A Non-Skorohod Topology on the Skorohod Space

Abstract

A new topology (called S) is defined on the space ID of functions x : [0, 1] → IR 1 which are right-continuous and admit limits from the left at each t > 0. Although S cannot be metricized, it is quite natural and shares many useful properties with the traditional Skorohod's topologies J 1 and M 1 . In particular, on the space P(ID) of laws of stochastic processes with trajectories in ID the topology S induces a sequential topology for which both the direct and the converse Prohorov's theorems are valid, the a.s. Skorohod representation for subsequences exists and finite dimensional convergence outside a countable set holds.

Homogenization of a Degenerate Pde With Non Linear Wentzell Boundary Condition

“…, R 2 , where we equip the first factor with the sup-norm topology, and the second factor with the S-topology of Jakubowski (see [1]). Considering the SDE and the BSDE satisfying respectively by X and Y :…”

Weak solutions of backward stochastic differential equations with continuous generator

“…are generated by the projection mappings π t : x → x(t) for t ∈ [0, T ]; we shall see later that these sets are the Borel sets of the topology S of A. Jakubowski [21]. )…”

“…Pseudopaths were invented by Dellacherie and Meyer [14], actually they are Young measures on the state space (see Subsection 3.4 for the definition of Young measures). The success of Meyer-Zheng topology comes from a tightness criterion which is easily satisfied and ensures that all limits have their trajectories in the Skorokhod space D. We use here the fact that Meyer-Zheng's criterion also yields tightness for Jakubowski's stronger topology S on D [21]. Note that the result of Buckdahn, Engelbert and Rȃşcanu [11,Theorem 4.6] is more general than ours in the sense that f in [11] depends functionally on Y , more precisely, their generator f (t, x, y) is defined on [0, T ] × D × D. Furthermore, in [11], W is only supposed to be a càdlàg martingale.…”

Homogenization of a semilinear parabolic PDE with locally periodic coefficients: a probabilistic approach

“…We can then solve the Poisson equation 9) and carry on with the usual line of proof (see e.g. the introduction in [2]), provided we have sufficient regularity.…”

“…Recall that in the linear case we first establish a tightness result for the family of processes X , > 0, in the space C( [0, T ] , R d ) endowed with the sup-norm and proceed to identify the limit via an ergodic theorem and a martingale problem formulation. In the non-linear case however, it seems difficult to work out tightness results for the process Y (and the related martingale M , see (2.17)) in C( [0, T ] , R d ) endowed with the sup-norm and it turns out that the weaker topology of Jakubowski [9] on D( [0, T ] , R d+1 ) is convenient, see also [11] where a tightness criterion is established (actually relaxed by Kurtz [10]). Moreover, it is important to note that given our formal assumptions on the coefficients, a natural stability argument, first devised in [5] and used below with a slight modification, seems to be necessary since the family of processes Z , > 0, does not seem to converge.…”