American Institute of Mathematical Sciences

We introduce G-Lévy processes which develop the theory of processes with independent and stationary increments under the framework of sublinear expectations. We then obtain the Lévy–Khintchine formula and the existence for G-Lévy processes. We also introduce G-Poisson processes.

Motivated by financial and empirical arguments and in order to introduce a more flexible methodology of pricing, we provide a new approach to asset pricing based on Backward Volterra equations. The approach relies on an arbitrage-free and incomplete market setting in continuous time by choosing non-unique pricing measures depending either on the time of evaluation or on the maturity of payoffs. We show that in the latter case the dynamics can be captured by a time-delayed backward stochastic Volterra integral equation here introduced which, to the best of our knowledge, has not yet been studied. We then prove an existence and uniqueness result for time-delayed backward stochastic Volterra integral equations. Finally, we present a Lucas-type consumption-based asset pricing model that justifies the emergence of stochastic discount factors matching the term structure of Sharpe ratios.

When addressing various financial problems, such as estimating stock portfolio risk, it is necessary to derive the distribution of the sum of the dependent random variables. Although deriving this distribution requires identifying the joint distribution of these random variables, exact estimation of the joint distribution of dependent random variables is difficult. Therefore, in recent years, studies have been conducted on the bound of the sum of dependent random variables with dependence uncertainty. In this study, we obtain an improved Hoeffding inequality for dependent bounded variables. Further, we expand the above result to the case of sub-Gaussian random variables.

We derive sufficient conditions for the convex and monotonic g-stochastic ordering of diffusion processes under nonlinear g-expectations and g-evaluations. Our approach relies on comparison results for forward-backward stochastic differential equations and on several extensions of convexity, monotonicity, and continuous dependence properties for the solutions of associated semilinear parabolic partial differential equations. Applications to contingent claim price comparison under different hedging portfolio constraints are provided.