
38 Doney s conjecture for scale functions For k = 0, 1, 2, If σ 0 (X has a Gaussian component) then W C k+3 (0, ) Π C k (0, ) 11/ 11
39 Doney s conjecture for scale functions For k = 0, 1, 2, If σ 0 (X has a Gaussian component) then W C k+3 (0, ) Π C k (0, ) If σ = 0 and R x Π(dx) = then ( 1,0) W C k+2 (0, ) Π C k (0, ) 11/ 11
40 Doney s conjecture for scale functions For k = 0, 1, 2, If σ 0 (X has a Gaussian component) then W C k+3 (0, ) Π C k (0, ) If σ = 0 and R x Π(dx) = then ( 1,0) W C k+2 (0, ) Π C k (0, ) If σ = 0 and R x Π(dx) < then ( 1,0) W C k+1 (0, ) Π C k (0, ) 11/ 11
Scale functions for spectrally negative Lévy processes and their appearance in economic models
Scale functions for spectrally negative Lévy processes and their appearance in economic models Andreas E. Kyprianou 1 Department of Mathematical Sciences, University of Bath 1 This is a review talk and
More informationSmoothness of scale functions for spectrally negative Lévy processes
Smoothness of scale functions for spectrally negative Lévy processes T. Chan, A. E. Kyprianou and M. Savov October 15, 9 Abstract Scale functions play a central role in the fluctuation theory of spectrally
More informationThe Theory of Scale Functions for Spectrally Negative Lévy Processes
The Theory of Scale Functions for Spectrally Negative Lévy Processes Alexey Kuznetsov, Andreas E. Kyprianou and Victor Rivero Abstract The purpose of this review article is to give an up to date account
More informationExponential functionals of Lévy processes
Exponential functionals of Lévy processes Víctor Rivero Centro de Investigación en Matemáticas, México. 1/ 28 Outline of the talk Introduction Exponential functionals of spectrally positive Lévy processes
More informationStable and extended hypergeometric Lévy processes
Stable and extended hypergeometric Lévy processes Andreas Kyprianou 1 Juan-Carlos Pardo 2 Alex Watson 2 1 University of Bath 2 CIMAT as given at CIMAT, 23 October 2013 A process, and a problem Let X be
More informationOn Optimal Stopping Problems with Power Function of Lévy Processes
On Optimal Stopping Problems with Power Function of Lévy Processes Budhi Arta Surya Department of Mathematics University of Utrecht 31 August 2006 This talk is based on the joint paper with A.E. Kyprianou:
More informationErik J. Baurdoux Some excursion calculations for reflected Lévy processes
Erik J. Baurdoux Some excursion calculations for reflected Lévy processes Article (Accepted version) (Refereed) Original citation: Baurdoux, Erik J. (29) Some excursion calculations for reflected Lévy
More informationPotentials of stable processes
Potentials of stable processes A. E. Kyprianou A. R. Watson 5th December 23 arxiv:32.222v [math.pr] 4 Dec 23 Abstract. For a stable process, we give an explicit formula for the potential measure of the
More informationErnesto Mordecki 1. Lecture III. PASI - Guanajuato - June 2010
Optimal stopping for Hunt and Lévy processes Ernesto Mordecki 1 Lecture III. PASI - Guanajuato - June 2010 1Joint work with Paavo Salminen (Åbo, Finland) 1 Plan of the talk 1. Motivation: from Finance
More informationHITTING DISTRIBUTIONS OF α-stable PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY
Submitted to the Annals of Probability arxiv: arxiv:2.369 HITTING DISTRIBUTIONS OF α-stable PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY By Andreas E. Kyprianou,,, Juan Carlos Pardo, and Alexander
More informationRIGHT INVERSES OF LÉVY PROCESSES: THE EXCURSION MEA- SURE IN THE GENERAL CASE
Elect. Comm. in Probab. 15 (21), 572 584 ELECTRONIC COMMUNICATIONS in PROBABILITY RIGHT INVERSES OF LÉVY PROCESSES: THE EXCURSION MEA- SURE IN THE GENERAL CASE MLADEN SAVOV 1 Department of Statistics,
More informationBernstein-gamma functions and exponential functionals of Lévy processes
Bernstein-gamma functions and exponential functionals of Lévy processes M. Savov 1 joint work with P. Patie 2 FCPNLO 216, Bilbao November 216 1 Marie Sklodowska Curie Individual Fellowship at IMI, BAS,
More informationAsymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Equivalent Cases
To appear in Insurance Math. Econom. Asymptotic Distributions of the Overshoot and Undershoots for the Lévy Insurance Risk Process in the Cramér and Convolution Euivalent Cases Philip S. Griffin, Ross
More information1 Infinitely Divisible Random Variables
ENSAE, 2004 1 2 1 Infinitely Divisible Random Variables 1.1 Definition A random variable X taking values in IR d is infinitely divisible if its characteristic function ˆµ(u) =E(e i(u X) )=(ˆµ n ) n where
More informationON SCALE FUNCTIONS OF SPECTRALLY NEGATIVE LÉVY PROCESSES WITH PHASE-TYPE JUMPS
ON SCALE FUNCTIONS OF SPECTRALLY NEGATIVE LÉVY PROCESSES WITH PHASE-TYPE JUMPS MASAHIKO EGAMI AND KAZUTOSHI YAMAZAKI ABSTRACT. We study the scale function of the spectrally negative phase-type Lévy process.
