Том 7

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Browsing Том 7 by Subject "Euler’s approach"

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    Fractional calculus and its application in financial mathematics
    (2024) Zubritska, Dariia; Shchestyuk, Nataliya; Sluchynskyi, Dmytro
    Fractional calculus extends classical calculus by allowing differentiation and integration of non-integer orders, providing valuable tools for analyzing complex systems. In this part of the paper we demonstrate the main methods of fractional calculus, including Euler’s, Riemann-Liouville, and Caputo approaches. The behavior of functions such as xn, eλx, and sin(x) is analyzed for fractional orders, demonstrating how fractional differentiation results in varying patterns of growth and decay. The second part explores the application of fractal derivatives in financial mathematics. We present the use of the Riemann-Liouville derivative to model stock prices in illiquid markets, where the price of an asset may remain unchanged for some time. For this, subdiffusion processes and a fractal integrodifferential equation with the Riemann-Liouville derivative are used. The idea of subdiffusion models is to replace the calendar time t in the risk-free bond motion and classical GBM by some stochastic process Ht, which represents a hitting time, which is interpreted as the first time at which Gt hits the barrier t. Next, we focus on the pricing of a European option when the underlying asset is illiquid. The option price is found as a solution to a fractal Dupire integro-differential equation, in which the time derivative is replaced by the Dzerbayshan–Caputo (D-K) derivative. The D–K derivative is a generalization of the Caputo approach. The form of the D–K derivative depends on a random process Gt, called the subordinate. We take a standard inverse Gaussian process with parameters (1,1) as the subordinate Gt and formulate the Proposition about the form of the fractal Dupire equation for the chosen subordinate. These approaches provide tools that allow the investor to take into account the illiquidity of the financial markets.