Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs

Camilli, Fabio; Marchi, Claudio

doi:10.1088/0951-7715/22/6/011

Cited by 31 publications

(50 citation statements)

References 32 publications

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“…and L is defined in (14). The prices V ε (t, x, y) converge locally uniformly, as ε → 0, to the unique viscosity solution V of the limit equation (37), due to our convergence result, Theorem 5.1 (see also Remark 5.1 describing the slight modifications to the argument in the proof needed to treat this case).…”

Section: Examples and Extensionsmentioning

confidence: 77%

“…where L is the differential operator defined in (14). If the claim is true, we can use Lemma 4.1, since V , V are bounded in y according to estimates (39), to conclude that the functions y → V (t, x, y), y → V (t, x, y) are constants for every (t, x) ∈ (0, T ) × R n + .…”

Section: Moreover If G Is Independent Of Y Then the Convergence Is mentioning

confidence: 94%

“…called the fast subsystem, and observe that its infinitesimal generator is Lw := L(y, D y w, D 2 yy w), with L defined by (14). We assume the following condition:…”

Section: The Main Assumptionmentioning

confidence: 99%

“…The problem of justifying further terms of the asymptotic expansion is wide open in stochastic control and fully nonlinear PDEs, even for the first corrector V 1 . The only related result we know is in the very recent paper by Camilli and Marchi [14] and concerns the rate of convergence in periodic homogenization. We plan to study this issue for particular models arising in applications.…”

Section: Merton Portfolio Optimization Problemmentioning

confidence: 99%

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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: Examples and Extensionsmentioning

confidence: 77%

Section: Moreover If G Is Independent Of Y Then the Convergence Is mentioning

confidence: 94%

“…called the fast subsystem, and observe that its infinitesimal generator is Lw := L(y, D y w, D 2 yy w), with L defined by (14). We assume the following condition:…”

Section: The Main Assumptionmentioning

confidence: 99%

Section: Merton Portfolio Optimization Problemmentioning

confidence: 99%

See 2 more Smart Citations

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Bardi

Cesaroni²,

Manca³

2010

SIAM J. Finan. Math.

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“…Due to the lack of variational structure for (P ) , it seems difficult -and perhaps not possible -to get an explicit expression of r(q). On the other hand some explicit results are available for problems with variational structure ( [6], [16]- [17]). Also see results for traveling waves for the flame propagation( [2]- [3]), which make use of the phase-field approximation.…”

mentioning

confidence: 99%

Homogenization and error estimates of free boundary velocities in periodic media

Kim¹

2012

Applicable Analysis

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In this note I describe a recent result ([14]- [15]) on homogenization and error estimates of a free boundary problem, which describes quasi-static contact angle dynamics on inhomogeneous surface. The method presented here also applies to more general class of free boundary problems with oscillating boundary velocities.Let us define e i ∈ IR n , i = 1, ..., n such that e 1 = (1, 0, .., 0), e 2 = (0, 1, 0, .., 0), ..., and e n = (0, ..., 0, 1), and consider a Lipschitz continuous functionwith Lipschitz constant L. In this paper we are interested in the behavior, as → 0, of the viscosity solutions u ≥ 0 of the following problem with K = {|x| ≤ 1} and with initial data u 0 :We refer to Γ t (u ) := ∂{u (·, t) > 0} − ∂K as the free boundary of u at time t. Note that if the free boundary of u is smooth, then the boundary moves with outward normal velocity V = u t |Du | , and therefore the second equation in (P ) implies thatwhere ν = ν x,t denotes the outward normal vector at x ∈ Γ t (u) with respect to Ω t (u).(P ) is a simplified model to describe contact line dynamics of quasi-static liquid droplets on an irregular surface ([11]). Here u(x, t) denotes the height of the droplet. Heterogeneities on the surface, represented by g( x ), result in contact lines with a fine scale structure that may lead to pinning of the interface and hysteresis of the overall fluid shape.The following is a summary of the main result in [14]: 1

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Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

Kim

Lee

2015

Arch Rational Mech Anal

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We establish higher order convergence rates in the theory of periodic homogenization of both linear and fully nonlinear uniformly elliptic equations of non-divergence form. The rates are achieved by involving higher order correctors which fix the errors occurring both in the interior and on the boundary layer of our physical domain. The proof is based on a viscosity method and a new regularity theory which captures the stability of the correctors with respect to the shape of our limit profile.

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Rates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs

Cited by 31 publications

References 32 publications

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Convergence by Viscosity Methods in Multiscale Financial Models with Stochastic Volatility

Homogenization and error estimates of free boundary velocities in periodic media

Higher Order Convergence Rates in Theory of Homogenization: Equations of Non-divergence Form

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