# 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

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: [email protected]

**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