We develop a martingale approach for studying continuous-time stochastic differential games of control and stopping, in a non-Markovian framework and with the control affecting only the drift term of the state-process. Under appropriate conditions, we show that the game has a value and construct a saddle pair of optimal control and stopping strategies. Crucial in this construction is a characte...

A martingale measure is constructed by using a mean correcting transform for the geometric Lévy processes model. It is shown that this measure is the mean correcting martingale measure if and only if, in the Lévy process, there exists a continuous Gaussian part. Although this measure cannot be equivalent to a physical probability for a pure jump Lévy process, we show that a European call option...

The paper develops a way of embedding general martingales in continuous ones in such a way that the quadratic variation of the continuous martingale has conditional cumulants (given the original martingale) that are explicitly given in terms of optional and predictable variations of the original process. Bartlett identities for the conditional cumulants are also found. A main corollary to these...

Consider G the progressive enlargement of a filtration F with a random time τ . Assuming that, in F, the martingale representation property holds, we examine conditions under which the martingale representation property holds also in G. It is noted that the classical results on this subject are no more sufficient to deal with all examples coming from credit risk modeling. In this paper, we intr...

A vector autoregression is singular when explosive characteristic roots have geometric multiplicity larger than one. The singular component is a mixingale. Martingale decompositions are constructed for sample moments involving the singular component. This permits weak and strong analysis in the case of martingale difference innovations. While least squares estimators are shown to be inconsisten...

The difference equations ξk = af(ξk−1) + εk, where (εk) is a square integrable difference martingale, and the differential equation dξ = −af(ξ)dt + dη, where η is a square integrable martingale, are considered. A family of estimators depending, besides the sample size n (or the observation period, if time is continuous) on some random Lipschitz functions is constructed. Asymptotic optimality of...