More informationRuin probabilities and decompositions for general perturbed risk processes
Ruin probabilities and decompositions for general perturbed risk processes Miljenko Huzak, Mihael Perman, Hrvoje Šikić, and Zoran Vondraček May 15, 23 Abstract We study a general perturbed risk process
More informationFluctuation Theory for Upwards Skip-Free Lévy Chains
risks Article Fluctuation Theory for Upwards Skip-Free Lévy Chains Matija Vidmar Department of Mathematics, Faculty of Mathematics and Physics, University of Ljubljana, 1 Ljubljana, Slovenia; matija.vidmar@fmf.uni-lj.si
More informationAsymptotic behaviour near extinction of continuous state branching processes
Asymptotic behaviour near extinction of continuous state branching processes G. Berzunza and J.C. Pardo August 2, 203 Abstract In this note, we study the asymptotic behaviour near extinction of sub- critical
More informationSOME REMARKS ON FIRST PASSAGE OF LÉVY PROCESSES, THE AMERICAN PUT AND PASTING PRINCIPLES
The Annals of Applied Probability 25, Vol. 15, No. 3, 262 28 DOI 1.1214/155165377 Institute of Mathematical Statistics, 25 SOME REMARKS ON FIRST PASSAGE OF LÉVY PROCESSES, THE AMERICAN PUT AND PASTING
More informationErnesto Mordecki. Talk presented at the. Finnish Mathematical Society
EXACT RUIN PROBABILITIES FOR A CLASS Of LÉVY PROCESSES Ernesto Mordecki http://www.cmat.edu.uy/ mordecki Montevideo, Uruguay visiting Åbo Akademi, Turku Talk presented at the Finnish Mathematical Society
More informationAn optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density
An optimal dividends problem with a terminal value for spectrally negative Lévy processes with a completely monotone jump density R.L. Loeffen Radon Institute for Computational and Applied Mathematics,
More informationPrecautionary Measures for Credit Risk Management in. Jump Models
Kyoto University, Graduate School of Economics Research Project Center Discussion Paper Series Precautionary Measures for Credit Risk Management in Jump Models Masahiko Egami and Kazutoshi Yamazaki Discussion
More informationQuasi-stationary distributions for Lévy processes
Bernoulli 2(4), 26, 57 58 Quasi-stationary distributions for Lévy processes A.E. KYPRIANOU and Z. PALMOWSKI 2,3 School of Mathematical and Computer Sciences, Heriot-Watt University, Edinburgh EH4 4AS,
More informationMEROMORPHIC LÉVY PROCESSES AND THEIR FLUCTUATION IDENTITIES
The Annals of Applied Probability 212, Vol. 22, No. 3, 111 1135 DOI: 1.1214/11-AAP787 Institute of Mathematical Statistics, 212 MEROMORPHIC LÉVY PROCESSES AND THEIR FLUCTUATION IDENTITIES BY A. KUZNETSOV,
More informationarxiv: v1 [math.oc] 22 Mar 2018
OPTIMALITY OF REFRACTION STRATEGIES FOR A CONSTRAINED DIVIDEND PROBLEM MAURICIO JUNCA 1, HAROLD MORENO-FRANCO 2, JOSÉ LUIS PÉREZ 3, AND KAZUTOSHI YAMAZAKI 4 arxiv:183.8492v1 [math.oc] 22 Mar 218 ABSTRACT.
