Jérôme Busca

@article{busca_1630,

title = {Nonlinear eigenvalues and bifurcation problems for Pucci's operators},

author = {Jérôme Busca},

url = {http://www.numdam.org/article/AIHPC_2005__22_2_187_0.pdf},

year  = {2005},

date = {2005-01-01},

journal = {Annales De L Institut Henri Poincare-Analyse Non Lineaire},

volume = {22},

number = {2},

pages = {187-206},

abstract = {In this paper we extend existing results concerning generalized eigenvalues of Pucci's extremal operators. In the radial case, we also give a complete description of their spectrum, together with an equivalent of Rabinowitz's Global Bifurcation Theorem. This allows us to solve nonlinear equations involving Pucci's operators.? 2005 Elsevier SAS. All rights reserved.},

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tppubtype = {article}

}

@article{busca_1623,

title = {Computing the Implied Volatility in Stochastic Volatility Models},

author = {Jérôme Busca},

url = {http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.483.7280&rep=rep1&type=pdf},

year  = {2004},

date = {2004-01-01},

journal = {Communications On Pure And Applied Mathematics},

volume = {57},

number = {10},

pages = {1352-1373},

abstract = {The Black-Scholes model has gained wide recognition on financial markets. One of its shortcomings, however, is that it is inconsistent with most observed

option prices. Although the model can still be used very efficiently, it has been proposed to relax its assumptions, and, for instance, to consider that the volatility of

the underlying asset S is no longer a constant but rather a stochastic process. There

are two well-known approaches to achieve this goal. In the first class of models, the

volatility is assumed to depend on the variables t (time) and S, giving rise to the

so-called local volatility models. The second one, conceptually more ambitious,

considers that the volatility has a stochastic component of its own. In the latter,

the number of factors is increased by the amount of stochastic factors entering the

volatility modeling. Both models are of practical interest.},

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}

@article{busca_1631,

title = {Harnack type estimates for nonlinear elliptic systems and applications},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S029414490300043X},

year  = {2004},

date = {2004-01-01},

journal = {Annales De L Institut Henri Poincare-Analyse Non Lineaire},

volume = {21},

number = {5},

pages = {543-590},

abstract = {In this paper we obtain a generalised maximum principle of Alexandrov-Bakelman-Pucci type for viscosity solutions of fully nonlinear cooperative elliptic systems. We also establish a Harnack estimate for such systems and give some applications. In particular, a Harnack type estimate for solutions of higher order equations is given.},

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tppubtype = {article}

}

@article{busca_1632,

title = {Classification of positive solutions of semilinear elliptic equations},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S1631073X03005351},

year  = {2004},

date = {2004-01-01},

journal = {Comptes Rendus Mathematique},

volume = {338},

number = {1},

pages = {7-11},

abstract = {We give a classification of all solutions of a general semilinear PDE in the positive quadrant of R 2. To cite this article: J. Busca et al., CR Acad. Sci. Paris, Ser. I 338 (2004).},

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pubstate = {published},

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@article{busca_1626,

title = {Application of large deviation methods to the pricing of index options in finance},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S1631073X03000323},

year  = {2003},

date = {2003-01-01},

journal = {Comptes Rendus Mathematique},

volume = {336},

number = {3},

pages = {263-266},

abstract = {We develop an asymptotic formula for calculating the implied volatility of European index options based on the volatility skews of the options on the underlying stocks and on a given correlation matrix for the basket. The derivation uses the steepest-descent approximation for evaluating the multivariate probability distribution function for stock prices, which is based on large-deviation estimates of diffusion processes densities by Varadhan.},

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pubstate = {published},

tppubtype = {article}

}

@article{busca_1627,

title = {Reconstructing volatility},

author = {Jérôme Busca},

url = {https://www.math.nyu.edu/~avellane/Avellaneda.pdf},

year  = {2002},

date = {2002-10-01},

journal = {Risk Magazine},

volume = {-},

pages = {91-95},

abstract = {Options on stock baskets have become a mainstay of the equity derivatives business, but pricing

and hedging of such products is highly sensitive to implied volatility and correlation assumptions.

Here, Marco Avellaneda, Dash Boyer-Olson, Jérôme Busca and Peter Friz address this vital

problem, presenting a new approach to basket option valuation from underlying volatilities using

the method of steepest descents and most-likely price configurations},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1624,

title = {Asymptotics and calibration in local volatility models},

author = {Jérôme Busca},

url = {http://www.fiquam.polytechnique.fr/finance/lyon/bbf.pdf},

year  = {2002},

date = {2002-01-01},

journal = {Quantitative Finance},

volume = {2},

number = {1},

pages = {61-69},

abstract = {We derive a direct link between local and implied volatilities in the form of a

quasilinear degenerate parabolic partial differential equation. Using this

equation we establish closed-form asymptotic formulae for the implied

volatility near expiry as well as for deep in- and out-of-the-money options.

