We introduce Bellman Conformal Inference (BCI), a framework that wraps around
any time series forecasting models and provides calibrated prediction
intervals. Unlike the existing methods, BCI is able to leverage multi-step
ahead forecasts and explicitly optimize the average interval lengths by solving
a one-dimensional stochastic control problem (SCP) at each time step. In
particular, we use the dynamic programming algorithm to find the optimal policy
for the SCP. We prove that BCI achieves long-term coverage under arbitrary
distribution shifts and temporal dependence, even with poor multi-step ahead
forecasts. We find empirically that BCI avoids uninformative intervals that
have infinite lengths and generates substantially shorter prediction intervals
on volatility forecasting problems when compared with existing methods.

介绍 Bellman Conformal Inference（BCI）框架，它能够包装任何时间序列预测模型并提供校准的预测区间，通过解决一维随机控制问题（SCP）从而利用多步预测并显式地优化平均区间长度。我们证明了 BCI 在任意分布转变和时间依赖的情况下能够实现长期覆盖，即使多步预测不佳。实证结果表明，与现有方法相比，BCI 避免了无限长度的无信息区间，并在波动率预测问题中生成较短的预测区间。

贝尔曼协同推理：时间序列预测区间校准

Bellman Conformal Inference: Calibrating Prediction Intervals For Time  Series

Motivated by applications in queueing theory, we consider a stochastic
control problem whose state space is the $d$-dimensional positive orthant. The
controlled process $Z$ evolves as a reflected Brownian motion whose covariance
matrix is exogenously specified, as are its directions of reflection from the
orthant's boundary surfaces. A system manager chooses a drift vector
$\theta(t)$ at each time $t$ based on the history of $Z$, and the cost rate at
time $t$ depends on both $Z(t)$ and $\theta(t)$. In our initial problem
formulation, the objective is to minimize expected discounted cost over an
infinite planning horizon, after which we treat the corresponding ergodic
control problem. Extending earlier work by Han et al. (Proceedings of the
National Academy of Sciences, 2018, 8505-8510), we develop and illustrate a
simulation-based computational method that relies heavily on deep neural
network technology. For test problems studied thus far, our method is accurate
to within a fraction of one percent, and is computationally feasible in
dimensions up to at least $d=30$.

通过深度神经网络技术开发和展示了一种基于模拟的计算方法来解决随机控制问题，将其准确性扩展到 $d=30$ 的维度。

高维 RBM 的漂移控制：基于神经网络的计算方法

Drift Control of High-Dimensional RBM: A Computational Method Based on  Neural Networks

We consider a framework for solving optimal liquidation problems in limit
order books. In particular, order arrivals are modeled as a point process whose
intensity depends on the liquidation price. We set up a stochastic control
problem in which the goal is to maximize the expected revenue from liquidating
the entire position held. We solve this optimal liquidation problem for
power-law and exponential-decay order book models and discuss several
extensions. We also consider the continuous selling (or fluid) limit when the
trading units are ever smaller and the intensity is ever larger. This limit
provides an analytical approximation to the value function and the optimal
solution. Using techniques from viscosity solutions we show that the discrete
state problem and its optimal solution converge to the corresponding quantities
in the continuous selling limit uniformly on compacts.

该论文考虑了极限订单簿中解决最优清算问题的框架，其中订单到达被建模为点过程，其强度取决于清算价格。通过构建随机控制问题来选择策略，目标是最大化清算所持有的全部头寸的预期收益，通过粘性解决方案的技术展示了异质性和连续出售极限的优化方案使离散状态问题及其最优解一致收敛于分段均匀的连续状态。

限价单簿中带有控制密度的清算

Liquidation in Limit Order Books with Controlled Intensity

In light of the growing market of Ad Exchanges for the real-time sale of
advertising slots, publishers face new challenges in choosing between the
allocation of contract-based reservation ads and spot market ads. In this
setting, the publisher should take into account the tradeoff between short-term
revenue from an Ad Exchange and quality of allocating reservation ads. In this
paper, we formalize this combined optimization problem as a stochastic control
problem and derive an efficient policy for online ad allocation in settings
with general joint distribution over placement quality and exchange bids. We
prove asymptotic optimality of this policy in terms of any trade-off between
quality of delivered reservation ads and revenue from the exchange, and provide
a rigorous bound for its convergence rate to the optimal policy. We also give
experimental results on data derived from real publisher inventory, showing
that our policy can achieve any pareto-optimal point on the quality vs. revenue
curve. Finally, we study a parametric training-based algorithm in which instead
of learning the dual variables from a sample data (as is done in non-parametric
training-based algorithms), we learn the parameters of the distribution and
construct those dual variables from the learned parameter values. We compare
parametric and non-parametric ways to estimate from data both analytically and
experimentally in the special case without the ad exchange, and show that
though both methods converge to the optimal policy as the sample size grows,
our parametric method converges faster, and thus performs better on smaller
samples.

本研究提出并解决了在线广告分配的优化问题，探讨了在 Ad Exchanges 中确定合同预定广告和现场市场广告的分配方式，通过参数训练算法进行实验，获得了广告质量与收入的 Pareto 最优解。