More informationThe optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture
The optimality of a dividend barrier strategy for Lévy insurance risk processes, with a focus on the univariate Erlang mixture by Javid Ali A thesis presented to the University of Waterloo in fulfilment
More informationGerber Shiu Risk Theory
Andreas E. Kyprianou Gerber Shiu Risk Theory Draft May 15, 213 Springer Preface These notes were developed whilst giving a graduate lecture course (Nachdiplomvorlesung) at the Forschungsinstitut für Mathematik,
More informationIntroduction to self-similar growth-fragmentations
Introduction to self-similar growth-fragmentations Quan Shi CIMAT, 11-15 December, 2017 Quan Shi Growth-Fragmentations CIMAT, 11-15 December, 2017 1 / 34 Literature Jean Bertoin, Compensated fragmentation
More informationCurriculum Vitae Dr. Mladen Savov
Curriculum Vitae Dr. Mladen Savov Basic and Contact Details Name: Mladen Svetoslavov Savov Address: j.k. Svoboda, bl.3, en.4, ap. 44, Sofia, Bulgaria Phone: 00359898204169 e-mail: mladensavov@math.bas.bg
More informationDefinition: Lévy Process. Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes. Theorem
Definition: Lévy Process Lectures on Lévy Processes and Stochastic Calculus, Braunschweig, Lecture 2: Lévy Processes David Applebaum Probability and Statistics Department, University of Sheffield, UK July
More informationHITTING DISTRIBUTIONS OF α-stable PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY. University of Bath, CIMAT and University of Bath
The Annals of Probability 24, Vol. 42, No., 398 43 DOI:.24/2-AOP79 Institute of Mathematical Statistics, 24 HITTING DISTRIBUTIONS OF α-stable PROCESSES VIA PATH CENSORING AND SELF-SIMILARITY BY ANDREAS
More informationThe McKean stochastic game driven by a spectrally negative Lévy process
E l e c t r o n i c J o u r n a l o f P r o b a b i l i t y Vol. 13 (28, Paper no. 8, pages 173 197. Journal URL http://www.math.washington.edu/~ejpecp/ The McKean stochastic game driven by a spectrally
More informationCritical branching Brownian motion with absorption. by Jason Schweinsberg University of California at San Diego
Critical branching Brownian motion with absorption by Jason Schweinsberg University of California at San Diego (joint work with Julien Berestycki and Nathanaël Berestycki) Outline 1. Definition and motivation
More informationarxiv: v2 [math.pr] 15 Jul 2015
STRIKINGLY SIMPLE IDENTITIES RELATING EXIT PROBLEMS FOR LÉVY PROCESSES UNDER CONTINUOUS AND POISSON OBSERVATIONS arxiv:157.3848v2 [math.pr] 15 Jul 215 HANSJÖRG ALBRECHER AND JEVGENIJS IVANOVS Abstract.
More informationLaw of the iterated logarithm for pure jump Lévy processes
Law of the iterated logarithm for pure jump Lévy processes Elena Shmileva, St.Petersburg Electrotechnical University July 12, 2010 limsup LIL, liminf LIL Let X (t), t (0, ) be a Lévy process. There are
More informationThe Contour Process of Crump-Mode-Jagers Branching Processes
The Contour Process of Crump-Mode-Jagers Branching Processes Emmanuel Schertzer (LPMA Paris 6), with Florian Simatos (ISAE Toulouse) June 24, 2015 Crump-Mode-Jagers trees Crump Mode Jagers (CMJ) branching
More informationNew families of subordinators with explicit transition probability semigroup
New families of subordinators with explicit transition probability semigroup J. Burridge, A. Kuznetsov, M. Kwaśnicki, A. E. Kyprianou This version: April 24, 214 Abstract There exist only a few known examples
More informationCELEBRATING 50 YEARS OF THE APPLIED PROBABILITY TRUST
J. Appl. Prob. Spec. Vol. 51A, 391 48 (214) Applied Probability Trust 214 CELEBRATING 5 YEARS OF THE APPLIED PROBABILITY TRUST Edited by S. ASMUSSEN, P. JAGERS, I. MOLCHANOV and L. C. G. ROGERS Part 8.
