IJPAM: Volume 55, No. 2 (2009)
STREAM BY THE JOINT LAW OF ITS FINAL VALUE AND
THE INTEREST RATE AT SOME FIXED TIME
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
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 
without knowing anything about the history
(during 
) 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