One stochastic process that is often used to model real phenomena is the compound Poisson process (CPP). CPP is a process in which a component in the process of the events occurred is assumed to be a Poisson process with a certain intensity function (homogeneous or nonhomogeneous). Thinning process algorithm is usually used to generate events that occurred in the Poisson process, but not yet in CPP. This study aims to find out the algorithm to produce CPP which has the function of Poisson nonhomogeneous (NHPP) intensity, where in addition to knowing the process of events that occur, it also takes into account the extent of the consequences of these events. The value of the load caused by the Poisson process is assumed to be a family of i.i.d random variables and the variables are also independent of the Poisson process. The results of this study have obtained the thinning process algorithm and its generalizations for compound Poisson process having nonhomogeneous Poisson process (NHPP) intensity functions. This algorithm is the result of theoretical development and analysis of computational simulations that can be applied in various fields of science such as the analysis of reliability and risk models.
Compound Poisson process (CPP) is one of the models of stochastic processes in which this model can model a real phenomenon that has an element of uncertainty in the process. CPP has two main components, those is a Poisson process on the component of the poses of an event that occurs and a sequence component of the magnitude as a result of the process of events that occur. This research aims to develop an algorithm to generate random numbers from CPP with a component in the Poisson process in the form of a nonhomogeneous Poisson process (NHPP) and a component in the magnitude of the effect in the form of an exponential distribution (CPP-NHPP-ED). The method used is using the acceptance and rejection method in the form of Thinning process techniques. The results of the study obtained several algorithms, namely the algorithm for CPP-HPP-ED, CPP-NHPP-ED type 1, and CPP-NHPP-ED type 2. These algorithms can be used for computer simulation analysis that can be applied to various fields of science and engineering.
The purpose of this research is to analyze the formation of the optimal portfolio on the Jakarta Islamic Index 30 (JII30) stocks during the new normal period. The model used is a single index model. This study uses secondary data for the period December 2020 - November 2021 from stocks listed on JII30 on the Indonesia Stock Exchange. The results showed that the selected model provides optimal benefits. The risk-free asset return does not provide a higher return than the simulation results. Of the 30 stocks listed in JII30, obtained 7 stocks with the most significant proportion of funds, namely ERAA (25.72%), MDKA (24.87%), TLKM (17.98%), EXCL (12.82%), AKRA (7.54%), ADRO (6.73 %) and ANTM (4.34%). The formation of the optimal portfolio in this study produces an expected return of 0.009974 or 1%, where the portfolio return is above the market rate of return and the average risk-free rate. Meanwhile, the optimal portfolio risk obtained is 0.003173 and the standard deviation is 0.056327 or 5.63%. The impact of the Covid-19 pandemic which has subsided and the economy is getting better has resulted in the formation of an optimal JII30 portfolio that can be carried out and reduce investment risk during the new normal period. Keywords: Jakarta Islamic Index 30 (JII30), Single Index model, new normal, Optimal portfolio
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