Impulse Control Problems and Adaptive Numerical Solution of Quasi-Variational Inequalities in Markovian Factor Models
- optimal control
- impulse control problems
- adaptive discretization
- quasi-variational inequalities
Project DescriptionImpulse control problems are ubiquitous when trading in financial markets. One striking example are portfolio optimization problems under transaction costs. Their mathematical description leads to quasi-variational inequalities, which typically do not admit analytical solutions. Current numerical methods, however, mostly do not incorporate adaptive techniques. Adaptive discretization is the key to an efficient and accurate determination of optimal trading strategies. In this project we shall develop adaptive methods and combine them with effective preconditioned solvers to obtain an efficient algorithm for the solution of quasi-variational inequalities. Integral terms, which occur in markets with jumps, will be included.
Besides this, the current turbulence in financial markets, such as the credit crisis and the European financial crisis, puts the focus on increasing credit and counterparty risk. This asks for an extension of existing models. Markovian factor models are an appropriate tool to achieve a low-dimensional representation of complex dynamics on financial markets. We will formulate and analyze the ensuing problems as impulse control problems, for instance in portfolio optimization. Their solution will ask for those efficient and adaptive numerical methods for quasi-variational inequalities, which are developed as part of the project.
Last but not least, the practical application requires the estimation of model parameters by means of historical data. This additional difficulty will be tackled by extending existing methodologies from incomplete information. The above-mentioned approaches will be extended in this direction. This is necessary to guarantee the practical applicability of the developed schemes.
- Blechschmidt: The Finite Volume Method and its Application for the Solution of HJB Equations, Chemnitzer Seminar zur Optimalsteuerung, Haus im Ennstal, Austria, February 2018
- Blechschmidt: Improving Policies for Hamilton-Jacobi-Bellman Equations by PostprocessingResearch Seminar, Freiburg, November 2017
- Blechschmidt: A semi-Lagrangian scheme for the solution of Hamilton-Jacobi-Bellman equations, SCAIM Seminar, Pacific Institute for the Mathematical Sciences, UBC, Vancouver, July 2017
- Herzog: Postprocessing for Finite Element Solutions of HJB Equations, Workshop on Numerical methods for Hamilton-Jacobi equations in optimal control and related fields, RICAM, Linz, November 2016
- Blechschmidt: New Computational Methods in Portfolio Optimization, Dublin City University, November 2016
- Blechschmidt: HJB Quasi-Variational Inequalities in Portfolio Optimization and their Discretization by Finite Elements, Research Seminar BTU Cottbus-Senftenberg, Cottbus, July 2015
- Blechschmidt: HJB Quasi-Variational Inequalities in Portfolio Optimization and their Discretization by Finite Elements, 27 th IFIP TC7 Conference 2015 on System Modelling and Optimization, Sophia Antipolis, July 2015
- Blechschmidt: Finite Element Discretization of Portfolio Optimization Problems, Chemnitzer Seminar zur Optimalsteuerung, Haus im Ennstal, Austria, February 2015
- Blechschmidt: Hamilton-Jacobi-Bellman Equations in Portfolio Optimization, Chemnitzer Seminar zur Optimalsteuerung, Haus im Ennstal, Austria, February 2014
- J. Blechschmidt and R. Herzog
Improving Policies for Hamilton-Jacobi-Bellman Equations by Postprocessing
De Gruyter Radon Series on Computational and Applied Mathematics, 2018