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## Absolutely continuous laws of Jump-Diffusions in finite and infinite dimensions with applications to mathematical Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Probability#Mathematics - Numerical Analysis#Quantitative Finance - Computational Finance#60H07, 60H15, 62P05

In mathematical Finance calculating the Greeks by Malliavin weights has
proved to be a numerically satisfactory procedure for finite-dimensional
It\^{o}-diffusions. The existence of Malliavin weights relies on absolute
continuity of laws of the projected diffusion process and a sufficiently
regular density. In this article we first prove results on absolute continuity
for laws of projected jump-diffusion processes in finite and infinite
dimensions, and a general result on the existence of Malliavin weights in
finite dimension. In both cases we assume H\"ormander conditions and hypotheses
on the invertibility of the so-called linkage operators. The purpose of this
article is to show that for the construction of numerical procedures for the
calculation of the Greeks in fairly general jump-diffusion cases one can
proceed as in a pure diffusion case. We also show how the given results apply
to infinite dimensional questions in mathematical Finance. There we start from
the Vasi\v{c}ek model, and add -- by pertaining no arbitrage -- a jump
diffusion component. We prove that we can obtain in this case an interest rate
model, where the law of any projection is absolutely continuous with respect to
Lebesgue measure on $\mathbb{R}^M $.; Comment: final version accepted for publication in SIAM Journal of
Mathematical Analysis

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## Degenerate elliptic operators in mathematical finance and Holder continuity for solutions to variational equations and inequalities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Analysis of PDEs#Mathematics - Probability#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#35J70, 35J86, 49J40, 35R45 (Primary) 35R35, 49J20, 60J60 (Secondary)

The Heston stochastic volatility process, which is widely used as an asset
price model in mathematical finance, is a paradigm for a degenerate diffusion
process where the degeneracy in the diffusion coefficient is proportional to
the square root of the distance to the boundary of the half-plane. The
generator of this process with killing, called the elliptic Heston operator, is
a second-order, degenerate-elliptic partial differential operator whose
coefficients have linear growth in the spatial variables and where the
degeneracy in the operator symbol is proportional to the distance to the
boundary of the half-plane. With the aid of weighted Sobolev spaces, we prove
supremum bounds, a Harnack inequality, and H\"older continuity near the
boundary for solutions to variational equations defined by the elliptic Heston
operator, as well as H\"older continuity up to the boundary for solutions to
variational inequalities defined by the elliptic Heston operator. In
mathematical finance, solutions to obstacle problems for the elliptic Heston
operator correspond to value functions for perpetual American-style options on
the underlying asset.; Comment: 68 pages, 3 figures

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## GPGPUs in computational finance: Massive parallel computing for American style options

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/01/2011
Português

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#Quantitative Finance - Computational Finance#Mathematics - Probability#Quantitative Finance - Pricing of Securities

The pricing of American style and multiple exercise options is a very
challenging problem in mathematical finance. One usually employs a Least-Square
Monte Carlo approach (Longstaff-Schwartz method) for the evaluation of
conditional expectations which arise in the Backward Dynamic Programming
principle for such optimal stopping or stochastic control problems in a
Markovian framework. Unfortunately, these Least-Square Monte Carlo approaches
are rather slow and allow, due to the dependency structure in the Backward
Dynamic Programming principle, no parallel implementation; whether on the Monte
Carlo levelnor on the time layer level of this problem. We therefore present in
this paper a quantization method for the computation of the conditional
expectations, that allows a straightforward parallelization on the Monte Carlo
level. Moreover, we are able to develop for AR(1)-processes a further
parallelization in the time domain, which makes use of faster memory structures
and therefore maximizes parallel execution. Finally, we present numerical
results for a CUDA implementation of this methods. It will turn out that such
an implementation leads to an impressive speed-up compared to a serial CPU
implementation.

