Relations between Markov Processes via Local Time and Coordinate Transformations

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

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Markov processes with spatial delay: Path space characterization, occupation time and properties

Stochastics and Dynamics ◽

10.1142/s0219493717500423 ◽

2017 ◽

Vol 17 (06) ◽

pp. 1750042 ◽

Cited By ~ 3

Author(s):

Michael Salins ◽

Konstantinos Spiliopoulos

Keyword(s):

Local Time ◽

Markov Processes ◽

Single Point ◽

Occupation Time ◽

One Dimensional ◽

Stochastic Dynamical System ◽

Second Derivatives ◽

Continuous Markov Process ◽

Positive Time ◽

Definition Of

In this paper, we study one-dimensional Markov processes with spatial delay. Since the seminal work of Feller, we know that virtually any one-dimensional, strong, homogeneous, continuous Markov process can be uniquely characterized via its infinitesimal generator and the generator’s domain of definition. Unlike standard diffusions like Brownian motion, processes with spatial delay spend positive time at a single point of space. Interestingly, the set of times that a delay process spends at its delay point is nowhere dense and forms a positive measure Cantor set. The domain of definition of the generator has restrictions involving second derivatives. In this paper we provide a pathwise characterization for processes with delay in terms of an SDE and an occupation time formula involving the symmetric local time. This characterization provides an explicit Doob–Meyer decomposition, demonstrating that such processes are semi-martingales and that all of stochastic calculus including Itô formula and Girsanov formula applies. We also establish an occupation time formula linking the time that the process spends at a delay point with its symmetric local time there. A physical example of a stochastic dynamical system with delay is lastly presented and analyzed.

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Nonparametric Estimation for a Class of Piecewise-Deterministic Markov Processes

Journal of Applied Probability ◽

10.1239/jap/1389370091 ◽

2013 ◽

Vol 50 (4) ◽

pp. 931-942

Author(s):

Takayuki Fujii

Keyword(s):

Local Time ◽

Markov Processes ◽

Nonparametric Estimation ◽

Sample Path ◽

Point Of View ◽

Statistical Point ◽

Jump Size ◽

Stationary Density ◽

Piecewise Deterministic Markov Processes ◽

Estimation Problems

In this paper we study nonparametric estimation problems for a class of piecewise-deterministic Markov processes (PDMPs). Borovkov and Last (2008) proved a version of Rice's formula for PDMPs, which explains the relation between the stationary density and the level crossing intensity. From a statistical point of view, their result suggests a methodology for estimating the stationary density from observations of a sample path of PDMPs. First, we introduce the local time related to the level crossings and construct the local-time estimator for the stationary density, which is unbiased and uniformly consistent. Secondly, we investigate other estimation problems for the jump intensity and the conditional jump size distribution.

Download Full-text