More informationInfinitely divisible distributions and the Lévy-Khintchine formula
Infinitely divisible distributions and the Cornell University May 1, 2015 Some definitions Let X be a real-valued random variable with law µ X. Recall that X is said to be infinitely divisible if for every
More informationA REFINED FACTORIZATION OF THE EXPONENTIAL LAW P. PATIE
A REFINED FACTORIZATION OF THE EXPONENTIAL LAW P. PATIE Abstract. Let ξ be a (possibly killed) subordinator with Laplace exponent φ and denote by I φ e ξs ds, the so-called exponential functional. Consider
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 1. Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt
More informationNumerical Methods with Lévy Processes
Numerical Methods with Lévy Processes 1 Objective: i) Find models of asset returns, etc ii) Get numbers out of them. Why? VaR and risk management Valuing and hedging derivatives Why not? Usual assumption:
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More informationECE 302 Solution to Homework Assignment 5
ECE 2 Solution to Assignment 5 March 7, 27 1 ECE 2 Solution to Homework Assignment 5 Note: To obtain credit for an answer, you must provide adequate justification. Also, if it is possible to obtain a numeric
More informationPotential theory of subordinate killed Brownian motions
Potential theory of subordinate killed Brownian motions Renming Song University of Illinois AMS meeting, Indiana University, April 2, 2017 References This talk is based on the following paper with Panki
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS 1. Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt
More informationThe genealogy of branching Brownian motion with absorption. by Jason Schweinsberg University of California at San Diego
The genealogy of branching Brownian motion with absorption by Jason Schweinsberg University of California at San Diego (joint work with Julien Berestycki and Nathanaël Berestycki) Outline 1. A population
More informationOptimal stopping, Appell polynomials and Wiener-Hopf factorization
arxiv:1002.3746v1 [math.pr] 19 Feb 2010 Optimal stopping, Appell polynomials and Wiener-Hopf factorization Paavo Salminen Åbo Aademi, Mathematical Department, Fänrisgatan 3 B, FIN-20500 Åbo, Finland, email:
More informationMOMENTS AND CUMULANTS OF THE SUBORDINATED LEVY PROCESSES
MOMENTS AND CUMULANTS OF THE SUBORDINATED LEVY PROCESSES PENKA MAYSTER ISET de Rades UNIVERSITY OF TUNIS 1 The formula of Faa di Bruno represents the n-th derivative of the composition of two functions
More informationYaglom-type limit theorems for branching Brownian motion with absorption. by Jason Schweinsberg University of California San Diego
Yaglom-type limit theorems for branching Brownian motion with absorption by Jason Schweinsberg University of California San Diego (with Julien Berestycki, Nathanaël Berestycki, Pascal Maillard) Outline
More informationConvergence of price and sensitivities in Carr s randomization approximation globally and near barrier
Convergence of price and sensitivities in Carr s randomization approximation globally and near barrier Sergei Levendorskĭi University of Leicester Toronto, June 23, 2010 Levendorskĭi () Convergence of
More informationGerber Shiu distribution at Parisian ruin for Lévy insurance risk processes
Gerber Shiu distribution at Parisian ruin for Lévy insurance risk processes arxiv:147.6785v3 math.pr 12 Mar 215 E.J. Baurdoux, J.C. Pardo, J.L. Pérez, J.-F. Renaud This version: March 13, 215 Abstract
More information9 Continuous-State Branching Processes
Lévy processes and continuous-state branching processes: part III Andreas E. Kyprianou, Department of Mathematical Sciences, University of Bath, Claverton Down, Bath, BA2 7AY. a.kyprianou@bath.ac.uk 9
More informationPoisson random measure: motivation
: motivation The Lévy measure provides the expected number of jumps by time unit, i.e. in a time interval of the form: [t, t + 1], and of a certain size Example: ν([1, )) is the expected number of jumps
More informationarxiv:math/ v1 [math.pr] 24 Mar 2005
