Martingale Additive Functionals Research Articles

Backward problems for stochastic differential equations on the Sierpinski gasket

In this paper, we study the non-linear backward problems (with deterministic or stochastic durations) of stochastic differential equations on the Sierpinski gasket. We prove the existence and uniqueness of solutions of backward stochastic differential equations driven by Brownian martingale (defined in Section 2) on the Sierpinski gasket constructed by S. Goldstein and S. Kusuoka. The exponential integrability of quadratic processes for martingale additive functionals is obtained, and as an application, a Feynman–Kac representation formula for weak solutions of semi-linear parabolic PDEs on the gasket is also established.

A Note on Time-Dependent Additive Functionals

This note develops shortly the theory of time-inhomogeneous additive functionals and is a useful support for the analysis of time-dependent Markov processes and related topics. It is a significant tool for the analysis of BSDEs in law. In particular we extend to a non-homogeneous setup some results concerning the quadratic variation and the angular bracket of Martin-gale Additive Functionals (in short MAF) associated to a homogeneous Markov processes.

Fukushima type decomposition for semi-Dirichlet forms

We present a Fukushima type decomposition in the setting of general quasi-regular semi-Dirichlet forms. The decomposition is then employed to give a transformation formula for martingale additive functionals. Applications of the results to some concrete examples of semi-Dirichlet forms are given at the end of the paper. We discuss also the uniqueness question about the Doob-Meyer decomposition on optional sets of interval type.

Stochastic calculus over symmetric Markov processes with time reversal

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

Stochastic calculus over symmetric Markov processes with time reversal

AbstractWe develop stochastic calculus for symmetric Markov processes in terms of time reversal operators. For this, we introduce the notion of the progressively additive functional in the strong sense with time-reversible defining sets. Most additive functionals can be regarded as such functionals. We obtain a refined formula between stochastic integrals by martingale additive functionals and those by Nakao's divergence-like continuous additive functionals of zero energy. As an application, we give a stochastic characterization of harmonic functions on a domain with respect to the infinitesimal generator of semigroup on L2-space obtained by lower-order perturbations.

Errata Stochastic calculus over symmetric Markov processes without time reversal

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao's divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized It\^{o} formula. (With Errata.)

Stochastic calculus over symmetric Markov processes without time reversal

We refine stochastic calculus for symmetric Markov processes without using time reverse operators. Under some conditions on the jump functions of locally square integrable martingale additive functionals, we extend Nakao’s divergence-like continuous additive functional of zero energy and the stochastic integral with respect to it under the law for quasi-everywhere starting points, which are refinements of the previous results under the law for almost everywhere starting points. This refinement of stochastic calculus enables us to establish a generalized Fukushima decomposition for a certain class of functions locally in the domain of Dirichlet form and a generalized Ito formula.

The law of iterated logarithm for additive functionals and martingale additive functionals of Harris recurrent Markov processes

We use Nummelin splitting in continuous time in order to prove laws of iterated logarithm for additive functionals of a Harris recurrent Markov process, with deterministic or random renormalization.

Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDE’s $$(\partial _{t}+L)u(t,x)+f(t,x,u,\nabla u\sigma )=0,\quad \forall 0\leqslant t\leqslant T,$$ $$u(T,x)=\Phi (x).$$ We solve this system in the framework of Dirichlet spaces and employ the symmetric Markov process of infinitesimal operator L in order to obtain a precised version of the solution u by solving the corresponding system of backward stochastic differential equations. This precised version verifies pointwise the so called “mild equation”, which is equivalent to the above PDE. As a technical ingrediend we prove a representation theorem for arbitrary martingales which generalises a result of Fukushima for martingale additive functionals. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.

Backward Stochastic Differential Equations Associated to a Symmetric Markov Process

We consider a second order semi-elliptic differential operator L with measurable coefficients, in divergence form, and the semilinear parabolic system of PDE’s $$\begin{gathered} \left( {\partial _t + L} \right)u\left( {t,x} \right) + f\left( {t,x,u,\nabla u\sigma } \right) = \forall 0 \leqslant t \leqslant T, \hfill \\ u\left( {T,x} \right) = \Phi \left( x \right). \hfill \\ \end{gathered} $$ We solve this system in the framework of Dirichlet spaces and employ the symmetric Markov process of infinitesimal operator L in order to obtain a precised version of the solution u by solving the corresponding system of backward stochastic differential equations. This precised version verifies pointwise the so called “mild equation”, which is equivalent to the above PDE. As a technical ingrediend we prove a representation theorem for arbitrary martingales which generalises a result of Fukushima for martingale additive functionals. The nonlinear term f satisfies a monotonicity condition with respect to u and a Lipschitz condition with respect to ∇u.

The Principle of Large Deviations for Martingale Additive Functionals of Recurrent Markov Processes

We give a principle of large deviations for a generalized version of the strong central limit theorem. This generalized version deals with martingale additive functionals of a recurrent Markov process.

Representation of martingale additive functionals on Banach spaces

Assume thatB is a separable real Banach space andX(t) is a diffusion process onB. In this paper, we will establish the representation theorem of martingale additive functionals ofX(t).

On a martingale representation theorem of m. h. a. davis-a markovian approach

In this paper, we give a Markovian proof of a well-known result of M. H. A. Davis on representation of square-integrable martingales associated with arbitrary pure-jump processes. The main tool used to replace the original (non-Markovian) set up by a Markovian one is an interesting idea originally due to F. B. Knight. The main analysis then follows the classical line of investigating structures of martingale additive functionals of a Markov process, wherein a Levy system of the Markov process (in the classical sense-due to S. Watanabe) plays a vital role

Integral of Differential Forms along the Path of Diffusion Processes

Let (Xt, S t, PX) be a Brownian motion on a Riemannian manifold M and a be a differential 1-form on M. In this paper we will be concerned with the integral of a along Xt. This integral is a stochastic version of the ordinary integral of the form a along a smooth curve on M and is defined by using the symmetric integral. We denote by A(t\a), tl>Q, the integrals of a along Xt. The one-parameter family A= {A(t a); £^>0} of random variables then defines a continuous additive functional of (Xt, 2^, P,) . In Section 3 we will show that A(t; a), tl>0, can be decomposed into a sum of a local martingale and a bounded variation process which is expressed by the divergence of a. The structure of the local martingale part will be analyzed by using the lifted diffusion (rt, 3t, Pr) on O(M) of (Xt, S t, PX) through the Riemannian connection where O(M) is the bundle of orthonormal frames. Next in Section 4, using some results in Section 3 we will give a representation theorem for continuous square integrable martingale additive functionals of (Xt9 31, PX) which was obtained, in some special cases, by a number of authors (cf. [11], [14], [15], [16], [17]). As an application of Theorem 3. 1, we discuss in Section 5 the Cameron-Martin formula. An approximation theorem similar to Nakao-Yamato [12] also holds in our case. Using this we will formulate and prove a stochastic version of Stokes' theorem. M. Yor [18] recently discussed a closely related subject independently in the case that M= C.