41,649 research outputs found
Optimal estimation for discrete time jump processes
Optimum estimates of nonobservable random variables or random processes which influence the rate functions of a discrete time jump process (DTJP) are obtained. The approach is based on the a posteriori probability of a nonobservable event expressed in terms of the a priori probability of that event and of the sample function probability of the DTJP. A general representation for optimum estimates and recursive equations for minimum mean squared error (MMSE) estimates are obtained. MMSE estimates are nonlinear functions of the observations. The problem of estimating the rate of a DTJP when the rate is a random variable with a probability density function of the form cx super K (l-x) super m and show that the MMSE estimates are linear in this case. This class of density functions explains why there are insignificant differences between optimum unconstrained and linear MMSE estimates in a variety of problems
Mixing it up: A general framework for Markovian statistics
Up to now, the nonparametric analysis of multidimensional continuous-time
Markov processes has focussed strongly on specific model choices, mostly
related to symmetry of the semigroup. While this approach allows to study the
performance of estimators for the characteristics of the process in the minimax
sense, it restricts the applicability of results to a rather constrained set of
stochastic processes and in particular hardly allows incorporating jump
structures. As a consequence, for many models of applied and theoretical
interest, no statement can be made about the robustness of typical statistical
procedures beyond the beautiful, but limited framework available in the
literature. To close this gap, we identify -mixing of the process and
heat kernel bounds on the transition density as a suitable combination to
obtain -norm and kernel invariant density estimation rates matching
the case of reversible multidimenisonal diffusion processes and outperforming
density estimation based on discrete i.i.d. or weakly dependent data. Moreover,
we demonstrate how up to -terms, optimal -norm adaptive invariant
density estimation can be achieved within our general framework based on tight
uniform moment bounds and deviation inequalities for empirical processes
associated to additive functionals of Markov processes. The underlying
assumptions are verifiable with classical tools from stability theory of
continuous time Markov processes and PDE techniques, which opens the door to
evaluate statistical performance for a vast amount of Markov models. We
highlight this point by showing how multidimensional jump SDEs with L\'evy
driven jump part under different coefficient assumptions can be seamlessly
integrated into our framework, thus establishing novel adaptive -norm
estimation rates for this class of processes
Extension and calibration of a Hawkes-based optimal execution model
We provide some theoretical extensions and a calibration protocol for our
former dynamic optimal execution model. The Hawkes parameters and the
propagator are estimated independently on financial data from stocks of the
CAC40. Interestingly, the propagator exhibits a smoothly decaying form with one
or two dominant time scales, but only so after a few seconds that the market
needs to adjust after a large trade. Motivated by our estimation results, we
derive the optimal execution strategy for a multi-exponential Hawkes kernel and
backtest it on the data for round trips. We find that the strategy is
profitable on average when trading at the midprice, which is in accordance with
violated martingale conditions. However, in most cases, these profits vanish
when we take bid-ask costs into account
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