A Model of Market Limit Orders By Stochastic PDE's, Parameter Estimation, and Investment Optimization

In this paper we introduce a completely continuous and time-variate model of the evolution of market limit orders based on the existence, uniqueness, and regularity of the solutions to a type of stochastic partial differential equations obtained in Zheng and Sowers (2012). In contrary to several models proposed and researched in literature, this model provides complete continuity in both time and price inherited from the stochastic PDE, and thus is particularly suitable for the cases where transactions happen in an extremely fast pace, such as those delivered by high frequency traders (HFT's). We first elaborate the precise definition of the model with its associated parameters, and show its existence and uniqueness from the related mathematical results given a fixed set of parameters. Then we statistically derive parameter estimation schemes of the model using maximum likelihood and least mean-square-errors estimation methods under certain criteria such as AIC to accommodate to variant number of parameters . Finally as a typical economics and finance use case of the model we settle the investment optimization problem in both static and dynamic sense by analysing the stochastic (It\^{o}) evolution of the utility function of an investor or trader who takes the model and its parameters as exogenous. Two theorems are proved which provide criteria for determining the best (limit) price and time point to make the transaction

Averaging of stochastic flows: Twist maps and escape from resonance

AbstractOur setup is a classical stochastic averaging one studied by Has’minskiĭ, which is a two-dimensional SDE (on a cylinder) consisting of a fast angular drift and a slow axial diffusion. We seek to understand the asymptotics of the flow generated by this SDE. To do so, we fix n initial points on the cylinder and consider the axial components of the trajectories evolving from these points. We conclude a propagation-of-chaos. There are two components of the limiting n-point motion: a common Brownian motion, and n independent Brownian motions, one for each initial point

Default clustering in large portfolios: Typical events

We develop a dynamic point process model of correlated default timing in a portfolio of firms, and analyze typical default profiles in the limit as the size of the pool grows. In our model, a firm defaults at a stochastic intensity that is influenced by an idiosyncratic risk process, a systematic risk process common to all firms, and past defaults. We prove a law of large numbers for the default rate in the pool, which describes the "typical" behavior of defaults.Comment: Published in at http://dx.doi.org/10.1214/12-AAP845 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Quantitative correlation between promoter methylation and messenger RNA levels of the reduced folate carrier

<p>Abstract</p> <p>Background</p> <p>Methotrexate (MTX) uptake is mediated by the reduced folate carrier (RFC). Defective drug uptake in association with decreased RFC expression is a common mechanism of MTX resistance in many tumor types. Heavy promoter methylation was previously identified as a basis for the complete silencing of RFC in MDA-MB-231 breast cancer cells, its role and prevalence in RFC transcription regulation are, however, not widely studied.</p> <p>Methods</p> <p>In the current study, RFC promoter methylation was assessed using methylation specific PCR in a panel of malignant cell lines (n = 8), including MDA-MB-231, and M805, a MTX resistant cell line directly established from the specimen of a patient with malignant fibrohistocytoma, whom received multiple doses of MTX. A quantitative approach of real-time PCR for measuring the extent of RFC promoter methylation was developed, and was validated by direct bisulfite genomic sequencing. RFC mRNA levels were determined by quantitative real-time RT-PCR and were related to the extent of promoter methylation in these cell lines.</p> <p>Results</p> <p>A partial promoter methylation and RFC mRNA down-regulation were observed in M805. Using the quantitative approach, a reverse correlation (correlation coefficient = -0.59, <it>p </it>< 0.05) was identified between the promoter methylation and RFC mRNA levels in this a panel of malignant cell lines.</p> <p>Conclusion</p> <p>This study further suggests that promoter methylation is a potential basis for MTX resistance. The quantitative correlation identified in this study implies that promoter methylation is possibly a mechanism involved in the fine regulation of RFC transcription.</p