60 research outputs found
Obstacle problem for nonlinear integro-differential equations arising in option pricing
No full textWe study the obstacle problem for a class of nonlinear integro-partial differential equations of second order, possibly degenerate, which includes the equation modeling American options in a jump-diffusion market with large investor. The viscosity solutions setting reveals appropriate, because of a monotonicity property with respect to the integral term. The same property allows to approximate the problem by penalization and to obtain the existence and uniqueness of solutions via a comparison principle. We also give uniform estimates of the solutions of the penalized problems which allow to prove further regularity
Uniqueness and comparison properties of the viscosity solution to some singular HJB equations
No full textWe study viscosity solutions to a class of HJB equations with singular coefficients near at the boundary: cases with either vanishing, or oscillating, or blowing-up diffusion coefficients are included. Because of proper structural conditions, strong comparison principle holds without assigning spatial boundary data, and unbounded initial data can be handled. The result applies to stochastic models for interest rate, and yields new results concerning Cauchy problems with unbounded coefficients
Global bifurcation for the Hénon problem
No full textWe prove the existence of nonradial solutions for the H'enon equation in the
ball with any given number of nodal zones, for arbitrary values of the exponent
. For sign-changing solutions, the case -- Lane-Emden
equation -- is included. The obtained solutions form global continua which
branch off from the curve of radial solutions , and the number of
branching points increases with both the number of nodal zones and the exponent
. The proof technique relies on the index of fixed points in cones and
provides information on the symmetry properties of the bifurcating solutions
and the possible intersection and/or overlapping between different branches,
thus allowing to separate them at least in some cases
On the asymptotically linear Hénon problem
No full textIn this paper we consider the Hénon problem in the ball with Dirichlet
boundary conditions. We study the asymptotic profile of radial solutions and
then deduce the exact computation of their Morse index when the exponent is
close to . Next we focus on the planar case and describe the asymptotic
profile of some solutions which minimize the energy among functions which are
invariant for reflection and rotations of a given angle . By
considerations based on the Morse index we see that, depending on the values of
and , such least energy solutions can be radial, or nonradial and
different one from another
Contour enhancement via a singular free boundary problem
No full textWe study a degenerate nonlinear parabolic equation with moving
boundaries which describes the technique of contour enhancement in image
processing. Such problem arises from the model by Malladi and Sethian after
an asymptotic expansion suggested by Barenblatt: in order to recover the
phenomenon of mass concentration, a singular data is imposed at the free
boundary
Some remarks about the Morse index for convex Hamiltonians systems
No full textWe investigate the (linearized) Morse index of solutions to Hamiltonan
systems, with a focus on convex Hamiltonians functions and sign-changing radial
solutions. For strongly coupled systems, we describe the profile of the radial
solutions and give an estimate of their Morse index
Quasi-variational inequalities with Dirichlet boundary condition related to exit time problems for impulse control
No full textWe study degenerate-elliptic quasi-variational inequalities with Dirichlet boundary condition, which are related to the value function of the exit time problem for stochastic impulse control by means of the dynamic programming principle. The boundary condition in the viscosity solutions sense does not identify a unique solution, because in this nonlocal problem the boundary layer gives rise to a loss of information also at the interior points. The eventual discontinuities of solutions at the boundary of the domain play an essential role and cannot be removed. Therefore we superimpose a selection criterion which, enforcing the information coming from the boundary datum, picks up the value function among all possible viscosity solutions. As a result, we attain the continuity of the value function up to the boundary. In addition, we produce a monotone iterative scheme approximating the value function
Nonradial sign changing solutions to Lane–Emden problem in an annulus
No full textIn this paper we prove the existence of continua of nonradial solutions for the Lane–Emden equation in the annulus. In a first result we show that there are infinitely many global continua detaching from the curve of radial solutions with any prescribed number of nodal zones. Next, using the fixed point index in cone, we produce nonradial solutions with a new type of symmetry. This result also applies to solutions with fixed signed, showing that the set of solutions to the Lane–Emden problem has a very rich and complex structure
Asymptotic profile and Morse index of nodal radial solutions to the Hénon problem
No full textWe compute the Morse index of nodal radial solutions to the H'enon problem
[left{egin{array}{ll}
-Delta u = |x|^alpha |u|^p-1 u qquad & ext{ in } B,
ewline
u= 0 & ext{ on } partial B,
end{array}
ight.
]
where stands for the unit ball in in dimension ,
alpha>0 and is near at the threshold exponent for existence of solutions
, obtaining that
egin{align*}
m(u_p) & = m sumlimits_j=0^1+left[alpha/2
ight] N_j quad &
mbox if is not an even integer, or
ewline
m(u_p)& = msumlimits_j=0^ alpha /2 N_j + (m-1) N_1+alpha/ 2 &
mbox if is an even number.
end{align*}
Here denotes the multiplicity of the spherical harmonics of order .
The computation builds on a characterization of the Morse index by means of a
one dimensional singular eigenvalue problem, and is carried out by a detailed
picture of the asymptotic behavior of both the solution and the singular
eigenvalues and eigenfunctions. In particular it is shown that nodal radial
solutions have multiple blow-up at the origin.
As side outcome we see that solutions are nondegenerate for near at
, and we give an existence result in perturbed balls
Solution of Optimal Control Problems by Hybridization
No full textOptimization problems for hybrid systems have attracted a lot of attention in recent years. This interest stimulated numerous scientific works and the development of tools to study optimal trajectories, such as necessary conditions. The idea of the present contribution is to use such results for hybrid systems to construct a Hybridization of an optimal control problem. The approximation via a hybrid system was already considered in the literature, but using different methods. Our procedure gives an efficient approximation, i.e. in many cases more efficient than standard discretization. ©2005 IEEE
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