The Annals of Applied Probability 24, Vol. 14, No. 4, 1766 181 DOI: 1.1214/155164927 c Institute of Mathematical Statistics, 24 arxiv:math/53539v1 [math.pr] 24 Mar 25 RUIN PROBABILITIES AND OVERSHOOTS
More informationCONTINUOUS-STATE BRANCHING PROCESSES AND SELF-SIMILARITY
Applied Probability Trust 2 October 28 CONTINUOUS-STATE BRANCHING PROCESSES AND SELF-SIMILARITY A. E. KYPRIANOU, The University of Bath J.C. PARDO, The University of Bath Abstract In this paper we study
More informationNEW FRONTIERS IN APPLIED PROBABILITY
J. Appl. Prob. Spec. Vol. 48A, 79 98 ) Applied Probability Trust NEW FRONTIERS IN APPLIED PROBABILITY A Festschrift for SØREN ASMUSSEN Edited by P. GLYNN, T. MIKOSCH and T. ROLSKI Part. Lévy processes
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 15. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationarxiv: v1 [math.pr] 30 Mar 2014
Binomial discrete time ruin probability with Parisian delay Irmina Czarna Zbigniew Palmowski Przemys law Świ atek October 8, 2018 arxiv:1403.7761v1 [math.pr] 30 Mar 2014 Abstract. In this paper we analyze
More informationSome remarks on special subordinators
Some remarks on special subordinators Renming Song Department of Mathematics University of Illinois, Urbana, IL 6181 Email: rsong@math.uiuc.edu and Zoran Vondraček Department of Mathematics University
More informationDIFFERENTIAL EQUATIONS
DIFFERENTIAL EQUATIONS Basic Terminology A differential equation is an equation that contains an unknown function together with one or more of its derivatives. 1 Examples: 1. y = 2x + cos x 2. dy dt =
More informationOn Dirichlet series and functional equations
On Dirichlet series and functional equations Alexey Kuznetsov April 11, 2017 arxiv:1703.08827v3 [math.nt] 7 Apr 2017 Abstract There exist many explicit evaluations of Dirichlet series. Most of them are
More informationHomework #6 : final examination Due on March 22nd : individual work
Université de ennes Année 28-29 Master 2ème Mathématiques Modèles stochastiques continus ou à sauts Homework #6 : final examination Due on March 22nd : individual work Exercise Warm-up : behaviour of characteristic
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationMixed Hitting-Time Models
Mixed Hitting-Time Models Jaap H. Abbring March 23, 2011 Abstract We study mixed hitting-time models that specify durations as the first time a Lévy process a continuous-time process with stationary and
More informationNash Type Inequalities for Fractional Powers of Non-Negative Self-adjoint Operators. ( Wroclaw 2006) P.Maheux (Orléans. France)
Nash Type Inequalities for Fractional Powers of Non-Negative Self-adjoint Operators ( Wroclaw 006) P.Maheux (Orléans. France) joint work with A.Bendikov. European Network (HARP) (to appear in T.A.M.S)
More informationGeometric projection of stochastic differential equations
Geometric projection of stochastic differential equations John Armstrong (King s College London) Damiano Brigo (Imperial) August 9, 2018 Idea: Projection Idea: Projection Projection gives a method of systematically
More informationMultilinear local Tb Theorem for Square functions
Multilinear local Tb Theorem for Square functions (joint work with J. Hart and L. Oliveira) September 19, 2013. Sevilla. Goal Tf(x) L p (R n ) C f L p (R n ) Goal Tf(x) L p (R n ) C f L p (R n ) Examples
More informationSums of exponentials of random walks
Sums of exponentials of random walks Robert de Jong Ohio State University August 27, 2009 Abstract This paper shows that the sum of the exponential of an oscillating random walk converges in distribution,
More informationBurgers equation in the complex plane. Govind Menon Division of Applied Mathematics Brown University
Burgers equation in the complex plane Govind Menon Division of Applied Mathematics Brown University What this talk contains Interesting instances of the appearance of Burgers equation in the complex plane
More informationHigher order weak approximations of stochastic differential equations with and without jumps
