Appl Math Optim 37:1–41 (1998)
1998 Springer-Verlag New York Inc.
Some Results on Risk-Sensitive Control with Full Observation
and H. Nagai
CNES, 2 place Maurice Quentin,
75001 Paris, France
Institut f¨ur Angewandte Mathematik, Universit¨at Bonn,
Department of Mathematical Science,
Faculty of Engineering Science, Osaka University,
Toyonaka, Osaka, Japan
Abstract. The Bellman equation of the risk-sensitive control problem with full
observation is considered. It appears as an example of a quasi-linear parabolic
equation in the whole space, and fairly general growth assumptions with respect to
the space variable x are permitted. The stochastic control problem is then solved,
making use of the analytic results. The case of large deviation with small noises is
then treated, and the limit corresponds to a differential game.
Key Words. Risk-sensitive control, Small noise limit, Differential game, Bellman
AMS Classiﬁcation. 93E20, 49L20, 35K, 90D.
Since the early work of Jacobson , introducing exponential criteria in stochastic con-
trol, followed by Whittle  and many others, there have been many lines of research.
An important and natural one has been to derive the full theory for this class of stochastic
control problems. Indeed, these problems generalize classical stochastic control theory.
For instance, the LEQG (linear exponential quadratic gaussian) model is the analogue
of the LQG (linear quadratic gaussian) model. The discrete-time LEQG problem was
solved by Whittle  and the continuous-time one by Bensoussan and van Schuppen
. See  for a more complete treatment.