Analytical Methods for Markov Semigroups

Lorenzi, Luca; Bertoldi, Marcello

doi:10.1201/9781420011586

Cited by 123 publications

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“…The condition (3) is easier to check and looks weaker than other known sufficient conditions for ergodicity [28,36]. It appears also in a remark of [34], where the proof of the existence of μ is different from ours.…”

Section: H(x Y D X V Dmentioning

confidence: 85%

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Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Bardi

Cesaroni²,

Manca³

2010

SIAM J. Finan. Math.

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Abstract. We study singular perturbations of a class of stochastic control problems under assumptions motivated by models of financial markets with stochastic volatilities evolving on a fast time scale. We prove the convergence of the value function to the solution of a limit (effective) Cauchy problem for a parabolic equation of HJB type. We use methods of the theory of viscosity solutions and of the homogenization of fully nonlinear PDEs. We test the result on some financial examples, such as Merton portfolio optimization problem.

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Section: H(x Y D X V Dmentioning

confidence: 85%

“…It is reminiscent of other similar conditions about ergodicity of diffusion processes in the whole space; see, for example, [28,9,34,12,36].…”

Section: The Main Assumptionmentioning

confidence: 99%

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Bardi

Cesaroni²,

Manca³

2010

SIAM J. Finan. Math.

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“…for a suitable constant C A , independent of i (see, e.g., [23,Appendix A]). So, from (6.5) and (6.3) we deduce that ϕ f ∈ H 2 loc (K), and (6.4) yields…”

Section: Appendixmentioning

confidence: 99%

“…On the other hand, unlike the semigroups that are associated with operators defined in the whole space R n (see e.g. [23] and the references therein), P t can have several invariant measures, which, moreover, need not be absolutely continuous with respect to Lebesgue's measure. In this paper, taking advantage of the interior invariance result described above, we prove that P t has at most one invariant measure on C(K), in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure.…”

Section: Introductionmentioning

confidence: 99%

Invariant measures associated to degenerate elliptic operators

Cannarsa¹,

Prato²,

Frankowska³

2010

Indiana Univ. Math. J.

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This paper is devoted to the study of the existence and uniqueness of the invariant measure associated to the transition semigroup of a diffusion process in a bounded open subset of R n . For this purpose, we investigate first the invariance of a bounded open domain with piecewise smooth boundary showing that such a property holds true under the same conditions that insure the invariance of the closure of the domain. A uniqueness result for the invariant measure is obtained in the class of all probability measures that are absolutely continuous with respect to Lebesgue's measure. A sufficient condition for the existence of such a measure is also provided.

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“…Parabolic PDEs with unbounded coefficients are studied, for example, in the monographs of Cerrai [12] and Lorenzi and Bertoldi [36]; see also the article of Da Prato and Lunardi [15] and references therein. We note in passing that, on letting b → +∞, the FENE potential converges to the (linear) Hookean spring potential U (s) = s while D then becomes the whole of R d -corresponding to a mathematically simple(r) albeit physically unrealistic scenario in which a polymer chain can have arbitrarily large elongation.…”

Section: Introductionmentioning

confidence: 99%

Finite element approximation of kinetic dilute polymer models with microscopic cut-off

Barrett¹,

Süli²

2010

ESAIM: M2AN

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Abstract. We construct a Galerkin finite element method for the numerical approximation of weak solutions to a coupled microscopic-macroscopic bead-spring model that arises from the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The model consists of the unsteady incompressible Navier-Stokes equations in a bounded domain Ω ⊂ R d , d = 2 or 3, for the velocity and the pressure of the fluid, with an elastic extra-stress tensor as right-hand side in the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, crucial features of which are the presence of a centre-of-mass diffusion term and a cut-off function β L (·) := min(·, L) in the drag and convective terms, where L 1. We focus on finitelyextensible nonlinear elastic, FENE-type, dumbbell models. We perform a rigorous passage to the limit as the spatial and temporal discretization parameters tend to zero, and show that a (sub)sequence of these finite element approximations converges to a weak solution of this coupled Navier-StokesFokker-Planck system. The passage to the limit is performed under minimal regularity assumptions on the data. Our arguments therefore also provide a new proof of global existence of weak solutions to Fokker-Planck-Navier-Stokes systems with centre-of-mass diffusion and microscopic cut-off. The convergence proof rests on several auxiliary technical results including the stability, in the Maxwellianweighted H 1 norm, of the orthogonal projector in the Maxwellian-weighted L 2 inner product onto finite element spaces consisting of continuous piecewise linear functions. We establish optimal-order quasi-interpolation error bounds in the Maxwellian-weighted L 2 and H 1 norms, and prove a new elliptic regularity result in the Maxwellian-weighted H 2 norm.

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Analytical Methods for Markov Semigroups

Cited by 123 publications

References 0 publications

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Invariant measures associated to degenerate elliptic operators

Finite element approximation of kinetic dilute polymer models with microscopic cut-off

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