This paper investigates a financial market where stock returns depend on a hidden Gaussian mean reverting drift process. Information on the drift is obtained from returns and expert opinions in the form of noisy signals about the current state of the drift arriving at the jump times of a homogeneous Poisson process. Drift estimates are based on Kalman filter techniques and described by the conditional mean and covariance matrix of the drift given the observations. We study the filter asymptotics for increasing arrival intensity of expert opinions and prove that the conditional mean is a consistent drift estimator, it converges in the mean-square sense to the hidden drift. Thus, in the limit as the arrival intensity goes to infinity investors have full information about the drift.

We introduce a new method to calculate the credit exposure of Bermudan, discretely monitored barrier and European options. Core of the approach is the application of the dynamic Chebyshev method of Glau et al. (2019). The dynamic Chebyshev method delivers a closed form approximation of the option prices along the paths together with the options' delta and gamma. Key advantage is the polynomial structure of the approximation, which allows us a highly efficient evaluation of the credit exposures, even for a large number of simulated paths. The approach is highly flexible in the model choice, payoff profiles and asset classes. We compute the exposure profiles for Bermudan and barrier options in three different equity models and compare them to the profiles of European options. The analysis reveals potential shortcomings of common simplifications in the exposure calculation. The proposed method is sufficiently simple and efficient to avoid such risk-bearing simplifications.

We investigate Monte Carlo based algorithms for solving stochastic control problems with probabilistic constraints. Our motivation comes from microgrid management, where the controller tries to optimally dispatch a diesel generator while maintaining low probability of blackouts. The key question we investigate are empirical simulation procedures for learning the admissible control set that is specified implicitly through a probability constraint on the system state. We propose a variety of relevant statistical tools including logistic regression, Gaussian process regression, quantile regression and support vector machines, which we then incorporate into an overall Regression Monte Carlo (RMC) framework for approximate dynamic programming. Our results indicate that using logistic or Gaussian process regression to estimate the admissibility probability outperforms the other options. Our algorithms offer an efficient and reliable extension of RMC to probability-constrained control. We illustrate our findings with two case studies for the microgrid problem.