Parameter-dependent linear evolution equations with a fractional noise in the boundary conditions are studied. Ergodic-type theorems for stationary and non-stationary solutions are verified and used to prove the strong consistency of a suitably defined family of estimators.

This paper considers a discrete-time Markovian model of asset prices with economic factors and transaction costs with proportional and fixed terms. Existence of optimal strategies maximizing average growth rate of portfolio is proved in the case of complete and partial observation of the process modelling the economic factors. The proof is based on a modification of the vanishing discount approach. The main difficulty is the discontinuity of the controlled transition operator of the underlying Markov process.

Many problems modeled by Markov decision processes (MDPs) have very large state and/or action spaces, leading to the well-known curse of dimensionality that makes solution of the resulting models intractable. In other cases, the system of interest is complex enough that it is not feasible to explicitly specify some of the MDP model parameters, but simulated sample paths can be readily generated (e.g., for random state transitions and rewards), albeit at a non-trivial computational cost. For these settings, we have developed various sampling and population-based numerical algorithms to overcome the computational difficulties of computing an optimal solution in terms of a policy and/or value function. Specific approaches presented in this survey include multi-stage adaptive sampling, evolutionary policy iteration and evolutionary random policy search.

A common practice for stock-selling decision making is often concerned with liquidation of the security in a short duration. This is feasible when a relative smaller number of shares of a stock is treated. Selling a large position during a short period of time in the market frequently depresses the market, resulting in poor filling prices. In this work, liquidation strategies are considered for selling much smaller number of shares over a longer period of time. By using a fluid model in which the number of shares are treated as fluid, and the corresponding liquidation is dictated by the rate of selling over time. Our objective is to maximize the expected overall return. The problem is formulated as a stochastic control problem with state constraints. Using the method of constrained viscosity solutions, we characterize the dynamics governing the value function and the associated boundary conditions. Numerical algorithms are also provided along with an illustrative example for demonstration purposes.