This in turn leads us to propose a new formulation near expiry of the

calibration problem for the local volatility model, which we show to be well

posed.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1634,

title = {A Liouville-type theorem for Lane-Emden systems},

author = {Jérôme Busca},

url = {https://www.jstor.org/stable/24902114},

year  = {2002},

date = {2002-01-01},

journal = {Indiana University Mathematics Journal},

volume = {51},

number = {1},

pages = {37-51},

abstract = {We provide a partial positive answer to a well-known conjecture about the nonexistence of positive solutions to Lane-Emden systems below the critical Sobolev hyperbola. Our proof is based on a monotonicity argument for suitable transformed functions. It relies on a special form of the Alexandrov-Serrin moving planes method, as well as some refined forms of the Maximum Principle for elliptic systems that we develop here. It is our hope that our method will shed some new light on this delicate problem.},

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pubstate = {published},

tppubtype = {article}

}

@article{busca_1633,

title = {CONVERGENCE TO EQUILIBRIUM FOR SEMILINEAR PARABOLIC PROBLEMS IN R^N},

author = {Jérôme Busca},

url = {https://www.tandfonline.com/doi/abs/10.1081/PDE-120016128?journalCode=lpde20},

year  = {2001},

date = {2001-01-01},

journal = {Communications In Partial Differential Equations},

volume = {27},

number = {9-10},

pages = {1793-1814},

abstract = {We consider the semilinear parabolic equation on , assuming that f is an arbitrary C 1 function satisfying and . We prove that any bounded positive solution that decays to zero at spatial infinity, uniformly with respect to t, converges to a (single) stationary solution as . Our proof combines energy and comparison techniques with dynamical system arguments. We first establish an asymptotic symmetrization result: as , approaches a set of steady states that are radially symmetric about a common origin in . To this aim we introduce a new tool that we call first moments of energy. Having established the symmetrization, we apply a general convergence result for gradient-like dynamical systems. This amounts to showing that the dimension of the kernel of the linearized operator around an equilibrium w matches the dimension of a manifold of equilibria passing through w.},

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pubstate = {published},

tppubtype = {article}

}

@article{busca_1636,

title = {Qualitative properties of some bounded positive solutions to scalar field equations},

author = {Jérôme Busca},

url = {https://link.springer.com/article/10.1007/PL00009928},

year  = {2001},

date = {2001-01-01},

journal = {Calculus Of Variations And Partial Differential Equations},

volume = {13},

number = {2},

pages = {191-211},

abstract = {Nonlinear scalar field equations serve as model in different areas of Applied Mathematics and their study has attracted the attention of numerous authors throughout the years. If nowadays existence as well as qualitative properties of ?ground-state? solutions of such equations are well understood, the full structure of the solutions set, whose study is relevant in many situations, is still a wide open and challenging problem. To be more specific, let us consider one of the simplest model cases

? u? u+ u 3= 0 in R2.(1.1)},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1642,

title = {Symmetry and nonexistence results for Emden-Fowler equations in cones},

author = {Jérôme Busca},

url = {https://projecteuclid.org/journals/differential-and-integral-equations/volume-14/issue-8/Symmetry-and-nonexistence-results-for-Emden-Fowler-equations-in-cones/die/1356123171.full},

year  = {2001},

date = {2001-01-01},

journal = {Differential And Integral Equations},

volume = {14},

number = {8},

pages = {897-912},

abstract = {The purpose of this paper is to state some qualitative properties of the solutions to the Emden-Fowler equation 

?

u

+

r

?

u

p

=

0

 in a cone with Dirichlet boundary conditions. Namely one can show that every solution has the same symmetry as the cone in some sense; furthermore it is possible to extend the nonexistence results for regular solutions to this equation already stated by C. Bandle and M. Essen in [2]. For this one needs to establish some asymptotics for the solutions as 

r

?

0

 or 

r

?

?

, relying on methods used by Veron in [25] for similar equations, but in different geometries, and then use a special form of the moving-planes method on a sphere in the spirit of [22].},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1643,

title = {First moments of energy and convergence to equilibrium},

author = {Jérôme Busca},

url = {https://ejde.math.txstate.edu/conf-proc/06/b3/busca.pdf},

year  = {2001},

date = {2001-01-01},

journal = {Electronic Journal Of Differential Equations},

volume = {6},

number = {1},

pages = {45-53},

abstract = {A basic question is to establish convergence to equilibrium for globally

defined solutions to evolution problems. The purpose here is to emphasize

the role of symmetry. In particular, it is proved that in some cases the first

moments of energy are constant on the ?-limit set of the solution. This key

property is used to prove convergence in two model evolution problems.

