Cox Point Processes Driven by Ornstein–Uhlenbeck Type Processes

The paper is devoted to the development of Cox point processes driven by nonnegative processes of Ornstein–Uhlenbeck (OU) type. Starting with multivariate temporal processes we develop formula for the cross pair correlation function. Further filtering problem is studied by means of two different approaches, either with discretization in time or through the point process densities with respect to the Poisson process. The first approach is described mainly analytically while in the second case we obtain numerical solution by means of MCMC. The Metropolis–Hastings birth–death chain for filtering can be also used when estimating the parameters of the model. In the second part we try to develop spatial and spatio-temporal Cox point processes driven by a stationary OU process. The generating functional of the point process is derived which enables evaluation of basic characteristics. Finally a simulation algorithm is given and applied.      

Convergence of the SMC Implementation of the PHD Filte

The probability hypothesis density (PHD) filter is a first moment approximation to the evolution of a dynamic point process which can be used to approximate the optimal filtering equations of the multiple-object tracking problem. We show that, under reasonable assumptions, a sequential Monte Carlo (SMC) approximation of the PHD filter converges in mean of order , and hence almost surely, to the true PHD filter. We also present a central limit theorem for the SMC approximation, show that the variance is finite under similar assumptions and establish a recursion for the asymptotic variance. This provides a theoretical justification for this implementation of a tractable multiple-object filtering methodology and generalises some results from sequential Monte Carlo theory.      

Risk minimizing hedging for a partially observed high frequency data model

Risk-minimizing hedging strategies for contingent claims are studied in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described throught a marked point process Y, whose local characteristics depend on some unobservable hidden state variable X. In the model presented the processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. The hedger is restricted to observing past asset prices. Thus, we are in presence not only of an incomplete market situation but also of partial information. Considering the case where the price of the risky asset is modeled directly under a martingale measure, the computation of the risk-minimizing hedging strategy under this partial information is obtained by using a projection result (M. Schweizer, Risk minimizing hedging strategies under restricted information, Mathematical Finance 4 (1994) 327–342). This approach leads to a filtering problem with marked point process observations whose solution, obtained via the Kushner-Stratonovich equation, allows us to provide a complete solution to the heding problem.     

Filtering of Finite-State Time-Nonhomogeneous Markov Processes, a Direct Approach

A filtering equation is derived for P(x _t =x|y _s ,s∈[0,t]) for a continuous-time finite-state two-component time-nonhomogeneous cadlag Markov process z _t =(x _t ,y _t ) . The derivation is based on some new ideas in the filtering theory and does not require any knowledge of stochastic integration. Accepted 10 August 1999. Online publication 13 November 2000.     

Generalizations of Rodrigues type formulas for hypergeometric difference equations on nonuniform lattices

ABSTRACT

By building a second-order adjoint difference equations on nonuniform lattices, the generalized Rodrigues type representation for the second kind solution of a second-order difference equation of hypergeometric type on nonuniform lattices is given. The general solution of the equation in the form of a combination of a standard Rodrigues formula and a ‘generalized’ Rodrigues formula is also established.     

Filtering of a Markov Jump Process with Counting Observations

This paper concerns the filtering of an R ^d -valued Markov pure jump process when only the total number of jumps are observed. Strong and weak uniqueness for the solutions of the filtering equations are discussed. Accepted 12 November 1999     

Risk Sensitive Filtering with Poisson Process Observations

In this paper we consider risk sensitive filtering for Poisson process observations. Risk sensitive filtering is a type of robust filtering which offers performance benefits in the presence of uncertainties. We derive a risk sensitive filter for a stochastic system where the signal variable has dynamics described by a diffusion equation and determines the rate function for an observation process. The filtering equations are stochastic integral equations. Computer simulations are presented to demonstrate the performance gain for the risk sensitive filter compared with the risk neutral filter. Accepted 23 July 1999     

Robust Dynamics and Control of a Partially Observed Markov Chain

In a seminal paper, Martin Clark (Communications Systems and Random Process Theory, Darlington, 1977, pp. 721–734, 1978) showed how the filtered dynamics giving the optimal estimate of a Markov chain observed in Gaussian noise can be expressed using an ordinary differential equation. These results offer substantial benefits in filtering and in control, often simplifying the analysis and an in some settings providing numerical benefits, see, for example Malcolm et al. (J. Appl. Math. Stoch. Anal., 2007, to appear). Clark’s method uses a gauge transformation and, in effect, solves the Wonham-Zakai equation using variation of constants. In this article, we consider the optimal control of a partially observed Markov chain. This problem is discussed in Elliott et al. (Hidden Markov Models Estimation and Control, Applications of Mathematics Series, vol. 29, 1995). The innovation in our results is that the robust dynamics of Clark are used to compute forward in time dynamics for a simplified adjoint process. A stochastic minimum principle is established.     

