Bessel Processes and Ray-Knight Theorems

1991 ◽  
pp. 409-434
Author(s):  
Daniel Revuz ◽  
Marc Yor
Keyword(s):  
Bessel Processes

scholarly journals Renewal series and square-root boundaries for Bessel processes

2008 ◽  
Vol 13 (0) ◽  
pp. 649-652 ◽  
Author(s):  
Nathanael Enriquez ◽  
Christophe Sabot ◽  
Marc Yor
Keyword(s):  
Square Root ◽  
Bessel Processes

MODELING PRIVATE EQUITY FUNDS AND PRIVATE EQUITY COLLATERALISED FUND OBLIGATIONS

2004 ◽  
Vol 07 (03) ◽  
pp. 193-230 ◽  
Author(s):  
Etienne de Malherbe
Keyword(s):  
Private Equity ◽  
Investment Horizon ◽  
Bessel Processes ◽  
Fund Managers ◽  
Investment Value ◽  
Equity Funds ◽  
Private Equity Funds ◽  
Three Stages ◽  
The Individual ◽  
Managed Funds

The recent development of the securitisation of funds of private equity funds poses the question of the individual and joint modelling of the underlying funds. Private equity funds are different from other managed funds because of their particular bounded life cycle: when the fund starts, the investment partners make an initial capital commitment, the fund managers gradually draw down the committed capital into investments, returns and proceeds are distributed as the investments are realised and the fund is eventually liquidated as the final investment horizon is reached. Modelling private equity funds therefore requires three stages: the modelling of the commitment drawdowns, the modelling of the investment value and the modelling of the return repayments. A standard lognormal process is utilised for the dynamics of the investment value. Squared Bessel processes are utilised for the dynamics of the rates of drawdowns and repayments. Résumé: Le récent développement de la titrisation de fonds de fonds de placements privés pose la question de la modélisation individuelle et jointe des fonds sous-jacents. Les fonds de placements privés sont différents des autres sociétés d'investissement à cause de leur cycle de vie particulier et limité: au démarrage du fonds, les associés s'engagent sur un apport initial en capital; puis les gérants du fonds opèrent des tirages progressifs sur le capital apporté pour procéder à des investissements; les revenus et les profits sont distribués à mesure que les investissements sont réalisés; enfin, le fonds est liquidé lorsque l'horizon d'investissement est atteint. La modélisation d'un fonds doit donc se faire en trois étapes: la modélisation des tirages sur l'apport en capital, la modélisation de la valeur des investissements et enfin la modélisation des paiements et remboursements des dividendes et retours sur investissements. Un processus lognormal standard est utilisé pour la dynamique de la valeur des investissements. Des processus de Bessel carré sont utilisés pour la dynamique des taux de tirage et de remboursement.

On functionals of excursions for Bessel processes with negative index

2019 ◽  
Vol 246 (3) ◽  
pp. 217-231
Author(s):  
Tomasz Byczkowski ◽  
Jacek Jakubowski ◽  
Maciej Wiśniewolski
Keyword(s):  
Negative Index ◽  
Bessel Processes

scholarly journals FROM BOUNDARY CROSSING OF NON-RANDOM FUNCTIONS TO BOUNDARY CROSSING OF STOCHASTIC PROCESSES

2015 ◽  
Vol 29 (3) ◽  
pp. 345-359 ◽  
Author(s):  
Mark Brown ◽  
Victor de la Peña ◽  
Tony Sit
Keyword(s):  
Stochastic Processes ◽  
Sequential Analysis ◽  
Form Solution ◽  
Boundary Crossing ◽  
Renewal Processes ◽  
Bessel Processes ◽  
Average Behavior ◽  
Random Functions ◽  
Close Form ◽  
Sharp Lower Bound

One problem of wide interest involves estimating expected crossing-times. Several tools have been developed to solve this problem beginning with the works of Wald and the theory of sequential analysis. Deriving the explicit close form solution for the expected crossing times may be difficult. In this paper, we provide a framework that can be used to estimate expected crossing times of arbitrary stochastic processes. Our key assumption is the knowledge of the average behavior of the supremum of the process. Our results include a universal sharp lower bound on the expected crossing times. Furthermore, for a wide class of time-homogeneous, Markov processes, including Bessel processes, we are able to derive an upper bound E[a(Tr)]≤2r, which implies that sup r>0|((E[a(Tr)]−r)/r)|≤1, where a(t)=E[sup tXt] with {Xt}t≥0 be a non-negative, measurable process. This inequality motivates our claim that a(t) can be viewed as a natural clock for all such processes. The cases of multidimensional processes, non-symmetric and random boundaries are handled as well. We also present applications of these bounds on renewal processes in Example 10 and other stochastic processes.

scholarly journals Occupation Times, Drawdowns, and Drawups for One-Dimensional Regular Diffusions

2015 ◽  
Vol 47 (1) ◽  
pp. 210-230 ◽  
Author(s):  
Hongzhong Zhang
Keyword(s):  
Brownian Motion ◽  
Diffusion Process ◽  
Time Horizon ◽  
Three Dimensional ◽  
Bessel Processes ◽  
Occupation Times ◽  
Exponential Time ◽  
One Dimensional ◽  
Brownian Motion With Drift ◽  
Running Maximum

The drawdown process of a one-dimensional regular diffusion process X is given by X reflected at its running maximum. The drawup process is given by X reflected at its running minimum. We calculate the probability that a drawdown precedes a drawup in an exponential time-horizon. We then study the law of the occupation times of the drawdown process and the drawup process. These results are applied to address problems in risk analysis and for option pricing of the drawdown process. Finally, we present examples of Brownian motion with drift and three-dimensional Bessel processes, where we prove an identity in law.

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