In this paper we present a variant of the well known Skorokhod Representation Theorem. In our main result, given $S$ a Polish space, to a given continous path $α$ in the space of probability measures on $S$, we associate a continuous path in the space of $S$-valued random variables on a nonatomic probability space (endowed with the topology of the convergence in probability). We call this associated path a lifting of $α$. an interesting feature of our result is that we can fix the endpoints ("boundary values") of the lifting of $α$, as long as their distribution correspond to the endpoints ("boundary values") of $α$. We also discuss an $n$-dimensional generalization of this result.
Let $Φ'_β$ denotes the strong dual of a nuclear space $Φ$ and let $D_{T}(Φ'_β)$ be the Skorokhod space of right-continuous with left limits (càdlàg) functions from $[0,T]$ into $Φ'_β$. In this article we introduce the concepts of cylindrical random variables and cylindrical measures on $D_{T}(Φ'_β)$, and prove analogues of the regularization theorem and Minlos theorem for extensions of these objects to bona fide random variables and probability measures on $D_{T}(Φ'_β)$ respectively. Later, we establish analogues of Lévy's continuity theorem to provide necessary and sufficient conditions for u…
We prove a robust super-hedging duality result for path-dependent options on assets with jumps, in a continuous time setting. It requires that the collection of martingale measures is rich enough and that the payoff function satisfies some continuity property. It is a by-product of a quasi-sure version of the optional decomposition theorem, which can also be viewed as a functional version of It{ô}'s Lemma, that applies to non-smooth functionals (of c{à}dl{à}g processes) which are only concave in space and non-increasing in time, in the sense of Dupire.
In this article, we introduce an infinite-dimensional analogue of the $α$-stable Lévy motion, defined as a Lévy process $Z=\{Z(t)\}_{t \geq 0}$ with values in the space $\mathbb{D}$ of càdlàg functions on $[0,1]$, equipped with Skorokhod's $J_1$ topology. For each $t \geq 0$, $Z(t)$ is an $α$-stable process with sample paths in $\mathbb{D}$, denoted by $\{Z(t,s)\}_{s\in [0,1]}$. Intuitively, $Z(t,s)$ gives the value of the process $Z$ at time $t$ and location $s$ in space. This process is closely related to the concept of regular variation for random elements in $\mathbb{D}$ introduced in de H…
We construct Skorokhod decompositions for diffusions with singular drift and reflecting boundary behavior on open subsets of $\mathbb R^d$ with $C^2$-smooth boundary except for a sufficiently small set. This decomposition holds almost surely under the path measures of the process for every starting point from an explicitly known set. This set is characterized by the boundary smoothness and the singularities of the drift term. We apply modern methods of Dirichlet form theory and $\mathcal L^p$-strong Feller processes. These tools have been approved as useful for the pointwise analysis of stocha…
We show that the sequential closure of a family of probability measures on the canonical space of c{à}dl{à}g paths satisfying Stricker's uniform tightness condition is a weak${}^*$ compact set of semimartingale measures in the pairing of the Riesz representation theorem under topological assumptions on the path space. Similar results are obtained for quasi- and supermartingales under analogous conditions. In particular, we give a full characterization of the strongest topology on the Skorokhod space for which these results are true.
We study a random walk on a point process given by an ordered array of points $(ω_k, \, k \in \mathbb{Z})$ on the real line. The distances $ω_{k+1} - ω_k$ are i.i.d. random variables in the domain of attraction of a $β$-stable law, with $β\in (0,1) \cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between $ω_k$ and $ω_\ell$ depend on $\ell-k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an $α$-stable law, with $α\in (0,1) \cup (1,2)$. Since the defining variables, for both the random walk and the po…
We provide a systematic, thorough treatment of the foundations of probability theory and stochastic processes along the lines of E. Bishop's constructive analysis. Every existence result presented shall be a construction; and the input data, the construction procedure, and the output objects shall be regarded as integral parts of the theorem. A brief description of this approach is in Part I of this book. Part II develops basic topics in probability theory in this constructive framework, expanding on [Bishop and Bridges 1985, Springer], and in terms familiar to probabilists. Part III, the main…
Martingale representation theorem for set-valued martingales was proposed by M. Kisielewicz [J. Math. Anal. Appl. 2014]. We shall prove that the result holds only for very special case: the set-valued martingale degenerates to the point-valued one. A revised representation theorem for a special kind of non-degenerate set-valued martingales is presented.
