etd@IISc Collection:http://hdl.handle.net/2005/412016-05-30T10:20:50Z2016-05-30T10:20:50ZAnalyzing Credit Risk Models In A Regime Switching MarketBanerjee, Tamalhttp://hdl.handle.net/2005/25172016-04-25T10:27:28Z2016-04-24T18:30:00ZTitle: Analyzing Credit Risk Models In A Regime Switching Market
Authors: Banerjee, Tamal
Abstract: Recently, the financial world witnessed a series of major defaults by several institutions and investment banks. Therefore, it is not at all surprising that credit risk analysis have turned out to be one of the most important aspect among the finance community. As credit derivatives are long term instruments, it is affected by the changes in the market conditions. Thus, it is a appropriate to take into consideration the effects of the market economy. This thesis addresses some of the important issues in credit risk analysis in a regime switching market. The main contribution in this thesis are the followings:
(1) We determine the price of default able bonds in a regime switching market for structural models with European type payoff. We use the method of quadratic hedging and minimal martingale measure to determine the defaultble bond prices. We also obtain hedging strategies and the corresponding residual risks in these models. The defaultable bond prices are obtained as solution to a system of PDEs (partial differential equations) with appropriate terminal and boundary conditions. We show the existence and uniqueness of the system of PDEs on an appropriate domain.
(2) We carry out a similar analysis in a regime switching market for the reduced form models. We extend some of the existing models in the literature for correlated default timings. We price single-name and multi-name credit derivatives using our regime switching models. The prices are obtained as solution to a system of ODEs(ordinary differential equations) with appropriate terminal conditions.
(3) The price of the credit derivatives in our regime switching models are obtained as solutions to a system of ODEs/PDEs subject to appropriate terminal and boundary conditions. We solve these ODEs/PDEs numerically and compare the relative behavior of the credit derivative prices with and without regime switching. We observe higher spread in our regime switching models. This resolves the low spread discrepancy that were prevalent in the classical structural models. We show further applications of our model by capturing important phenomena that arises frequently in the financial market. For instance, we model the business cycle, tight liquidity situations and the effects of firm restructuring. We indicate how our models may be extended to price various other credit derivatives.2016-04-24T18:30:00ZRiesz Transforms Associated With Heisenberg Groups And Grushin OperatorsSanjay, P Khttp://hdl.handle.net/2005/24962015-12-08T10:07:31Z2015-12-07T18:30:00ZTitle: Riesz Transforms Associated With Heisenberg Groups And Grushin Operators
Authors: Sanjay, P K
Abstract: We characterise the higher order Riesz transforms on the Heisenberg group and also show that they satisfy dimension-free bounds under some assumptions on the multipliers. We also prove the boundedness of the higher order Riesz transforms associated to the Hermite operator. Using transference theorems, we deduce boundedness theorems for Riesz transforms on the reduced Heisenberg group and hence also for the Riesz transforms associated to special Hermite and Laguerre expansions.
Next we study the Riesz transforms associated to the Grushin operator G = - Δ - |x|2@t2 on Rn+1. We prove that both the first order and higher order Riesz transforms are bounded on Lp(Rn+1): We also prove that norms of the first order Riesz transforms are independent of the dimension n.2015-12-07T18:30:00ZRigidity And Regularity Of Holomorphic MappingsBalakumar, G Phttp://hdl.handle.net/2005/24472015-07-16T07:34:08Z2015-07-15T18:30:00ZTitle: Rigidity And Regularity Of Holomorphic Mappings
Authors: Balakumar, G P
Abstract: We deal with two themes that are illustrative of the rigidity and regularity of holomorphic
mappings.
The first one concerns the regularity of continuous CR mappings between smooth pseudo convex, finite type hypersurfaces which is a well studied subject for it is linked with the problem of studying the boundary behaviour of proper holomorphic mappings between domains bounded by such hypersurfaces. More specifically, we study the regularity of Lipschitz CR mappings from an h-extendible(or semi-regular) hypersurface in Cn .Under various assumptions on the target hypersurface, it is shown that such mappings must be smooth. A rigidity result for proper holomorphic mappings from strongly pseudo convex domains is also proved.
The second theme dealt with, is the classification upto biholomorphic equivalence of model domains with abelian automorphism group in C3 .It is shown that every model domain i.e.,a hyperbolic rigid polynomial domainin C3 of finite type, with abelian automorphism group is equivalent to a domain that is balanced with respect to some weight.2015-07-15T18:30:00ZRicci Flow And Isotropic CurvatureGururaja, H Ahttp://hdl.handle.net/2005/23762014-09-03T05:54:21Z2014-09-02T18:30:00ZTitle: Ricci Flow And Isotropic Curvature
Authors: Gururaja, H A
Abstract: This thesis consists of two parts. In the first part, we study certain Ricci flow invariant nonnegative curvature conditions as given by B. Wilking. We begin by proving that any such nonnegative curvature implies nonnegative isotropic curvature in the Riemannian case and nonnegative orthogonal bisectional curvature in the K¨ahler case. For any closed AdSO(n,C) invariant subset S so(n, C) we consider the notion of positive curvature on S, which we call positive S- curvature. We show that the class of all such subsets can be naturally divided into two subclasses:
The first subclass consists of those sets S for which the following holds: If two Riemannian manifolds have positive S- curvature then their connected sum also admits a Riemannian metric of positive S- curvature.
The other subclass consists of those sets for which the normalized Ricci flow on a closed Riemannian manifold with positive S-curvature converges to a metric of constant positive sectional curvature.
In the second part of the thesis, we study the behavior of Ricci flow for a manifold having positive S - curvature, where S is in the first subclass. More specifically, we study the Ricci flow for a special class of metrics on Sp+1 x S1 , p ≥ 4, which have positive isotropic curvature.2014-09-02T18:30:00Z