Theory of Probability and Mathematical Statistics
On fluctuation theory for spectrally negative Levy processes with Parisian reflection below, and applications
Florin Avram, Xiaowen Zhou.
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Abstract: As well known, all functionals of a Markov process may be expressed in terms of the generator operator, modulo some analytic work. In the case of spectrally negative Markov processes however, it is conjectured that everything can be expressed in a more direct way using the $W$ scale function which intervenes in the two-sided first passage problem, modulo performing various integrals. This conjecture arises from work on Levy processes, where the $W$ scale function has explicit Laplace transform, and is therefore easily computable;furthermore it was found in the papers above that a second scale function $Z$ introduced in \cite{AKP} (this is an exponential transform \eqref{Z} of $W$) greatly simplifies \fp laws, especially for reflected processes. $Z$ is an harmonic function of the L\'evy process (like $W$), corresponding to exterior boundary conditions $w(x)=e^{\th x}$ \eqref{hext}, and is also a particular case of a "smooth \GS function" $\mS_w$. The concept of \GS function was introduced in \cite{gerber1998time}; we will use it however here in the more restricted sense of \cite{APP15}, who define this to be a "smooth" harmonic function of the process, which fits the exterior boundary condition $w(x)$ and solves simultaneously the problems \eqref{serui}, \eqref{rserui}. It has been conjectured that similar laws govern other classes of \sn processes, but it is quite difficult to find assumptions which allow proving this for general classes of Markov processes.However, we show below that in the particular case of \sn \lev processes with Parisian absorption and reflection from below \cite{AIZ,BPPR,APY}, this conjecture holds true, once the appropriate $W$ and $Z$ are identified (this observation seems new).This paper gathers a collection of first passage formulas for spectrally negative Parisian L\'evy processes, expressed in terms of $W,Z$ and $\mS_w$, which may serve as an "instruction kit" for computing quantities of interest in applications, for example in risk theory and mathematical finance.\iffalse An important point to notice is that the \fp laws reviewed concern equally absorbed, reflected, or doubly reflected processes. The benefit of presenting together all the possible boundary behaviors, as is done already in the papers cited above, is that the results suggest interesting relationships %between the different boundary behaviors -- see for example \eqref{proof}.\fi To illustrate the usefulness of our list, we construct a new index for the valuation of financial companies modeled by spectrally negative L\'evy processes, based on a Dickson-Waters modifications of the de Finetti optimal expected discounted dividends objective. We offer as well an index for the valuation of conglomerates of inancial companies. %Analogue results for spectrally negative Markov additive processes are listed in the web version of the paper; An implicit question arising is to investigate analog results for other classes of \sn Markovian processes.
Keywords: spectrally negative Levy process, scale functions, capital injections, dividend optimization, valuation problem, Parisian absorbtion and reflection
Bibliography: [AA10] Hansj¨org Albrecher and S¨oren Asmussen. Ruin probabilities, volume 14. World Scientific, 2010.
[ABPR16] F. Avram, A. Badescu, M. Pistorius, and L. Rabehasaina. On a class of dependent sparre andersen risk models and a bailout application. IME, (71):27–39, 2016.
[AI13] Hansj¨org Albrecher and Jevgenijs Ivanovs. A risk model with an observer in a markov environment. Risks, 1(3):148–161, 2013.
[AI14] Hansj¨org Albrecher and Jevgenijs Ivanovs. Power identities for l´evy risk models under taxation and capital injections. Stochastic Systems, 4(1):157–172, 2014.
[AI15] Hansjoerg Albrecher and Jevgenijs Ivanovs. Strikingly simple identities relating exit problems for l´evy processes under continuous and poisson observations. arXiv preprint arXiv:1507.03848, 2015.
[AIZ16] Hansjoerg Albrecher, Jevgenijs Ivanovs, and Xiaowen Zhou. Exit identities for l´evy processes observed at poisson arrival times. Bernoulli, 22(3):1364–1382, 2016.
[AKP04] Florin Avram, Andreas Kyprianou, and Martijn Pistorius. Exit problems for spectrally negative l´evy processes and applications to (canadized) russian options. The Annals of Applied Probability, 14(1):215–238, 2004.
[ALR09] Florin Avram, Nikolai Leonenko, and Landy Rabehasaina. Series expansions for the first passage distribution of wong–pearson jump-diffusions. Stochastic Analysis and Applications, 27(4):770– 796, 2009.
