投资消费模型是研究投资者如何利用初始财富在投资和消费过程中，最大化贴现消费效用的期望值。我们主要用偏微分方程的工具，辅助随机分析的技术来研究一类有限时间、带有某些投资限制和消费限制的连续型投资消费模型。目的是通过研究方程的解的性质和自由边界的性质，来定性定量分析最优的投资策略，为投资消费决策提供依据。

The investment-consumption problems are concerning how to exploit initial wealth to maximize the expected discounted utility of consumption over time. We mainly use the techniques of partial differential equations, supplemented by stochastic analysis, to study a class of continuous investment-consumption models with some constraints on investment and consumption. The aim is by studying the solutions of equations and the free boundaries to quantitative and qualitative research the optimal investment strategy, to provide reference for investment and consumption decisions.