This communication is based on two joint works with P. Felmer [3] and

M.A. Jendoubi, P. Polacik [4].},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1625,

title = {An inverse parabolic problem arising in finance},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S0764444200017493},

year  = {2000},

date = {2000-01-01},

journal = {Comptes Rendus Mathematique},

volume = {331},

number = {12},

pages = {965-969},

abstract = {We derive a quasilinear degenerate parabolic partial differential equation linking the implied and local volatility in the extended Black-Scholes model. Using this equation we establish a closed-form asymptotic formula for the implied volatility near expiry. This in turn leads us to propose a new formulation of the calibration problem near expiry which we show to be well-posed.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1635,

title = {EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM*},

author = {Jérôme Busca},

url = {https://www.tandfonline.com/doi/abs/10.1081/PDE-100107824?journalCode=lpde20},

year  = {2000},

date = {2000-01-01},

journal = {Communications In Partial Differential Equations},

volume = {26},

number = {11-12},

pages = {2323-2337},

abstract = {We are interested in this article in fully nonlinear second-order, possibly degenerate, elliptic equations of the following form},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1641,

title = {Symmetry Results for Semilinear Elliptic Systems in the Whole Space},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S0022039699937014},

year  = {2000},

date = {2000-01-01},

journal = {Journal Of Differential Equations},

volume = {163},

number = {1},

pages = {41-56},

abstract = {We establish symmetry results for solutions to semilinear PDEs in the whole space.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1637,

title = {Existence results for bellman equations and maximum principles in unbounded domains},

author = {Jérôme Busca},

url = {https://www.tandfonline.com/doi/abs/10.1080/03605309908821493},

year  = {1999},

date = {1999-01-01},

journal = {Communications In Partial Differential Equations},

volume = {24},

number = {11-12},

pages = {2023-2042},

abstract = {The purpose of this paper is to extend known existence results for Hamilton-Jacobi-Bellman equations. The classical results give existence, uniqueness and Holder regularity when all elliptic operators involved have nonpositive zero-order term. We want to handle here the more general case where they have principal eigenvalues bounded below by a positive constant. As a motivation for this work, we give an application to the study of the Maximum Principle in infinite cylinders, following a work by Berestycki, Caffarelli and Nirenberg [2]. This is used to extend the cylindrical symmetry result in [2] to a more general class of operators.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1640,

title = {Approximate radial symmetry for overdetermined boundary value problems},

author = {Jérôme Busca},

url = {https://projecteuclid.org/journals/advances-in-differential-equations/volume-4/issue-6/Approximate-radial-symmetry-for-overdetermined-boundary-value-problems/ade/1366030751.full},

year  = {1999},

date = {1999-01-01},

journal = {Advances In Differential Equations},

volume = {4},

number = {6},

pages = {907-932},

abstract = {In this paper, we study the stability of Serrin's classical symmetry result for overdetermined boundary value problems [13]. We prove that if there exists a positive solution of 

?

u

+

f

(

u

)

=

0

 in 

?

 with 

u

=

0

 on 

?

?

 and if 

?

u

/

?

?

 on 

?

?

 is close to a constant, then the domain 

?

 is close to a ball. Additionally, we give an explicit estimate for the distance of the domain to a circumscribed and inscribed ball. The proof relies on the method of moving planes and new quantitative versions of the Hopf Lemma and Serrin's corner Lemma.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1638,

title = {A Serrin type result for infinite cylinders},

author = {Jérôme Busca},

url = {https://scholar.google.com/scholar?hl=en&as_sdt=0,5&cluster=9161995215848178304},

year  = {1998},

date = {1998-01-01},

volume = {PITMAN RESEARCH NOTES IN MATHEMATICS SER},

number = {1},

pages = {1-6},

abstract = {We establish symmetry results for solutions to semilinear partial differential equations in unbounded cylinders.},

keywords = {},

pubstate = {published},

tppubtype = {article}

}

@article{busca_1639,

title = {Symétrie radiale pour des problèmes elliptiques surdéterminés posés dans des domaines extérieurs},

author = {Jérôme Busca},

url = {https://www.sciencedirect.com/science/article/pii/S0764444297869807},

year  = {1997},

date = {1997-01-01},

journal = {Comptes Rendus de l'Académie des Sciences - Series I - Mathematics},

volume = {324},

number = {6},

pages = {633-638},

abstract = {In this_Note, we extend a classical result by J. Serrin,(see [10]), to exterior domains ?"?¯. where ? is a bounded domain. We prove, under some hypotheses on ?, that if there exists a solution of ? u+ f (u)= 0 in ?"?¯ satisfying the overdetermined boundary conditions? u/? ? and u constant on? ?, with 0? u? u¦? ?, then the domain ? is a ball. Under different assumptions on ?, the result has been obtained by W. Reichel in [9]. The main result here covers new cases like (u)= u p with n/u-2< p? n+ 2/n-2. When ? is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition u constant on? ?. Our method relies on Kelvin transforms, various forms of the maximum principle and on the device of moving planes up to a critical position.},

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}