Continuous-time Diffusion Processes Research Articles

Factor Overnight GARCH-Itô Models

Abstract This article introduces a unified factor overnight GARCH-Itô model for large volatility matrix estimation and prediction. To account for whole-day market dynamics, the proposed model has two different instantaneous factor volatility processes for the open-to-close and close-to-open periods, while each embeds the discrete-time multivariate GARCH model structure. To estimate latent factor volatility, we assume the low rank plus sparse structure and employ nonparametric estimation procedures. Then, based on the connection between the discrete-time model structure and the continuous-time diffusion process, we propose a weighted least squares estimation procedure with the non-parametric factor volatility estimator and establish its asymptotic theorems.

Unadjusted Langevin algorithm for sampling a mixture of weakly smooth potentials

Discretization of continuous-time diffusion processes is a widely recognized method for sampling. However, it seems to be a considerable restriction when the potentials are often required to be smooth (gradient Lipschitz). This paper studies the problem of sampling through Euler discretization, where the potential function is assumed to be a mixture of weakly smooth distributions and satisfies weakly dissipative. We establish the convergence in Kullback–Leibler (KL) divergence with the number of iterations to reach ϵ-neighborhood of a target distribution in only polynomial dependence on the dimension. We relax the degenerated convex at infinity conditions of (Erdogdu and Hosseinzadeh (2020)) and prove convergence guarantees under Poincaré inequality or non-strongly convex outside the ball. In addition, we also provide convergence in Lβ-Wasserstein metric for the smoothing potential.

Multivariate Overnight GARCH-Ito Model with Applications in Large Volatility Matrix Estimation and Prediction

This paper introduces a unified multivariate overnight GARCH-It\^o model for volatility matrix estimation and prediction both in the low- and high-dimensional set-up. To account for whole-day market dynamics in the financial market, the proposed model has two different instantaneous volatility processes for the open-to-close and close-to-open periods, while each embeds the discrete-time multivariate GARCH model structure. We call it the multivariate overnight GARCH-It\^o (MOGI) model. Based on the connection between the discrete-time model structure and the continuous-time diffusion process, we propose a weighted least squares estimation procedure for estimating model parameters with open-to-close high-frequency and close-to-open low-frequency financial data, and establish its asymptotic theorems. We further discuss the prediction of future vast volatility matrices and study its asymptotic properties. A simulation study is conducted to check the finite sample performance of the proposed estimation and prediction methods. The empirical analysis is carried out to compare the performance of the proposed MOGI model with other benchmark methods in portfolio allocation problems.

Parametric Estimation of Diffusion Processes: A Review and Comparative Study

This paper provides an in-depth review about parametric estimation methods for stationary stochastic differential equations (SDEs) driven by Wiener noise with discrete time observations. The short-term interest rate dynamics are commonly described by continuous-time diffusion processes, whose parameters are subject to estimation bias, as data are highly persistent, and discretization bias, as data are discretely sampled despite the continuous-time nature of the model. To assess the role of persistence and the impact of sampling frequency on the estimation, we conducted a simulation study under different settings to compare the performance of the procedures and illustrate the finite sample behavior. To complete the survey, an application of the procedures to real data is provided.

Credit Scoring with Ego-Network Data

This article investigates a stochastic filtering problem whereby the bor-rower’s hidden credit quality is estimated using ego-network signals. The hidden credit quality process is modeled as a mean reverting Ornstein-Ulehnbeck process. The lender observes the borrower’s behavior modeled as a continuous time diffusion process. The drift of the diffusion process is driven by the hidden credit quality. At discrete fixed times, the lender gets ego-network signals from the borrower and the borrower’s direct friends. The observation filtration thus contains continuous time borrower data augmented with discrete time ego-network signals. Combining the continuous time observation data and ego-network information, we derive filter equations for the hidden process and the properties of the conditional variance. Further, we study the asymptotic properties of the conditional variance when the frequency of arrival of ego-network signals is increased.

