IJPAM: Volume 55, No. 2 (2009)

RETRIEVAL OF THE LAW OF A RANDOM PAYMENT
STREAM BY THE JOINT LAW OF ITS FINAL VALUE AND
THE INTEREST RATE AT SOME FIXED TIME

Daniel NeuenschwanderDepartment of Mathematical Statistics and Actuarial Science, University of Bern, Sidlerstrasse 5, Bern, CH-3012, SWITZERLAND
Department of Mathematical Statistics and Actuarial Science
University of Bern
Sidlerstrasse 5, Bern, CH-3012, SWITZERLAND
École des Hautes Etudes Commerciales
Institut de Sciences Actuarielles
Université de Lausanne
Lausanne, CH-1015, SWITZERLAND
Institut de Science Financi ${\rm\grave{e}}$re et d'Assurances
Université de Lyon
Université Claude Bernard Lyon 1
50, Avenue Tony Garnier, Lyon, F-69007, FRANCE
e-mail: daniel.neuenschwander@bluewin.ch


Abstract.Consider a random payment stream (where retirements from the account and overdrawing are allowed) the balance of which obeys some Lévy process and a stochastic interest rate model (or numéraire) where the logarithm of the accumulation factor obeys some independent Lévy process whose non-Gaussian part is concentrated on the non-negative half-axis. In this setting we will show that if we can observe the joint distribution of the final value of the payment stream and the interest rate at some fixed time point $t=1$ without knowing anything about the history (during $0\leq t<1$) nor of the distribution of this history. Then the law of the balance of the payment stream is uniquely determined among all models with such Lévy processes (in the Brownian case) resp. among all compound Poisson processes (in certain compound Poisson cases). In statistical language, this means that the joint distribution of the final value and the accumulation factor (if it can be observed e.g. by sampling from several i.i.d. models) is a sufficient statistic for the accountholder's payment policy.

Received: July 6, 2009

AMS Subject Classification: 60B15, 60G51, 62B05, 91B28

Key Words and Phrases: random payment stream, sufficient statistics, stochastic interest, Lévy processes

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2009
Volume: 55
Issue: 2