Showing papers by "David Criens published in 2023"
18 Apr 2023
TL;DR: In this paper , a convergence theorem for Crandall-Lions viscosity solutions to path-dependent Hamilton-Jacobi-Bellman PDEs is established based on dynamic sublinear expectations.
Abstract: We establish a convergence theorem for Crandall-Lions viscosity solutions to path-dependent Hamilton-Jacobi-Bellman PDEs. Our proof is based on a novel convergence theorem for dynamic sublinear expectations and the stochastic representation of viscosity solutions as value functions.
TL;DR: In this paper , the propagation of chaos property for weakly interacting mild solutions to semilinear stochastic partial differential equations whose coefficients might not satisfy Lipschitz conditions was investigated.
Abstract: Abstract This article investigates the propagation of chaos property for weakly interacting mild solutions to semilinear stochastic partial differential equations whose coefficients might not satisfy Lipschitz conditions. Furthermore, we establish existence and uniqueness results for mild solutions to SPDEs with distribution dependent coefficients, so-called McKean–Vlasov SPDEs.
TL;DR: In this article , the authors established a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the sets of locally finite speed measures for suitable topologies, and proved a continuous dependence of the speed measures on their diffusions.
Abstract: Abstract It is well known that the law of a one-dimensional diffusion on natural scale is fully characterized by its speed measure. Stone proved a continuous dependence of such diffusions on their speed measures. In this paper we establish the converse direction, i.e., we prove a continuous dependence of the speed measures on their diffusions. Furthermore, we take a topological point of view on the relation. More precisely, for suitable topologies, we establish a homeomorphic relation between the set of regular diffusions on natural scale without absorbing boundaries and the set of locally finite speed measures.
TL;DR: In this paper , it was shown that the Feller-dynkin and martingale properties are equivalent for regular continuous strong Markov processes on natural scale with open state space.
Abstract: We show that a one-dimensional regular continuous strong Markov process \(X\) with scale function \(s\) is a Feller-Dynkin process precisely if the space transformed process \(s (X)\) is a martingale when stopped at the boundaries of its state space. As a consequence, the Feller-Dynkin and the martingale property are equivalent for regular diffusions on natural scale with open state space. Furthermore, for Ito diffusions we discuss relations to existence and uniqueness properties of Cauchy problems, and we identify the infinitesimal generator.
20 Jun 2023
TL;DR: In this article , it was shown that the no-free-lunch with vanishing risk (NFLVR) condition is equivalent to the weaker no unbounded profit with bounded risk (NUPBR) condition in the canonical framework.
Abstract: Consider a single asset financial market whose discounted asset price process is a stochastic integral with respect to a continuous regular strong Markov semimartingale (a so-called general diffusion semimartingale) that is parameterized by a scale function and a speed measure. In a previous paper, we established a characterization of the no free lunch with vanishing risk (NFLVR) condition for a canonical framework of such a financial market in terms of the scale function and the speed measure. Ioannis Karatzas (personal communication) asked us whether it is also possible to prove a characterization for the weaker no unbounded profit with bounded risk (NUPBR) condition, which is the main question we treat in this paper. Here, we do not restrict our attention to canonical frameworks but we allow a general setup with a general filtration that preserves the strong Markov property. Our main results are precise characterizations of NUPBR and NFLVR which only depend on the scale function and the speed measure. In particular, we prove that NUPBR forces the scale function to be continuously differentiable with absolutely continuous derivative. The latter extends our previous result, that, in the canonical framework, NFLVR implies such a property, in two directions (a weaker no-arbitrage notion and a more general framework). We also make the surprising observation that NUPBR and NFLVR are equivalent whenever finite boundary points are accessible for the driving diffusion.