UNIFORM CONVERGENCE RATES OF KERNEL-BASED NONPARAMETRIC ESTIMATORS FOR CONTINUOUS TIME DIFFUSION PROCESSES: A DAMPING FUNCTION APPROACH

In this paper, we derive uniform convergence rates of nonparametric estimators for continuous time diffusion processes. In particular, we consider kernel-based estimators of the Nadaraya–Watson type, introducing a new technical device called adamping function. This device allows us to derive sharp uniform rates over an infinite interval with minimal requirements on the processes: The existence of the moment of any order is not required and the boundedness of relevant functions can be significantly relaxed. Restrictions on kernel functions are also minimal: We allow for kernels with discontinuity, unbounded support, and slowly decaying tails. Our proofs proceed by using the covering-number technique from empirical process theory and exploiting the mixing and martingale properties of the processes. We also present new results on the path-continuity property of Brownian motions and diffusion processes over an infinite time horizon. These path-continuity results, which should also be of some independent interest, are used to control discretization biases of the nonparametric estimators. The obtained convergence results are useful for non/semiparametric estimation and testing problems of diffusion processes.

Particle Smoothing for Hidden Diffusion Processes: Adaptive Path Integral Smoother

Particle smoothing methods are used for inference of stochastic processes based on noisy observations. Typically, the estimation of the marginal posterior distribution given all observations is cumbersome and computational intensive. In this paper, we propose a simple algorithm based on path integral control theory to estimate the smoothing distribution of continuous-time diffusion processes with partial observations. In particular, we use an adaptive importance sampling method to improve the effective sampling size of the posterior over processes given the observations and the reliability of the estimation of the marginals. This is achieved by estimating a feedback controller to sample efficiently from the joint smoothing distributions. We compare the results with estimations obtained from the standard Forward Filter/Backward Simulator for two diffusion processes of different complexity. We show that the proposed method gives more reliable estimations than the standard FFBSi when the smoothing distribution is poorly represented by the filter distribution.

Long time scaling behaviour for diffusion with resetting and memory

We consider a continuous-space and continuous-time diffusion process under resetting with memory. A particle resets to a position chosen from its trajectory in the past according to a memory kernel. Depending on the form of the memory kernel, we show analytically how different asymptotic behaviours of the variance of the particle position emerge at long times. These range from standard diffusive () all the way to anomalous ultraslow growth .

Sequential Likelihood Ratio Test for Cognitive Radios

This paper presents a sequential likelihood ratio test (SLRT) detector for spectrum sensing scenarios in cognitive radios (CRs). Similar to other CR detectors, we exploit the structure of the sample covariance matrix of the received signal to achieve detection with minimal information regarding the signal. Unlike the majority of covariance-based CR detectors, the SLRT is a sequential detector that allows for smaller detection delays, which is a prized asset in CR systems. Using methods borrowed from the theory of continuous-time diffusion processes, we derive the statistical properties of the SLRT detector and compare it with an eigenvalue-based sequential detector, which has been presented in previous work for CR systems. The comparison also includes detection scenarios with non-Gaussian noise to illustrate the robustness of the proposed detector in these situations.

CONVERGENCE RATES OF SUMS OF α-MIXING TRIANGULAR ARRAYS: WITH AN APPLICATION TO NONPARAMETRIC DRIFT FUNCTION ESTIMATION OF CONTINUOUS-TIME PROCESSES

The convergence rates of the sums of α-mixing (or strongly mixing) triangular arrays of heterogeneous random variables are derived. We pay particular attention to the case where central limit theorems may fail to hold, due to relatively strong time-series dependence and/or the nonexistence of higher-order moments. Several previous studies have presented various versions of laws of large numbers for sequences/triangular arrays, but their convergence rates were not fully investigated. This study is the first to investigate the convergence rates of the sums of α-mixing triangular arrays whose mixing coefficients are permitted to decay arbitrarily slowly. We consider two kinds of asymptotic assumptions: one is that the time distance between adjacent observations is fixed for any sample size n; and the other, called the infill assumption, is that it shrinks to zero as n tends to infinity. Our convergence theorems indicate that an explicit trade-off exists between the rate of convergence and the degree of dependence. While the results under the infill assumption can be seen as a direct extension of those under the fixed-distance assumption, they are new and particularly useful for deriving sharper convergence rates of discretization biases in estimating continuous-time processes from discretely sampled observations. We also discuss some examples to which our results and techniques are useful and applicable: a moving-average process with long lasting past shocks, a continuous-time diffusion process with weak mean reversion, and a near-unit-root process.

