We propose a heterogeneous simultaneous graphical dynamic linear model (H-SGDLM), which extends the standard SGDLM framework to incorporate a heterogeneous autoregressive realised volatility (HAR-RV) model. This novel approach creates a GPU-scalable multivariate volatility estimator, which decomposes multiple time series into economically-meaningful variables to explain the endogenous and exogenous factors driving the underlying variability. This unique decomposition goes beyond the classic one step ahead prediction; indeed, we investigate inferences up to one month into the future using stocks, FX futures and ETF futures, demonstrating its superior performance according to accuracy of large moves, longer-term prediction and consistency over time.

There are various types of pyramid schemes which have inflicted or are inflicting losses on many people in the world. We propose a pyramid scheme model which has the principal characters of many pyramid schemes appeared in recent years: promising high returns, rewarding the participants recruiting the next generation of participants, and the organizer will take all the money away when he finds the money from the new participants is not enough to pay the previous participants interest and rewards. We assume the pyramid scheme carries on in the tree network, ER random network, SW small-world network or BA scale-free network respectively, then give the analytical results of how many generations the pyramid scheme can last in these cases. We also use our model to analyse a pyramid scheme in the real world and we find the connections between participants in the pyramid scheme may constitute a SW small-world network.

We propose a new data-driven method to select the optimal number of relevant components in Principal Component Analysis (PCA). This new method applies to correlation matrices whose time autocorrelation function decays more slowly than an exponential, giving rise to long memory effects. In comparison with other available methods present in the literature, our procedure does not rely on subjective evaluations and is computationally inexpensive. The underlying basic idea is to use a suitable factor model to analyse the residual memory after sequentially removing more and more components, and stopping the process when the maximum amount of memory has been accounted for by the retained components. We validate our methodology on both synthetic and real financial data, and find in all cases a clear and computationally superior answer entirely compatible with available heuristic criteria, such as cumulative variance and cross-validation.

This paper develops original models to study interacting agents in financial markets. The key feature of these models is how interactions are formulated and analysed. Agents learn from their observations and learning ability to interpret news or private information. Central limit theorems are developed but they arise rather unexpectedly. Under certain type of conditions governing the learning, agents beliefs converge in distribution that can be even fractal. The underlying randomness in the systems is not restricted to be of a certain class. Fresh insights are gained not only from developing new non-linear social learning models but also from using different techniques to study discrete time random linear dynamical systems.

We formalize one aspect of reliability in the context of Mobility-on-Demand (MoD) systems by acknowledging the uncertainty in the pick-up time of these services. This study answers two key questions: i) how the difference between the stated and actual pick-up times affect the propensity of a passenger to choose an MoD service? ii) how an MoD service provider can leverage this information to increase its ridership? We conduct a discrete choice experiment in New York to answer the former question and adopt a micro-simulation-based optimization method to answer the latter question. In our experiments, the ridership of an MoD service could be increased by up to 10\% via displaying the predicted wait time strategically.

Using the most comprehensive source of commercially available data on the US National Market System, we analyze all quotes and trades associated with Dow 30 stocks in 2016 from the vantage point of a single and fixed frame of reference. Contrary to prevailing academic and popular opinion, we find that inefficiencies created in part by the fragmentation of the equity marketplace are widespread and potentially generate substantial profit for agents with superior market access. Information feeds reported different prices for the same equity---violating the commonly-supposed economic behavior of a unified price for an indistinguishable product---more than 120 million times, with "actionable" latency arbitrage opportunities totaling almost 64 million. During this period, roughly 22% of all trades occurred while the SIP and aggregated direct feeds were dislocated. The current market configuration resulted in a realized opportunity cost totaling over $160 million when compared with a single feed, single exchange alternative---a conservative estimate that does not take into account intra-day offsetting events.

