The classical concept of martingales and compensators bases on the monotony of filtrations. This paper looks at the situation where innovations can have an expiry date such that the information dynamics becomes non-monotone. The central idea is to focus only on those properties that martingales and compensators show on infinitesimally short intervals. Infinitesimal martingale representations are derived that extend classical martingale representations to non-monotone information. While the classical representations describe innovations only, the extended representations have an additional symmetric counterpart that quantifies the effect of information loss.

In this article, we tackle the problem of a market maker in charge of a book of options on a single liquid underlying asset. By using an approximation of the portfolio in terms of its vega, we show that the seemingly high-dimensional stochastic optimal control problem of an option market maker is in fact tractable. More precisely, when volatility is modeled using a classical stochastic volatility model -- e.g. the Heston model -- the problem faced by an option market maker is characterized by a low-dimensional functional equation that can be solved numerically using a Euler scheme along with interpolation techniques, even for large portfolios. In order to illustrate our findings, numerical examples are provided.

Detecting structure in noisy time series is a difficult task. One intuitive feature is the notion of trend. From theoretical hints and using simulated time series, we empirically investigate the efficiency of standard recurrent neural networks (RNNs) to detect trends. We show the overall superiority and versatility of certain standard RNNs structures over various other estimators. These RNNs could be used as basic blocks to build more complex time series trend estimators.

In this paper we study a continuous time equilibrium model of limit order book (LOB) in which the liquidity dynamics follows a non-local, reflected mean-field stochastic differential equation (SDE) with evolving intensity. Generalizing the basic idea of Ma et.al. (2015), we argue that the frontier of the LOB (e.g., the best asking price) is the value function of a mean-field stochastic control problem, as the limiting version of a Bertrand-type competition among the liquidity providers. With a detailed analysis on the $N$-seller static Bertrand game, we formulate a continuous time limiting mean-field control problem of the representative seller. We then validate the dynamic programming principle (DPP), and show that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman (HJB) equation. We argue that the value function can be used to obtain the equilibrium density function of the LOB, following the idea of Ma et.al. (2015).

The paper proposes a new algorithm for the high-dimensional financial data -- the Groupwise Interpretable Basis Selection (GIBS) algorithm, to estimate a new Adaptive Multi-Factor (AMF) asset pricing model, implied by the recently developed Generalized Arbitrage Pricing Theory, which relaxes the convention that the number of risk-factors is small. We first obtain an adaptive collection of basis assets and then simultaneously test which basis assets correspond to which securities, using high-dimensional methods. The AMF model, along with the GIBS algorithm, is shown to have a significantly better fitting and prediction power than the Fama-French 5-factor model.

How much should you receive in a week to be indifferent to \$ 100 in six months? Note that the indifference requires a rule to ensure the similarity between early and late payments. Assuming that rational individuals have low accuracy, then the following rule is valid: if the amounts to be paid are much less than the personal wealth, then the $q$-exponential discounting guarantees indifference in several periods. Thus, the discounting can be interpolated between hyperbolic and exponential functions due to the low accuracy to distinguish time averages when the payments have low impact on personal wealth. Therefore, there are physical conditions that allow the hyperbolic discounting regardless psycho-behavioral assumption.

We prove that a large class of discrete-time insurance surplus processes converge weakly to a generalized Ornstein-Uhlenbeck process, under a suitable re-normalization and when the time-step goes to 0. Motivated by ruin theory, we use this result to obtain approximations for the moments, the ultimate ruin probability and the discounted penalty function of the discrete-time process.