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Rajat Subhra Hazra (ISI, Kolkata)
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An inverse problem on randomly wighted series
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24 Dec
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Suppose X1 , X2 , . . . are independent and identically distributed
positive random variables. If X1 has regularly varying tail of index
-α, where
α > 0, and if
{Θt , t≥ 1} is a positive sequence of random variables independent
of
{Xt }, then it is known that under some appropriate moment conditions
on {Θt , t≥ 1}, X(∞) =
∑t=1 ∞ &Thetat Xt converges with probability 1 and has
regularly varying tail of index -α. We ask the question that, if X(∞) has
regularly varying tail, then does X1 have regularly varying tail under some
appropriate conditions? We obtain appropriate suffcient moment conditions,
including nonvanishing Mellin transform of ∑t=1∞ &Thetat along some vertical line in
the complex plane, so that the above is true. We also show that the condition on the Mellin transform cannot be dropped.
This is a joint work with Krishanu Maulik.
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Ali Saeb (University of Mysore)
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Extreme Value Modeling For Flood Data - A Case Study
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24 Dec
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We present a case-study wherein we model annual maximum flood data using
the generalized extreme value and the log generalized extreme value distributions. Joint work with S. Ravi.
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Sunil Kumar Gauttam (Indian Institute of Technology, Mumbai)
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Constrained Risk Sensitive Control Problem
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24 Dec
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We consider a general risk-sensitive control problem with state
being constraint to non-negative
orthrant. We have assumed a stability condition on the state dynamics which
gives asymptotic flatness of the controlled
reflecting diffusion. We prove the existence of optimal risk-sensitive
control.
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Ghurumuruhan Ganesan (ISI, Delhi)
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Critical probabilities of bond percolation in
Delaunay Triangulation and Voronoi Tessellation
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26 Dec
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Let pc and p*c denote the critical probabilities
for the bond percolation in Delaunay Triangulation (DT)
and its dual the Voronoi Tessellation (VT). In many regular
lattices, we know that pc + p*c = 1. In random lattices,
very few exact critical probabilities have been established. Pimentel
has studied the critical probability in DT and VT for general d-dimensions.
Recently, Bollobas and Riordan have proved that the critical
probability for the Voronoi site percolation is 1/2.
Using a crucial RSW result of Bollobas and Riordan, we prove that
bond percolation in DT and VT satisfies pc + p*c = 1.
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Arunangshu Biswas (Presidency College, Kolkata)
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Diffusive limits for Adaptive MCMC
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26 Dec
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Adaptive MCMC algorithms are designed to tune the chain prop-
erly to decrease the time to convergence to stationary. However since
the transition kernel is changed at each step of iteration proper ergod-
icity conditions has to be ensured. Here we try to define a toy discrete
time Adaptive MCMC algorithm and study its convergence properties. We apply the diffusion approximation method to a discrete time
Adaptive MCMC procedure to obtain the limiting stochastic differen-
tial equation governing the dynamics of the adaptation parameter θ
and the state space variable X . The solution to the coupled equation
will give the stationary distribution of the chain, and this could be
one way to show that the stationary distribution of the chain will be
the distribution of interest.
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Koushik Saha (Bidhannagar Government College, Kolkata)
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Some observations on joint convergence of patterned matrices
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26 Dec
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One of the most important examples of pattern matrices is the Wigner
matrix in physics. Voiculescu had shown that the joint distribution of Wigner
matrices are asymptotically free. This freeness of the limit is very special to Wigner
type matrices. We study the joint convergence of a few other patterned matrices.
In particular, the matricial limits of symmetric circulants and reverse circulants
satisfy respectively the classical independence and the half independence. The
matricial limits of Toeplitz and Hankel matrices do not seem to submit to any easy
or explicit independence/dependence notions. Their limits are not independent,
free or half independent.
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Santanu Dey (TIFR, Mumbai)
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Multidimensional Fourier Inversion using Importance Sampling with
Application to Option Pricing.
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26 Dec
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In this poster we demonstrate that importance sampling can be used to
develop unbiased, efficient estimators of densities, distribution
functions and expectations
of functions of a random vector, when the characteristic function of the
random vector is available in analytic or semi-analytic form.
Specifically, using this method we develop efficient algorithms to design
unbiased estimators of densities and distribution functions of sample mean
of iid non-lattice light tailed random variables when the characteristic
function of the random variable (Fourier transform of the density)
satisfy mild integrability conditions.In particular, we propose an efficient
algorithm for estimation of the tail probability P[\bar{X} ≥ x] where
X1,X2,...Xn are iid and x>E[X1].
All our algorithms are efficient in the sense that the number of simulation
runs needed to come up with the estimate goes to zero as n tends to
infinity. This efficiency is achieved by estimating the relative error in conventional
saddle point approximation through importance sampling (IS) technique with
an IS density which is Gaussian near centre but inverse power in tail.
This is especially of interest in options pricing as stochastic processes
such as affine jump processes and Levy processes are ubiquitous in financial
modeling and typically have characteristic functions
(of their value at a given time) that are easily evaluated while their
density or distribution functions have no readily computable closed form.
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Santanu Dey (TIFR, Mumbai)
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Entropy Approach to Incorporate Constraints involving Fat Tailed RV in Financial Models
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27 Dec
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In the existing financial literature, entropy based ideas have been proposed in portfolio optimisation and in model calibration for options pricing.
The abstracted problem corresponds to finding a probability measure that minimises Kullbach- Leibler (KL) distance with respect to a known measure
while it satisfies certain moment constraints on functions of underlying assets.Under KL distance, the optimal solution may not exist when constraints
involve fat tailed distributions ubiquitous in financial practice. We note that this drawback may be corrected if `polynomial-divergence' entropy distance is used.
We discuss existence and uniqueness issues related to this new optimisation problem as well as the nature of the optimal solution under di erent objectives. We also identify the optimal solution structure under KL distance as well as polynomial divergence when the associated constraints include those on marginal
distribution of functions of underlying assets. These results are applied to a simple problem of model calibration to options prices as well as to portfolio
modeling in Markowitz framework, where we note that a reasonable view that a particular portfolio of assets has heavy tailed losses may lead to fatter and more reasonable tail distributions of all assets.
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Debleena Thacker (ISI, Delhi)
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Urn Models on One-dimensional Integer Lattice.
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27 Dec
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In this ongoing work, we study urn models with infinite color of balls.
At every time, a single ball is drawn at random and then balls of in-
???nitely many colors are added to the urn in proportion determined
by a speci???ed stochastic replacement matrix. We consider two special
replacement matrices, one arriving from the right shift operator, and
the other arriving from the simple symmetric random walk on Z. If
Un =
{Un1 , Un2 , ...........}, where Uni denotes the number of balls of color i
in the urn at time n, then we are interested in the asymptotic behavior of
Un as well the individual colors Uni with the specified replacement matri-
ces. We also plan to study the urn models with generalised replacement
matrices.
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Farkhondeh Sajadi (ISI, Delhi)
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Greedy algorithm for traveling salesman problem (TSP) in mean field models
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27 Dec
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In his work we study asymptotic of the total length from greedy
tour in the mean field setup. In the case of the lengths are i.i.d.
Exponential , the length of shortest tour grows like log n. We conjectured
this can be true for other distribution under fairly general sufficient
conditions.
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