Abstract 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(R 2 n) = o(n), then g is said to admit a martinga...

We introduce a new measure notion on small complexity classes (called F -measure), based on martingale families, that gets rid of some drawbacks of previous measure notions: it can be used to define dimension because martingale families can make money on all strings, and it yields random sequences with an equal frequency of 0’s and 1’s. As applications to F -measure, we answer a question raised...

We give a unified characterization of q-optimal martingale measures for q ∈ [0,∞) in an incomplete market model, where the dynamics of asset prices are described by a continuous semimartingale. According to this characterization the variance-optimal, the minimal entropy and the minimal martingale measures appear as the special cases q = 2, q = 1 and q = 0 respectively. Under assumption that the...

A Martingale Representation for Matching Estimators Matching estimators are widely used in statistical data analysis. However, the distribution of matching estimators has been derived only for particular cases (Abadie and Imbens, 2006). This article establishes a martingale representation for matching estimators. This representation allows the use of martingale limit theorems to derive the asym...

In this paper, a nontrivial generalization of a scalar martingale convergence theorem to the vector case is derived. In particular, a theorem is derived concerning the almxat sure convergence to zero of a (vector) martingale normalized by its (ntatrix) process vaziance. This theorem allows a derivation of almost a~n-e convergence results for least squarea identification algorithms applicable to...

We investigate the effect of martingale control as a smoother for MC/QMC methods. Numerical results of estimating low-biased solutions of the American put option price under the Black-Scholes model demonstrate the unreliability of using QMC methods. But it can be fixed by considering a martingale control variate estimator. In another example of estimating European option prices under stochastic...

In this paper we extend recent results on the existence and uniqueness of solutions of ODEs with non-smooth vector fields to the case of martingale solutions, in the StroockVaradhan sense, of SDEs with non-smooth coefficients. In the first part we develop a general theory, which roughly speaking allows to deduce existence, uniqueness and stability of martingale solutions for L-almost every init...

General martingale theory shows that every martingale can be decomposed into continuous and purely discontinuous parts. In this paper specify a filtration for which the continuous part of the decomposition is 0 a.s. for any Ft martingale. It is a well-known fact that every martingale can be decomposed into continuous and purely discontinuous parts. It is of interest to study the filtrations tha...

I discuss models which allow the local level model, which rationalised exponentially weighted moving averages, to have a time-varying signal/noise ratio. I call this a martingale component model. This makes the rate of discounting of data local. I show how to handle such models effectively using an auxiliary particle filter which deploys M Kalman filters run in parallel competing against one an...

We discuss optimal portfolio selection with respect to utility functions of type −e−αx, α > 0 (exponential problem) and −|1 − αx p |p (p-th problem). We consider N risky assets and a risk-free bond. Risky assets are modeled by continuous semimartingales or exponential Lévy processes. These dynamic expected utility maximization problems are solved by transforming the model into a constrained sta...