In this paper we propose and analyze a class of stochastic $N$-player games that includes finite fuel stochastic games as a special case. We first derive sufficient conditions for the Nash equilibrium (NE) in the form of a verification theorem, which reveals an essential game component regarding the interaction among players. It is an analytical representation of the conditional optimality condition for NEs, largely missing in the existing literature on stochastic games. The derivation of NEs involves first solving a multi-dimensional free boundary problem and then a Skorokhod problem, where the boundary is "moving" in that it depends on both the changes of the system and the control strategies of other players. Finally, we reformulate NE strategies in the form of controlled rank-dependent stochastic differential equations.

Several systematic studies have suggested that a large fraction of published research is not reproducible. One probable reason for low reproducibility is insufficient sample size, resulting in low power and low positive predictive value. It has been suggested that insufficient sample-size choice is driven by a combination of scientific competition and 'positive publication bias'. Here we formalize this intuition in a simple model, in which scientists choose economically rational sample sizes, balancing the cost of experimentation with income from publication. Specifically, assuming that a scientist's income derives only from 'positive' findings (positive publication bias) and that individual samples cost a fixed amount, allows to leverage basic statistical formulas into an economic optimality prediction. We find that if effects have i) low base probability, ii) small effect size or iii) low grant income per publication, then the rational (economically optimal) sample size is small. Furthermore, for plausible distributions of these parameters we find a robust emergence of a bimodal distribution of obtained statistical power and low overall reproducibility rates, both matching empirical findings. Finally, we explore conditional equivalence testing as a means to align economic incentives with adequate sample sizes. Overall, the model describes a simple mechanism explaining both the prevalence and the persistence of small sample sizes, and is well suited for empirical validation. It proposes economic rationality, or economic pressures, as a principal driver of irreproducibility and suggests strategies to change this.

In this paper, a statistical analysis of high frequency fluctuations of the IPC, the Mexican Stock Market Index, is presented. A sample of tick-to-tick data covering the period from January 1999 to December 2002 was analyzed, as well as several other sets obtained using temporal aggregation. Our results indicates that the highest frequency is not useful to understand the Mexican market because almost two thirds of the information corresponds to inactivity. For the frequency where fluctuations start to be relevant, the IPC data does not follows any alpha-stable distribution, including the Gaussian, perhaps because of the presence of autocorrelations. For a long range of lower-frequencies, but still in the intra-day regime, fluctuations can be described as a truncated L\'evy flight, while for frequencies above two-days, a Gaussian distribution yields the best fit. Thought these results are consistent with other previously reported for several markets, there are significant differences in the details of the corresponding descriptions.

We extend the Deep Galerkin Method (DGM) introduced in Sirignano and Spiliopoulos (2018) to solve a number of partial differential equations (PDEs) that arise in the context of optimal stochastic control and mean field games. First, we consider PDEs where the function is constrained to be positive and integrate to unity, as is the case with Fokker-Planck equations. Our approach involves reparameterizing the solution as the exponential of a neural network appropriately normalized to ensure both requirements are satisfied. This then gives rise to a partial integro-differential equation (PIDE) where the integral appearing in the equation is handled using importance sampling. Secondly, we tackle a number of Hamilton-Jacobi-Bellman (HJB) equations that appear in stochastic optimal control problems. The key contribution is that these equations are approached in their unsimplified primal form which includes an optimization problem as part of the equation. We extend the DGM algorithm to solve for the value function and the optimal control simultaneously by characterizing both as deep neural networks. Training the networks is performed by taking alternating stochastic gradient descent steps for the two functions, a technique similar in spirit to policy improvement algorithms.

We call a given American option representable if there exists a European claim which dominates the American payoff at any time and such that the values of the two options coincide in the continuation region of the American option. This concept has interesting implications from a probabilistic, analytic, financial, and numeric point of view. Relying on methods from Jourdain and Martini (2001, 2002), Chrsitensen (2014) and convex duality, we make a first step towards verifying representability of American options.