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## The financial framework of the sustainability of health universal coverage in Italy. A quantitative financial model for the assessment of the italian stability and reform program of public health financing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/07/2012
Português

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Italy and the Eurozone are heading in the year 2012 into a financial
depression of unprecedented magnitude, with a forthcoming multitude of often
contradictory public economic and financial stability emergency interventions
whose ultimate endogenous and exogenous effects on public and private health
spending and on the sustainability of universal coverage are difficult to
predict ex ante. The research question is to assess whether it is possible to
synthesise into a single and simple quantitative index such multitude of public
economic and financial stability interventions and assess their magnitude and
direction towards increasing or decreasing sustainability of publicly funded
health care and universal coverage. We have analyzed the Italian Economic and
Stability Reform Program 2011-2014 and we have proposed a quantitative
synthetic sustainability index {\sigma} based on simple partial and absolute
differential equations. The sustainability index {\sigma} highlights that in
case the growth of the GDP in the period 2011-2014 be insufficient - as is
already the case in the first semester of 2012 - all the assumptions on which
the Italian Economic and Stability Reform Program 2011-2014 rests will fall,
and Universal Coverage will become unsustainable. Health and Public Health
professionals should intervene immediately with Italian and Eurozone national
budgets planners and financial health regulators before unselective exogenously
induced health financing and provision shortages produce irreparable
epidemiological effects.; Comment: 28 pages

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## To the problem of turbulence in quantitative easing transmission channels and transactions network channels at quantitative easing policy implementation by central banks

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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In agreement with the recent research findings in the econophysics, we
propose that the nonlinear dynamic chaos can be generated by the turbulent
capital flows in both the quantitative easing transmission channels and the
transaction networks channels, when there are the laminar turbulent capital
flows transitions in the financial system. We demonstrate that the capital
flows in both the quantitative easing transmission channels and the transaction
networks channels in the financial system can be accurately characterized by
the Reynolds numbers. We explain that the transition to the nonlinear dynamic
chaos regime can be realized through the cascade of the Landau, Hopf
bifurcations in the turbulent capital flows in both the quantitative easing
transmission channels and the transaction networks channels in the financial
system. We completed the computer modeling, using both the Nonlinear Dynamic
Stochastic General Equilibrium Theory (NDSGET) and the Hydrodynamics Theory
(HT), to accurately characterize the US economy in the conditions of the QE
policy implementation by the US Federal Reserve. We found that the ability of
the US financial system to adjust to the different levels of liquidity depends
on the nonlinearities appearance in the QE transmission channels...

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## Robust Quantitative Comparative Statics for a Multimarket Paradox

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Computer Science - Computer Science and Game Theory#Quantitative Finance - General Finance#91B66#J.4.1

We introduce a quantitative approach to comparative statics that allows to
bound the maximum effect of an exogenous parameter change on a system's
equilibrium. The motivation for this approach is a well known paradox in
multimarket Cournot competition, where a positive price shock on a monopoly
market may actually reduce the monopolist's profit. We use our approach to
quantify for the first time the worst case profit reduction for multimarket
oligopolies exposed to arbitrary positive price shocks. For markets with affine
price functions and firms with convex cost technologies, we show that the
relative profit loss of any firm is at most 25% no matter how many firms
compete in the oligopoly. We further investigate the impact of positive price
shocks on total profit of all firms as well as on social welfare. We find tight
bounds also for these measures showing that total profit and social welfare
decreases by at most 25% and 16.6%, respectively. Finally, we show that in our
model, mixed, correlated and coarse correlated equilibria are essentially
unique, thus, all our bounds apply to these game solutions as well.; Comment: 23 pages, 1 figure

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## A New Kind of Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/10/2012
Português

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Finance has benefited from the Wolfram's NKS approach but it can and will
benefit even more in the future, and the gains from the influence may actually
be concentrated among practitioners who unintentionally employ those principles
as a group.; Comment: 13 pages; Forthcoming in "Irreducibility and Computational
Equivalence: 10 Years After Wolfram's A New Kind of Science," Hector Zenil,
ed., Springer Verlag, 2013

