In this paper, we present a theoretical framework for studying coherent acceptability indices (CAIs) in a dynamic setup. We study dynamic CAIs (DCAIs) and dynamic coherent risk measures (DCRMs), and we establish a duality between them. We derive a representation theorem for DCRMs in terms of a so‐called dynamically consistent sequence of sets of probability measures. Based on these results, we give a specific construction of DCAIs. We also provide examples of DCAIs, both abstract and also some that generalize selected classical financial measures of portfolio performance.

The main goal of this paper is to build consistent and asymptotically normal estimators for the drift and volatility parameter of the stochastic heat equation driven by an additive space-only white noise when the solution is sampled discretely in the physical domain. We consider both the full space R and the bounded domain (0, π). First, we establish the exact regularity of the solution and its spatial derivative, which in turn, using power-variation arguments, allows building the desired estimators. Second, we propose two sets of estimators, based on sampling either the spatial derivative or the solution itself on a discrete space-time grid. Using the so-called Malliavin-Stein's method, we prove that these estimators are consistent and asymptotically normal as the spatial mesh-size vanishes. More importantly, we show that naive approximations of the derivatives appearing in the power-variation based estimators may create nontrivial biases, which we compute explicitly. We conclude with some numerical experiments that illustrate the theoretical results.

This paper provides a unified framework, which allows, in particular, to study the structure of dynamic monetary risk measures and dynamic acceptability indices. The main mathematical tool, which we use here, and which allows us to significantly generalize existing results is the theory of L 0 -modules. In the first part of the paper we develop the general theory and provide a robust representation of conditional assessment indices, and in the second part we apply this theory to dynamic acceptability indices acting on stochastic processes.

In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations. Using an implicit time stepping scheme, we transform stochastic parabolic equations into stochastic elliptic equations. Our main attention is concentrated on the numerical solution of the elliptic equations at each time step. The estimator of the solution of the elliptic equations is given as a linear combination of reproducing kernels derived from the differential and boundary operators of the PDE centered at collocation points to be chosen by the user. The random expansion coefficients are computed by solving a random system of linear equations. Numerical experiments demonstrate the feasibility of the method.

The aim of this work is to give an overview of the recent developments in the area of statistical inference for parabolic stochastic partial differential equations. Significant part of the paper is devoted to the spectral approach, which is the most studied sampling scheme under which the observations are done in the Fourier space over some finite time interval. We also discuss into details the practically important case of discrete sampling of the solution. Other relevant methodologies and some open problems are briefly discussed over the course of the manuscript.