The main goal of this paper is to study the parameter estimation problem, using the Bayesian methodology, for the drift coefficient of some linear (parabolic) SPDEs driven by a multiplicative noise of special structure. We take the spectral approach by assuming that one path of the first N Fourier modes of the solution is continuously observed over a finite time interval. First, we show that the model is regular and fits into classical local asymptotic normality framework, and thus the MLE and the Bayesian estimators are weakly consistent, asymptotically normal, efficient, and asymptotically equivalent in the class of loss functions with polynomial growth. Secondly, and mainly, we prove a Bernstein-Von Mises type result, that strengthens the existing results in the literature, and that also allows to investigate the Bayesian type estimators with respect to a larger class of priors and loss functions than that covered by classical asymptotic theory. In particular, we prove strong consistency and asymptotic normality of Bayesian estimators in the class of loss functions of at most exponential growth. Finally, we present some numerical examples that illustrate the obtained theoretical results.

The Kusuoka representation of proper lower semi-continuous law invariant coherent risk measures allows one to cast them in terms of average value-at-risk. Here, we introduce the notion of Kusuoka-type conditional risk-mappings and use it to define a dynamic risk measure. We use such dynamic risk measures to study infinite horizon Markov decision processes with random costs and random actions. Under mild assumptions, we derive a dynamic programming principle and prove the existence of an optimal policy. We also derive the Q-learning version of the dynamic programming principle, which is important for applications. Furthermore, we provide a sufficient condition for when deterministic actions are optimal.

This work contributes to the theory of Wiener-Hopf type factorization for finite Markov chains. This theory originated in the seminal paper [BRW80], which treated the case of finite time-homogeneous Markov chains. Since then, several works extended the results of [BRW80] in many directions. However, all these extensions were dealing with time-homogeneous Markov case. The first work dealing with the time-inhomogeneous situation was [BCGH18], where Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with piecewise constant generator matrix function was derived. In the present paper we go further: we derive and study Wiener-Hopf type factorization for time-inhomogeneous finite Markov chain with the generator matrix function being a fairly general matrix valued function of time.

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In this paper, we use mean field games (MFGs) to investigate approximations of N -player games with uniformly symmetrically continuous heterogeneous closed-loop actions. To incorporate agents' risk aversion (beyond the classical expected total costs), we use an abstract evaluation functional for their performance criteria. Centered around the notion of regret, we conduct non-asymptotic analysis on the approximation capability of MFGs from the perspective of state-action distribution without requiring the uniqueness of equilibria. Under suitable assumptions, we first show that scenarios in the N -player games with large N and small average regrets can be well approximated by approximate solutions of MFGs with relatively small regrets. We then show that δ-mean field equilibria can be used to construct ε-equilibria in N -player games. Furthermore, in this general setting, we prove the existence of mean field equilibria.