Sumit Sourabh

Algorithmic correspondence for intuitionistic modal mu-calculus

In the present paper, the algorithmic correspondence theory developed in Conradie and Palmigiano [9] is extended to mu-calculi with a non-classical base. We focus in particular on the language of bi-intuitionistic modal mu-calculus. We enhance the algorithm ALBA introduced in Conradie and Palmigiano [9] so as to guarantee its success on the class of recursive mu-inequalities, which we introduce in this paper. Key to the soundness of this enhancement are the order-theoretic properties of the algebraic interpretation of the fixed point operators. We show that, when restricted to the Boolean setting, the recursive mu-inequalities coincide with the “Sahlqvist mu-formulas” defined in van Benthem, Bezhanishvili and Hodkinson [22].

Algebraic modal correspondence: Sahlqvist and beyond

The present paper proposes a new introductory treatment of the very well known Sahlqvist correspondence theory for classical modal logic. The first motivation for the present treatment is {\em pedagogical}: classical Sahlqvist correspondence is presented in a uniform and modular way, and, unlike the existing textbook accounts, extends itself to a class of formulas laying outside the Sahlqvist class proper. The second motivation is {\em methodological}: the present treatment aims at highlighting the {\em algebraic} and {\em order-theoretic} nature of the correspondence mechanism. The exposition remains elementary and does not presuppose any previous knowledge or familiarity with the algebraic approach to logic. However, it provides the underlying motivation and basic intuitions for the recent developments in the Sahlqvist theory of nonclassical logics, which compose the so-called unified correspondence theory.

Sahlqvist theory for impossible worlds

We extend unified correspondence theory to Kripke frames with impossible worlds and their associated regular modal logics. These are logics the modal connectives of which are not required to be normal: only the weaker properties of additivity ◊x∨◊y=◊(x∨y) and multiplicativity □x∧□y=□(x∧y) are required. Conceptually, it has been argued that their lacking necessitation makes regular modal logics better suited than normal modal logics at the formalization of epistemic and deontic settings. From a technical viewpoint, regularity proves to be very natural and adequate for the treatment of algebraic canonicity Jonsson-style. Indeed, additivity and multiplicativity turn out to be key to extend Jonsson’s original proof of canonicity to the full Sahlqvist class of certain regular distributive modal logics naturally generalizing distributive modal logic. Most interestingly, additivity and multiplicativity are key to Jonsson-style canonicity also in the original (i.e. normal DML. Our contributions include: the definition of Sahlqvist inequalities for regular modal logics on a distributive lattice propositional base; the proof of their canonicity following Jonsson’s strategy; the adaptation of the algorithm ALBA to the setting of regular modal logics on two non-classical (distributive lattice and intuitionistic) bases; the proof that the adapted ALBA is guaranteed to succeed on a syntactically defined class which properly includes the Sahlqvist one; finally, the application of the previous results so as to obtain proofs, alternative to Kripke’s, of the strong completeness of Lemmon’s epistemic logics E2-E5 with respect to elementary classes of Kripke frames with impossible worlds.

Jónsson-style canonicity for ALBA-inequalities

The theory of canonical extensions typically considers extensions of maps A→B to maps Aδ→Bδ. In the present article, the theory of canonical extensions of maps A→Bδ to maps Aδ→Bδ is developed, and is applied to obtain a new canonicity proof for those inequalities in the language of Distributive Modal Logic (DML) on which the algorithm ALBA [9] is successful.

Sahlqvist preservation for topological fixed-point logic

We introduce a new order-topological semantics for the positive modal mu-calculus over modal compact Hausdorff spaces, which are generalizations of descriptive frames. We define Sahlqvist sequents in this language, prove Esakias lemma and Sahlqvist preservation theorem for this semantics. We show that every Sahlqvist sequent has a frame correspondent in first-order logic with fixed-point operators.

Generalized Vietoris bisimulations

We introduce and study bisimulations for coalgebras on Stone spaces (Kupke et al., 2004, Theoretical Computer Science, 327, 109-134), motivated by previous work on ultrafilter extensions for coalge ...

Incorporating Contagion in Portfolio Credit Risk Models Using Network Theory

Portfolio credit risk models estimate the range of potential losses due to defaults or deteriorations in credit quality. Most of these models perceive default correlation as fully captured by the dependence on a set of common underlying risk factors. In light of empirical evidence, the ability of such a conditional independence framework to accommodate for the occasional default clustering has been questioned repeatedly. Thus, financial institutions have relied on stressed correlations or alternative copulas with more extreme tail dependence. In this paper, we propose a different remedy—augmenting systematic risk factors with a contagious default mechanism which affects the entire universe of credits. We construct credit stress propagation networks and calibrate contagion parameters for infectious defaults. The resulting framework is implemented on synthetic test portfolios wherein the contagion effect is shown to have a significant impact on the tails of the loss distributions.

Irreducible Equivalence Relations, Gleason Spaces, and de Vries Duality

By de Vries duality, the category of compact Hausdorff spaces is dually equivalent to the category of de Vries algebras (complete Boolean algebras endowed with a proximity-like relation). We provide an alternative “modal-like” duality by introducing the concept of a Gleason space, which is a pair (X,R), where X is an extremally disconnected compact Hausdorff space and R is an irreducible equivalence relation on X. Our main result states that the category of Gleason spaces is equivalent to the category of compact Hausdorff spaces, and is dually equivalent to the category of de Vries algebras.

Liquidity Risk in Derivatives Valuation: An Improved Credit Proxy Method

The models used to calculate post-crisis valuation adjustments, market risk and capital measures for derivatives are subject to liquidity risk due to severe lack of available information to obtain market implied model parameters. The European Banking Authority has proposed an intersection methodology to calculate a proxy CDS or Bond spread. Due to practical issues of this method, Chourdakis et al. introduce a cross-section approach. In this paper, we extend the cross-section methodology using equity returns, and show that our methodology is significantly more accurate compared to both existing methodologies, and produces more reliable, stable and robust market risk and capital measures, and credit valuation adjustment.