Existence results for Isaacs equations with local conditions and related semilinear Cauchy problems

Our goal is to prove existence results for classical solutions to some general nondegenerate Cauchy problems which are natural generalizations of Isaacs equations. For the latter we are able to extend our results by admitting local conditions for coefficients. Such equations appear naturally for instance in robust control theory. Using our general results, we can solve not only Isaacs equations, but also equations for other sophisticated control problems, for instance models with state dependent constraints on the control set

A note on the worst case approach for a market with a stochastic interest rate

We solve a robust optimization problem and show an example of a market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction. To protect against the model uncertainty he uses the worst case measure approach. The problem is formulated as a stochastic game between the investor and the market. PDE methods are used to find a saddle point and a precise verification argument is provided

The investor problem based on the HJM model

We consider a consumption-investment problem (both on finite and infinite time horizon) in which the investor has an access to the bond market. In our approach prices of bonds with different maturities are described by the general HJM factor model. We assume that the bond market consists of entire family of rolling bonds and the investment strategy is a general signed measure distributed on all real numbers representing time to maturity specifications for different rolling bonds. In particular, we can consider portfolio of coupon bonds. The investor's objective is to maximize time-additive utility of the consumption process. We solve the problem by means of the HJB equation for which we prove required regularity of its solution and all required estimates to ensure applicability of the verification theorem. Explicit calculations for affine models are presented.Comment: v2 - 26 pages, detailed calculations of G2++ model, extended proof of theorem 4.1, two references added( [2] and [33]), v3 - 28 pages, revised version after reviews, (v4) - 30 pages, language corrections, (v5),(v6) - 29 pages, final correction

Continuous time portfolio choice under monotone mean-variance preferences : stochastic factor case

We consider an incomplete market with a nontradable stochastic factor and a continuous time investment problem with an optimality criterion based on monotone mean-variance preferences. We formulate it as a stochastic differential game problem and use Hamilton-Jacobi-Bellman-Isaacs equations to find an optimal investment strategy and the value function. What is more, we show that our solution is also optimal for the classical Markowitz problem and every optimal solution for the classical Markowitz problem is optimal also for the monotone mean-variance preferences. These results are interesting because the original Markowitz functional is not monotone, and it was observed that in the case of a static one-period optimization problem the solutions for those two functionals are different. In addition, we determine explicit Markowitz strategies in the square root factor models.Comment: Major revision, the same model but the main result is strenghtened, the square root factor model added (Heston model