We explore in this chapter questions of existence and uniqueness for solutions to stochastic differential equations and offer a study of their properties. This endeavor is really a study of diffusion processes. Loosely speaking, the term diffusion is attributed to a Markov process which has continuous sample paths and can be characterized in terms of its infinitesimal generator.

Stochastic Differential Equations

In technical applications, uncertainties are a topic of increasing interest. During the last years the Polynomial Chaos of Wiener (Amer. J. Math. 60(4), 897–936, 1938) was revealed to be a cheap alternative to Monte Carlo simulations. In this paper we apply Polynomial Chaos to stationary and transient problems, both from academics and from industry. For each of the applications, chances and limits of Polynomial Chaos are discussed. The presented problems show the need for new theoretical results.

Polynomial chaos for the approximation of uncertainties: Chances and limits

Modelling and simulation of autonomous oscillators with random parameters

Original article: Further properties of random orthogonal matrix simulation

The high-frequency trading (HFT) uses the discrete and finite precision Gaussian cumulative distribution table, which leads to the erroneous computation of modern option pricing models. In this context, this paper provides two fast and tight approximations for the Gaussian cumulative distribution function which takes continuous input values and provides a wide range of precision values correct up to $$O(10^{-5})$$
 . These approximations eliminate the existing finite precision and discrete tabulated values with the real-valued computations. We derive accurate and tight option price by exploiting these approximations, which are in excellent agreement with the exact option price under a wide range of volatilities and time periods. It also operates well on the risk hedging metrics such as option greeks. Further, this framework can be employed for accurate and fast computation of pricing, designing of replicating portfolio, and dynamic hedging purposes which in turn reduces arbitrage in HFT. Finally, we discuss time-dependent Fokker–Planck equation corresponding to geometric brownian motion stochastic differential equation and derive steady-state solution in terms of lognormal probability density function which helps to compute the aforementioned option price models.

A new tight approximation towards the computation of option price

https://www.imacm.uni-wuppertal.de/fileadmin/imacm/preprints/amna_07_01.pdf

Polynomial chaos for simulating random volatilities

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Polynomial chaos for simulating random volatilities