STOCHASTIC PROGRAMMING METHOD FOR MULTIPERIOD CONSUMPTION AND INVESTMENT PROBLEMS WITH TRANSACTIONS COSTS

Using the GARCH model to describe the risky asset's return process so thatits time-varying moments and conditional heteroskedasticity can be properly reflected,general multiperiod optimal investment and consumption problems with both fixed andproportional transactions costs are investigated in this paper. We model this kind ofdifficult problems as a dynamic stochastic optimization problem, which can cope withdifferent utility functions and any number of time periods. The procedure to solve theresulting complex nonlinear stochastic optimization problem is discussed in detail and abranch-decomposition algorithm is devised.

The Convergence Rate from Discrete to Continuous Optimal Investment Stopping Problem

The author studies the optimal investment stopping problem in both continuous and discrete cases, where the investor needs to choose the optimal trading strategy and optimal stopping time concurrently to maximize the expected utility of terminal wealth.Based on the work of Hu et al.(2018) with an additional stochastic payoff function,the author characterizes the value function for the continuous problem via the theory of quadratic reflected backward stochastic differential equations(BSDEs for short) with unbounded terminal condition. In regard to the discrete problem, she gets the discretization form composed of piecewise quadratic BSDEs recursively under Markovian framework and the assumption of bounded obstacle, and provides some useful a priori estimates about the solutions with the help of an auxiliary forward-backward SDE system and Malliavin calculus. Finally, she obtains the uniform convergence and relevant rate from discretely to continuously quadratic reflected BSDE, which arise from corresponding optimal investment stopping problem through above characterization.

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.

Optimal consumption and portfolio selection with Epstein–Zin utility under general constraints

The paper investigates the consumption–investment problem for an investor with Epstein–Zin utility in an incomplete market.Closed but not necessarily convex constraints are imposed on strategies.The optimal consumption and investment strategies are characterized via a quadratic backward stochastic differential equation(BSDE).Due to the stochastic market environment,solutions to this BSDE are unbounded,so the BMO argument breaks down.After establishing the martingale optimality criterion and carefully selecting Lyapunov functions,the verification theorem is ultimately obtained.In addition,several examples and numerical simulations of optimal strategies are provided and illustrated.