Reflexion Discontinue et Systemes Stochastiques

Regularizing effect for integro-differential parabolic equations

“…This theorem can be regarded as a generalization of the construction of reflected diffusion processes with jumps reported on Anulova [1,2], ChaleyatMaurel et al [5], Menaldi and Robin [24]. [7]).…”

Probabilistic aspects of finite-fuel, reflected follower problems

“…The above problem was studied by Chaleyat-Maurel, El Karoui and Marchal (1980). We summarize below the principal results of these authors, in the form that we shall find useful for our purposes.…”

“…In particular, one needs to consider a family of optimal stopping problems of the form We shall show that the analogue of the relation (1.6), in the new context, is given by DQ(r, x, y) = u(r, x; DQ(., 0, y)) (1. 10) i.e., by a functional equation for the directional derivative in (1.9). We shall use refinements of the direct, probabilistic methodology of Karatzas and Shreve (1985), in order to establish (1.10); in particular, the device of 'switching paths at appropriate random times' will be used quite often, to compare expected costs at nearby points.…”

On some cheap control problems for diffusion processes

“…Let us consider Taking X -a = 1 for the sake of simplicity, we look for x E R and u(x) solution of (5.5) such that we get a regular solution of (5.5), and one can show, using Itô's formula for semimartingales as in §1, that u(x) < Jx(v) for any adapted process v with bounded variation. Moreover, using the work of Chaleyat-Maurel, El Karoui and Marchai [9] on reflected diffusion with jump, we can say that there exists an increasing process v(t) corresponding to the generator (5.8) Lw(x) = -X(w(x-1) -w(x)), x^x, wx(x) = 0, and thus,…”