Showing papers on "Semimartingale published in 1984"

Tightness criteria for laws of semimartingales

Abstract: On montre que si l'on munit l'espace des trajectoires cādlāg de la topologie de la convergence en mesure (beaucoup plus faible que la topologie des Skorohod), on obtient des criteres tres commodes de compacite etroite pour des ensembles bornes de lois de quasimartingales et de semimartingales. On etudie la stabilite de diverses classes de processus pour ce mode de convergence

A stopped Doob inequality for stochastic convolution integrals and stochastic evolution equations

Abstract: A stopped Doob inequality is proved for stochastic convolution integrals in Hilbert space, where M is a square integrable Hilbert space valued cadlag martingale, ⊘ an operator valued predictable function and U(t, s) a contraction-type evolution operator. This allows to obtain the mild solution for evolution equations (with memory) where A(t) is the quasigenerator of U(t,s), V(t) a bounded variation process, and Z(t) a semimartingale, under the same weak assumptions on B and D as for stochastic differential equations with bounded coefficients, i .e ., (A( t) = 0) . Moreover, a weak convergence criterion for is derived.

Integral representation in the theory of continuous trading

Abstract: This paper deals with predictable representation in continuous trading. First of all we give a positive answer to a conjecture of Harrison and Pliska [2]. Then we prove the completeness of a model of mixed type (see[2]). A slight modificationof this example shows that the continuous and purely discontinuous parts of a local martingale depend on the filtration.

Synonymity, Generalized Martingales, and Subfiltration

Abstract: Aldous recently introduced the notion of synonymity of stochastic processes, a notion of equivalence for processes on a stochastic basis which generalizes the notion of "having the same distribution". We show that generalized martingale properties, such as the semimartingale property, are preserved under synonymity, and that synonymous semimartingales have decompositions with the same distribution law. A variation of our method yields a relatively elementary proof of the theorem of Stricker that semimartingale remains a semimartingale with respect to any subfiltration to which it is adapted.

Stochastic integrals on general topological measurable spaces

Abstract: A general theory of stochastic integral in the abstract topological measurable space is established. The martingale measure is defined as a random set function having some martingale property. All square integrable martingale measures constitute a Hilbert space M 2. For each μ∈M 2, a real valued measure 〈μ〉 on the predictable σ-algebra ℘ is constructed. The stochastic integral of a random function \(\mathfrak{h} \in L^2 \left( {\left\langle \mu \right\rangle } \right)\) with respect to μ is defined and investigated by means of Riesz's theorem and the theory of projections. The stochastic integral operator I μis an isometry from L 2(〈μ〉) to a stable subspace of M 2, its inverse is defined as a random Radon-Nikodym derivative. Some basic formulas in stochastic calculus are obtained. The results are extended to the cases of local martingale and semimartingale measures as well.

Hazard Rates and Probability Distributions: Representation of Random Intensities

Abstract: Recent attempts to apply the results of martingale theory in probability theory have shown that it is first necessary to interpret this abstract mathematical theory in more conventional terms. One example of this is the need to obtain a representation of the dual predictable projections (compensators) used in martingale theory in terms of probability distributions. However, up to now a representation of this type has been derived only for one special case. In this paper, the author gives probabilistic representations of the dual predictable projection of integer-valued random measures that correspond to jumps in a semimartingale with respect to the sigma-algebras generated by this process. The results are of practical importance because such dual predictable projections are usually interpreted as random intensities or hazard rates related to jumps in trajectories: applications are found in such fields as mathematical demography and risk analysis.