Continuous-time perpetuities and time reversal of diffusions

Abstract

We consider the problem of estimating the joint distribution of a continuous-time perpetuity and the underlying factors which govern the cash flow rate, in an ergodic Markovian model. Two approaches are used to obtain the distribution. The first identifies a partial differential equation for the conditional cumulative distribution function of the perpetuity given the initial factor value, which under certain conditions ensures the existence of a density for the perpetuity. The second (and more general) approach, using techniques of time reversal, identifies the joint law as the stationary distribution of an ergodic multidimensional diffusion. This latter approach allows efficient use of Monte Carlo simulation, as the distribution is obtained by sampling a single path of the reversed process.

On Exponential Functionals of Processes with Independent Increments

“…Our aim now is to obtain a recurrent integral equation for the Mellin transform of I t . For condition (18) below see Remarks 1 and 2.…”

“…In the mathematical statistics the exponential functionals appear, for exemple, in the study of Pitman estimators (see [23]). In the mathematical finance the question is related to the perpetuities containing the liabilities, the perpetuities subjected to the influence of economical factors (see, for example, [18]), and also with the prices of Asian options and related questions (see, for instance, [16] and references therein). In the insurance, this connection is made via the ruin problem, the problem in which the exponential functionals appear very naturally (see, for exemple [25], [1], [17] and references therein).…”

On distributions of exponential functionals of the processes with independent increments

“…In the relation (19), the process M is the martingale component of the semi-martingale decomposition of Y : Y s = Bs + Ms , and µ Y is the measure of jumps of the process Y . It should be noticed that since Y is a process with independent increments and B is deterministic, M is a martingale (see [30], Th.…”

“…This study was inspired by the questions arising in mathematical finance, namely by the questions related with the perpetuities containing the liabilities, perpetuities subjected to the influence of the economical factors (see, for example, [19]), and also with the price of Asian options and similar questions (see, for instance, [17], [31], and references there). The study of exponential functionals are also important in insurance, since the insurace companies invest the money on risky assets.…”

On distributions of exponential functionals of the processes with independent increments

“…This study was inspired by the questions arising in mathematical finance, namely by the questions related to perpetuities containing the liabilities, perpetuities subjected to the influence of economical factors (see, for example, Kardaras, Robertson [23]), and also with the price of Asian options and similar questions (see, for instance, Jeanblanc, Yor, Chesnay [21], Vecer [38] and references there). The study of exponential functionals is also important in the insurance, since the distributions of these functionals appear very naturally in the ruin problem (see, for example, Asmussen [2], Paulsen [29], Kabanov, Pergamentshchikov [22], Spielmann, Vostrikova [37] and the references there).…”