Quasi-Maximum Likelihood (QML) procedures are theoretically appealing and widely used for statistical inference. While there are extensive references on QML estimation in batch settings, the QML estimation in streaming settings has attracted little attention until recently. An investigation of the convergence properties of the QML procedure in a general conditionally heteroscedastic time series model is conducted, and the classical batch optimization routines extended to the framework of streaming and large-scale problems. An adaptive recursive estimation routine for GARCH models named AdaVol is presented. The AdaVol procedure relies on stochastic approximations combined with the technique of Variance Targeting Estimation (VTE). This recursive method has computationally efficient properties, while VTE alleviates some convergence difficulties encountered by the usual QML estimation due to a lack of convexity. Empirical results demonstrate a favorable trade-off between AdaVol's stability and the ability to adapt to time-varying estimates for real-life data.

This scientific research paper presents an innovative approach based on deep reinforcement learning (DRL) to solve the algorithmic trading problem of determining the optimal trading position at any point in time during a trading activity in stock markets. It proposes a novel DRL trading strategy so as to maximise the resulting Sharpe ratio performance indicator on a broad range of stock markets. Denominated the Trading Deep Q-Network algorithm (TDQN), this new trading strategy is inspired from the popular DQN algorithm and significantly adapted to the specific algorithmic trading problem at hand. The training of the resulting reinforcement learning (RL) agent is entirely based on the generation of artificial trajectories from a limited set of stock market historical data. In order to objectively assess the performance of trading strategies, the research paper also proposes a novel, more rigorous performance assessment methodology. Following this new performance assessment approach, promising results are reported for the TDQN strategy.

Reinforcement learning is a machine learning approach concerned with solving dynamic optimization problems in an almost model-free way by maximizing a reward function in state and action spaces. This property makes it an exciting area of research for financial problems. Asset allocation, where the goal is to obtain the weights of the assets that maximize the rewards in a given state of the market considering risk and transaction costs, is a problem easily framed using a reinforcement learning framework. It is first a prediction problem for expected returns and covariance matrix and then an optimization problem for returns, risk, and market impact. Investors and financial researchers have been working with approaches like mean-variance optimization, minimum variance, risk parity, and equally weighted and several methods to make expected returns and covariance matrices' predictions more robust. This paper demonstrates the application of reinforcement learning to create a financial model-free solution to the asset allocation problem, learning to solve the problem using time series and deep neural networks. We demonstrate this on daily data for the top 24 stocks in the US equities universe with daily rebalancing. We use a deep reinforcement model on US stocks using different architectures. We use Long Short Term Memory networks, Convolutional Neural Networks, and Recurrent Neural Networks and compare them with more traditional portfolio management. The Deep Reinforcement Learning approach shows better results than traditional approaches using a simple reward function and only being given the time series of stocks. In Finance, no training to test error generalization results come guaranteed. We can say that the modeling framework can deal with time series prediction and asset allocation, including transaction costs.

In this paper, we In this paper, we obtain the existence, uniqueness and positivity of the solution to delayed stochastic differential equations with jumps. This equation is then applied to model the price movement of the risky asset in a financial market and the Black-Scholes formula for the price of European option is obtained together with the hedging portfolios. The option price is evaluated analytically at the last delayed period by using the Fourier transformation technique. But in general there is no analytical expression for the option price. To evaluate the price numerically we then use the Monte-Carlo method. To this end we need to simulate the delayed stochastic differential equations with jumps. We propose a logarithmic numerical scheme to approximate the equation and prove that all the approximations remain positive and the rate of convergence of the scheme is proved to be 0.5.

Savage (1972) lays down the foundation of Bayesian decision theory, but asserts that it is not applicable in big worlds where the environment is complex. Using the theory of finite automaton to model belief formation, this paper studies the characteristics of optimal learning behavior in small and big worlds, where the complexity of the environment is low and high, respectively, relative to the cognitive ability of the decision maker. Confirming Savage's claim, optimal learning behavior is closed to Bayesian in small worlds but significantly different in big worlds. In addition, I show that in big worlds, the optimal learning behavior could exhibit a wide range of well-documented non-Bayesian learning behavior, including the use of heuristic, correlation neglect, persistent over-confidence, inattentive learning, and other behaviors of model simplification or misspecification. These results establish a clear and testable relationship between the prominence of non-Bayesian learning behavior, complexity and cognitive ability.

In the face of a pandemic, urban protests, and an affordability crisis, is the desirability of dense urban settings at a turning point? Assessing cities' long term trends remains challenging. The first part of this chapter describes the short-run dynamics of the housing market in 2020. Evidence from prices and price-to-rent ratios suggests expectations of resilience. Zip-level evidence suggests a short-run trend towards suburbanization, and some impacts of urban protests on house prices. The second part of the chapter analyzes the long-run dynamics of urban growth between 1970 and 2010. It analyzes what, in such urban growth, is explained by short-run shocks as opposed to fundamentals such as education, industrial specialization, industrial diversification, urban segregation, and housing supply elasticity. This chapter's original results as well as a large established body of literature suggest that fundamentals are the key drivers of growth. The chapter illustrates this finding with two case studies: the New York City housing market after September 11, 2001; and the San Francisco Bay Area in the aftermath of the 1989 Loma Prieta earthquake. Both areas rebounded strongly after these shocks, suggesting the resilience of the urban metropolis.

In this paper, we develop a generalized method of moments approach for joint estimation of the parameters of a fractional log-normal stochastic volatility model. We show that with an arbitrary Hurst exponent an estimator based on integrated variance is consistent. Moreover, under stronger conditions we also derive a central limit theorem. These results stand even when integrated variance is replaced with a realized measure of volatility calculated from discrete high-frequency data. However, in practice a realized estimator contains sampling error, the effect of which is to skew the fractal coefficient toward "roughness". We construct an analytical approach to control this error. In a simulation study, we demonstrate convincing small sample properties of our approach based both on integrated and realized variance over the entire memory spectrum. We show that the bias correction attenuates any systematic deviance in the estimated parameters. Our procedure is applied to empirical high-frequency data from numerous leading equity indexes. With our robust approach the Hurst index is estimated around 0.05, confirming roughness in integrated variance.

Using the most comprehensive, commercially-available dataset of trading activity in U.S. equity markets, we catalog and analyze quote dislocations between the SIP National Best Bid and Offer (NBBO) and a synthetic BBO constructed from direct feeds. We observe a total of over 3.1 billion dislocation segments in the Russell 3000 during trading in 2016, roughly 525 per second of trading. However, these dislocations do not occur uniformly throughout the trading day. We identify a characteristic structure that features more dislocations near the open and close. Additionally, around 23% of observed trades executed during dislocations. These trades may have been impacted by stale information, leading to estimated opportunity costs on the order of $ 2 billion USD. A subset of the constituents of the S&P 500 index experience the greatest amount of opportunity cost and appear to drive inefficiencies in other stocks. These results quantify impacts of the physical structure of the U.S. National Market System.