We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C except at the stopping boundary and that it is C everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each o...

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. AMS 2000 Mathematics Subject Classification: Primary 60J45, 60J75, Secondary 60J25.

The statistical learning theory of risk minimization depends heavily on probability bounds for uniform deviations of the empirical risks. Classical probability bounds using Hoeffding’s inequality cannot accommodate more general situations with unbounded loss and dependent data. The current paper introduces an inequality that extends Hoeffding’s inequality to handle these more general situations...

Let X = (X0, . . . , Xn) be a discrete-time martingale taking values in any real Euclidean space such that X0 = 0 and for all n, ‖Xn − Xn−1‖ ≤ 1. We prove the large deviation bound Pr [‖Xn‖ ≥ a] < 2e1−(a−1) 2/2n. This upper bound is within a constant factor, e2, of the AzumaHoeffding Inequality for real-valued martingales. This improves an earlier result of O. Kallenberg and R. Sztencel (1992)....

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C except at the stopping boundary and that it is C everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each o...

The role of correlation inequalities and martingale arguments in establishing conditional exponential bounds is reviewed. Applications to the computation of the Onsager Machlup functional for diiusions under non supremum norms follow.

In this paper non-asymptotic exponential estimates are derived for tail of maximum martingale distribution by naturally norm-ing in the spirit of the classical Law of Iterated Logarithm.

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C except at the stopping boundary and that it is C everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each o...

We prove that the perpetual American put option price of an exponential Lévy process whose jumps come from a compound Poisson process is the classical solution of its associated quasi-variational inequality, that it is C except at the stopping boundary and that it is C everywhere (i.e. the smooth pasting condition always holds). We prove this fact by constructing a sequence of functions, each o...