in this paper, we extend some famous maximal inequalities and obtain strong laws of largenumbers for arbitrary random variables by use of these inequalities and martingale techniques.

We give sufficient criteria for the Doléans-Dade exponential of a stochastic integral with respect to a counting process local martingale to be a true martingale. The criteria are sufficiently weak to be useful and verifiable, as illustrated by several non-trivial examples, without introducing artificial constraints. In particular, they make it possible to construct nonexplosive point processes...

We provide conditions that guarantee for a discrete time martingale Mn, either limn→∞Mn exists, or both lim supn→∞Mn = ∞ and lim infn→∞Mn = −∞, a.s. on sample paths. A sufficient condition on the martingale is a uniform control on the ratio of the L2 and L1 norms of the increments. Near-symmetry of the increments provides an alternative condition. We also discuss a counterexample when these con...

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...

•All have the property that E[θn+1 | θn] = θn. •Called the martingale property •Philosophically desirable because it means that information is preserved as we move down the hierarchy •Theorem (Doob): All non-negative martingale sequences have a limit with probability 1. θn+1 | θn ∼ DP(cθn), θn+1 | θn ∼ BP(cθn), θn+1 | θn ∼ GammaP(cθn), θn+1 | θn ∼ PYP(cθn) Pathological Properties of Deep Bayesi...

We study large population stochastic dynamic games where the so-called Nash certainty equivalence based control laws are implemented by the individual players. We first show a martingale property for the limiting control problem of a single agent and then perform averaging across the population; this procedure leads to a constant value for the martingale which shows an invariance property of th...

Consider additive functionals of a Markov chain Wk , with stationary (marginal) distribution and transition function denoted by π and Q, say Sn = g(W1) + · · · + g(Wn), where g is square integrable and has mean 0 with respect to π . If Sn has the form Sn =Mn + Rn, where Mn is a square integrable martingale with stationary increments and E(R2 n) = o(n), then g is said to admit a martingale appro...

In this paper we investigate the properties of the free Sheffer systems, which are certain families of martingale polynomials with respect to the free Lévy processes. First, we classify such families that consist of orthogonal polynomials; these are the free analogs of the Meixner systems. Next, we show that the fluctuations around free convolution semigroups have as principal directions the po...

A nonnegative 1-periodic multifractal measure on R is obtained as infinite random product of harmonics of a 1-periodic function W(t). Such infinite products are statistically self-affine and generalize certain Riesz products with random phases. They are martingale structures, therefore converge. The criterion on W for nondegeneracy is provided. It differs completely from those for other known r...