[AM15] Florin Avram and Andreea Minca. Steps towards a management toolkit for central branch risk networks, using rational approximations and matrix scale functions. In A. B. Piunovskyi, editor, Modern trends in controlled stochastic processes: theory and applications, page 263:285, 2015.
[AM16] Florin Avram and Andreea Minca. On the management of central branch risk networks. Advances in Applied probability, in print, 2016.
[AMP16] Pablo Azcue, Nora Muler, and Zbigniew Palmowski. Optimal dividend payments for a twodimensional insurance risk process. arXiv preprint arXiv:1603.07019, 2016.12. O. Baranovskyi, M. Pratsiovytyi, G. Torbin, Ostrogradsky-Sierpinski-Pierce series and their applications, Naukova Dumka, Kyiv, 2013.
[APP07] Florin Avram, Zbigniew Palmowski, and Martijn R Pistorius. On the optimal dividend problem for a spectrally negative l´evy process. The Annals of Applied Probability, pages 156–180, 2007.
[APP08a] Florin Avram, Zbigniew Palmowski, and Martijn Pistorius. A two-dimensional ruin problem on the positive quadrant. Insurance: Mathematics and Economics, 42(1):227–234, 2008.
[APP08b] Florin Avram, Zbigniew Palmowski, and Martijn R Pistorius. Exit problem of a twodimensional risk process from the quadrant: exact and asymptotic results. The Annals of Applied Probability, 18(6):2421–2449, 2008.
[APP15] Florin Avram, Zbigniew Palmowski, and Martijn R Pistorius. On gerber–shiu functions and optimal dividend distribution for a l´evy risk process in the presence of a penalty function. The Annals of Applied Probability, 25(4):1868–1935, 2015. [APY16] Florin Avram, Jose-Luiz Perez, and Kazutoshi Yamazaki. Spectrally negative l´evy processes with parisian reflection below and classical reflection above. SPA, 2016.
[AzMul14] Pablo Azcue and Nora Muler. Stochastic Optimization in Insurance: A Dynamic Programming Approach. Springer, 2014.
[BB11] Nicole B¨auerle and Anja Blatter. Optimal control and dependence modeling of insurance portfolios with l´evy dynamics. Insurance: Mathematics and Economics, 48(3):398–405, 2011.
[Ber98] Jean Bertoin. L´evy processes, volume 121. Cambridge university press, 1998.
[Bog03] Elena Boguslavskaya. On optimization of dividend flow for a company in a presence of liquidation value. Can be downloaded from http://www. boguslavsky. net/fin/dividendflow. pdf, 2003.
[BPPR16] Erik Baurdoux, Juan Carlos Pardo, Jos´e Luis P´erez, and Jean-Francois Renaud. Gerber-shiu distribution at parisian ruin for l´evy insurance risk processes. Journal of Applied probability, 2016.
[BS12] Andrei N Borodin and Paavo Salminen. Handbook of Brownian motion-facts and formulae. Birkh¨auser, 2012.
[CP16] Michael Choi, and Pierre Patie. Skip-free Markov chains. Preprint, Research gate, 2016.
[DF57] Bruno De Finetti. Su un’impostazione alternativa della teoria collettiva del rischio. In Transactions of the XVth international congress of Actuaries, volume 2, pages 433–443, 1957.
[DW04] David Dickson and Howard R Waters. Some optimal dividends problems. Astin Bulletin, 34(01):49–74, 2004.
[GS98] Hans Gerber and Elias Shiu. On the time value of ruin. North American Actuarial Journal, 2, (1), 48–72, 1998.
[GLY06] Hans U Gerber, X Sheldon Lin, and Hailiang Yang. A note on the dividends-penalty identity and the optimal dividend barrier. Astin Bulletin, 36(02):489–503, 2006.
[IP12] Jevgenijs Ivanovs and Zbigniew Palmowski. Occupation densities in solving exit problems for markov additive processes and their reflections. Stochastic Processes and their Applications, 122(9):3342–3360, 2012.
[Iva11] Jevgenijs Ivanovs. PhD thesis: One-sided Markov additive processes and related exit problems. 2011.
[Iva13] Jevgenijs Ivanovs. Potential measures of one-sided markov additive processes with reflecting and terminating barriers. arXiv preprint arXiv:1309.4987, 2013.
[Iva13a] Jevgenijs Ivanovs. Spectrally-negative Markov additive processes, a Mathematica 8.0 package. https://sites.google.com/site/jevgenijsivanovs/files, 2013.
[Ken76] Douglas Kennedy. Some martingales related to cumulative sum tests and single-server queues. Stochastic processes and their applications, 4, 3, pages 261–269, 1976.
