We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

Stochastic differential equations and diffusion processes

1 Introduction.- 2 Experiments, Deficiencies, Distances v.- 2.1 Comparing Risk Functions.- 2.2 Deficiency and Distance between Experiments.- 2.3 Likelihood Ratios and Blackwell's Representation.- 2.4 Further Remarks on the Convergence of Distri butions of Likelihood Ratios.- 2.5 Historical Remarks.- 3 Contiguity - Hellinger Transforms.- 3.1 Contiguity.- 3.2 Hellinger Distances, Hellinger Transforms.- 3.3 Historical Remarks.- 4 Gaussian Shift and Poisson Experiments.- 4.1 Introduction.- 4.2 Gaussian Experiments.- 4.3 Poisson Experiments.- 4.4 Historical Remarks.- 5 Limit Laws for Likelihood Ratios.- 5.1 Introduction.- 5.2 Auxiliary Results.- 5.2.1 Lindeberg's Procedure.- 5.2.2 Levy Splittings.- 5.2.3 Paul Levy's Symmetrization Inequalities.- 5.2.4 Conditions for Shift-Compactness.- 5.2.5 A Central Limit Theorem for Infinitesimal Arrays.- 5.2.6 The Special Case of Gaussian Limits.- 5.2.7 Peano Differentiable Functions.- 5.3 Limits for Binary Experiments.- 5.4 Gaussian Limits.- 5.5 Historical Remarks.- 6 Local Asymptotic Normality.- 6.1 Introduction.- 6.2 Locally Asymptotically Quadratic Families.- 6.3 A Method of Construction of Estimates.- 6.4 Some Local Bayes Properties.- 6.5 Invariance and Regularity.- 6.6 The LAMN and LAN Conditions.- 6.7 Additional Remarks on the LAN Conditions.- 6.8 Wald's Tests and Confidence Ellipsoids.- 6.9 Possible Extensions.- 6.10 Historical Remarks.- 7 Independent, Identically Distributed Observations.- 7.1 Introduction.- 7.2 The Standard i.i.d. Case: Differentiability in Quadratic Mean.- 7.3 Some Examples.- 7.4 Some Nonparametric Considerations.- 7.5 Bounds on the Risk of Estimates.- 7.6 Some Cases Where the Number of Observations Is Random.- 7.7 Historical Remarks.- 8 On Bayes Procedures.- 8.1 Introduction.- 8.2 Bayes Procedures Behave Nicely.- 8.3 The Bernstein-von Mises Phenomenon.- 8.4 A Bernstein-von Mises Result for the i.i.d. Case.- 8.5 Bayes Procedures Behave Miserably.- 8.6 Historical Remarks.- Author Index.

Asymptotics in Statistics–Some Basic Concepts

We apply the techniques of stochastic integration with respect to the fractional Brownian motion and the Gaussian theory of regularity and supremum estimation to study the maximum likelihood estimator (MLE) for the drift parameter of stochastic processes satisfying stochastic equations driven by fractional Brownian motion with any level of H\"{o}lder-regularity (any \emph{Hurst} parameter) We prove existence and strong consistency of the MLE for linear and nonlinear equations We\ also prove that a basic discretized version of the MLE, is still a strongly consistent estimator

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Statistical aspects of the fractional stochastic calculus

This book is fundamentally about the estimation of risk. At an intuitive level, risk is easy to understand: given an asset with a current price of say $100, what is the likelihood that at some future time—the risk horizon—its price will be less than $90? Or more than $120? And how does the probability of observing a price below $90 change as a function of the risk horizon? Such heuristic notions can be formalized in several different ways. Suppose that the (log) price of the asset follows Xt “ X0 ` σWt where Wt is a standard Brownian motion and σ ą 0 is a parameter quantifying the risk of X, which needs to be estimated. Suppose also that the price is observed at a sequence of fixed points in time, say at the end of each trading day. Then a simple measure of risk is the squared change in the observed price from one day to the next, possibly averaged over several adjacent intervals to obtain a smoothed estimate—this is, of course, the (daily) variance. As the number of intervals increases, and assuming the statistical properties of the underlying price process remain unchanged, this estimator eventually yields the true value of σ (or in fact its square). But asymptotic results can be considered in another sense: suppose that the total length of the observation interval is fixed (say one day) but the distance between observations within the interval becomes increasingly small—that is, one is in the realm of high-frequency data. Then another way to estimate risk on the interval is to sum the squared incremental price changes. This estimator is called the realized volatility, and it is this estimator, along with its variants and applications, that form the main focus of the book. Formally, if the observation interval has length T and the distance between observations is ∆ then the realized volatility is:

High-Frequency Financial Econometrics

Continuous Sampling- Parametric Stochastic Differential Equations- Rates of Weak Convergence of Estimators in Homogeneous Diffusions- Large Deviations of Estimators in Homogeneous Diffusions- Local Asymptotic Mixed Normality for Nonhomogeneous Diffusions- Bayes and Sequential Estimation in Stochastic PDEs- Maximum Likelihood Estimation in Fractional Diffusions- Discrete Sampling- Approximate Maximum Likelihood Estimation in Nonhomogeneous Diffusions- Rates of Weak Convergence of Estimators in the Ornstein-Uhlenbeck Process- Local Asymptotic Normality for Discretely Observed Homogeneous Diffusions- Estimating Function for Discretely Observed Homogeneous Diffusions

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Parameter estimation in stochastic differential equations

The paper shows that the distribution of the normalized minimum contrast estimator of the drift parameter in the fractional Ornstein-Uhlenbeck process observed over [0, T] converges to the standard normal distribution with an uniform error rate of the order O(T−1/2) for the case H > 1/2 where H is the Hurst exponent of the fractional Brownian motion driving the Ornstein-Uhlenbeck process. Then based on discrete observations, it introduces several approximate minimum contrast estimators and studies their rate of of weak convergence to normal distribution.

Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling

Sequential maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes

The paper is concerned with the approximation of the maximum likelihood estimate of the parameter in the linear drift coefficient of the Ito stochastic differential equation by “trapezodial rule” approximations when observations are made at regularly spaced discrete but very dense time points

Approximate maximum likelihood estimation for diffusion processes from discrete observations

Berry-Esseen bounds, with random and nonrandom normings, and large deviation probability bounds for two approximate maximum likelihood estimators of the drift parameter in the Ornstein-Uhlenbeck process are obtained when the process is observed at equally spaced dense time points. Also obtained are the rates at which these estimators converge to the maximum likelihood estimator based on continuous observation.

Jaya P. N. Bishwal

Papers

Parameter estimation in stochastic differential equations

Minimum contrast estimation in fractional Ornstein-Uhlenbeck process: Continuous and discrete sampling

Sequential maximum likelihood estimation for reflected Ornstein–Uhlenbeck processes

Approximate maximum likelihood estimation for diffusion processes from discrete observations

Rates of convergence of approximate maximum likelihood estimators in the Ornstein-Uhlenbeck process