Bernoulli

  • Bernoulli
  • Volume 16, Number 1 (2010), 258-273.

Viscosity solutions for systems of parabolic variational inequalities

Lucian Maticiuc, Etienne Pardoux, Aurel Răşcanu, and Adrian Zălinescu

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Abstract

In this paper, we first define the notion of viscosity solution for the following system of partial differential equations involving a subdifferential operator:

\[\catcode`\*=4\cases{\dfrac{\partial u}{\partial t}(t,x )+\mathcal{L}_{t}u(t,x )+f (t,x,u (t,x ))\in\partial\varphi (u (t,x)),*\quad $t\in[ 0,T) ,x\in\mathbb{R}^{d}$,\cr u( T,x ) =h(x),*\quad $x\in\mathbb{R}^{d}$,}\]

where ∂φ is the subdifferential operator of the proper convex lower semicontinuous function φ : ℝk→(−∞, +∞] and $\mathcal{L}_{t}$ is a second differential operator given by $\mathcal{L}_{t}v_{i}(x)=\frac{1}{2}\operatorname{Tr}[\sigma(t,x)\sigma^{\ast}(t,x)\mathrm{D}^{2}v_{i}(x)]+\langle b(t,x),\nabla v_{i}(x)\rangle$, $i\in\overline{1,k}$.

We prove the uniqueness of the viscosity solution and then, via a stochastic approach, prove the existence of a viscosity solution u : [0, T]×ℝd→ℝk of the above parabolic variational inequality.

Article information

Source
Bernoulli Volume 16, Number 1 (2010), 258-273.

Dates
First available in Project Euclid: 12 February 2010

Permanent link to this document
https://projecteuclid.org/euclid.bj/1265984711

Digital Object Identifier
doi:10.3150/09-BEJ204

Mathematical Reviews number (MathSciNet)
MR2648757

Zentralblatt MATH identifier
05815971

Keywords
Feynman–Kac formula systems of variational inequalities viscosity solutions

Citation

Maticiuc, Lucian; Pardoux, Etienne; Răşcanu, Aurel; Zălinescu, Adrian. Viscosity solutions for systems of parabolic variational inequalities. Bernoulli 16 (2010), no. 1, 258--273. doi:10.3150/09-BEJ204. https://projecteuclid.org/euclid.bj/1265984711.


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