In this paper we prove optimal interior regularity for solutions to the obstacle problem for a class of second order differential operators of Kolmogorov type. We treat smooth obstacles as well as non-smooth obstacles. All our proofs follow the same line of thought and are based on blow-ups, compactness, barriers and arguments by contradiction. This problem arises in financial mathematics, when considering path-dependent derivative contracts with the early exercise feature.2000 Mathematics Subject classification.

The main result established in this paper is the existence and uniqueness of strong solutions to the obstacle problem for a class of subelliptic operators in non-divergence form. The operators considered are structured on a set of smooth vector fields in R n , X = {X0, X1, ..., Xq}, q ≤ n, satisfying Hörmander's finite rank condition. In this setting, X0 is a lower order term while {X1, ..., X q} are building blocks of the subelliptic part of the operator. In order to prove this, we establish an embedding theorem under the assumption that the set {X0, X1, ..., Xq} generates a homogeneous Lie group. Furthermore, we prove that any strong solution belongs to a suitable class of Hölder continuous functions.

The classical ν-metric introduced by Vinnicombe in robust control theory for rational plants was extended to classes of nonrational transfer functions in [1]. In [11], an extension of the classical ν-metric was given when the underlying ring of stable transfer functions is the Hardy algebra, H ∞ . However, this particular extension to H ∞ did not directly fit in the abstract framework given in [1]. In this paper we show that the case of H ∞ also fits into the general abstract framework in [1] and that the ν-metric defined in this setting is identical to the extension of the ν-metric defined in [11]. This is done by introducing a particular Banach algebra, which is the inductive limit of certain C * -algebras.2010 Mathematics Subject Classification. Primary 93B36; Secondary 93D09, 46J15.

In a cylinder T = × (0, T ) ⊂ R n+1 + we study the boundary behavior of nonnegative solutions of second order parabolic equations of the formwhere X = {X 1 , . . . , X m } is a system of C ∞ vector fields in R n satisfying Hörmander's rank condition (1.2), and is a non-tangentially accessible domain with respect to the Carnot-Carathéodory distance d induced by X. Concerning the matrix-valued function A = {a i j }, we assume that it is real, symmetric and uniformly positive definite. Furthermore, we suppose that its entries a i j are Hölder continuous with respect to the parabolic distance associated with d. Our main results are: 1) a backward Harnack inequality for nonnegative solutions vanishing on the lateral boundary (Theorem 1.1); 2) the Hölder continuity up to the boundary of the quotient of two nonnegative solutions which vanish continuously on a portion of the lateral boundary (Theorem 1.2); 3) the doubling property for the parabolic measure associated with the operator H (Theorem 1.3). These results generalize to the subelliptic setting of the present paper, those in Lipschitz cylinders by Fabes, Safonov and Yuan in [20,39]. With one proviso: in those papers the authors assume that the coefficients a i j be only bounded and measurable, whereas we assume Hölder continuity with respect to the intrinsic parabolic distance.

In this paper we continue the study initiated in [FGN12] concerning the obstacle problem for a class of parabolic nondivergence operators structured on a set of vector fields X = {X1, ..., Xq} in R n with C ∞ -coefficients satisfying Hörmander's finite rank condition, i.e., the rank of Lie[X1, ..., Xq] equals n at every point in R n . In [FGN12] we proved, under appropriate assumptions on the operator and the obstacle, the existence and uniqueness of strong solutions to a general obstacle problem. The main result of this paper is that we establish further regularity, in the interior as well as at the initial state, of strong solutions. Compared to [FGN12] we in this paper assume, in addition, that there exists a homogeneous Lie group G = (R n , •, δ λ ) such that X1, . . . , Xq are left translation invariant on G and such that X1, . . . , Xq are δ λ -homogeneous of degree one.