**Preface to "Applications of Stochastic Optimal Control to Economics and Finance"**

In a world dominated by uncertainty, the modeling and understanding of the optimal behavior of agents are of the utmost importance. Many problems in economics, finance, and actuarial science naturally require decision-makers to undertake choices in stochastic environments. Examples include optimal individual consumption and retirement choices, optimal management of portfolios and risk, hedging, optimal timing issues in pricing American options, or in investment decisions. Stochastic control theory provides the methods and results to tackle all such problems. This book collects the papers published in the 2019 Special Issue of *Risks* "Applications of Stochastic Optimal Control to Economics and Finance" and contains 7 peer-reviewed papers dealing with stochastic control models motivated by important questions in Economics and Finance. Each model is mathematically rigorously funded and treated, and numerical methods are also possibly employed to derive the optimal solution. The topics of the book's chapters range from optimal public debt management, to optimal reinsurance, real options in energy markets, and optimal portfolio choice in partial and complete information setting, just to mention a few. From a mathematical point of view, techniques and arguments from dynamic programming theory, filtering theory, optimal stopping, one-dimensional diffusions, and multi-dimensional jump processes are used.

The paper by Anquandah and Bogachev deals with unemployment insurance and proposes a simple model of optimal agent entrance in a scheme. The paper analyzes individual decisions regarding optimal entry time, the dependence of the optimal solution on macroeconomic variables, and individual preferences.

In the second paper of the Special Issue, Rotondi documents a bias that may arise when American options are monetarily valued using the traditional least square method. The estimates on which this method is based utilize a regression run of the in-the-money paths of the Monte Carlo simulation. Therefore, large biases may occur if, for instance, the option is far out of the money. The paper proposes two approaches that completely overcome this problem and evaluates their performance, using the standard least square method.

In the third paper, Ceci and Brachetta study optimal excess-of-loss reinsurance problems when both the intensity of the claims' arrival and the claims' size are influenced by an exogenous stochastic factor. The model allows for stochastic risk premia, which takes into account risk fluctuations. Using stochastic control theory, based on the Hamilton–Jacobi– Bellman equation, the authors analyze the optimal reinsurance strategy under the criterion of maximizing the expected exponential utility of terminal wealth.

In the fourth paper, Moriarty and Palczewski study a real option problem arising in the context of energy markets. They assess the real option value of an arrangement under which an autonomous energy-limited storage unit sells incremental balancing reserve. The problem is set as a perpetual American swing put option with random refraction times.

In the fifth paper, Palmowski, Stettner, and Sulima study a portfolio selection problem in a continuous-time Itô–Markov additive market, where the prices of financial assets are described by the Markov additive processes that combines Lévy processes and regimeswitching models. The model takes into account two sources of risk: the jump-diffusion risk and the regime-switching risk. The resulting market is incomplete and the authors give conditions under which the market is asymptotic arbitrage-free. The portfolio selection problem is explicitly solved in the case of power and logarithmic utility function.

The work by De Franco, Nicolle, and Pham considers the Markowitz portfolio problem in a setting in which the drift of the underlying asset prices is uncertain. The authors use a Bayesian learning approach to solve the problem and provide a simple and practical procedure to implement the optimal policy. The strategy obtained via the Bayesian learning approach is then compared in three different investment universes to a naive non-learning strategy in which the drift is kept constant at all times.

The last paper in our Special Issue proposes a control-theoretic model for the optimal reduction of debt-to-GDP ratio. In particular, Cadenillas and Huamán-Aguilar consider a government that has limited ability in generating primary surpluses to optimally reduce the level of the stochastic time-dependent debt ratio. The authors explicitly solve the resulting stochastic control problem and identify the endogenous debt ceiling at which a reduction policy should be implemented. A detailed study of the effects of the model's parameters on the optimal policy is also provided.

We hope that the contents of this book might be helpful for senior scholars working in mathematical economics and finance, practitioners of the financial industry, and for younger researchers approaching this exciting field of research. We would once more like to thank all the authors of this book's chapters and we hope that our readers find this work both enjoyable and useful.

> **Salvatore Federico, Giorgio Ferrari, Luca Regis**  *Special Issue Editors*
