We compare equilibrium jump diffusion option prices with endogenously determined stochastic dominance (SD) option bounds. We use model parameters from earlier studies and find that most equilibrium model prices consistent with SD bounds yield economically meaningless results. Further, the implied distributions of the SD bounds exhibit a tail risk comparable to that of the underlying return data, thus shedding light on the dark matter of the inconsistency of physical and risk‐neutral tail probabilities. Since the SD bound assumptions are weaker, we conclude that these bounds should either replace or be used to verify the equilibrium model results.

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This letter proposes a multiple parametric dictionary learning algorithm for direction of arrival (DOA) estimation in presence of array gain-phase error and mutual coupling. It jointly solves both the DOA estimation and array imperfection problems to yield a robust DOA estimation in presence of array imperfection errors and off-grid. In the proposed method, a multiple parametric dictionary learning-based algorithm with an steepest-descent iteration is used for learning the parametric perturbation matrices and the steering matrix simultaneously. It also exploits the multiple snapshots information to enhance the performance of DOA estimation. Simulation results show the efficiency of the proposed algorithm when both off-grid problem and array imperfection exist.

N-heterocyclic carbenes [1,3-bis(2,4,6-trimethylphenyl)imidazol-2-ylidene 1a (IMes) or 1,3-bis(2,6-di-iso-propylphenyl)imidazolidene 1b (SIPr)] react with the polarized alkenes 2 and 4 to form the crystalline betaines 3a, 3b and 5a. Furthermore, a one-pot reaction between an aldehyde, malonitrile, and an imidazolium salt of an N-heterocyclic carbene has been developed for the efficient preparation of betaine 5a without isolation of the free carbene. Full characterization data, including X-ray crystal structures, is reported for the three synthesized betaines. The structures of the betaines 3a, 3b and 5a shed new light on the initial products formed in the reactions between N-heterocyclic carbenes and compounds containing polarized double bonds.