1,721,002 research outputs found
N-player games and mean-field games with smooth dependence on past absorptions.
Full text linkMean-field games with absorption is a class of games that has been introduced in (Ann. Appl. Probab. 28 (2018) 2188–2242) and that can be viewed as natural limits of symmetric stochastic differential games with a large number of players who, interacting through a mean-field, leave the game as soon as their private states hit some given boundary. In this paper, we push the study of such games further, extending their scope along two main directions. First, we allow the state dynamics and the costs to have a very general, possibly infinite-dimensional, dependence on the (non-normalized) empirical sub-probability measure of the survivors’ states. This includes the particularly relevant case where the mean-field interaction among the players is done through the empirical measure of the survivors together with the fraction of absorbed players over time. Second, the boundedness of coefficients and costs has been considerably relaxed including drift and costs with linear growth in the state variables, hence allowing for more realistic dynamics for players’ private states. We prove the existence of solutions of the MFG in strict as well as relaxed feedback form, and we establish uniqueness of the MFG solutions under monotonicity conditions of Lasry–Lions type. Finally, we show in a setting with finite-dimensional interaction that such solutions induce approximate Nash equilibria for the N-player game with vanishing error as N → ∞
The Yoccoz-Birkeland livestock population model coupled with random price dynamics
Full text linkWe study a random version of the population-market model proposed by Arlot,
Marmi and Papini in Arlot et al. (2019). The latter model is based on the
Yoccoz-Birkeland integral equation and describes a time evolution of livestock
commodities prices which exhibits endogenous deterministic stochastic
behaviour. We introduce a stochastic component inspired from the Black-Scholes
market model into the price equation and we prove the existence of a random
attractor and of a random invariant measure. We compute numerically the fractal
dimension and the entropy of the random attractor and we show its convergence
to the deterministic one as the volatility in the market equation tends to
zero. We also investigate in detail the dependence of the attractor on the
choice of the time-discretization parameter. We implement several statistical
distances to quantify the similarity between the attractors of the discretized
systems and the original one. In particular, following a work by Cuturi (2013),
we use the Sinkhorn distance. This is a discrete and penalized version of the
Optimal Transport Distance between two measures, given a transport cost matrix
Continuous time mean-variance portfolio optimization through the mean field approach
Full text linkA simple mean-variance portfolio optimization problem in continuous time is solved using the mean field approach. In this approach, the original optimal control problem, which is time inconsistent, is viewed as the McKean–Vlasov limit of a family of controlled many-component weakly interacting systems. The prelimit problems are solved by dynamic programming, and the solution to the original problem is obtained by passage to the limit
Asymptotic results for the Fourier estimator of the integrated quarticity
Full text linkIn this paper we prove a central limit theorem for the Fourier quarticity estimator. We obtain a new consistency result and we show that the estimator reaches the parametric rate 1/2. The optimal variance is obtained, with a suitable choice of the number of frequencies employed to compute the Fourier coefficients of the volatility, while the limiting distribution has a bias. As a by-product, thanks to the Fourier methodology, we obtain consistent estimators of any even power of the volatility function and an estimator of the spot quarticity. We assess the finite sample performance of the Fourier quarticity estimator in a numerically exercise with different market micro-structure frictions
One-Shot Learning of Stochastic Differential Equations with Data Adapted Kernels
Full text linkWe consider the problem of learning Stochastic Differential Equations of the
form from one sample trajectory. This
problem is more challenging than learning deterministic dynamical systems
because one sample trajectory only provides indirect information on the unknown
functions , , and stochastic process representing the drift,
the diffusion, and the stochastic forcing terms, respectively. We propose a
method that combines Computational Graph Completion and data adapted kernels
learned via a new variant of cross validation. Our approach can be decomposed
as follows: (1) Represent the time-increment map as
a Computational Graph in which , and appear as unknown
functions and random variables. (2) Complete the graph (approximate unknown
functions and random variables) via Maximum a Posteriori Estimation (given the
data) with Gaussian Process (GP) priors on the unknown functions. (3) Learn the
covariance functions (kernels) of the GP priors from data with randomized
cross-validation. Numerical experiments illustrate the efficacy, robustness,
and scope of our method.Comment: 22 pages, 21 figure
Volatility of volatility estimation: central limit theorems for the Fourier transform estimator and empirical study of the daily time series stylized facts
Full text linkWe study the asymptotic normality of two feasible estimators of the
integrated volatility of volatility based on the Fourier methodology, which
does not require the pre-estimation of the spot volatility. We show that the
bias-corrected estimator reaches the optimal rate , while the
estimator without bias-correction has a slower convergence rate and a smaller
asymptotic variance. Additionally, we provide simulation results that support
the theoretical asymptotic distribution of the rate-efficient estimator and
show the accuracy of the latter in comparison with a rate-optimal estimator
based on the pre-estimation of the spot volatility. Finally, using the
rate-optimal Fourier estimator, we reconstruct the time series of the daily
volatility of volatility of the S\&P500 and EUROSTOXX50 indices over long
samples and provide novel insight into the existence of stylized facts about
the volatility of volatility dynamics
Adding Cycles into the Neoclassical Growth Model
Full text linkWe propose a stochastic Solow growth model where a cyclical component is added to the total factor productivity process. Theoretically, an important feature of the model is that its main equation takes a state space representation where key parameters can be estimated via an unobserved component approach without involving capital stock measures. In addition, the dynamic properties of the model are mostly unaffected by the newly introduced cyclical component. Empirically, our novel framework is consistent with secular U.S. empirical evidence
Uncertainty in firm valuation and a cross-sectional misvaluation measure
Full text linkThe degree of uncertainty associated with the value of a company plays a relevant role in valuation analysis. We propose an original and robust methodology for company market valuation, which replaces the traditional point estimate of the conventional Discounted Cash Flow model with a probability distribution of fair values that convey information about both the expected value of the company and its intrinsic uncertainty. Our methodology depends on two main ingredients: an econometric model for company revenues and a set of firm-specific balance sheet relations that are estimated using historical data. We explore the effectiveness and scope of our methodology through a series of statistical exercises on publicly traded U.S. companies. At the firm level, we show that the fair value distribution derived with our methodology constitutes a reliable predictor of the company’s future abnormal returns. At the market level, we show that a long-short valuation (LSV) factor, built using buy-sell recommendations based on the fair value distribution, contains information not accessible through the traditional market factors. The LSV factor significantly increases the explanatory and the predictive power of factor models estimated on portfolios and individual stock returns
A closed-form formula characterization of the Epps effect
No full textIn this study we provide an analytical characterization of the impact of zero returns on the popular realized covariance estimator of Barndorff-Nielsen and Shephard [Econometric analysis of realized covariation: High frequency based covariance, regression, and correlation in financial economics. Econometrica, 2004, 72(3), 885–925]. In our framework, efficient price processes evolve as a semimartingale with some likelihood of repeated prices. We show that the standard realized covariance estimator is asymptotically affected by a downward bias, and the size of the bias depends on these likelihoods. We demonstrate that this result can be used to construct a consistent estimator of the integrated covariance of a vector semimartingale. The advantages with respect to other estimators are discussed with data
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