Michaela Szölgyenyi

A numerical method for SDEs with discontinuous drift

In this paper we introduce a transformation technique, which can on the one hand be used to prove existence and uniqueness for a class of SDEs with discontinuous drift coefficient. One the other hand we present a numerical method based on transforming the Euler–Maruyama scheme for such a class of SDEs. We prove convergence of order

Bayesian Dividend Optimization and Finite Time Ruin Probabilities

1/2

1/2

1/2. Finally, we present numerical examples.

method for multidimensional SDEs with discontinuous drift

The classical result by Ito on the existence of strong soluti ons of stochastic differential equations (SDEs) with Lipschitz coefficients can be extended to the cas e where the drift is only measurable and bounded. These generalizations are based on techniques presented by Zvonkin [14] and Veretennikov [12], which rely on the uniform ellipticity of the diff usion coefficient. In this paper we study the case of degenerate ellipticity and give sufficient conditions for the existence of a solution. The conditions are significantly more ge neral than previous results and we gain fundamental insight into the geometric properties of the discontinuity of the drift on the one hand and the diffusion vector field on the other hand. Besides presenting existence results for the degenerate el liptic situation, we give an example illustrating the difficulties in obtaining more general results t han those given. The particular types of SDEs considered arise naturally in the framework of combined optimal control and filtering problems.

OPTIMAL CONTROL OF AN ENERGY STORAGE FACILITY UNDER A CHANGING ECONOMIC ENVIRONMENT AND PARTIAL INFORMATION

We consider the valuation problem of an (insurance) company under partial information. Therefore, we use the concept of maximizing discounted future dividend payments. The firm value process is described by a diffusion model with constant and observable volatility and constant but unknown drift parameter. For transforming the problem to a problem with complete information, we derive a suitable filter. The optimal value function is characterized as the unique viscosity solution of the associated Hamilton-Jacobi-Bellman equation. We state a numerical procedure for approximating both the optimal dividend strategy and the corresponding value function. Furthermore, threshold strategies are discussed in some detail. Finally, we calculate the probability of ruin in the uncontrolled and controlled situation.

Convergence of the Euler-Maruyama method for multidimensional SDEs with discontinuous drift and degenerate diffusion coefficient

In this paper we consider multidimensional stochastic differential equations (SDEs) with discontinuous drift and possibly degenerate diffusion coefficient. We prove an existence and uniqueness result for this class of SDEs and we present a numerical method that converges with strong order 1/2. Our result is the first one that shows strong convergence for such a general class of SDEs. The proof is based on a transformation technique that removes the discontinuity from the drift such that the coefficients of the transformed SDE are Lipschitz continuous. Thus the Euler-Maruyama method can be applied to this transformed SDE. The approximation can be transformed back, giving an approximation to the solution of the original SDE. As an illustration, we apply our result to an SDE the drift of which has a discontinuity along the unit circle.

Modeling and Performance of Certain Put-WriteStrategies

In this paper we consider an energy storage optimization problem in finite time in a model with partial information that allows for a changing economic environment. The state process consists of the storage level controlled by the storage manager and the energy price process, which is a diffusion process the drift of which is assumed to be unobservable. We apply filtering theory to find an alternative state process which is adapted to our observation filtration. For this alternative state process we derive the associated Hamilton-Jacobi-Bellman equation and solve the optimization problem numerically. This results in a candidate for the optimal policy for which it is a-priori not clear whether the controlled state process exists. Hence, we prove an existence and uniqueness result for a class of time-inhomogeneous stochastic differential equations with discontinuous drift and singular diffusion coefficient. Finally, we apply our result to prove admissibility of the candidate optimal control.

Utility indifference pricing of insurance catastrophe derivatives

We prove strong convergence of order