Higher order weak approximations of stochastic differential equations with and without jumps Hideyuki TANAKA Graduate School of Science and Engineering, Ritsumeikan University Rough Path Analysis and Related
More informationOn a family of critical growth-fragmentation semigroups and refracted Lévy processes
On a family of critical growth-fragmentation semigroups and refracted Lévy processes Benedetta Cavalli arxiv:1812.7951v1 [math.pr] 19 Dec 218 December 2, 218 Abstract The growth-fragmentation equation
More informationSubordinated Brownian Motion: Last Time the Process Reaches its Supremum
Sankhyā : The Indian Journal of Statistics 25, Volume 77-A, Part, pp. 46-64 c 24, Indian Statistical Institute Subordinated Brownian Motion: Last Time the Process Reaches its Supremum Stergios B. Fotopoulos,
More informationExistence Theory: Green s Functions
Chapter 5 Existence Theory: Green s Functions In this chapter we describe a method for constructing a Green s Function The method outlined is formal (not rigorous) When we find a solution to a PDE by constructing
More informationA slow transient diusion in a drifted stable potential
A slow transient diusion in a drifted stable potential Arvind Singh Université Paris VI Abstract We consider a diusion process X in a random potential V of the form V x = S x δx, where δ is a positive
More informationAnomalous Lévy diffusion: From the flight of an albatross to optical lattices. Eric Lutz Abteilung für Quantenphysik, Universität Ulm
Anomalous Lévy diffusion: From the flight of an albatross to optical lattices Eric Lutz Abteilung für Quantenphysik, Universität Ulm Outline 1 Lévy distributions Broad distributions Central limit theorem
More informationA direct method for solving optimal stopping problems for Lévy processes
A direct method for solving optimal stopping problems for Lév processes arxiv:1303.3465v1 [math.pr] 14 Mar 2013 E.J. Baurdoux This version: March 15, 2013 Abstract We propose an alternative approach for
More informationTwo viewpoints on measure valued processes
Two viewpoints on measure valued processes Olivier Hénard Université Paris-Est, Cermics Contents 1 The classical framework : from no particle to one particle 2 The lookdown framework : many particles.
More informationMixed Hitting-Time Models
Mixed Hitting-Time Models Jaap H. Abbring First draft, July 2007 Preliminary second draft, April 7, 2009 Abstract We study a mixed hitting-time (MHT) model that specifies durations as the first time a
More informationLECTURE 2: LOCAL TIME FOR BROWNIAN MOTION
LECTURE 2: LOCAL TIME FOR BROWNIAN MOTION We will define local time for one-dimensional Brownian motion, and deduce some of its properties. We will then use the generalized Ray-Knight theorem proved in
More informationDISTRIBUTIONAL STUDY OF DE FINETTI S DIVIDEND PROBLEM FOR A GENERAL LÉVY INSURANCE RISK PROCESS
J. Appl. Prob. 44, 428 443 (27) Printed in England Applied Probability Trust 27 DISTRIBUTIONAL STUDY OF DE FINETTI S DIVIDEND PROBLEM FOR A GENERAL LÉVY INSURANCE RISK PROCESS A. E. KYPRIANOU, The University
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationAn Introduction to Probability Theory and Its Applications
An Introduction to Probability Theory and Its Applications WILLIAM FELLER (1906-1970) Eugene Higgins Professor of Mathematics Princeton University VOLUME II SECOND EDITION JOHN WILEY & SONS Contents I
More informationMultiple Random Variables
Multiple Random Variables Joint Probability Density Let X and Y be two random variables. Their joint distribution function is F ( XY x, y) P X x Y y. F XY ( ) 1, < x
More informationSome explicit identities associated with positive self-similar Markov processes
Stochastic Processes and their Applications 9 29 98 www.elsevier.com/locate/spa Some explicit identities associated with positive self-similar Markov processes L. Chaumont a, A.E. Kyprianou b, J.C. Pardo
More informationExam 3 Solutions. Multiple Choice Questions
MA 4 Exam 3 Solutions Fall 26 Exam 3 Solutions Multiple Choice Questions. The average value of the function f (x) = x + sin(x) on the interval [, 2π] is: A. 2π 2 2π B. π 2π 2 + 2π 4π 2 2π 4π 2 + 2π 2.