Nonlinear Filtering for Diffusions in Random Environments

Suppose that the signal X to be estimated is a diffusion process in a random medium W and the signal is correlated with the observation noise. We study the historical filtering problem concerned with estimating the signal path up until the current time based upon the back observations. Using Dirichlet form theory, we introduce a filtering model for general rough signal X ^W and establish a multiple Wiener integrals representation for the unnormalized pathspace filtering process. Then, we construct a precise nonlinear filtering model for the process X itself and give the corresponding Wiener chaos decomposition.     

On a Boundary Value Problem in the Filtering Theory for Discrete Volterra Equations

Abstract

A boundary value problem that arises in the filtering theory for discrete Volterra equations is considered. An important dependence between primal and adjoint variables is obtained.     

Stochastic integral representations,stochastic derivatives and minimal variance hedging

The stochastic integral representation for an arbitrary random variable in a standard L 2 -space is considered in the case of the integrator as a martingale. In relation to this, a certain stochastic derivative is defined. It is shown that this derivative determines the integrand in the stochastic integral which serves as the best L 2 - approximation to the random variable considered. For a general Lévy process as integrator some specification of the suggested stochastic derivative is given. In the case of the Wiener process the considered specification reduces to the well-known Clark-Haussmann-Ocone formula. This result provides a general solution to the problem of minimal variance hedging in incomplete markets.     

On the Feynman-Kac formula and its applications to filtering theory

In this article the Feynman-Kac formula is obtained for a Markov process (X _t) whose transition probability function is not stationary. A converse to the Feynman-Kac formula is also obtained. This is used to prove the uniqueness of the solution to a measure-valued equation satisfied by the optimal filter in the white-noise approach to nonlinear filtering theory.Research partially supported by the Air Force Office of Scientific Research Contract No. F49620 85 C 0144 and by the Indian Statistical Institute.     

Martingale Representation of Functionals of Lévy Processes

Abstract

The main focus of the paper is a Clark–Ocone–Haussman formula for Lévy processes. First a difference operator is defined via the Fock space representation of L ²(P), then from this definition a Clark–Ocone–Haussman type formula is derived. We also derive some explicit chaos expansions for some common functionals. Later we prove that the difference operator defined via the Fock space representation and the difference operator defined by Picard [Picard, J. Formules de dualitésur l'espace de Poisson. Ann. Inst. Henri Poincaré 1996, 32 (4), 509–548] are equal. Finally, we give an example of how the Clark–Ocone–Haussman formula can be used to solve a hedging problem in a financial market modelled by a Lévy process.     

Stochastic Volatility Model with Filtering

Abstract

We generalize the stochastic volatility model by allowing the volatility to follow different dynamics in different states of the world. The dynamics of the “states of the world” are represented by a Markov chain. We estimate all the parameters by using the filtering and the EM algorithms. Closed form estimates for all parameters are derived in this paper. These estimates can be updated using new information as it arrives.     

Implied Filtering Densities on the Hidden State of Stochastic Volatility

Abstract

We formulate and analyse an inverse problem using derivative prices to obtain an implied filtering density on volatility’s hidden state. Stochastic volatility is the unobserved state in a hidden Markov model (HMM) and can be tracked using Bayesian filtering. However, derivative data can be considered as conditional expectations that are already observed in the market, and which can be used as input to an inverse problem whose solution is an implied conditional density on volatility. Our analysis relies on a specification of the martingale change of measure, which we refer to as separability. This specification has a multiplicative component that behaves like a risk premium on volatility uncertainty in the market. When applied to SPX options data, the estimated model and implied densities produce variance-swap rates that are consistent with the VIX volatility index. The implied densities are relatively stable over time and pick up some of the monthly effects that occur due to the options’ expiration, indicating that the volatility-uncertainty premium could experience cyclic effects due to the maturity date of the options.     

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.     

Bayes Formula for Optimal Filter with n-ple Markov Gaussian Errors

We consider the nonlinear filtering problem where the observation noise process is n-ple Markov Gaussian. A Kallianpur–Striebel type Bayes formula for the optimal filter is obtained.     

Methoden der Differentialgeometrie in Himmelsmechanik und Astronomie

With methods of differential geometry we can find three laws of Kepler type for the restricted 3-body problem. The first law is a geodesic equation, the third law is comparable with the formula of Gauß-Bonnet. Applications to astronomy are mentioned. For proofs and applications see [1].     

Modelling a Multitype Branching Brownian Motion: Filtering of a Measure-Valued Process

In the framework of marked trees, a multitype branching brownian motion, described by measure-valued processes, is studied. By applying the strong branching property, the Markov property and the expression of the generator are derived for the process whose components are the measure-valued processes associated to each type particles. The conditional law of the measure-valued process describing the whole population observing the cardinality of the subpopulation of a given type particles is characterized as the unique weak solution of the Kushner‐Stratonovich equation. An explicit representation of the filter is obtained by Feyman–Kac formula using the linearized filtering equation.     

Large Deviation Principle for Optimal Filtering

Abstract. We derive a large deviation principle for the optimal filter where the signal and the observation processes take values in conuclear spaces. The approach follows from the framework established by the author in an earlier paper. The key is the verification of the exponential tightness for the optimal filtering process and the exponential continuity of the coefficients in the Zakai equation.