XV: 36, 529-546, LNM 850 (1981)
JACOD, Jean;
MÉMIN, Jean
Sur un type de convergence intermédiaire entre la convergence en loi et la convergence en probabilité (
Measure theory)
For simplicity we consider only real valued r.v.'s, but it is essential that the paper considers general Polish spaces instead of $
R$. Let us define a fuzzy r.v. $X$ on $(\Omega, {\cal F},P)$ as a probability measure on $\Omega\times
R$ whose projection on $\Omega$ is $P$. In particular, a standard r.v. $X$ defines such a measure as the image of $P$ under the map $\omega\mapsto (\omega,X(\omega))$. The space of fuzzy r.v.'s is provided with a weak topology, associated with the bounded functions $f(\omega,x)$ which are continuous in $x$ for every $\omega$, or equivalently with the functions $I_A(\omega)\,f(x)$ with $f$ bounded continuous. The main topic of this paper is the study of this topology
Comment: From this description, it is clear that this paper extends to general Polish spaces the topology of Baxter-Chacon (forgetting about the filtration), for which see
1228Keywords: Fuzzy random variables,
Convergence in lawNature: Original
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XV: 38, 561-586, LNM 850 (1981)
PELLAUMAIL, Jean
Solutions faibles et semi-martingales (
Stochastic calculus,
General theory of processes)
From the author's summary: ``we consider a stochastic differential equation $dX=a(X)\,dZ$ where $Z$ is a semimartingale and $a$ is a previsible functional which is continuous for the uniform norm. We prove the existence of a weak solution for such an equation''. The important point is the definition of a weak solution: it turns out to be a ``fuzzy process'' in the sense of
1536, i.e., a fuzzy r.v. taking values in the Polish space of cadlag sample functions
Keywords: Stochastic differential equations,
Weak solutions,
Fuzzy random variablesNature: Original Retrieve article from Numdam