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## Agent-based Versus Macroscopic Modeling of Competition and Business Processes in Economics and Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We present examples of agent-based and stochastic models of competition and
business processes in economics and finance. We start from as simple as
possible models, which have microscopic, agent-based, versions and macroscopic
treatment in behavior. Microscopic and macroscopic versions of herding model
proposed by Kirman and Bass diffusion of new products are considered in this
contribution as two basic ideas. Further we demonstrate that general herding
behavior can be considered as a background of nonlinear stochastic model of
financial fluctuations.; Comment: 20 pages, 8 figures

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## Pricing, liquidity and the control of dynamic systems in finance and economics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 27/05/2011
Português

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The paper discusses various practical consequences of treating economics and
finance as an inherently dynamic and chaotic system. On the theoretical side
this looks at the general applicability of the market-making pricing approach
to economics in general. The paper also discuses the consequences of the
endogenous creation of liquidity and the role of liquidity as a state variable.
On the practical side, proposals are made for reducing chaotic behaviour in
both housing markets and stock markets.

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## On three filtering problems arising in mathematical finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/12/2008
Português

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Three situations in which filtering theory is used in mathematical finance
are illustrated at different levels of detail. The three problems originate
from the following different works: 1) On estimating the stochastic volatility
model from observed bilateral exchange rate news, by R. Mahieu, and P.
Schotman; 2) A state space approach to estimate multi-factors CIR models of the
term structure of interest rates, by A.L.J. Geyer, and S. Pichler; 3)
Risk-minimizing hedging strategies under partial observation in pricing
financial derivatives, by P. Fischer, E. Platen, and W. J. Runggaldier; In the
first problem we propose to use a recent nonlinear filtering technique based on
geometry to estimate the volatility time series from observed bilateral
exchange rates. The model used here is the stochastic volatility model. The
filters that we propose are known as projection filters, and a brief derivation
of such filters is given. The second problem is introduced in detail, and a
possible use of different filtering techniques is hinted at. In fact the
filters used for this problem in 2) and part of the literature can be
interpreted as projection filters and we will make some remarks on how more
general and possibly more suitable projection filters can be constructed. The
third problem is only presented shortly.; Comment: A short version appeared in "Insurance. Mathematics and Economics"...

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## Schr\"odinger group and quantum finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/04/2013
Português

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Using the one dimensional free particle symmetries, the quantum finance
symmetries are obtained. Namely, it is shown that Black-Scholes equation is
invariant under Schr\"odinger group. In order to do this, the one dimensional
free non-relativistic particle and its symmetries are revisited. To get the
Black-Scholes equation symmetries, the particle mass is identified as the
inverse of square of the volatility. Furthermore, using financial variables, a
Schr\"odinger algebra representation is constructed.; Comment: 10 page, non figures

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## Magic points in finance: Empirical integration for parametric option pricing

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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We propose an interpolation method for parametric option pricing tailored to
the persistently recurring task of pricing liquid financial instruments. The
method supports the acceleration of such essential tasks of mathematical
finance as model calibration, real-time pricing, and, more generally, risk
assessment and parameter risk estimation. We adapt the empirical magic point
interpolation method of Barrault et al. (2004) to parametric Fourier pricing.
For a large class of combinations of option types, models and free parameters
the approximation converges exponentially in the degrees of freedom and
moreover has explicit error bounds. Numerical experiments confirm our
theoretical findings and show a significant gain in efficiency, even for
examples beyond the scope of the theoretical results. This is especially
promising for further applications of the method.

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## L\'evy Processes For Finance: An Introduction In R

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/03/2015
Português

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This brief manuscript provides an introduction to L\'evy processes and their
applications in finance as the random process that drives asset models.
Characteristic functions and random variable generators of popular L\'evy
processes are presented in R.; Comment: 18 pages, 9 figures

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## Degenerate-elliptic operators in mathematical finance and higher-order regularity for solutions to variational equations

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Analysis of PDEs#Mathematics - Probability#Quantitative Finance - Computational Finance#Quantitative Finance - Pricing of Securities#35J70, 49J40, 35R45 (Primary), 60J60 (Secondary)

