Rank-dependent ppredictableforward performance processes

Predictable forward performance processes(PFPPs)are stochastic optimal control frameworks for an agent who controls a randomly evolving system but can only prescribe the system dynamics for a short period ahead.This is a common scenario in which a controlling agent frequently re-calibrates her model.We introduce a new class of PFPPs based on rank-dependent utility,generalizing existing models that are based on expected utility theory(EUT).We establish existence of rank-dependent PFPPs under a conditionally complete market and exogenous probability distortion functions which are updated periodically.We show that their construction reduces to solving an integral equation that generalizes the integral equation obtained under EUT in previous studies.We then propose a new approach for solving the integral equation via theory of Volterra equations.We illustrate our result in the special case of conditionally complete Black-Scholes model.

Now decision theory

The spirit of now in nowcasting suggests expanding the current to include the near future.Decision theory is then developed by incorporating the consequences of actions into the present.With the future falling into the present discounting it is no longer permitted.Value functions are then observed to be determinate only up to scale and shift that are then locked down by fixing values arbitrarily in two selected states,much like declaring water to freeze and boil at zero and a hundred degrees celsius.The locked down value functions associated policy functions are seen to exist in decision contexts in where the only time is now.Examples are studied in univariate and multivariate dimensions for the decision state space and the dimension of shocks delivering state transitions.The policy functions are expanded from realisitic training sets to the full state space using Gaussian Process Regression.They are implemented on real data with reported performances.

Lower and upper pricing of financial assets

Modeling of uncertainty by probability errs by ignoring the uncertainty in probability.When financial valuation recognizes the uncertainty of probability,the best the market may offer is a two price framework of a lower and upper valuation.The martingale theory of asset prices is then replaced by the theory of nonlinear martingales.When dealing with pure jump compensators describing probability,the uncertainty in probability is captured by introducing parametric measure distortions.The two price framework then alters asset pricing theory by requiring two required return equations,one each for the lower upper valuation.Proxying lower and upper valuations by daily lows and highs,the paper delivers the first empirical study of nonlinear martingales via the modeling and simultaneous estimation of the two required return equations.