Free-knot spline approximation of stochastic processes

We study optimal approximation of stochastic processes by polynomial splines with free knots. The number of free knots is either a priori fixed or may depend on the particular trajectory. For the s-fold integrated Wiener process as well as for scalar diffusion processes we determine the asymptotic behavior of the average L_p-distance to the splines spaces, as the (expected) number of free knots tends to infinity.     

Representations of general stochastic processes

In this paper we carry on our study [4] of the algebraic representations of general stochastic processes. We give methods for constructing the algebraic representation of a stochastic process from the distribution of the process at a fixed finite number of times, we develope some techniques of integration, and we introduce the notion of a fibre bundle representation of a stochastic process. We then use this fibre bundle representation to study existence, methods of computation and the geometry of Markov process representations of the general stochastic process; thus extending [4] where existence was only discussed for discrete time or simple stochastic processes.     

Position dependent and stochastic thinning of point processes

A short probabilistic proof of Kallenberg's theorem [2] on thinning of point processes is given. It is extended to the case where the probability of deletion of a point depends on the position of the point and is itself random. The proof also leads easily to a statement about the rate of convergence in Renyi's theorem on thinning a renewal process.     

Vector-valued stochastic processes. I. Vector measures and vector-valued stochastic processes with finite variation

In this paper we study the relationship _V (M)=E(1 _M dV _S ) between operatorvalued processesV with finite variation V and operator-valued stochastic measures _V with finite variation | _V |. The variations satisfy the inequality | _V | _|V|, which, under certain conditions, is an equality (for example, ifV is measurable).     

Local times for stochastic processes which are subordinate to Gaussian processes

Let X and Y be random vectors of the same dimension such that Y has a normal distribution with mean vector O and covariance matrix R. Let g(x), x≥0, be a bounded nonincreasing function. X is said to be g-subordinate to Y if |Ee^iu′X| ≤ g(u′Ru) for all real vectors u of the same dimension as X. This is used to define the g-subordination of a real stochastic process X(t), 0 ≤ t ≤ 1, to a Gaussian process Y(t), 0 ≤ t ≤ 1. It is shown that the basic local time properties of a given Gaussian process are shared by all the processes that age g-subordinate to it. It is shown in particular that certain random series, including some random Fourier series, are g-subordinate to Gaussian processes, and so have their local time properties.     

On stochastic processes associated with relativistic stable distributions

Lévy processes with marginal relativistic α-stable distributions are described. Strictly stationary Ornstein-Uhlenbeck type processes with one-dimentional relativistic α-stable distributions are constructed. The exponential family as Esscher transforms of distributions on D _[0,∞)(R ^d ) of relativistic α-stable Lévy processes is obtained and the corresponding mixed exponential processes are characterized.     

Principal component analysis for multivariate stochastic processes

The problem of characterizing a /c-dimensional statistic contained in the past cj>f a discrete-time stochastic process y, which allows the best linear least-squares prediction of th^ future of y, is considered. The solution is provided in terms of the Schmidt pairs and singular values of an infinite matrix, and of the linear innovations of y. In the stationary case, the spectral characteristic of the optimal statistic, and of the corresponding prediction estimate, is obtained. In the base of a rational spectrum, the results are shown to assume a form particularly attractive from the algorithmic point of view. The results admit a straightforward extension to multivariate stochastic processes.     

Composition of stochastic processes governed by higher-order parabolic and hyperbolic equations

We consider compositions of stochastic processes that are governed by higherorder partial differential equations. The processes studied include compositions of Brownian motions, stable-like processes with Brownian time, Brownian motion whose time is an integrated telegraph process, and an iterated integrated telegraph process. The governing higher-order equations that are obtained are shown to be either of the usual parabolic type or, as in the last example, of hyperbolic type.     

Optimal stochastic control of the intensity of point processes

We consider an intensity control problem for a point process to maximize the expectation of a function of the time when the nth event occurs. We find the optimal control policy when the objective function is unimodal. Moreover, if the objective function is log-concave, so is the value function. As an application, we completely solve an intensity control problem that generalizes the problem studied by Brémaud (1976) and Defourny (2018). Also, we resolve the two conjectures made by Defourny (2018).     

Heuristic anytime approaches to stochastic decision processes

This paper proposes a set of methods for solving stochastic decision problems modeled as partially observable Markov decision processes (POMDPs). This approach (Real Time Heuristic Decision System, RT-HDS) is based on the use of prediction methods combined with several existing heuristic decision algorithms. The prediction process is one of tree creation. The value function for the last step uses some of the classic heuristic decision methods. To illustrate how this approach works, comparative results of different algorithms with a variety of simple and complex benchmark problems are reported. The algorithm has also been tested in a mobile robot supervision architecture.     

Generalized Grassmann algebras and applications to stochastic processes

In this paper, we present the groundwork for an Itô/Malliavin stochastic calculus and Hida's white noise analysis in the context of a supersymmetry with ${?}_3$-graded algebras. To this end, we establish a ternary Fock space and the corresponding strong algebra of stochastic distributions and present its application in the study of stochastic processes in this context.     