[KL10] Andreas Kyprianou and Ronnie Loeffen. Refracted l´evy processes. Ann. Inst. H. Poincar´e, 46(1):24–44, 2010.
[KP08] Andreas Kyprianou and Zbigniew Palmowski. Fluctuations of spectrally negative markov additive processes. In S´eminaire de probabilit´es XLI, pages 121–135. Springer, 2008.
[KPP14] Andreas Kyprianou, Juan Carlos Pardo, and Jos´e Luis P´erez. Occupation times of refracted l´evy processes. Journal of Theoretical Probability, 27(4):1292–1315, 2014.
[Kyp14] Andreas Kyprianou. Fluctuations of L´evy Processes with Applications: Introductory Lectures. Springer Science & Business Media, 2014.
[LRZ11] David Landriault, Jean-Francois Renaud, and Xiaowen Zhou. Occupation times of spectrally negative l´evy processes with applications. Stochastic processes and their applications, 121(11):2629– 2641, 2011.
[LRZ14] David Landriault, Jean-Francois Renaud, and Xiaowen Zhou. An insurance risk model with Parisian implementation delays. Methodology and Computing in Applied Probability, 16(3):583–607, 2014.
[LST14] Gunther Leobacher, Michaela Sz¨olgyenyi, and Stefan Thonhauser. Bayesian dividend optimization and finite time ruin probabilities. Stochastic Models, 30(2):216–249, 2014.
[LZ14] Yingqiu Li and Xiaowen Zhou. On pre-exit joint occupation times for spectrally negative l´evy processes. Statistics & Probability Letters, 94:48–55, 2014.
[LZZ15] Yingqiu Li, Xiaowen Zhou, and Na Zhu. Two-sided discounted potential measures for spectrally negative l´evy processes. Statistics & Probability Letters, 100:67–76, 2015.
[LR10] Ronnie Loeffen and Jean-Francois Renaud. De finetti’s optimal dividends problem with an affine penalty function at ruin. Insurance: Mathematics and Economics, 46(1):98–108, 2010.
[Loi05] St´ephane Loisel. Differentiation of some functionals of risk processes, and optimal reserve allocation. Journal of applied probability, pages 379–392, 2005.
[LZ08] Arne Løkka and Mihail Zervos. Optimal dividend and issuance of equity policies in the presence of proportional costs. Insurance: Mathematics and Economics, 42(3):954–961, 2008.
[MM61] Merton H Miller and Franco Modigliani. Dividend policy, growth, and the valuation of shares. the Journal of Business, 34(4):411–433, 1961.
[NY05] Nguyen-Ngoc and Marc Yor. Some martingales associated to reflected L´evy processes. S´eminaire de probabilit´es XXXVIII, pages=42–69, 2005.
[PY15] Jos´e Luis P´erez and Kazutoshi Yamazaki. On the refracted-reflected spectrally negative l´evy processes. Preprint, 2015.
[Pic94] Philippe Picard. On some measures of the severity of ruin in the classical poisson model. Insurance: Mathematics and Economics, 14(2):107–115, 1994.
[Pis03] Martijn Pistorius. On doubly reflected completely asymmetric L´evy processes Stochastic Processes and their Applications, 107, pages 131–143, 2003.
[Pis05] Martijn Pistorius. A potential-theoretical review of some exit problems of spectrally negative l´evy processes. S´eminaire de Probabilit´es XXXVIII, pages 30–41, 2005.
[Pis07] Martijn Pistorius. An excursion-theoretical approach to some boundary crossing problems and the Skorokhod embedding for reflected L´evy processes. S´eminaire de Probabilit´es XL, pages 287–307, 2007.
[Ren14] Jean-Francois Renaud. On the time spent in the red by a refracted L´evy risk process. Journal of Applied Probability, 51, pages 1171–1188, 2014.
[RSST09] Tomasz Rolski, Hanspeter Schmidli, Volker Schmidt, and Jozef Teugels. Stochastic processes for insurance and finance, volume 505. John Wiley & Sons, 2009.
[Sat99] K.I. Sato. L´evy processes and infinitely divisible distributions. Cambridge university press, 1999.
[Sch07] Hanspeter Schmidli. Stochastic control in insurance. Springer Science & Business Media, 2007.
[SLG84] Steven E Shreve, John P Lehoczky, and Donald P Gaver. Optimal consumption for general diffusions with absorbing and reflecting barriers. SIAM Journal on Control and Optimization, 22(1):55– 75, 1984.
