## About this Research Topic

Stochastic calculus is the area of mathematics that deals with processes containing a stochastic component and thus allows the modeling of random systems. Many stochastic processes are based on functions which are continuous, but nowhere differentiable. This rules out differential equations that require the use of derivative terms, since they are unable to be defined on non-smooth functions. Instead, a theory of integration is required where integral equations do not need the direct definition of derivative terms. In quantitative finance, the theory is known as Ito calculus.

Over the past four decades, stochastic calculus has represented a rapidly growing area of research, both in terms of the theory and its application to practical problems arising in such varied fields as econophysics and mathematical finance, in which self-similar processes are used – including Brownian motion, stable Lévy processes, and fractional Brownian motion. Brownian motion was first applied in finance by Bachelier in 1900. In 1973, the log-price of a stock was modeled using Brownian motion in an approach named the Black–Scholes–Merton model. Stable Lévy processes are widely used in financial econometrics to model the dynamics of stock, commodities, and currency exchange prices, etc. Fractional Brownian motion is a centered Gaussian process that extends Brownian motion, and has attracted interest from researchers in a number of fields due to, among other things, its long-range dependence.

The primary use of stochastic calculus in finance is for modeling the random motion of an asset price in the Black–Scholes model. The physical process of Brownian motion (specifically geometric Brownian motion) is used to model asset prices via the Weiner process. This process is represented by a stochastic differential equation, which despite its name is in fact an integral equation.

This Research Topic encourages the submission of original research and review articles exploring stochastic processes, stochastic partial differential equations and integration, and their application to finance. Potential topics of interest include, but are not limited to:

- The rough volatility model;

- Stochastic processes applied in finance and other fields;

- Stochastic differential equations;

- Stochastic partial differential equations;

- Fractional diffusion;

- Transform methods applied in stochastic differential equations;

- Numerical methods for stochastic partial differential equations;

- Fractional operators;

- Asset pricing;

- Energy market pricing by stochastic models.

**Keywords**:
mixed Gaussian processes, fractional Brownian motion, sub-fractional Brownian motion, stochastic partial differential equations, financial applications

**Important Note**:
All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.

Over the past four decades, stochastic calculus has represented a rapidly growing area of research, both in terms of the theory and its application to practical problems arising in such varied fields as econophysics and mathematical finance, in which self-similar processes are used – including Brownian motion, stable Lévy processes, and fractional Brownian motion. Brownian motion was first applied in finance by Bachelier in 1900. In 1973, the log-price of a stock was modeled using Brownian motion in an approach named the Black–Scholes–Merton model. Stable Lévy processes are widely used in financial econometrics to model the dynamics of stock, commodities, and currency exchange prices, etc. Fractional Brownian motion is a centered Gaussian process that extends Brownian motion, and has attracted interest from researchers in a number of fields due to, among other things, its long-range dependence.

The primary use of stochastic calculus in finance is for modeling the random motion of an asset price in the Black–Scholes model. The physical process of Brownian motion (specifically geometric Brownian motion) is used to model asset prices via the Weiner process. This process is represented by a stochastic differential equation, which despite its name is in fact an integral equation.

This Research Topic encourages the submission of original research and review articles exploring stochastic processes, stochastic partial differential equations and integration, and their application to finance. Potential topics of interest include, but are not limited to:

- The rough volatility model;

- Stochastic processes applied in finance and other fields;

- Stochastic differential equations;

- Stochastic partial differential equations;

- Fractional diffusion;

- Transform methods applied in stochastic differential equations;

- Numerical methods for stochastic partial differential equations;

- Fractional operators;

- Asset pricing;

- Energy market pricing by stochastic models.

**Keywords**:
mixed Gaussian processes, fractional Brownian motion, sub-fractional Brownian motion, stochastic partial differential equations, financial applications

**Important Note**:
All contributions to this Research Topic must be within the scope of the section and journal to which they are submitted, as defined in their mission statements. Frontiers reserves the right to guide an out-of-scope manuscript to a more suitable section or journal at any stage of peer review.