We study the evolution of the Arrow–Pratt measure of risk-tolerance in the framework of discrete-time predictable forward utility processes in a complete semimartingale financial market. An agent starts with an initial utility function, which is then sequentially updated forward at discrete times under the guidance of a martingale optimality principle. We mostly consider a one-period framework and first show that solving the associated inverse investment problem is equivalent to solving some generalised integral equations for the inverse marginal function or for the conjugate function, both associated with the forward utility. We then completely characterise the class of forward utility pairs that can have a time-invariant measure of risk-tolerance and thus a preservation of preferences in time. Next, we show that in general, preferences vary over time and that whether the agent becomes more or less tolerant to risk is related to the curvature of the measure of risk-tolerance of the forward utility pair. Finally, to illustrate the obtained general results, we present an example in a binomial market model where the initial utility function belongs to the SAHARA class, and we find that this class is analytically tractable and stable in the sense that all the subsequent utility functions belong to the same class as the initial one.

我们研究了在完整的半市场金融市场中离散时间可预测的远期效用过程的框架中，Arrow–Pratt风险容忍度测度的演变。代理从初始效用函数开始，然后在a优化原则的指导下在不连续的时间顺序更新。我们主要考虑一个周期的框架，首先表明解决相关的逆向投资问题等同于求解一些与逆效用的边际函数或共轭函数的广义积分方程。然后，我们完全表征了可以对风险承受能力进行时不变度量的前向效用对的类别，从而可以及时保留偏好。接下来，我们证明 偏好会随时间而变化，并且代理对风险的耐受性或高或低与正向效用对的风险容忍度的度量标准有关。最后，为了说明获得的一般结果，我们在一个二项式市场模型中给出了一个示例，其中初始效用函数属于SAHARA类，并且从所有后续效用函数都属于该意义上说，我们发现该类在分析上易于处理且稳定与最初的班级相同。