On the foundations of Lévy finance. Equilibrium for a single-agent financial market with jumps

TLDR: In this article, for a continuous-time financial market with a single agent, the authors established equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process, and the resulting equilibrium prices depend on the agent's risk-aversion through the felicity functions.

Abstract: For a continuous-time financial market with a single agent, we establish equilibrium pricing formulae under the assumption that the dividends follow an exponential Levy process. The agent is allowed to consume a lump at the terminal date; before, only flow consumption is allowed. The agent's utility function is assumed to be additive, defined via strictly increasing, strictly concave smooth felicity functions which are bounded below (thus, many CRRA and CARA utility functions are included). For technical reasons we require that only pathwise continuous trading strategies are permitted in the demand set. The resulting equilibrium prices depend on the agent's risk-aversion through the felicity functions. It turns out that these prices will be the (stochastic) exponential of a Levy process essentially only if this process is geometric Brownian motion.