Bond and Option Prices under Skew Vasicek Model with Transaction Cost

This paper studies the European option pricing on the zero-coupon bond in which the Skew Vasicek model uses to predict the interest rate amount. To do this, we apply the skew Brownian motion as the random part of the model and show that results of the model predictions are better than other types of the model. Besides, we obtain an analytical formula for pricing the zero-coupon bond and find the European option price by constructing a portfolio that contains the option and a share of the bond. Since the skew Brownian motion is not a martingale, thus we add transaction costs to the portfolio, where the time between trades follows the exponential distribution. Finally, some numerical results are presented to show the efficiency of the proposed model.

Strong Solutions of Stochastic Differential Equations with Generalized Drift and Multidimensional Fractional Brownian Initial Noise

In this paper, we prove the existence of strong solutions to an stochastic differential equation with a generalized drift driven by a multidimensional fractional Brownian motion for small Hurst parameters $$H<\frac{1}{2}.$$ H < 1 2 . Here, the generalized drift is given as the local time of the unknown solution process, which can be considered an extension of the concept of a skew Brownian motion to the case of fractional Brownian motion. Our approach for the construction of strong solutions is new and relies on techniques from Malliavin calculus combined with a “local time variational calculus” argument.

On some properties of sticky Brownian motion

In this paper, we investigate a generalization of Brownian motion, called sticky skew Brownian motion, which has two interesting characteristics: stickiness and skewness. This kind of processes spends a lot more time at its sticky points so that the time they spend at the sticky points has positive Lebesgue measure. By using time change, we obtain an SDE for the sticky skew Brownian motion. Then, we present the explicit relationship between symmetric local time and occupation time. Some basic probability properties, such as transition density, are studied and we derive the explicit expression of Laplace transform of transition density for the sticky skew Brownian motion. We also consider the first hitting time problems over a constant boundary and a random jump boundary, respectively, and give some corollaries based on the results above.

On Weak Limiting Distributions for Random Walks on a Spider

In this article, we study random walks on a spider that can be established from the classical case of simple symmetric random walks. The primary purpose of this article is to establish a functional central limit theorem for random walks on a spider and to define Brownian spider as the resulting weak limit. In special case, random walks on a spider can be characterized as skew random walks. It is well known for skew Brownian motion as the resulting weak limit of skew random walks. We first will study the tightness and then it will be shown for the convergence of finite dimensional distribution for random walks on a spider.

Two consistent estimators for the skew Brownian motion

The skew Brownian motion (SBm) 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 an SBm 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.