Although recent studies have shown that electricity systems with shares of wind and solar above 80% can be affordable, economists have raised concerns about market integration. Correlated generation from variable renewable sources depresses market prices, which can cause wind and solar to cannibalize their own revenues and prevent them from covering their costs from the market. This cannibalization appears to set limits on the integration of wind and solar, and thus contradict studies that show that high shares are cost effective. Here we show from theory and with numerical examples how policies interact with prices, revenue and costs for renewable electricity systems. The decline in average revenue seen in some recent literature is due to an implicit policy assumption that technologies are forced into the system, whether it be with subsidies or quotas. If instead the driving policy is a carbon dioxide cap or tax, wind and solar shares can rise without cannibalising their own market revenue, even at penetrations of wind and solar above 80%. Policy is thus the primary factor driving lower market values; the variability of wind and solar is only a secondary factor that accelerates the decline if they are subsidised. The strong dependence of market value on the policy regime means that market value needs to be used with caution as a measure of market integration.

We consider an SPDE description of a large portfolio limit model where the underlying asset prices evolve according to certain stochastic volatility models with default upon hitting a lower barrier. The asset prices and their volatilities are correlated via systemic Brownian motions, and the resulting SPDE is defined on the positive half-space with Dirichlet boundary conditions. We study the convergence of the loss from the system, a function of the total mass of a solution to this stochastic initial-boundary value problem under fast mean reversion of the volatility. We consider two cases. In the first case the volatility converges to a limiting distribution and the convergence of the system is in the sense of weak convergence. On the other hand, when only the mean reversion of the volatility goes to infinity we see a stronger form of convergence of the system to its limit. Our results show that in a fast mean-reverting volatility environment we can accurately estimate the distribution of the loss from a large portfolio by using an approximate constant volatility model which is easier to handle.

Stochastic simulation has been widely used to analyze the performance of complex stochastic systems and facilitate decision making in those systems. Stochastic simulation is driven by the input model, which is a collection of probability distributions that model the stochasticity in the system. The input model is usually estimated using a finite amount of data, which introduces the so-called input model uncertainty to the simulation output. How to quantify input uncertainty has been studied extensively, and many methods have been proposed for the batch data setting, i.e., when all the data are available at once. However, methods for "streaming data" arriving sequentially in time are still in demand, despite that streaming data have become increasingly prevalent in modern applications. To fill this gap, we propose a two-layer importance sampling framework that incorporates streaming data for online input uncertainty quantification. Under this framework, we develop two algorithms that suit different application scenarios: the first scenario is when data come at a fast speed and there is no time for any new simulation in between updates; the second is when data come at a moderate speed and a few but limited simulations are allowed at each time stage. We prove the consistency and asymptotic convergence rate results, which theoretically show the efficiency of our proposed approach. We further demonstrate the proposed algorithms on a numerical example of the news vendor problem.

This work studies a stochastic optimal control problem for a pension scheme which provides an income-drawdown policy to its members after their retirement. To manage the scheme efficiently, the manager and members agree to share the investment risk based on a pre-decided risk-sharing rule. The objective is to maximise both sides' utilities by controlling the manager's investment in risky assets and members' benefit withdrawals. We use stochastic affine class models to describe the force of mortality of the members' population and consider a longevity bond whose coupon payment is linked to a survival index. In our framework, we also investigate the longevity basis risk, which arises when the members' and the longevity bond's reference populations show different mortality behaviours. By applying the dynamic programming principle to solve the corresponding HJB equations, we derive optimal solutions for the single- and sub-population cases. Our numerical results show that by sharing the risk, both manager and members increase their utility. Moreover, even in the presence of longevity basis risk, we demonstrate that the longevity bond acts as an effective hedging instrument.

Most doctors in the NRMP are matched to one of their most-preferred internship programs. Since various surveys indicate similarities across doctors' preferences, this suggests a puzzle. How can nearly everyone get a position in a highly-desirable program when positions in each program are scarce? We provide one possible explanation for this puzzle. We show that the patterns observed in the NRMP data may be an artifact of the interview process that precedes the match. Our analysis highlights the importance of interactions occurring outside of a matching clearinghouse for resulting outcomes, and casts doubts on analysis of clearinghouses that take reported preferences at face value.