We investigate solving partial integro-differential equations (PIDEs) using unsupervised deep learning in this paper. To price options, assuming underlying processes follow \levy processes, we require to solve PIDEs. In supervised deep learning, pre-calculated labels are used to train neural networks to fit the solution of the PIDE. In an unsupervised deep learning, neural networks are employed as the solution, and the derivatives and the integrals in the PIDE are calculated based on the neural network. By matching the PIDE and its boundary conditions, the neural network gives an accurate solution of the PIDE. Once trained, it would be fast for calculating options values as well as option \texttt{Greeks}.

The birth order literature emphasizes the role of parental investments in explaining why firstborns have higher human capital outcomes than their laterborn siblings. We use birth order as a proxy for investments and interact it with genetic endowments. Exploiting only within-family variation in both ensures they are exogenous as well as orthogonal to each other. As such, our setting is informative about the existence of dynamic complementarity in skill production. Our empirical analysis exploits data from 15,019 full siblings in the UK Biobank. We adopt a family-fixed effects strategy combined with instrumental variables to deal with endogeneity issues arising from omitted variables and measurement error. We find that birth order and genetic endowments interact: those with above-average genetic endowments benefit disproportionally more from being firstborn compared to those with below-average genetic endowments. This finding is a clean example of how genetic endowments (nature) and the environment (nurture) interact in producing educational attainment. Moreover, our results are consistent with the existence of dynamic complementarity in skill formation: additional parental investments associated with being firstborn are more effective for those siblings who randomly inherited higher genetic endowments for educational attainment.

We obtain error estimates for strong approximations of a diffusion with a diffusion matrix $\sigma$ and a drift b by the discrete time process defined recursively X_N((n+1)/N) = X_N(n/N)+N^{1/2}\sigma(X_N(n/N))\xi(n+1)+N^{-1}b(XN(n/N)); where \xi(n); n\geq 1 are i.i.d. random vectors, and apply this in order to approximate the fair price of a game option with a diffusion asset price evolution by values of Dynkin's games with payoffs based on the above discrete time processes. This provides an effective tool for computations of fair prices of game options with path dependent payoffs in a multi asset market with diffusion evolution.

We discuss a powerful, geometric representation of financial portfolios and stock markets, which identifies the space of portfolios with the points lying in a simplex convex polytope. The ambient space has dimension equal to the number of stocks, or assets. Although our statistical tools are quite general, in this paper we focus on the problem of portfolio scoring. Our contribution is to introduce an original computational framework to model portfolio allocation strategies, which is of independent interest for computational finance. To model asset allocation strategies, we employ log-concave distributions centered on portfolio benchmarks. Our approach addresses the crucial question of evaluating portfolio management, and is relevant to the individual private investors as well as financial organizations. We evaluate portfolio performance, in a certain time period, by providing a new portfolio score, based on the aforementioned framework and concepts. In particular, it relies on the expected proportion of allocations that the portfolio outperforms when a certain set of strategies take place in that time period. We also discuss how this set of strategies -- and the knowledge one may have about them -- could vary in our framework, and we provide additional versions of our score in order to obtain a more complete picture of its performance. In all cases, we show that the score computations can be performed efficiently. Last but not least, we expect this framework to be useful in portfolio optimization and in automatically identifying extreme phenomena in a stock market.

Combinations of neural ODEs with recurrent neural networks (RNN), like GRU-ODE-Bayes or ODE-RNN are well suited to model irregularly observed time series. While those models outperform existing discrete-time approaches, no theoretical guarantees for their predictive capabilities are available. Assuming that the irregularly-sampled time series data originates from a continuous stochastic process, the $L^2$-optimal online prediction is the conditional expectation given the currently available information. We introduce the neural jump ODE (NJ-ODE) that provides a data-driven approach to learn, continuously in time, the conditional expectation of a stochastic process. Our approach models the conditional expectation between two observations with a neural ODE and jumps whenever a new observation is made. We define a novel training framework, which allows us to prove theoretical guarantees for the first time. In particular, we show that the output of our model converges to the $L^2$-optimal prediction. We provide experiments showing that the theoretical results also hold empirically. Moreover, we experimentally show that our model outperforms the baselines in more complex learning tasks and give comparisons on real-world datasets.

We study general classes of parametric measures of variability with applications in risk management. Particular focus is put on variability measures induced by three classes of popular risk measures: the Value-at-Risk, the Expected Shortfall, and the expectiles. Properties of these variability measures are explored in detail, and a characterization result is obtained via the mixture of inter-ES differences. Convergence properties and asymptotic normality of their empirical estimators are established. We provide an illustration of the three classes of variability measures applied to financial data and analyze their relative advantages.

We study the conditions under which input-output networks can dynamically attain competitive equilibrium, where markets clear and profits are zero. We endow a classical firm network model with simple dynamical rules that reduce supply/demand imbalances and excess profits. We show that the time needed to reach equilibrium diverges as the system approaches an instability point beyond which the Hawkins-Simons condition is violated and competitive equilibrium is no longer realisable. We argue that such slow dynamics is a source of excess volatility, through accumulation and amplification of exogenous shocks. Factoring in essential physical constraints, such as causality or inventory management, we propose a dynamically consistent model that displays a rich variety of phenomena. Competitive equilibrium can only be reached after some time and within some region of parameter space, outside of which one observes periodic and chaotic phases, reminiscent of real business cycles. This suggests an alternative explanation of the excess volatility that is of purely endogenous nature. Other regimes include deflationary equilibria and intermittent crises characterised by bursts of inflation. Our model can be calibrated using highly disaggregated data on individual firms and prices, and may provide a powerful tool to describe out-of-equilibrium economies.

Literature about the scholarly impact of scientific research offers very few contributions on private sector research, and the comparison with public sector. In this work, we try to fill this gap examining the citation-based impact of Italian 2010-2017 publications distinguishing authorship by the private sector from the public sector. In particular, we investigate the relation between different forms of collaboration and impact: how intra-sector private publications compare to public, and how private-public joint publications compare to intra-sector extramural collaborations. Finally, we assess the different effect of international collaboration on private and public research impact, and whether there occur differences across research fields.

The dynamical evolution of multiscaling in financial time series is investigated using time-dependent Generalized Hurst Exponents (GHE), $H_q$, for various values of the parameter $q$. Using $H_q$, we introduce a new visual methodology to algorithmically detect critical changes in the scaling of the underlying complex time-series. The methodology involves the degree of multiscaling at a particular time instance, the multiscaling trend which is calculated by the Change-Point Analysis method, and a rigorous evaluation of the statistical significance of the results. Using this algorithm, we have identified particular patterns in the temporal co-evolution of the different $H_q$ time-series. These GHE patterns, distinguish in a statistically robust way, not only between time periods of uniscaling and multiscaling, but also among different types of multiscaling: symmetric multiscaling (M) and asymmetric multiscaling (A). We apply the visual methodology to time-series comprising of daily close prices of four stock market indices: two major ones (S\&P~500 and NIKKEI) and two peripheral ones (Athens Stock Exchange general Index and Bombay-SENSEX). Results show that multiscaling varies greatly with time: time periods of strong multiscaling behavior and time periods of uniscaling behavior are interchanged while transitions from uniscaling to multiscaling behavior occur before critical market events, such as stock market bubbles. Moreover, particular asymmetric multiscaling patterns appear during critical stock market eras and provide useful information about market conditions. In particular, they can be used as 'fingerprints' of a turbulent market period as well as provide warning signals for an upcoming stock market 'bubble'. The applied visual methodology also appears to distinguish between exogenous and endogenous stock market crises, based on the observed patterns before the actual events.

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.