[Paper Review] On Deterministic Markov Processes: Expandability and Related Topics

Alexander Schnurr|arXiv (Cornell University)|Nov 8, 2011

Stochastic processes and financial applications17 references3 citations

TL;DR

This paper establishes criteria for determining whether a given deterministic function $ f:[0,\infty)\to\mathbb{R}^d $ can be a path of a deterministic Markov process, particularly focusing on semimartingale and Itô process structures. The key contribution is a characterization of the semimartingale property for one-dimensional deterministic Markov processes in terms of their jump behavior, revealing that such processes are more complex than Hunt processes with jump structures.

ABSTRACT

We treat the class of universal Markov processes on the d-dimensional Euklidean space which do not depend on random. For these, as well as for several subclasses, we prove criteria whether a function f, defined on the positive half-line, can be a path of a process in the respective class. This is useful in particular in the construction of (counter-)examples. Furthermore we characterize the processes of this kind, which are homogeneous in space and time. The semimartingale property is characterized in terms of the jumps of a one-dimensional deterministic Markov process. We emphasize the differences between the time homogeneous and the time inhomogeneous case and we show that a deterministic Markov process is in general more complicated than a Hunt process plus 'jump structure'.

Motivation & Objective

• To determine when a deterministic function $ f:[0,\infty)\to\mathbb{R}^d $ can be a path of a deterministic Markov process.
• To clarify the structural differences between time-homogeneous and time-inhomogeneous deterministic Markov processes.
• To investigate whether deterministic Markov processes can be semimartingales and, if so, under what conditions.
• To address the limitations of modeling such processes as Hunt processes with added jumps.

Proposed method

• The paper uses the concept of expandability to determine whether a given path can be embedded into a deterministic Markov process.
• It characterizes the semimartingale property for one-dimensional deterministic Markov processes via their jump behavior and path regularity.
• The analysis relies on the symbol of the generator and the time-inhomogeneous structure of the process, particularly in the space-time extension.
• It introduces injective path mappings $ \Phi^x(t) = (X_t^x, t) $ to study the time-inhomogeneous case and derive the generator symbol.
• The paper constructs explicit counterexamples using piecewise linear and self-similar paths to demonstrate non-Markovian behavior and path complexity.
• It applies techniques from stochastic analysis, including characteristics of semimartingales and the martingale problem, to analyze path properties.

Experimental results

Research questions

• RQ1 Can a given deterministic path $ f:[0,\infty)\to\mathbb{R}^d $ be the path of a deterministic Markov process in a given class (e.g., semimartingale, Itô process)?
• RQ2 What characterizes the semimartingale property for one-dimensional deterministic Markov processes in terms of their path structure and jumps?
• RQ3 How do time-homogeneous and time-inhomogeneous deterministic Markov processes differ in their path behavior and structural complexity?
• RQ4 Can the sum of two time-homogeneous deterministic Markov processes remain time-homogeneous and Markovian?
• RQ5 To what extent can deterministic Markov processes be modeled as Hunt processes with additional jump components?

Key findings

• A deterministic Markov process is not necessarily a semimartingale, and the semimartingale property can be characterized by the jump behavior of its one-dimensional paths.
• There exist deterministic Markov processes that are not of finite variation and do not exhibit monotonicity, even in one dimension.
• The sum of two time-homogeneous deterministic Markov processes is not necessarily time-homogeneous or Markovian, as demonstrated by explicit counterexamples.
• A deterministic process that reaches infinity in finite time can be constructed via a combination of linear drift and jump structures, even under time homogeneity.
• The space-time extension of a time-inhomogeneous deterministic Markov process results in a time-homogeneous process, but the generator symbol becomes path-dependent and non-trivial.
• The symbol of the generator for the space-time process depends on the right-hand derivative of the path at the corresponding point, and this expression is well-defined due to the Markov property.