Motivated by Kyprianou and Zhou (2009), Wang and Hu (2012), Avram et al. (2017), Li et al. (2017) and Wang and Zhou (2018), we consider in this paper the problem of maximizing the expected accumulated discounted tax payments of an insurance company, whose reserve process (before taxes are deducted) evolves as a spectrally negative L\'{e}vy process with the usual exclusion of negative subordinator or deterministic drift. Tax payments are collected according to the very general loss-carry-forward tax system introduced in Kyprianou and Zhou (2009). To achieve a balance between taxation optimization and solvency, we consider an interesting modified objective function by considering the expected accumulated discounted tax payments of the company until the general draw-down time, instead of until the classical ruin time. The optimal tax return function together with the optimal tax strategy is derived, and some numerical examples are also provided.

Motivated by successes of fast reverting volatility models, and the implicit dependence of `rough' processes on infinitesimal reversionary timescales, we establish a pathwise volatility framework which leads to a complete understanding of volatility trajectories' behaviour in the limit of infinitely-fast reversion. Towards this, we first establish processes that are weakly equivalent to Cox-Ingersoll-Ross (CIR) processes, but in contrast prove well-defined without reference to a probability measure. This provides an unusual example of Skorokhod's representation theorem. In particular, we become able to generalise Heston's model of volatility to an arbitrary degree, by sampling drivers $\omega$ under any probability measure; a rough one if so desired.

Our main analysis relates to separable initial-value problems of type $$x'= \omega(x) + t - x + 1,\quad x(0)=0,$$ with $\omega$ only assumed continuous (not Lipschitz, nor H\"older), solutions of which $\varphi$ correspond to time-averages of volatility trajectories $\varphi'$. Such solutions are shown to exist, be unique and bijective for any $\omega$, essentially placing no constraints on corresponding volatility trajectories, except for their non-negativity. After bounding these solutions in time, we prove a rare type of convergence result, towards c\`adl\`ag exit-time limits, on Skorokhod's $M_1$ topology. One immediate corollary of this limiting result is a weak connection between the time-averaged CIR process and the inverse-Gaussian L\'evy subordinator.

Using the most comprehensive, commercially-available dataset of trading activity in U.S. equity markets, we catalog and analyze latency arbitrage opportunities and realized opportunity costs (ROC) incurred by market participants. We find that latency arbitrage opportunities are common, observing a total of over 3.1 billion latency arbitrage opportunities in the Russell 3000 during trading in 2016, or roughly 525 per second of trading. Up to 23% of observed trades may have contributed the the measured inefficiencies, leading to a ROC greater than $2 billion USD. A subset of the constituents of the S&P 500 index experience the greatest amount of ROC and may drive inefficiencies in other stocks. In addition, we identify fine structure and self-similarity in the intra-day distribution of latency arbitrage opportunity start times. These results point to universal underlying market mechanisms arising from the physical structure of the U.S. National Market System.

We presented Bayesian portfolio selection strategy, via the $k$ factor asset pricing model. If the market is information efficient, the proposed strategy will mimic the market; otherwise, the strategy will outperform the market. The strategy depends on the selection of a portfolio via Bayesian multiple testing methodologies. We present the "discrete-mixture prior" model and the "hierarchical Bayes model with horseshoe prior." We define the Oracle set and prove that asymptotically the Bayes rule attains the risk of Bayes Oracle up to $O(1)$. Our proposed Bayes Oracle test guarantees statistical power by providing the upper bound of the type-II error. Simulation study indicates that the proposed Bayes oracle test is suitable for the efficient market with few stocks inefficiently priced. However, as the model becomes dense, i.e., the market is highly inefficient, one should not use the Bayes oracle test. The statistical power of the Bayes Oracle portfolio is uniformly better for the $k$-factor model ($k>1$) than the one factor CAPM. We present the empirical study, where we considered the 500 constituent stocks of S\&P 500 from the New York Stock Exchange (NYSE), and S\&P 500 index as the benchmark for thirteen years from the year 2006 to 2018. We showed the out-sample risk and return performance of the four different portfolio selection strategies and compared with the S\&P 500 index as the benchmark market index. Empirical results indicate that it is possible to propose a strategy which can outperform the market.