The generalized Beurling–Ahlfors operator S on L(R; Λ), where Λ := Λ(R) is the exterior algebra with its natural Hilbert space norm, satisfies the estimate ‖S‖L (Lp(Rn;Λ)) ≤ (n/2 + 1)(p ∗ − 1), p∗ := max{p, p′}. This improves on earlier results in all dimensions n ≥ 3. The proof is based on the heat extension and relies at the bottom on Burkholder’s sharp inequality for martingale transforms.

The value function of an optimal stopping problem for a process with Lévy jumps is known to be a generalized solution of a variational inequality. Assuming the diffusion component of the process is nondegenerate and a mild assumption on the singularity of the Lévy measure, this paper shows that the value function of problems on an unbounded domain with infinite activity jumps is W 2,1 p,loc . A...

Optimal fictitious completions of an incomplete financial market are shown to be associated with exponential martingales (not just local martingales) and, therefore, to "an optimal equivalent martingale measure'. Results of independent interest, in the theory of continuous-time martingales, are derived as well. iirshurgb, PA 15213-3890

Abstract. The paper gives a Banach space -valued extension of the Tb theorem of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure μ, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces L(μ;X) of functions with val...

Abstract. The paper gives a Banach space -valued extension of the “Tb theorem” of Nazarov, Treil and Volberg (2003) concerning the boundedness of singular integral operators with respect to a measure μ, which only satisfies an upper control on the size of balls. Under the same assumptions as in their result, such operators are shown to be bounded on the Bochner spaces L(μ;X) of functions with v...

Let L be a multidimensional Lévy process under P in its own filtration. The f-minimal martingale measure Qq is defined as that equivalent local martingale measure for E(L) which minimizes the f-divergence E [ (dQ/dP ) ] for fixed q ∈ (−∞, 0) ∪ (1,∞). We give necessary and sufficient conditions for the existence of Qq and an explicit formula for its density. For q = 2, we relate the sufficient c...

This letter derives some new exponential bounds for discrete time, real valued, conditionally symmetric martingales with bounded jumps. The new bounds are extended to conditionally symmetric sub/ supermartingales, and are compared to some existing bounds.