We propose a method for numerical approximation of backward stochastic differential equations. Our method allows the final condition of the equation to be quite general and simple to implement. It relies on an approximation of Brownian motion by simple random walk.
The Skew Brownian motion is of primary importance in modeling diffusion in media with interfaces which arise in many domains ranging from population ecology to geophysics and finance. We show that the maximum likelihood procedure estimates consistently the parameter of a Skew Brownian motion observed at discrete times. The difficulties arise because the observed process is only null recurrent and has a singular distribution with respect to the one of the Brownian motion. Finally, using the idea of the Expectation-Maximization algorithm, we show that the maximum likelihood estimator can be naturally interpreted as the expected total number of positive excursions divided by the expected number of excursions given the observations. The theoretical results are illustrated by numerical simulations.
We previously reported that the vesicular monoamine transporter 2 (VMAT2) is physically and functionally coupled with Hsc70 as well as with the dopamine synthesis enzymes tyrosine hydroxylase (TH) and aromatic amino acid decarboxylase, providing a novel mechanism for dopamine homeostasis regulation. Here we expand those findings to demonstrate that Hsc70 physically and functionally interacts with TH to regulate the enzyme activity and synaptic vesicle targeting. Co-immunoprecipitation assays performed in brain tissue and heterologous cells demonstrated that Hsc70 interacts with TH and aromatic amino acid decarboxylase. Furthermore, in vitro binding assays showed that TH directly binds the substrate binding and carboxyl-terminal domains of Hsc70. Immunocytochemical studies indicated that Hsc70 and TH co-localize in midbrain dopaminergic neurons. The functional significance of the Hsc70-TH interaction was then investigated using TH activity assays. In both dopaminergic MN9D cells and mouse brain synaptic vesicles, purified Hsc70 facilitated an increase in TH activity. Neither the closely related protein Hsp70 nor the unrelated Hsp60 altered TH activity, confirming the specificity of the Hsc70 effect. Overexpression of Hsc70 in dopaminergic MN9D cells consistently resulted in increased TH activity whereas knockdown of Hsc70 by short hairpin RNA resulted in decreased TH activity and dopamine levels. Finally, in cells with reduced levels of Hsc70, the amount of TH associated with synaptic vesicles was decreased. This effect was rescued by addition of purified Hsc70. Together, these data demonstrate a novel interaction between Hsc70 and TH that regulates the activity and localization of the enzyme to synaptic vesicles, suggesting an important role for Hsc70 in dopamine homeostasis.Dopaminergic neurons within the substantia nigra and ventral tegmental area are the primary sources of the catecholamine neurotransmitter dopamine (DA).2 Despite the fact that dopaminergic neurons account for less than 0.01% of all neurons, they play a significant role in brain function (1-3). Consequently, DA homeostasis is crucial for the preservation and regulation of physiological functions such as locomotion, cognition, neuroendocrine secretion, and motivated behaviors (4). Thus, it is not surprising that disruptions in the DA system have been implicated in several neurological and psychiatric disorders, including Parkinson disease, depression, schizophrenia, attention deficit hyperactivity disorder, Tourette syndrome, and drug addiction (4 -8).The DA life cycle consists of a series of highly regulated molecular events that are ultimately responsible for controlling DA homeostasis. Synthesis of DA occurs in the presynaptic terminals via two enzymatic reactions. First, tyrosine is converted into L-3,4-dihydroxyphenylalanine through the actions of the rate-limiting enzyme tyrosine hydroxylase (TH) (9, 10). Subsequently, aromatic amino acid decarboxylase (AADC) converts L-DOPA into DA (11). When synthesized, DA is packaged and stored withi...
We study the asymptotic behaviour of the maximum likelihood estimator corresponding to the observation of a trajectory of a skew Brownian motion, through a uniform time discretization. We characterize the speed of convergence and the limiting distribution when the step size goes to zero, which in this case are non-classical, under the null hypothesis of the skew Brownian motion being an usual Brownian motion. This allows to design a test on the skewness parameter. We show that numerical simulations can be easily performed to estimate the skewness parameter and provide an application in Biology.
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