We establish higher-order weighted Sobolev and Holder regularity for
solutions to variational equations defined by the elliptic Heston operator, a
linear second-order degenerate-elliptic operator arising in mathematical
finance. Furthermore, given $C^\infty$-smooth data, we prove
$C^\infty$-regularity of solutions up to the portion of the boundary where the
operator is degenerate. In mathematical finance, solutions to obstacle problems
for the elliptic Heston operator correspond to value functions for perpetual
American-style options on the underlying asset.; Comment: 55 pages, 1 figure. To appear in Advances in Differential Equations.
Incorporates final galley proof corrections corresponding to published
version

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## Quantitative easing is an incomplete strategy that must be accompanied by the nullification of debt

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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Compound interest as well as inflation grows exponentially with time, whereas
other means to repay debt grow polynomially. For this and other, mostly
political, reasons, debt without inflation is unsustainable. We suggest a
discontinuous way to eliminate debt by nullifying it. This scenario is
preferable to current central bank strategies of quantitative easing because it
allows the disposal of debt without hyperinflation or bloated balance sheets.; Comment: 8 pages, some revisions

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## Adjoints and Automatic (Algorithmic) Differentiation in Computational Finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/07/2011
Português

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Two of the most important areas in computational finance: Greeks and,
respectively, calibration, are based on efficient and accurate computation of a
large number of sensitivities. This paper gives an overview of adjoint and
automatic differentiation (AD), also known as algorithmic differentiation,
techniques to calculate these sensitivities. When compared to finite difference
approximation, this approach can potentially reduce the computational cost by
several orders of magnitude, with sensitivities accurate up to machine
precision. Examples and a literature survey are also provided.; Comment: 23 pages

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## Multilevel Monte Carlo methods for applications in finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/12/2012
Português

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Since Giles introduced the multilevel Monte Carlo path simulation method
[18], there has been rapid development of the technique for a variety of
applications in computational finance. This paper surveys the progress so far,
highlights the key features in achieving a high rate of multilevel variance
convergence, and suggests directions for future research.; Comment: arXiv admin note: text overlap with arXiv:1202.6283; and with
arXiv:1106.4730 by other authors

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## Quantitative relations between corruption and economic factors

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/05/2007
Português

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#Quantitative Finance - General Finance#Physics - Data Analysis, Statistics and Probability#Physics - Physics and Society

We report quantitative relations between corruption level and economic
factors, such as country wealth and foreign investment per capita, which are
characterized by a power law spanning multiple scales of wealth and investments
per capita. These relations hold for diverse countries, and also remain stable
over different time periods. We also observe a negative correlation between
level of corruption and long-term economic growth. We find similar results for
two independent indices of corruption, suggesting that the relation between
corruption and wealth does not depend on the specific measure of corruption.
The functional relations we report have implications when assessing the
relative level of corruption for two countries with comparable wealth, and for
quantifying the impact of corruption on economic growth and foreign
investments.; Comment: 10 pages, 9 figures

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## Convex duality in stochastic programming and mathematical finance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 21/06/2010
Português

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#Quantitative Finance - Computational Finance#Mathematics - Optimization and Control#52A41, 46A20, 91B25, 90C15, 52A07

This paper proposes a general duality framework for the problem of minimizing
a convex integral functional over a space of stochastic processes adapted to a
given filtration. The framework unifies many well-known duality frameworks from
operations research and mathematical finance. The unification allows the
extension of some useful techniques from these two fields to a much wider class
of problems. In particular, combining certain finite-dimensional techniques
from convex analysis with measure theoretic techniques from mathematical
finance, we are able to close the duality gap in some situations where
traditional topological arguments fail.

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## Nonlinear PDEs risen when solving some optimization problems in finance, and their solutions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 16/10/2015
Português

Relevância na Pesquisa

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#Quantitative Finance - Computational Finance#Quantitative Finance - Mathematical Finance#Quantitative Finance - Pricing of Securities

We consider a specific type of nonlinear partial differential equations (PDE)
that appear in mathematical finance as the result of solving some optimization
problems. We review some existing in the literature examples of such problems,
and discuss the properties of these PDEs. We also demonstrate how to solve them
numerically in a general case, and analytically in some particular case.; Comment: 20 pages, 3 figs

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