Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆

Rubio, Gerardo. Emilia Caballero, Ma., et al, eds. "Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆." ESAIM: Proceedings 31 (2011): 73-100. <http://eudml.org/doc/251228>.

@article{Rubio2011,
abstract = {We consider the Cauchy problem in ℝd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains.Finally, we apply the results to a stochastic optimal consumption problem.},
author = {Rubio, Gerardo},
editor = {Emilia Caballero, Ma., Chaumont, Loïc, Hernández-Hernández, Daniel, Rivero, Víctor},
journal = {ESAIM: Proceedings},
keywords = {Cauchy problem; semilinear parabolic partial differential equations; stochastic control problems; approximation with linear parabolic equations},
language = {eng},
month = {3},
pages = {73-100},
publisher = {EDP Sciences},
title = {Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆},
url = {http://eudml.org/doc/251228},
volume = {31},
year = {2011},
}

TY - JOUR
AU - Rubio, Gerardo
AU - Emilia Caballero, Ma.
AU - Chaumont, Loïc
AU - Hernández-Hernández, Daniel
AU - Rivero, Víctor
TI - Existence and uniqueness to the Cauchy problem for linear and semilinear parabolic equations with local conditions⋆
JO - ESAIM: Proceedings
DA - 2011/3//
PB - EDP Sciences
VL - 31
SP - 73
EP - 100
AB - We consider the Cauchy problem in ℝd for a class of semilinear parabolic partial differential equations that arises in some stochastic control problems. We assume that the coefficients are unbounded and locally Lipschitz, not necessarily differentiable, with continuous data and local uniform ellipticity. We construct a classical solution by approximation with linear parabolic equations. The linear equations involved can not be solved with the traditional results. Therefore, we construct a classical solution to the linear Cauchy problem under the same hypotheses on the coefficients for the semilinear equation. Our approach is using stochastic differential equations and parabolic differential equations in bounded domains.Finally, we apply the results to a stochastic optimal consumption problem.
LA - eng
KW - Cauchy problem; semilinear parabolic partial differential equations; stochastic control problems; approximation with linear parabolic equations
UR - http://eudml.org/doc/251228
ER -