Inhalt
| Kommentar |
TOPICS COVERED:
Convex extensions of the Markowitz portfolio selection problem
- Basic notions from convex optimization
- Criteria of convexity
- Classes of convex problems: LP, SOCP, SDP
- Duality
- Vector optimization: Pareto optimal solutions, Scalarization
- Introduction to the CVX solver (as extension to Matlab) as special solver for convex optimization problems
- Examples from finance and their numerical solution via CVX:
- Extensions of Markowitz portfolio selection problem and their convexification: incomplete information about covariance, probability constraints, robust optimization
- Factor models
- Log-optimal investment strategy
- Sharpe ratio
Risk measures and decision models
- Basic notions and concepts
- Single-stage decision models
- Risk deviation functionals and their properties
- Standard and less standard risk measures
- More on Conditional value-at-risk
- Formulation of decision optimization models with different risk measures as objectives and their transformation to LP
- Multi-stage decision models
- Multi-stage risk deviation functionals
- Formulation of multi-stage decision models, their transformation to linear programs, large-scale optimization
Hamilton-Jacobi-Bellman equation for dynamic portfolio optimization problems
- Expected utility maximization problem
- Bellman’s optimality principle and its application to the utility maximization problem
- Deriving a Hamilton-Jacobi-Bellman (HJB) nonlinear partial differential equation
- Notes on standard ways of solving HJB equations
- Riccati-type of transformation to HJB equations
- Advantages of the transformed HJB equations
- Worst-case portfolio optimization using techniques from Topic 1
- Inter-temporal utility optimization in the HJB context
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| Literatur |
Topic 1
- S. Boyd and L. Vandenberghe: Convex Optimization. Cambridge University Press, March 2004
Topic 2
- G.Ch. Pflug and W. Römisch: Modeling, Measuring and Managing Risk. World Scientific Pub Co Inc, April 2008
Topic 3
- Any literature on optimal control, and papers
- S. Kilianová and D. Sevcovic: A Transformation Method for Solving the Hamilton-Jacobi-Bellman Equation for a Constrained Dynamic Stochastic Optimal Allocation Problem, ANZIAM Journal (55) 2013, 14–38.
- S. Kilianová and D. Sevcovic: Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton-Jacobi Bellman equation, Japan Journal of Industrial and Applied Mathematics, 36(2) 2019, 497–517.
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