Optimal solution of investment problems via linear parabolic equations generated by Kalman filter

Maximin setting for investment problems and fixed income management with observable but non-predictable parameters

We study optimal investment problem for a diffusion market consisting of a finite number of risky assets (for example, bonds, stocks and options). Risky assets evolution is described by Ito's equation, and the number of risky assets can be larger than the number of driving Brownian motions. We assume that the risk-free rate, the appreciation rates and the volatility of the stocks are all random; they are not necessary adapted to the driving Brownian motion, and their distributions are unknown, but they are supposed to be currently observable. Admissible strategies are based on current observat…

Optimal investment for all time horizons and Martin boundary of space-time diffusions

This paper is concerned with the axiomatic foundation and explicit construction of a general class of optimality criteria that can be used for investment problems with multiple time horizons, or when the time horizon is not known in advance. Both the investment criterion and the optimal strategy are characterized by the Hamilton-Jacobi-Bellman equation on a semi-infinite time interval. In the case when this equation can be linearized, the problem reduces to a time-reversed parabolic equation, which cannot be analyzed via the standard methods of partial differential equations. Under the additio…

An Explicit Example Of Optimal Portfolio-Consumption Choices With Habit Formation And Partial Observations

We consider a model of optimal investment and consumption with both habit formation and partial observations in incomplete Itô processes market. The investor chooses his consumption under the addictive habits constraint while only observing the market stock prices but not the instantaneous rate of return. Applying the Kalman-Bucy filtering theorem and the Dynamic Programming arguments, we solve the associated Hamilton-Jacobi-Bellman (HJB) equation explicitly for the path dependent stochastic control problem in the case of power utilities. We provide the optimal investment and consumption polic…

一个奇异控制问题的数值方案：在比例交易成本下的投资消费问题

A Numerical Scheme for A Singular control problem: Investment-Consumption Under Proportional Transaction Costs

本文提出一种数值方法，结合蒙特卡洛模拟和有限差分法，解决具有比例交易成本的有限投资组合选择问题，以最大化有限时间折扣消费效用。

On optimal arbitrage

In a Markovian model for a financial market, we characterize the best arbitrage with respect to the market portfolio that can be achieved using nonanticipative investment strategies, in terms of the smallest positive solution to a parabolic partial differential inequality; this is determined entirely on the basis of the covariance structure of the model. The solution is intimately related to properties of strict local martingales and is used to generate the investment strategy which realizes the best possible arbitrage. Some extensions to non-Markovian situations are also presented.

Optimal arbitrage under model uncertainty

In an equity market model with "Knightian" uncertainty regarding the relative risk and covariance structure of its assets, we characterize in several ways the highest return relative to the market that can be achieved using nonanticipative investment rules over a given time horizon, and under any admissible configuration of model parameters that might materialize. One characterization is in terms of the smallest positive supersolution to a fully nonlinear parabolic partial differential equation of the Hamilton--Jacobi--Bellman type. Under appropriate conditions, this smallest supersolution is …