CLASSIFICATION OF TWO- AND THREE-FACTOR TIME-HOMOGENEOUS SEPARABLE LMMs

The flexibility of parametrizations of the LIBOR market model (LMM) comes at a cost, namely the LMM is high-dimensional, which makes it cumbersome to use when pricing derivatives with early exercise features. One way to overcome this issue for short- and medium-term time horizons is by imposing the separability condition on the volatility functions and approximating the model using a single time-step approximation. In this paper, we examine the flexibility of separable LMMs under the relaxed assumption that the driving Brownian motions can be correlated. In particular, we are interested in how the separability condition interacts with time-homogeneity, a desirable property of a LMM. We show that the two concepts can be related using a Levi-Civitá equation and provide a characterization of two- and three-factor separable and time-homogeneous LMMs and show that they are of practical interest. The results presented in this paper are also applicable to local-volatility LMMs. These separable volatility structures can be used for the driver of a two- or three-dimensional Markov-functional model — in which case no (single time-step) approximation is needed and the resultant model is both time-homogeneous and arbitrage-free.

Incomparable actions of free groups

Suppose that $X$ is a Polish space, $E$ is a countable Borel equivalence relation on $X$, and $\unicode[STIX]{x1D707}$ is an $E$-invariant Borel probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\unicode[STIX]{x1D6E4}$, there is a Borel sequence $(\cdot _{r})_{r\in \mathbb{R}}$ of free actions of $\unicode[STIX]{x1D6E4}$ on $X$, generating subequivalence relations $E_{r}$ of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic, with the further property that $(E_{r})_{r\in \mathbb{R}}$ is an increasing sequence of relations which are pairwise incomparable under $\unicode[STIX]{x1D707}$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free Borel action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\unicode[STIX]{x1D707}$ is ergodic.

Separability Conditions for a Subclass of Circulant States in Two-Qutrit System and Quasi-Distillation of Entanglement

We will show the necessary and sufficient condition of separability for a sub-class of circulant states in the ℂ3 ⊗ ℂ3 system. This sub-class includes the models proposed by several authors as special cases. Moreover, it will be proved that our separability condition equals to the failure condition of quasi-distillation of entanglement.

ON THE EXISTENCE OF MAXIMAL ORDERS

We generalize the existence of maximal orders in a semi-simple algebra for general ground rings. We also improve several statements in Chaps. 5 and 6 of Reiner's book [Maximal Orders, London Mathematical Society Monographs, Vol. 5 (Academic Press, London, 1975), 395 pp.] concerning separable algebras by removing the separability condition, provided the ground ring is only assumed to be Japanese, a very mild condition. Finally, we show the existence of maximal orders as endomorphism rings of abelian varieties in each isogeny class.

MAKING DECISIONS IN AN INTELLIGENT TUTORING SYSTEM

In computerized tutoring, the pace of instruction is related to the student's mastery levels of the learning objectives. The observable student's behavior that can be used to measure his knowledge is usually his responses to test items. Unobservable variables that are related to learner's motivation can affect learning but are difficult to quantify. In comparison with other decision-theoretic tutoring systems, the novelties of this research are: (1) the efficiency-centric approach to develop the Bayesian networks; (2) the formulation of utility values for different tutoring outcomes that are independent of past actions and to satisfy the separability condition; (3) the development of a common measure for student's mastery levels and item difficulties; and (4) the generation of optimal policies in polynomial time. A prototype web-based tutoring system, known as iTutor, incorporating the novelties has been developed for engineering mechanics. Formative evaluations of iTutor have shown encouraging results.