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

In this paper we study the system $$begin{aligned}&min biggl {-mathcal H u_i(x,t)-psi _i(x,t),u_i(x,t)-max _{jne i}(-c_{i,j}(x,t)+u_j(x,t))biggr }=0,&u_i(x,T)=g_i(x), iin {1,ldots ,d}, end{aligned}$$ where ((x,t)in mathbb R ^{N}times [0,T]) . 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 (mathcal H ) and on continuous functions (psi _i) , (c_{i,j}) , and (g_i) . A key aspect is that we make no sign assumption on the switching costs ({c_{i,j}}) and that (c_{i,j}) is allowed to depend on (x) as well as (t) . Using the comparison principle, the existence of a unique viscosity solution ((u_1,ldots ,u_d)) 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 ((u_1,ldots ,u_d)) beyond continuity. In this context, in particular, we assume that (mathcal H ) belongs to a class of second-order differential operators of Kolmogorov type of the form: $$begin{aligned} mathcal H =sum _{i,j=1}^m a_{i,j}(x,t)partial _{x_i x_j}+sum _{i=1}^m a_i(x,t)partial _{x_i} +sum _{i,j=1}^N b_{i,j}x_ipartial _{x_j}+partial _t, end{aligned}$$ where (1le mle N) . The matrix ({a_{i,j}(x,t)}_{i,j=1,ldots ,m}) is assumed to be symmetric and uniformly positive definite in (mathbb R ^m) . In particular, uniform ellipticity is only assumed in the first (m) coordinate directions, and hence, (mathcal H ) may be degenerate.