We examine the household-specific effects of the introduction of Time-of-Use (TOU) electricity pricing schemes. Using a causal forest (Athey and Imbens, 2016; Wager and Athey, 2018; Athey et al., 2019), we consider the association between past consumption and survey variables, and the effect of TOU pricing on household electricity demand. We describe the heterogeneity in household variables across quartiles of estimated demand response and utilise variable importance measures.

Household-specific estimates produced by a causal forest exhibit reasonable associations with covariates. For example, households that are younger, more educated, and that consume more electricity, are predicted to respond more to a new pricing scheme. In addition, variable importance measures suggest that some aspects of past consumption information may be more useful than survey information in producing these estimates.

We introduce a new meta-learning procedure, called non-Euclidean upgrading (NEU), which learns algorithm-specific geometries by deforming the ambient space until the algorithm can achieve optimal performance. We prove that these deformations have several novel and semi-classical universal approximation properties. These deformations can be used to approximate any continuous, Borel, or modular-Lebesgue integrable functions to arbitrary precision. Further, these deformations can transport any data-set into any other data-set in a finite number of iterations while leaving most of the space fixed. The NEU meta-algorithm embeds these deformations into a wide range of learning algorithms. We prove that the NEU version of the original algorithm must perform better than the original learning algorithm. Moreover, by quantifying model-free learning algorithms as specific unconstrained optimization problems, we find that the NEU version of a learning algorithm must perform better than the model-free extension of the original algorithm. The properties and performance of the NEU meta-algorithm are examined in various simulation studies and applications to financial data.

We consider a discrete-time, linear state equation with delay which arises as a model for a trader's account value when buying and selling a risky asset in a financial market. The state equation includes a nonnegative feedback gain $\alpha$ and a sequence $v(k)$ which models asset returns which are within known bounds but otherwise arbitrary. We introduce two thresholds, $\alpha_-$ and $\alpha_+$, depending on these bounds, and prove that for $\alpha < \alpha_-$, state positivity is guaranteed for all time and all asset-return sequences; i.e., bankruptcy is ruled out and positive solutions of the state equation are continuable indefinitely. On the other hand, for $\alpha > \alpha_+$, we show that there is always a sequence of asset returns for which the state fails to be positive for all time; i.e., along this sequence, bankruptcy is certain and the solution of the state equation ceases to be meaningful after some finite time. Finally, this paper also includes a conjecture which says that for the "gap" interval $\alpha_- \leq \alpha \leq \alpha_+,$ state positivity is also guaranteed for all time. Support for the conjecture, both theoretical and computational, is provided.

An unconventional approach for optimal stopping under model ambiguity is introduced. Besides ambiguity itself, we take into account how ambiguity-averse an agent is. This inclusion of ambiguity attitude, via an $\alpha$-maxmin nonlinear expectation, renders the stopping problem time-inconsistent. We look for subgame perfect equilibrium stopping policies, formulated as fixed points of an operator. For a one-dimensional diffusion with drift and volatility uncertainty, we show that every equilibrium can be obtained through a fixed-point iteration. This allows us to capture much more diverse behavior, depending on an agent's ambiguity attitude, beyond the standard worst-case (or best-case) analysis. In a concrete example of real options valuation under volatility uncertainty, all equilibrium stopping policies, as well as the best one among them, are fully characterized. It demonstrates explicitly the effect of ambiguity attitude on decision making: the more ambiguity-averse, the more eager to stop---so as to withdraw from the uncertain environment. The main result hinges on a delicate analysis of continuous sample paths in the canonical space and the capacity theory. To resolve measurability issues, a generalized measurable projection theorem, new to the literature, is also established.

Investment returns naturally reside on irregular domains, however, standard multivariate portfolio optimization methods are agnostic to data structure. To this end, we investigate ways for domain knowledge to be conveniently incorporated into the analysis, by means of graphs. Next, to relax the assumption of the completeness of graph topology and to equip the graph model with practically relevant physical intuition, we introduce the portfolio cut paradigm. Such a graph-theoretic portfolio partitioning technique is shown to allow the investor to devise robust and tractable asset allocation schemes, by virtue of a rigorous graph framework for considering smaller, computationally feasible, and economically meaningful clusters of assets, based on graph cuts. In turn, this makes it possible to fully utilize the asset returns covariance matrix for constructing the portfolio, even without the requirement for its inversion. The advantages of the proposed framework over traditional methods are demonstrated through numerical simulations based on real-world price data.

We study an optimization problem for a portfolio with a risk-free, a liquid risky, and an illiquid asset which is sold in an exogenous random moment of time with a prescribed liquidation time distribution. Problems of such type lead to three dimensional nonlinear partial differential equations (PDEs) on the value function. We study the optimization problem with a utility function of a CARA type, i.e. with negative and positive exponential utility functions (EXPn and EXPp). It is well known that both the LOG and the EXPn utility functions are connected with the HARA utility function by means of a limiting procedure: in the first case the parameter of a HARA utility function is going to zero and in the second case to infinity. In our previous papers devoted to the optimization problem with a HARA and LOG utility functions we proved that also the corresponding analytical and Lie algebraic structures are connected with the same limiting procedure. In this paper we show that the case of EXPn utility function differs from the case of the HARA utility and is not connected to the HARA case by the limiting procedure. We carry out the Lie group analysis of the PDEs for the cases EXPn and EXPp utility functions and proved that they are connected by a one-to-one analytical substitution and are identical from the economical, analytical or Lie algebraic point of view. The complete set of nonequivalent group invariant reductions to two dimensional PDEs is provided for the three dimensional PDE with the EXPn utility function in accordance with an optimal system of sub algebras of the admitted Lie algebra. We prove that in one case the invariant reduction is consistent with the boundary condition. We can use the reduced two dimensional PDE to study the properties of the optimal solution and the investment - consumption strategies.

Let ${\bf X}$ and ${\bf X}$ be two $n$-dimensional elliptical random vectors, we establish an identity for $E[f({\bf Y})]-E[f({\bf X})]$, where $f: \Bbb{R}^n \rightarrow \Bbb{R}$ fulfilling some regularity conditions. Using this identity we provide a unified derivation of sufficient and necessary conditions for classifying multivariate elliptical random vectors according to several main integral stochastic orders. As a consequence we obtain new inequalities by applying it to multivariate elliptical distributions. The results generalize the corresponding ones for multivariate normal random vectors in the literature.

We propose a methodology for computing single and multi-asset European option prices, and more generally expectations of scalar functions of (multivariate) random variables. This new approach combines the ability of Monte Carlo simulation to handle high-dimensional problems with the efficiency of function approximation. Specifically, we first generalize the recently developed method for multivariate integration in [arXiv:1806.05492] to integration with respect to probability measures. The method is based on the principle "approximate and integrate" in three steps i) sample the integrand at points in the integration domain, ii) approximate the integrand by solving a least-squares problem, iii) integrate the approximate function. In high-dimensional applications we face memory limitations due to large storage requirements in step ii). Combining weighted sampling and the randomized extended Kaczmarz algorithm we obtain a new efficient approach to solve large-scale least-squares problems. Our convergence and cost analysis along with numerical experiments show the effectiveness of the method in both low and high dimensions, and under the assumption of a limited number of available simulations.