Narrow Results

In this work, we consider a stock price process subjected to idiosyncratic Lévy jumps and global structural changes attributed to interventions due to a semi-Markov process. The semi-Markov process decomposes both the time and state domains of the price process into sub-intervals and price state sub-domains respectively, where a Lévy–Ito process operates. The Lévy jumps decompose the space domain of the currently operating Lévy process. We derive an infinitesimal generator for a stock price process and a closed form expression for the conditional characteristic function of a log price. The former result is used to derive a PIDE satisfied by option prices, while the latter could be used to retrieve risk neutral densities via Fourier transform and price European vanilla options. In the sequel, we derive the characteristic function of the residence time of a semi-Markov process. Incompleteness of the market is exhibited through a general change of measure. For pricing purpose, the minimum entropy martingale measure is defined as an Esscher transform.

The study of minimum entropy of a natural language has been an interesting research subject. For English, great progress has been made, but few reports on other languages have been found in literature. Based on two hypotheses on the conservation of information quantity, we proposed a method which can be used to estimate the minimum entropy of characters in natural languages. With a large quantity of translation corpus, this method enables us to estimate the minimum entropy without calculating the probability. Besides, as the scale of translation corpus increases, the fluctuation of the ratio between character quantities in any two languages becomes negligible. In this paper, we apply this method to the study of two languages of a large character total — Japanese and Chinese.