The constant r will change depending on the species. One of the most basic examples of differential equations is the Malthusian Law of population growth dp/dt = rp shows how the population (p) changes with respect to time. What are some real life examples of differential equations? In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, and the differential equation defines a relationship between the two. In mathematics, a differential equation is an equation that relates one or more functions and their derivatives. I'd keep an eye on non-parametric black-box models like neural networks too.Frequently Asked Questions What exactly are differential equations? With recent leaps in computational power (cloud computing, super-computers for hire), brute force computational methods are becoming more important.
Martingales are non-anticipating processes, but to a model that cannot account for things like insider information, trader intuition, self-fulfilling herd behaviour etc, financial markets may look like anticipating systems. For example, the Markov property almost never holds in human systems - people have memory. In my view, these objects are too nice for the real world. One thing to keep in mind is that SDEs and related technology was built around mathematically "nice" objects like Brownian motion, Markov processes and martingales, just because in these cases, theoretical calculations could be done by hand. Also have a look at Oksendal's book on Jump Diffusions.įor numerical treatment of SDEs: Numerical Solution of Stochastic Differential Equations by Platen and Kloedenįor generalization of stochastic calculus to Lévy processes: Lévy Processes and Stochastic Calculus by David Applebaum The 2013 paper referred to above notes that the application of fractional Brownian motion to financial modeling still has several unsolved problems of a foundational nature, so this might a fruitful area of research for someone entering the field (it seems a less mature topic than others).įor basic theory: Stephen Shreve's books (Stochastic Calculus for Finance I and II) and Martingale Methods in Financial Modelling by Marek Musiela and Marek Rutkowski. Hedging in fractional Black-Scholes model with transaction costs
BEST DIFFERENTIAL EQUATIONS TEXTBOOK SERIES
Series representation of the pricing formula for the European option driven by space-time fractional diffusion The evaluation of geometric Asian power options under time changed mixed fractional Brownian motion
Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications Pricing European option with the short rate under Subdiffusive fractional Brownian motion regime Modeling the price of Bitcoin with geometric fractional Brownian motion To get a feel for recent research on this topic, here are some arXiv contributions from the last year or so: Here are some overviews:įractional Brownian Motion in Finance (2003)įractional Brownian Motion and applications to financial modelling (2011)Ī note on the use of fractional Brownian motion for financial modeling (2013) As indicated in the comments, the field is very wide, but I understand from the comment of the OP to zab's answer that there is a specific interest in the more narrow subtopic of applications of fractional Brownian motion to quantitative finance.
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