Abstracts
Series of lectures presented during Simons Semester:
Vivek Borkar (IIT Mumbai), Controlled Markov Processes
1. controlled diffusions : solution concepts,
types of control policies
2. existence of optimal controls
3. dynamic programming
4. ergodic control
5. risk-sensitive control
6. control with partial observations
7. special topics
Andrzej Świech (Georgia Tech), Viscosity solutions: Theory and applications in stochastic optimal control
The minicourse is designed to give a short introduction to the theory of
viscosity solutions of partial differential equations
and its basic techniques and results. The topics presented will include:
comparison theorems and uniqueness of viscosity
solutions, Perron's method and existence of viscosity solutions, relaxed
limits and consistency of viscosity solutions,
regularity of viscosity solutions, generalized boundary conditions, and
Hamilton-Jacobi-Bellman equations and their connection with stochastic
optimal control. In particular, we will discuss the value function of a
stochastic optimal control problem and the fact that it is the viscosity
solution of the associated Hamilton-Jacobi-Bellman equation. We also
plan to discuss briefly obstacle problems and their connection to
optimal stopping, and Hamilton-Jacobi-Bellman equations for
risk-sensitive control problems and singular perturbation problems.
Teemu Pennanen (Kings College), Incomplete markets
The minicourse gives an introduction to financial economics in terms of basic optimization theory. We provide a unified treatment of financial risk management, accounting, and asset pricing in a simple discrete-time model that avoids many of the technicalities associated with traditional continuous-time models. This leaves room for practical considerations that are often neglected in more mathematical texts. In particular, the approach allows for nonlinear illiquidity effects and portfolio constraints which are significant in practice but invalidate much of the classical theory of financial mathematics.
Miklos Rasonyi (Renyi Institute HAS), Utility maximization without using the dual problem
Portfolio choice for risk-averse investors is an exciting instance of
convex optimization on infinite dimensional spaces. The standard
route for solving such problems exploits duality in an essential way (see e.g. [5]):
in the case of frictionless markets, instead of maximizing a concave functional over terminal values of admissible portfolios, one minimizes a conjugate (convex) functional over the family of martingale measures for the given price process (i.e. the dual problem is solved) and then the primal optimizer is retrieved from the dual one by a simple transformation.
There is unquestionable elegance and depth in this approach. However, it turns out to become cumbersome in certain situations. First, when the investor receives a random endowment, the corresponding dual problem becomes complicated (see e.g. [3]). Second, the same is true when portfolio constraints are enforced. Third, this approach requires a solid grip on the concrete trading mechanism (frictionless, with proportional transaction costs, etc.) so it would be desirable to find a method that works well for a rather general market structure. We will present techniques from [6] that can easily handle random endowments, constraints and work well under minimal assumptions about the trading mechanism, e.g. even for infinitely many assets.
We do not rely on duality and perform all arguments uniquely on the primal problem, based on results of [4]. Such a method has already been developed in [7]
in the case where the utility function is defined on the positive axis. We can now
treat utilities defined on the whole real line as well.
In the case of markets with transaction costs we can even tackle model ambiguity,
exploiting further novel ideas of [1]. Finally, we will also consider the case of non-concave utilities where the convex optimization setting can no longer serve and one needs to employ advanced weak convergence techniques to find an optimizer, see
[2].
References:
[1] H. N. Chau, M. Rasonyi. Robust utility maximization in markets with transaction costs.
Preprint, 2018. arXiv:1803.04213
[2] H. N. Chau, M. Rasonyi. Skorohod's representation theorem and optimal strategies for
markets with frictions. SIAM Journal on Control and Optimization, 55:3592--3608, 2017.
[3] J. Cvitanic, W. Schachermayer and H. Wang. Utility Maximization in Incomplete Markets with Random Endowment. Finance and Stochastics, 5:259--272, 2001.
[4] F. Delbaen and K. Owari. On convex functions on the duals of Delta- Orlicz spaces. Preprint, 2018. arXiv:1611.06218
[5] D. O. Kramkov and W. Schachermayer. The asymptotic elasticity of utility functions and optimal investment in incomplete markets. Annals of Applied Probability}, 9:904--950, 1999.
[6] M. Rasonyi. On utility maximization without passing by the dual problem. Published online by Stochastics, 2018. arXiv:1702.00982
[7] W. Schachermayer.Portfolio Optimization in Incomplete Financial Markets. Notes of the Scuola Normale Superiore Cattedra Galileiana, Pisa, 2004. ISBN 8876421416
Michal Zajac (University of Tartu/Clearmatics Ltd.)