Wishart processes

We propose some matrix generalizations of square Bessel processes and we indicate their first properties: hitting time of 0 of the smallest eigenvalue, additivity property, associated Martingales, distributions, which mainly extend the real-valued classical results. We explain why these processes are indecomposable and therefore differ from the real-valued ones. We conclude with some formulae concerning matrix quadratic functionals analogous to the Cameron Martin formula.     

Boson fock representations of stochastic processes

A classification theory of quantum stationary processes similar to the corresponding theory for classical stationary processes is presented. Our main result is the classification of those pairs of classical stationary processes that admit a joint boson Fock canonical representation. Translated fromMatematicheskie Zametki, Vol. 67, No. 1, pp. 3–14, January, 2000.     

Hilbert space representations of general discrete time stochastic processes

We show that under mild conditions the joint densities Px₁,…,x_n) of the general discrete time stochastic process X_n on $\text{pH}$ can be computed via

$\text{Px}{}_1\text{,…,x}{}_{\mathrm{n}}\text{(x}{}_1\text{,…,x}{}_{\mathrm{n}}\text{) = 6?T(x}{}_1\text{)…T(x}{}_{\mathrm{n}}\text{)6}{}^2$

where ? is in a Hilbert space $\text{pH}$, and T (x), x ? $\text{pH}$ are linear operators on $\text{pH}$. We then show how the Central Limit Theorem can easily be derived from such representations.     

Curved exponential families of stochastic processes and their envelope families

Exponential families of stochastic processes are usually curved. The full exponential families generated by the finite sample exponential families are called the envelope families to emphasize that their interpretation as stochastic process models is not straightforward. A general result on how to calculate the envelope families is given, and the interpretation of these families as stochastic process models is considered. For Markov processes rather explicit answers are given. Three examples are considered some in detail: Gaussian autoregressions, the pure birth process and the Ornstein-Uhlenbeck process. Finally, a goodness-of-fit test for censored data is discussed.     

The influence of the nonrecent past in prediction for stochastic processes

Consider the stochastic processes X₁, X₂,… and Λ₁, Λ₂,… where the X process can be thought of as observations on the Λ process. We investigate the asymptotic behavior of the conditional distributions of X_t+v given X₁,…, X_t and Λ_t+v given X₁,…, X_t with regard to their dependency on the “early” part of the X process. These distributions arise in various time series and sequential decision theory problems. The results support the intuitively reasonable and often used (as a basic tenet of model building) assumption that only the more recent past is needed for near optimal prediction.     

Locally most powerful sequential tests for stochastic processes

For a continuous time stochastic process with distribution P_? depending on a one-dimensional parameter ? the problem of sequentially testing ? = 0 against ? > 0 is treated. We assume that the process of likelihood ratios has a certain representation which allows to obtain identities of the Wald type for stopping times. These identities are then used to derive a result on locally most powerful tests for which a problem of optimal stopping is solved.     

Parametric estimation of discretely sampled Gamma-OU processes

The stationary Gamma-OU processes are recommended to be the volatility of the financial assets. A parametric estimation for the Gamma-OU processes based on the discrete observations is considered in this paper. The estimator of an intensity parameter A and its convergence result are given, and the simulations show that the estimation is quite accurate. Assuming that the parameter A is estimated, the maximum likelihood estimation of shape parameter c and scale parameter a, whose likelihood function is not explicitly computable, is considered. By means of the Gaver-Stehfest algorithm, we construct an explicit sequence of approximations to the likelihood function and show that it converges the true (but unkown) one. Maximizing the sequence results in an estimator that converges to the true maximum likelihood estimator and the approximation shares the asymptotic properties of the true maximum likelihood estimator. Some simulation experiments reveal that this method is still quite accurate in most of rational situations for the background of volatility.     

Linearized filtering of affine processes using stochastic Riccati equations

We consider an affine process $X$ which is only observed up to an additive white noise, and we ask for the law of $X_t$, for some $t>0$, conditional on all observations up to time $t$. This is a general, possibly high dimensional filtering problem which is not even locally approximately Gaussian, whence essentially only particle filtering methods remain as solution techniques. In this work we present an efficient numerical solution by introducing an approximate filter for which conditional characteristic functions can be calculated by solving a system of generalized Riccati differential equations depending on the observation and the process characteristics of $X$. The quality of the approximation can be controlled by easily observable quantities in terms of a macro location of the signal in state space. Asymptotic techniques as well as maximization techniques can be directly applied to the solutions of the Riccati equations leading to novel very tractable filtering formulas. The efficiency of the method is illustrated with numerical experiments for Cox–Ingersoll–Ross and Wishart processes, for which Gaussian approximations usually fail.     

Discrete-space partial dynamic equations on time scales and applications to stochastic processes

We consider a general class of discrete-space linear partial dynamic equations. The basic properties of solutions are provided (existence and uniqueness, sign preservation, maximum principle). Above all, we derive the following main results: first, we prove that the solutions depend continuously on the choice of the time scale. Second, we show that, under certain conditions, the solutions describe probability distributions of nonhomogeneous Markov processes, and that their time integrals remain the same for all underlying regular time scales.