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Register a new userInterest rate models and Whittaker functions
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Authors
Dmitry Muravey
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I present the technique which can analyse some interest rate models: Constantinides-Ingersoll, CIR-model, geometric CIR and Geometric Brownian Motion. All these models have the unified structure of Whittaker function. The main focus of this text is closed-form solutions of the zero-coupon bond value in these models. In text I emphasize the specific details of mathematical methods of their determination such as Laplace transform and hypergeometric functions.
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High future discounting rates favor inaction on present expending while lower rates advise for a more immediate political action. A possible approach to this key issue in global economy is to take historical time series for nominal interest rates and inflation, and to construct then real interest rates and finally obtaining the resulting discount rate according to a specific stochastic model. Extended periods of negative real interest rates, in which inflation dominates over nominal rates, are commonly observed, occurring in many epochs and in all countries. This feature leads us to choose a well-known model in statistical physics, the Ornstein-Uhlenbeck model, as a basic dynamical tool in which real interest rates randomly fluctuate and can become negative, even if they tend to revert to a positive mean value. By covering 14 countries over hundreds of years we suggest different scenarios and include an error analysis in order to consider the impact of statistical uncertainty in our results. We find that only 4 of the countries have positive long-run discount rates while the other ten countries have negative rates. Even if one rejects the countries where hyperinflation has occurred, our results support the need to consider low discounting rates. The results provided by these fourteen countries significantly increase the priority of confronting global actions such as climate change mitigation. We finally extend the analysis by first allowing for fluctuations of the mean level in the Ornstein-Uhlenbeck model and secondly by considering modifi
We give a recursive method for computing all values of a basis of Whittaker functions for unramified principal series invariant under an Iwahori or parahoric subgroup of a split reductive group $G$ over a nonarchimedean local field $F$. Structures in the proof have surprising analogies to features of certain solvable lattice models. In the case $G=mathrm{GL}_r$ we show that there exist solvable lattice models whose partition functions give precisely all of these values. Here `solvable means that the models have a family of Yang-Baxter equations which imply, among other things, that their partition functions satisfy the same recursions as those for Iwahori or parahoric Whittaker functions. The R-matrices for these Yang-Baxter equations come from a Drinfeld twist of the quantum group $U_q(widehat{mathfrak{gl}}(r|1))$, which we then connect to the standard intertwining operators on the unramified principal series. We use our results to connect Iwahori and parahoric Whittaker functions to variations of Macdonald polynomials.
In this paper we consider Iwahori Whittaker functions on $n$-fold metaplectic covers $widetilde{G}$ of $mathbf{G}(F)$ with $mathbf{G}$ a split reductive group over a non-archimedean local field $F$. For every element $phi$ of a basis of Iwahori Whittaker functions, and for every $ginwidetilde{G}$, we evaluate $phi(g)$ by recurrence relations over the Weyl group using vector Demazure-Whittaker operators. Specializing to the case of $mathbf{G} = mathbf{GL}_r$, we exhibit a solvable lattice model whose partition function equals $phi(g)$. These models are of a new type associated with the quantum affine super group $U_q(widehat{mathfrak{gl}}(r|n))$. The recurrence relations on the representation theory side then correspond to solutions to Yang-Baxter equations for the lattice models. Remarkably, there is a bijection between the boundary data specifying the partition function and the data determining all values of the Whittaker functions.
A term structure model in which the short rate is zero is developed as a candidate for a theory of cryptocurrency interest rates. The price processes of crypto discount bonds are worked out, along with expressions for the instantaneous forward rates and the prices of interest-rate derivatives. The model admits functional degrees of freedom that can be calibrated to the initial yield curve and other market data. Our analysis suggests that strict local martingales can be used for modelling the pricing kernels associated with virtual currencies based on distributed ledger technologies.
We show that spherical Whittaker functions on an $n$-fold cover of the general linear group arise naturally from the quantum Fock space representation of $U_q(widehat{mathfrak{sl}}(n))$ introduced by Kashiwara, Miwa and Stern (KMS). We arrive at this connection by reconsidering solvable lattice models known as `metaplectic ice whose partition functions are metaplectic Whittaker functions. First, we show that a certain Hecke action on metaplectic Whittaker coinvariants agrees (up to twisting) with a Hecke action of Ginzburg, Reshetikhin, and Vasserot. This allows us to expand the framework of KMS by Drinfeld twisting to introduce Gauss sums into the quantum wedge, which are necessary for connections to metaplectic forms. Our main theorem interprets the row transfer matrices of this ice model as `half vertex operators on quantum Fock space that intertwine with the action of $U_q(widehat{mathfrak{sl}}(n))$. In the process, we introduce new symmetric functions termed textit{metaplectic symmetric functions} and explain how they relate to Whittaker functions on an $n$-fold metaplectic cover of GL$_r$. These resemble textit{LLT polynomials} introduced by Lascoux, Leclerc and Thibon; in fact the metaplectic symmetric functions are (up to twisting) specializations of textit{supersymmetric LLT polynomials} defined by Lam. Indeed Lam constructed families of symmetric functions from Heisenberg algebra actions on the Fock space commuting with the $U_q(widehat{mathfrak{sl}}(n))$-action. We explain that half vertex operators agree with Lams construction and this interpretation allows for many new identities for metaplectic symmetric and Whittaker functions, including Cauchy identities. While both metaplectic symmetric functions and LLT polynomials can be related to vertex operators on the $q$-Fock space, only metaplectic symmetric functions are connected to solvable lattice models.
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