More informationGaussian Processes. Le Song. Machine Learning II: Advanced Topics CSE 8803ML, Spring 2012
Gaussian Processes Le Song Machine Learning II: Advanced Topics CSE 8803ML, Spring 01 Pictorial view of embedding distribution Transform the entire distribution to expected features Feature space Feature
More informationExtremes of Markov-additive processes with one-sided jumps, with queueing applications
Extremes of Markov-additive processes with one-sided jumps, with queueing applications A. B. Dieker and M. Mandjes December 2008 Abstract Through Laplace transforms, we study the extremes of a continuous-time
More informationA LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES.
A LOGARITHMIC EFFICIENT ESTIMATOR OF THE PROBABILITY OF RUIN WITH RECUPERATION FOR SPECTRALLY NEGATIVE LEVY RISK PROCESSES Riccardo Gatto Submitted: April 204 Revised: July 204 Abstract This article provides
More information6. Brownian Motion. Q(A) = P [ ω : x(, ω) A )
6. Brownian Motion. stochastic process can be thought of in one of many equivalent ways. We can begin with an underlying probability space (Ω, Σ, P) and a real valued stochastic process can be defined
More informationLecture Characterization of Infinitely Divisible Distributions
Lecture 10 1 Characterization of Infinitely Divisible Distributions We have shown that a distribution µ is infinitely divisible if and only if it is the weak limit of S n := X n,1 + + X n,n for a uniformly
More informationarxiv: v2 [math.pr] 4 Jul 2008
Applied Probability Trust (7 January 4) OLD AND NEW EXAMPLES OF SCALE FUNCTIONS FOR SPECTRALLY NEGATIVE LÉVY PROCESSES F. HUBALEK, Vienna University of Technology A. E. KYPRIANOU, The University of Bath
More informationExit identities for Lévy processes observed at Poisson arrival times
Bernoulli 223, 216, 1364 1382 DOI: 1.315/15-BEJ695 arxiv:143.2854v2 [math.pr] 17 Mar 216 Exit identities for Lévy processes observed at Poisson arrival times HANSJÖRG ALBRECHER1, JEVGENIJS IVANOVS 2 and
More informationMATH 317 Fall 2016 Assignment 5
MATH 37 Fall 26 Assignment 5 6.3, 6.4. ( 6.3) etermine whether F(x, y) e x sin y îı + e x cos y ĵj is a conservative vector field. If it is, find a function f such that F f. enote F (P, Q). We have Q x
More informationLectures on Lévy Processes, Stochastic Calculus and Financial Applications, Ovronnaz September 2005
Lectures on Lévy Processes, Stochastic Calculus and Financial Applications, Ovronnaz September 005 David Applebaum Probability and Statistics Department, University of Sheffield, Hicks Building, Hounsfield
More informationEQUITY MARKET STABILITY
EQUITY MARKET STABILITY Adrian Banner INTECH Investment Technologies LLC, Princeton (Joint work with E. Robert Fernholz, Ioannis Karatzas, Vassilios Papathanakos and Phillip Whitman.) Talk at WCMF6 conference,
More informationTheory of fractional Lévy diffusion of cold atoms in optical lattices
Theory of fractional Lévy diffusion of cold atoms in optical lattices, Erez Aghion, David Kessler Bar-Ilan Univ. PRL, 108 230602 (2012) PRX, 4 011022 (2014) Fractional Calculus, Leibniz (1695) L Hospital:
More informationarxiv:cond-mat/ v2 [cond-mat.stat-mech] 23 Feb 1998
Multiplicative processes and power laws arxiv:cond-mat/978231v2 [cond-mat.stat-mech] 23 Feb 1998 Didier Sornette 1,2 1 Laboratoire de Physique de la Matière Condensée, CNRS UMR6632 Université des Sciences,
More informationON THE FIRST PASSAGE TIME FOR BROWNIAN MOTION SUBORDI- NATED BY A LÉVY PROCESS
Applied Probability Trust (5 December 28) ON THE FIST PASSAGE TIME FO BOWNIAN MOTION SUBODI- NATED BY A LÉVY POCESS T.. HUD, McMaster University A. KUZNETSOV, York University Abstract This paper considers
More informationPotential Theory on Wiener space revisited
Potential Theory on Wiener space revisited Michael Röckner (University of Bielefeld) Joint work with Aurel Cornea 1 and Lucian Beznea (Rumanian Academy, Bukarest) CRC 701 and BiBoS-Preprint 1 Aurel tragically
More information