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 establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide “sequential” (non-i.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between thes...

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...

*Correspondence: [email protected] College of Mathematics and Computer Science, Fuzhou University, Fuzhou, 350116, P.R. China Abstract Choosing space Cg as the phase space, the existence, uniqueness and stability of the solution to neutral stochastic functional differential equations with infinite delay (short for INSFDEs) are studied in this paper. Under non-Lipschitz condition, weakened ...

A simple bound is presented for the probability that the sum of nonnegative independent random variables is exceeded by its expectation by more than a positive number t. If the variables have the same expectation the bound is slightly weaker than the Bennett and Bernstein inequalities, otherwise it can be significantly stronger. The inequality extends to one-sidedly bounded martingale differenc...

For any strictly positive martingale S = e for which X has an analytically tractable characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in log(K/S0). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martinga...

This paper discusses the H∞ control problem for a class of nonlinear stochastic systems with Markovian jumps subjected to both stateand disturbance-dependent noise. We establish the equivalent conditions among Hamilton-Jacobi inequality (HJI), Hamilton-Jacobi equality (HJE), the dissipative inequality and L2-gain property for this class of systems. As to the infinite time horizon case, we synth...