We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integ... more We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finitecodimension affine subspace in infinite dimensions.
We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to H... more We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to Hermite polynomials in the large-N limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the large-N limit.
This is the written version of a set of four lectures given at the CIB in Lausanne in April 2015.... more This is the written version of a set of four lectures given at the CIB in Lausanne in April 2015. The aim of these lectures was to present some of the mathematical-physical ideas underlying low-dimensional gauge theories, and some of the mathematical results which were obtained in the last twenty years on 2-dimensional Yang–Mills and 3-dimensional Chern–Simons theories.
We prove that integration over the moduli space of flat connections can be obtained as a limit of... more We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang– Mills measure defined in terms of the heat kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space. 2000MSC: 81T13
An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, wit... more An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the large-N limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops.
Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner produc... more Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt {\det A}} {\pi^n}e^{-{\rm Re} } dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure setting.
Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner produc... more Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt{\det A}} {\pi^n}e^{-\Re\langle Az,z\rangle}\,dz$ is described in terms of the linear and antilinear decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal-Bargmann transform, which we also study in the Gaussian-measure setting.
We examine general copula models for the valuation of CDOs and provide explicit formulas for the ... more We examine general copula models for the valuation of CDOs and provide explicit formulas for the sensitivities with respect to spreads. In the case of Gaussian copulas with non-uniform correlations, we prove a functional relation between spread sensitivities and correlation sensitivities.
The securitization of subprime mortgages in instruments like mortgage-backed securities and colla... more The securitization of subprime mortgages in instruments like mortgage-backed securities and collateralized debt obligations is one of the key ingredients to the current financial crisis. During 2007 and 2008, subprime defaults increased sharply, displaying high serial correlation in their arrival. Subprime default events depend on house price changes. We establish a link between the dynamics of house price changes and the dynamics of default rates in the Gaussian copula framework by specifying a time series model for a common risk factor. We show analytically and in simulations that serial correlation propagates from the common risk factor to default rates. We simulate prices of mortgage-backed securities, which are securitized from pools of mortgages using a waterfall structure. We find that subsequent vintages of these securities inherit temporal correlation from the common risk factor. The findings in this paper formalize one important dynamic of the subprime crisis: transmission of the decline in housing prices after 2006 into financial derivatives based on subprime mortgages.
We derive explicit formulas for CDO tranche sensitivity to parameter variations, and prove result... more We derive explicit formulas for CDO tranche sensitivity to parameter variations, and prove results concerning the qualitative behavior of such tranche sensitivities, for a homogeneous portfolio governed by the onefactor Gaussian copula. Similar results are also derived for a Poisson-mixture model. 2000 Mathematics Subject Classification. AMS Primary 60G35.
We show that for a suitable class of functions of finitely-many variables, the limit of integrals... more We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.
We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integ... more We show that the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finitecodimension affine subspace in infinite dimensions.
We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to H... more We show that a natural class of orthogonal polynomials on large spheres in N dimensions tend to Hermite polynomials in the large-N limit. We determine the behavior of the spherical Laplacian as well as zonal harmonic polynomials in the large-N limit.
This is the written version of a set of four lectures given at the CIB in Lausanne in April 2015.... more This is the written version of a set of four lectures given at the CIB in Lausanne in April 2015. The aim of these lectures was to present some of the mathematical-physical ideas underlying low-dimensional gauge theories, and some of the mathematical results which were obtained in the last twenty years on 2-dimensional Yang–Mills and 3-dimensional Chern–Simons theories.
We prove that integration over the moduli space of flat connections can be obtained as a limit of... more We prove that integration over the moduli space of flat connections can be obtained as a limit of integration with respect to the Yang– Mills measure defined in terms of the heat kernel for the gauge group. In doing this we also give a rigorous proof of Witten’s formula for the symplectic volume of the moduli space of flat connections. Our proof uses an elementary identity connecting determinants of matrices along with a careful accounting of certain dense subsets of full measure in the moduli space. 2000MSC: 81T13
An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, wit... more An overview of mathematical aspects of U(N) pure gauge theory in two dimensions is presented, with focus on the large-N limit of the theory. Examples are worked out expressing Wilson loop expectation values in terms of areas enclosed by loops.
Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner produc... more Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt {\det A}} {\pi^n}e^{-{\rm Re} } dz$ is described in terms of the holomorphic--antiholomorphic decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal--Bargmann transform, which we also study in the Gaussian-measure setting.
Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner produc... more Suppose $A$ is a positive real linear transformation on a finite dimensional complex inner product space $V$. The reproducing kernel for the Fock space of square integrable holomorphic functions on $V$ relative to the Gaussian measure $d\mu_A(z)=\frac {\sqrt{\det A}} {\pi^n}e^{-\Re\langle Az,z\rangle}\,dz$ is described in terms of the linear and antilinear decomposition of the linear operator $A$. Moreover, if $A$ commutes with a conjugation on $V$, then a restriction mapping to the real vectors in $V$ is polarized to obtain a Segal-Bargmann transform, which we also study in the Gaussian-measure setting.
We examine general copula models for the valuation of CDOs and provide explicit formulas for the ... more We examine general copula models for the valuation of CDOs and provide explicit formulas for the sensitivities with respect to spreads. In the case of Gaussian copulas with non-uniform correlations, we prove a functional relation between spread sensitivities and correlation sensitivities.
The securitization of subprime mortgages in instruments like mortgage-backed securities and colla... more The securitization of subprime mortgages in instruments like mortgage-backed securities and collateralized debt obligations is one of the key ingredients to the current financial crisis. During 2007 and 2008, subprime defaults increased sharply, displaying high serial correlation in their arrival. Subprime default events depend on house price changes. We establish a link between the dynamics of house price changes and the dynamics of default rates in the Gaussian copula framework by specifying a time series model for a common risk factor. We show analytically and in simulations that serial correlation propagates from the common risk factor to default rates. We simulate prices of mortgage-backed securities, which are securitized from pools of mortgages using a waterfall structure. We find that subsequent vintages of these securities inherit temporal correlation from the common risk factor. The findings in this paper formalize one important dynamic of the subprime crisis: transmission of the decline in housing prices after 2006 into financial derivatives based on subprime mortgages.
We derive explicit formulas for CDO tranche sensitivity to parameter variations, and prove result... more We derive explicit formulas for CDO tranche sensitivity to parameter variations, and prove results concerning the qualitative behavior of such tranche sensitivities, for a homogeneous portfolio governed by the onefactor Gaussian copula. Similar results are also derived for a Poisson-mixture model. 2000 Mathematics Subject Classification. AMS Primary 60G35.
We show that for a suitable class of functions of finitely-many variables, the limit of integrals... more We show that for a suitable class of functions of finitely-many variables, the limit of integrals along slices of a high dimensional sphere is a Gaussian integral on a corresponding finite-codimension affine subspace in infinite dimensions.
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Papers by Ambar Sengupta