Existence and uniqueness of risk-sensitive estimates

Risk-sensitive criteria have been used to derive robust filters, identifiers, and controllers. The fundamental issues of existence and uniqueness of an estimate of a random variable given a random vector with respect to an order-(λ, p) risk-sensitive criterion are studied in this note. More precisely, we prove the existence of a unique risk-sensitive estimate provided λ>0 and p>1. For the remaining cases, a general existence result is not available at this time. We do, however, prove the existence in certain special cases. Moreover, we present examples with uncountably many optimal risk-sensitive estimates, i.e., exhibiting an extremely high level of nonuniqueness.