## Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations

In optimal control this corresponds to the case when the optimal trajectory of the Bolza problem is unique for every initial state and the optimal feedback is an upper semicontinuous set-valued map with convex, compact images. Research Overview Annual Report. Science into Policy Science and policy must work together to achieve a sustainable future for humanity. Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations.

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Robust and optimal control. Optimal control and estimation. Optimal control. Optimization and Optimal Control. Numerical methods for viscosity solutions and applications.

## Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations

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.

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There is unquestion able 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.

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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.

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We will present techniques from [6] that can easily handle random endowments, constraints and work well under minimal assumptions about the trading mechanism, e. 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.