The second-order Esscher martingale densities for continuous-time market models

In this paper, we introduce the second-order Esscher pricing notion for continuous-time models. Depending whether the stock price process $ S $ or its logarithm is the main driving noise/shock in the Esscher definition, we obtained two classes of second-order Esscher densities called linear class and exponential class. Using the semimartingale characteristics to parametrize $ S $, we characterize the second-order Esscher densities (exponential and linear) using pointwise equations. The role of the second-order concept is highlighted in many manners and the relationship between the two classes is singled out for the one-dimensional case. Furthermore, when $ S $ is a compound Poisson model, we show how both classes are related to the Delbaen-Haenzendonck's risk-neutral measure. Afterwards, we restrict our model $ S $ to follow the jump-diffusion model, for simplicity only, and address the bounds of the stochastic Esscher pricing intervals. In particular, no matter what is the Esscher class, we prove that both bounds (upper and lower) are solutions to the same linear backward stochastic differential equation (BSDE hereafter for short) but with two different constraints. This shows that BSDEs with constraints appear also in a setting beyond the classical cases of constraints on gain-processes or constraints on portfolios. We prove that our resulting constrained BSDEs have solutions in our framework for a large class of claims' payoffs including bounded claims, in contrast to the literature, and we single out the monotonic sequence of BSDEs that 'naturally' approximate them as well.