The non-locality of Markov chain approximations to two-dimensional diffusions

In this short paper, we consider discrete-time Markov chains on lattices as approximations to continuous-time diffusion processes. The approximations can be interpreted as finite difference schemes for the generator of the process. We derive conditions on the diffusion coefficients which permit transition probabilities to match locally first and second moments. We derive a novel formula which expresses how the matching becomes more difficult for larger (absolute) correlations and strongly anisotropic processes, such that instantaneous moves to more distant neighbours on the lattice have to be allowed. Roughly speaking, for non-zero correlations, the distance covered in one timestep is proportional to the ratio of volatilities in the two directions. We discuss the implications to Markov decision processes and the convergence analysis of approximations to Hamilton–Jacobi–Bellman equations in the Barles–Souganidis framework.

Theoretical Guarantees for Approximate Sampling from Smooth and Log-Concave Densities

Summary Sampling from various kinds of distribution is an issue of paramount importance in statistics since it is often the key ingredient for constructing estimators, test procedures or confidence intervals. In many situations, exact sampling from a given distribution is impossible or computationally expensive and, therefore, one needs to resort to approximate sampling strategies. However, there is no well-developed theory providing meaningful non-asymptotic guarantees for the approximate sampling procedures, especially in high dimensional problems. The paper makes some progress in this direction by considering the problem of sampling from a distribution having a smooth and log-concave density defined on Rp, for some integer p > 0. We establish non-asymptotic bounds for the error of approximating the target distribution by the distribution obtained by the Langevin Monte Carlo method and its variants. We illustrate the effectiveness of the established guarantees with various experiments. Underlying our analysis are insights from the theory of continuous time diffusion processes, which may be of interest beyond the framework of log-concave densities that are considered in the present work.

Seeing the Unseen Network: Inferring Hidden Social Ties from Respondent-Driven Sampling

Learning about the social structure of hidden and hard-to-reach populations — such as drug users and sex workers — is a major goal of epidemiological and public health research on risk behaviors and disease prevention. Respondent-driven sampling (RDS) is a peer-referral process widely used by many health organizations, where research subjects recruit other subjects from their social network. In such surveys, researchers observe who recruited whom, along with the time of recruitment and the total number of acquaintances (network degree) of respondents. However, due to privacy concerns, the identities of acquaintances are not disclosed. In this work, we show how to reconstruct the underlying network structure through which the subjects are recruited. We formulate the dynamics of RDS as a continuous-time diffusion process over the underlying graph and derive the likelihood of the recruitment time series under an arbitrary inter-recruitment time distribution. We develop an efficient stochastic optimization algorithm called RENDER (REspoNdent-Driven nEtwork Reconstruction) that finds the network that best explains the collected data. We support our analytical results through an exhaustive set of experiments on both synthetic and real data.

Explicit form of approximate transition probability density functions of diffusion processes

A continuous-time diffusion process is very popular in modeling and provides useful tools to analyze particularly, but not restricted to, a variety of economic and financial variables. The transition probability density function (TPDF) of a diffusion process plays an important role in understanding and explaining the dynamics of the process. A new way to find closed-form approximate TPDFs for multivariate diffusions is proposed in this paper. This method can be applied to general multivariate time-inhomogeneous diffusion processes, as long as, roughly speaking, they have smooth drift and volatility functions. A diffusion process is said to be reducible if it can be converted into a unit diffusion process where the volatility is the identity matrix. We have established how to obtain the approximate TPDF of a reducible diffusion explicitly. When a diffusion process is not reducible, an explicit form of approximate TPDF can be obtained by using the results in Aït-Sahalia (2008) and Choi (2013). The TPDF expansion suggested here enables us to obtain a recursive formula for the coefficient of the approximate TPDF for a multivariate jump diffusion. Monte Carlo simulation studies of conducting maximum likelihood estimation (MLE) using our approximations provide convincing evidence that our TPDF expansion can be used for the MLE when the true TPDF is unavailable. We also applied our approximate TPDFs to option pricing. The differences between our option prices and those from the Extended Black–Scholes formula are shown to be quite small. This implies that our methods can be employed to price assets whose underlying state variables follow general diffusion models.

Sequential Monte Carlo with Highly Informative Observations

We propose sequential Monte Carlo (SMC) methods for sampling the posterior distribution of state-space models under highly informative observation regimes, a situation in which standard SMC methods can perform poorly. A special case is simulating bridges between given initial and final values. The basic idea is to introduce a schedule of intermediate weighting and resampling times between observation times, which guide particles towards the final state. This can always be done for continuous-time models, and may be done for discrete-time models under sparse observation regimes; our main focus is on continuous-time diffusion processes. The methods are broadly applicable in that they support multivariate models with partial observation, do not require simulation of the backward transition (which is often unavailable), and, where possible, avoid pointwise evaluation of the forward transition. When simulating bridges, the last characteristic cannot be avoided entirely without concessions, and we suggest an $\...

