We introduce the entropic measure transform (EMT) problem for a general
process and prove the existence of a unique optimal measure characterizing the
solution. The density process of the optimal measure is characterized using a
semimartingale BSDE under general conditions. The EMT is used to reinterpret
the conditional entropic risk-measure and to obtain a convenient formula for
the conditional expectation of a process which admits an affine representation
under a related measure. The entropic measure transform is then used provide a
new characterization of defaultable bond prices, forward prices, and futures
prices when the asset is driven by a jump diffusion. The characterization of
these pricing problems in terms of the EMT provides economic interpretations as
a maximization of returns subject to a penalty for removing financial risk as
expressed through the aggregate relative entropy. The EMT is shown to extend
the optimal stochastic control characterization of default-free bond prices of
Gombani and Runggaldier (Math. Financ. 23(4):659-686, 2013). These methods are
illustrated numerically with an example in the defaultable bond setting.