  This paper builds a multivariate L\'evy-driven Ornstein-Uhlenbeck process for
the management of non-maturing deposits, that are a major source of funding for
banks. The contribution of the paper is both theoretical and operational. On
the theoretical side, the novelty of this model is to include three independent
sources of randomness in a L\'evy framework: market interest rates, deposit
rates and deposit volumes. The choice of a L\'evy background driving process
allows us to model rare but severe events. On the operational side, we propose
a procedure to include severe volume outflows with positive probability in
future scenarios simulation, explaining its implementation with an illustrative
example using Italian banking sector data.
