Systems of variational inequalities in the context of optimal switching problems and operators of Kolmogorov type

• In this paper we study the system where . A special case of this type of system of variational inequalities with terminal data occurs in the context of optimal switching problems. We establish a general comparison principle for viscosity sub- and supersolutions to the system under mild regularity, growth, and structural assumptions on the data, i.e., on the operator and on continuous functions , , and . A key aspect is that we make no sign assumption on the switching costs and that is allowed to depend on as well as . Using the comparison principle, the existence of a unique viscosity solution to the system is constructed as the limit of an increasing sequence of solutions to associated obstacle problems. Having settled the existence and uniqueness, we subsequently focus on regularity of beyond continuity. In this context, in particular, we assume that belongs to a class of second-order differential operators of Kolmogorov type of the form: where . The matrix is assumed to be symmetric and uniformly positive definite in . In particular, uniform ellipticity is only assumed in the first coordinate directions, and hence, may be degenerate.

Ämnesord

NATURVETENSKAP  -- Matematik (hsv//swe)

NATURAL SCIENCES  -- Mathematics (hsv//eng)

NATURVETENSKAP  -- Matematik -- Matematisk analys (hsv//swe)

NATURAL SCIENCES  -- Mathematics -- Mathematical Analysis (hsv//eng)

Nyckelord

System

Variational inequality

Existence

Viscosity solution

Obstacle problem

Regularity

Kolmogorov equation

Ultraparabolic

Hypoelliptic

Backward stochastic differential equation

Reflected backward stochastic differential equation

Optimal switching problem

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