On characterisation of Markov processes via martingale problems

A general diffusion semimartingale is a one-dimensional path-continuous semimartingale that is also a regular strong Markov process. We say that a continuous semimartingale has the representation property if all local martingales w.r.t. its canonical filtration have an integral representation w.r.t. its continuous local martingle part. The representation property is of fundamental interest in the field of mathematical finance, where it is strongly connected to market complete...

In this paper we explain how the notion of ''weak Dirichlet process'' is the suitable generalization of the one of semimartingale with jumps. For such a process we provide a unique decomposition which is new also for semimartingales: in particular we introduce ''characteristics'' for weak Dirichlet processes. We also introduce a weak concept (in law) of finite quadratic variation. We investigate a set of new useful chain rules and we discuss a general framework of (possibly p...

This paper gives a complete characterization of infinitely divisible semimartingales, i.e., semimartingales whose finite dimensional distributions are infinitely divisible. An explicit and essentially unique decomposition of such semimartingales is obtained. A new approach, combining series decompositions of infinitely divisible processes with detailed analysis of their jumps, is presented. As an ilustration of the main result, the semimartingale property is explicitely deter...

This work shows how exponential concentration inequalities for additive functionals of stochastic processes over a finite time interval can be derived from concentration inequalities for martingales. The approach is entirely probabilistic and naturally includes time-inhomogeneous and non-stationary processes as well as initial laws concentrated on a single point. The class of processes studied includes martingales, Markov processes and general square integrable processes. The...

We provide verification theorems (at different levels of generality) for infinite horizon stochastic control problems in continuous time for semimartingales. The control framework is given as an abstract "martingale formulation", which encompasses a broad range of standard control problems. Under appropriate conditions we show that the set of admissible controls gives rise to a certain class of controlled special semimartingales. Our results generalise both the standard contr...

The information dynamics in finance and insurance applications is usually modeled by a filtration. This paper looks at situations where information restrictions apply such that the information dynamics may become non-monotone. A fundamental tool for calculating and managing risks in finance and insurance are martingale representations. We present a general theory that extends classical martingale representations to non-monotone information generated by marked point processes....

We develop a general method for extending Markov processes to a larger state space such that the added points form a polar set. The so obtained extension is an improvement on the standard trivial extension in which case the process is made stuck in the added points, and it renders a new technique of constructing extended solutions to S(P)DEs from all starting points, in such a way that they are solutions at least after any strictly positive time. Concretely, we adopt this str...

We identify the linear space spanned by the real-valued excessive functions of a Markov process with the set of those functions which are quasimartingales when we compose them with the process. Applications to semi-Dirichlet forms are given. We provide a unifying result which clarifies the relations between harmonic, co-harmonic, invariant, co-invariant, martingale and co-martingale functions, showing that in the conservative case they are all the same. Finally, using the co-...

Let $A$ be a pseudo-differential operator with negative definite symbol $q$. In this paper we establish a sufficient condition such that the well-posedness of the $(A,C_c^{\infty}(\mathbb{R}^d))$-martingale problem implies that the unique solution to the martingale problem is a Feller process. This provides a proof of a former claim by van Casteren. As an application we prove new existence and uniqueness results for L\'evy-driven stochastic differential equations and stable-l...

We give a collection of explicit sufficient conditions for the true martingale property of a wide class of exponentials of semimartingales. We express the conditions in terms of semimartingale characteristics. This turns out to be very convenient in financial modeling in general. Especially it allows us to carefully discuss the question of well-definedness of semimartingale Libor models, whose construction crucially relies on a sequence of measure changes.