Nonparametric Estimation of Volatility Function with Variable Bandwidth Parameter

In this article, we introduce a new method for the volatility function estimation of continuous-time diffusion process dX t  = μ(X t )dt + σ(X t )dW t , which is based on combining the idea of local linear smoother and variable bandwidth. We give the expressions for the conditional MSE and MISE of the estimator and obtain the optimal variable bandwidth. An explicit formula for the optimal variable bandwidth is presented by minimizing the MISE, which extends the related results in Fan and Gijbels (1992), etc. Finally, some simulations show that the performance of the proposed estimator with optimal variable bandwidth is often much better than that of the local linear estimator with invariable bandwidth.

A nonparametric model for spot price dynamics and pricing of futures contracts in electricity markets

AbstractThis paper deals with the estimation of continuous time diffusion processes describing the dynamics of electricity spot prices. Different parametric models have been proposed in the literature, each attempting to capture empirical characteristics and stylized facts of the electricity market like the spiky behavior of the spot prices. Although jump-diffusion and regime-switching models perform reasonably well, there is always a trade-off between model parsimony and adequacy. The results in the literature indicate that none of the models seem to consistently outperform its counterparts. This paper avoids making parametric assumption about the drift and the diffusion coefficient functions of the underlying electricity spot prices, and estimates these functions together with the market price of risk in a nonparametric way. The latter allows us to price futures contracts written on electricity spots. Using electricity spot prices and futures data from the regional electricity markets in Australia, we show that besides offering a convenient way of estimating the continuous-time models for electricity spot prices, our nonparametric estimation procedure performs well in- and out-of-sample when dealing with pricing of future contracts.

A Nonparametric Model for Spot Price Dynamics and Pricing of Futures Contracts in Electricity Markets

This paper deals with the estimation of continuous time diffusion processes describing the dynamics of electricity spot prices. Different parametric models have been proposed in the literature, each attempting to capture empirical characteristics and stylized facts of the electricity market like the spiky behavior of the spot prices. Although jump-diffusion and regime-switching models perform reasonably well, there is always a trade-off between model parsimony and adequacy. The results in the literature indicate that none of the models seem to consistently outperform its counterparts. This paper avoids making parametric assumption about the drift and the diffusion coefficient functions of the underlying electricity spot prices, and estimates these functions together with the market price of risk in a nonparametric way. The latter allows us to price futures contracts written on electricity spots. Using electricity spot prices and futures data from the regional electricity markets in Australia, we show that besides offering a convenient way of estimating the continuous-time models for electricity spot prices, our nonparametric estimation procedure performs well in- and out-of-sample when dealing with pricing of future contracts.

Realized Volatility or Price Range: Evidence from a Discrete Simulation of the Continuous Time Diffusion Process

The study provides evidence in favour of the price range as a proxy estimator of volatility in financial time series, in the cases that either intra-day datasets are unavailable or they are available at a low sampling frequency. A stochastic differential equation with time varying volatility of the instantaneous logreturns process is simulated, in order to mimic the continuous time diffusion analogue of the discrete time volatility process. The simulations provide evidence that the price range measures are superior to the realized volatility constructed at low sampling frequency. The high-low price range volatility estimator is more accurate than the realized volatility estimator based on five, or less, equidistance points in time. The open-high-low-close price range is more accurate than the realized volatility estimator based on eight, or less, intraperiod log-returns.

Inference for Interest Rate Models Using Milstein’s Approximation

A class of martingale estimating functions based on the first two moments of the observed process provides a convenient framework for estimating the parameters of diffusion processes [1]. In the Bayesian set up, combined estimating functions had been studied for diffusion processes in [2] with filtering applications. However, when the conditional mean and the conditional variance are functions of parameters of interest in a diffusion process model, the basic martingales generating components of quadratic estimating functions are such that one is an absolute continuous function with respect to the other [3, p. 94]. Hence, the combined martingale estimating functions cannot be constructed for continuous-time diffusion processes. In this paper, a general framework for parameter estimation of discretely observed interest rate models is developed by using the Milstein approximation and closed form expressions for the information gain are also obtained. The method is used to study the estimates of the parameters for an extended version of the CoxIngersoll-Ross interest rate model.