New results and improvements in the study of nonparametric exponential and mixture models are proposed. In particular, different equivalent characterizations of maximal exponential models, in terms of open exponential arcs and Orlicz spaces, are given. Our theoretical results are supported by several examples and counterexamples and provide an answer to some open questions in the literature.

We illustrate some financial applications of the Tsallis and Kaniadakis deformed exponential. The minimization of the corresponding deformed divergence is discussed as a criterion to select a pricing measure in the valuation problems of incomplete markets. Moreover, heavy-tailed models for price processes are proposed, which generalized the well-known Black and Scholes model.

We consider the problem of maximizing the expected utility of terminal wealth with a terminal random liability when the underlying asset price process is a continuous semimartingale. The optimal strategy is characterized in terms of a forward backward semimartingale system of equations. The results cover the cases of exponential, logarithmic, and power utilities, which we analyze as illustrative examples.

We introduce the notion of κ-entropy (κ ∈ ℝ, |κ| ≤ 1), starting from Kaniadakis' (2001, 2002, 2005) one-parameter deformation of the ordinary exponential function. The κ-entropy is in duality with a new class of utility functions which are close to the exponential utility functions, for small values of wealth, and to the power law utility functions, for large values of wealth. We give conditions on the existence and on the equivalence to the basic measure of the minimal κ-entropy martingale measure. Moreover, we provide characterizations of its density as a κ-exponential function. We show that the minimal κ-entropy martingale measure is closely related to both the standard entropy martingale measure and the well known q-optimal martingale measures. We finally establish the convergence of the minimal κ-entropy martingale measure to the minimal entropy martingale measure as κ tends to 0.