标题： 基于遍历倒向随机微分方程的前向效用深度学习方案 显示英文标题

标题： Deep learning scheme for forward utilities using ergodic BSDEs

摘要： 在本文中，我们提出了一种概率数值方法，用于随机因子模型中的一类前向效用函数。 为此，我们使用了由Liang和Zariphopoulou在[27]中引入的与遍历倒向随机微分方程（eBSDEs）的均值相关的动态一致效用表示。 我们建立了遍历BSDE的解与一个关联的具有随机终端时间$\tau$的BSDE之间的联系，该时间定义为正遍历随机因子V的到达时间。 基于具有随机终止时间的BSDE的观点，给出了遍历成本$\lambda$的新表征，这是eBSDEs解的一部分。 特别是，对于某种具有二次生成器的eBSDEs，Cole-Hopf变换导致了解的半显式表示以及遍历成本$\lambda$的新表达式。 后者可以利用蒙特卡洛方法进行估计。 我们还提出了两种新的深度学习数值方案用于eBSDEs，其中遍历成本$\lambda$根据在随机终止时间$\tau$处的损失函数进行优化，或考虑整个轨迹。 最后，我们展示了不同eBSDEs和前向效用函数的数值结果以及相关的投资策略。 显示英文摘要

摘要： In this paper, we present a probabilistic numerical method for a class of forward utilities in a stochastic factor model. For this purpose, we use the representation of dynamic consistent utilities with mean of ergodic Backward Stochastic Differential Equations (eBSDEs) introduced by Liang and Zariphopoulou in [27]. We establish a connection between the solution of the ergodic BSDE and the solution of an associated BSDE with random terminal time $\tau$ , defined as the hitting time of the positive recurrent stochastic factor V . The viewpoint based on BSDEs with random horizon yields a new characterization of the ergodic cost $\lambda$ which is a part of the solution of the eBSDEs. In particular, for a certain class of eBSDEs with quadratic generator, the Cole-Hopf transform leads to a semi-explicit representation of the solution as well as a new expression of the ergodic cost $\lambda$. The latter can be estimated with Monte Carlo methods. We also propose two new deep learning numerical schemes for eBSDEs, where the ergodic cost $\lambda$ is optimized according to a loss function at the random horizon $\tau$ or taking into account the whole trajectory. Finally, we present numerical results for different examples of eBSDEs and forward utilities along with the associated investment strategies.