Nonsmooth analysis approach to Isaac's equation

We study Isaacs' equation (∗)wt(t,x)+H(t,x,wx(t,x))=0 (H is a highly nonlinear function) whose natural solution is a value W(t,x) of a suitable differential game. It has been felt that even though Wx(t,x) may be a discontinuous function or it may not exist everywhere, W(t,x) is a solution of (∗) in some generalized sense. Several attempts have been made to overcome this difficulty, including viscosity solution approaches, where the continuity of a prospective solution or even slightly less than that is required rather than the existence of the gradient Wx(t,x). Using ideas from a very recent paper of Subbotin, we offer here an approach which, requiring literally no regularity assumptions from prospective solutions of (∗), provides existence results. To prove the uniqueness of solutions to (∗), we make some lower- and upper-semicontinuity assumptions on a terminal set Γ. We conclude with providing a close relationship of the results presented on Isaacs' equation with a differential games theory

Financial Engineering in the Complete Markets

Celem rozważań jest pokazanie, że algebra liniowa, wykładana we wszystkich politechnikach oraz w większości szkół ekonomicznych, oferuje bardzo adekwatne narzędzia do prezentacji najważniejszych pojęć z inżynierii finansowej. Daje możliwość klarownego wyjaśnienia, na czym polegają podstawowe problemy badawcze w inżynierii finansowej, oraz ukazuje sposoby ich rozwiązywania. Chodzi tu przede wszystkim o zabezpieczanie się firm i, ogólnie mówiąc, inwestorów przed ryzykiem na rynkach kapitałowych przez tworzenie syntetycznych instrumentów replikujących pożądane (nieobecne na rynku kapitałowym) papiery wartościowe. Jest to niezwykle ważne dla dużych firm, takich jak np. firmy z WIG 30, w których nieumiejętne zabezpieczanie się przed ryzykiem spadków cen surowców (np. miedzi) lub, co gorsza, brak zabezpieczania, prowadzi do dużych strat (KGHM).An aim of considerations is to show that the linear algebra lectured at all technical universities and in the majority of economic schools, offers very adequate tools for presentation of the most important notions in the field of financial engineering. It provides the opportunity to clearly explain in what the basic research problems in financial engineering consist as well as shows the ways of resolution thereof. Th e matter is here, first of all, with safeguarding by companies and, generally speaking, investors against the risk in capital markets by way of formation of synthetic instruments replicating the desired (absent in the capital market) securities. It is extremely important for big firms such as e.g. companies from WIG 30, in which incompetent hedging against the risk of fall in prices of commodities (e.g. copper) or, what’s worse, lack of hedge leads to great losses (KGHM)

Modification of Shapley Value and its Implementation in Decision Making

The article presents a solution of a problem that is critical from a practical point of view: how to share a higher than usual discount of $10 million among 5 importers. The discount is a result of forming a coalition by 5 current, formerly competing, importers. The use of Shapley value as a concept for co-operative games yielded a solution that was satisfactory for 4 lesser importers and not satisfactory for the biggest importer. Appropriate modification of Shapley value presented in this article allowed to identify appropriate distribution of the saved purchase amount, which according to each player accurately reflects their actual strength and position on the importer market. A computer program was used in order to make appropriate calculations for 325 permutations of all possible coalitions. In the last chapter of this paper, we recognize the lasting contributions of Lloyd Shapley to the cooperative game theory, commemorating his recent (March 12, 2016) descent from this world

Linear time-invariant control problems with perturbations

We deal with a class of control problems whose (uncertain) mathematical model is given by a linear differential equation with control parameters and perturbation parameters influencing the dynamics of an object under consideration. All we know about the perturbations is that their values belong to a given compact set Q. We show how to find both lower and upper bounds for the value function of our original perturbed system in terms of the value functions of two simplified unperturbed control problems. We also give explicit formulae for [var epsilon]-optimal policies in the class of so-called step-guided strategies. In the last section the results above are extended to the infinite horison setting.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/28796/1/0000630.pd