CONSISTENCY OF THE DRIFT PARAMETERS ESTIMATES IN THE FRACTIONAL BROWNIAN DIFFUSION MODEL AND ESTIMATION OF THE HURST PARAMETER BY MAXIMUM LIKELIHOOD METHOD

In this paper, we study the problem of estimating the unknow parameters in a long memory process based on the maximum likelihood method. We consider again a diffusion model involving fractional Brownian motion. Our goal is to study the consistency of the drift parameter estimates depending on the form of the model.

Mixingale Estimation Function for SPDEs with Random Sampling

We study the mixingale estimation function estimator of the drift parameter in the stochastic partial differential equation when the process is observed at the arrival times of a Poisson process. We use a two stage estimation procedure. We first estimate the intensity of the Poisson process. Then we substitute this estimate in the estimation function to estimate the drift parameter. We obtain the strong consistency and the asymptotic normality of the mixingale estimation function estimator.

A Special Study of the Mixed Weighted Fractional Brownian Motion

In this work, we present the analysis of a mixed weighted fractional Brownian motion, defined by ηt:=Bt+ξt, where B is a Brownian motion and ξ is an independent weighted fractional Brownian motion. We also consider the parameter estimation problem for the drift parameter θ>0 in the mixed weighted fractional Ornstein–Uhlenbeck model of the form X0=0;Xt=θXtdt+dηt. Moreover, a simulation is given of sample paths of the mixed weighted fractional Ornstein–Uhlenbeck process.

Option pricing: a yet simpler approach

We provide a lean, non-technical exposition on the pricing of path-dependent and European-style derivatives in the Cox–Ross–Rubinstein (CRR) pricing model. The main tool used in this paper for simplifying the reasoning is applying static hedging arguments. In applying the static hedging principle, we consider Arrow–Debreu securities and digital options, or backward random processes. In the last case, the CRR model is extended to an infinite state space which leads to an interesting new phenomenon not present in the classical CRR model. At the end, we discuss the paradox involving the drift parameter $$\mu $$ μ in the Black–Scholes–Merton model pricing. We provide sensitivity analysis and an approximation of the speed of convergence for the asymptotically vanishing effect of drift in prices.

Least squares estimation for non-ergodic weighted fractional Ornstein-Uhlenbeck process of general parameters

<abstract><p>Let $ B^{a, b}: = \{B_t^{a, b}, t\geq0\} $ be a weighted fractional Brownian motion of parameters $ a &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. We consider a least square-type method to estimate the drift parameter $ \theta &gt; 0 $ of the weighted fractional Ornstein-Uhlenbeck process $ X: = \{X_t, t\geq0\} $ defined by $ X_0 = 0; \ dX_t = \theta X_tdt+dB_t^{a, b} $. In this work, we provide least squares-type estimators for $ \theta $ based continuous-time and discrete-time observations of $ X $. The strong consistency and the asymptotic behavior in distribution of the estimators are studied for all $ (a, b) $ such that $ a &gt; -1 $, $ |b| &lt; 1 $, $ |b| &lt; a+1 $. Here we extend the results of ^{[<xref ref-type="bibr" rid="b1">1</xref>,<xref ref-type="bibr" rid="b2">2</xref>]} (resp. ^{[<xref ref-type="bibr" rid="b3">3</xref>]}), where the strong consistency and the asymptotic distribution of the estimators are proved for $ -\frac12 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $ (resp. $ -1 &lt; a &lt; 0 $, $ -a &lt; b &lt; a+1 $). Simulations are performed to illustrate the theoretical results.</p></abstract>

The Combined Estimator for Stochastic Equations on Graphs with Fractional Noise

In the present paper, we study the problem of estimating a drift parameter in stochastic evolution equations on graphs. We focus on equations driven by fractional Brownian motions, which are particularly useful e.g., in biology or neuroscience. We derive a novel estimator (the combined estimator) and prove its strong consistency in the long-span asymptotic regime with a discrete-time sampling scheme. The promising performance of the combined estimator for finite samples is examined under various scenarios by Monte Carlo simulations.