Stochastic calculus is to do with mathematics that operates on stochastic processes.
The best known stochastic process is the Wiener process used for modelling Brownian motion.
Other key components are Ito calculus & Malliavin calculus.
Stochastic calculus is used in finance where prices can be modelled to follow SDEs. In the Black-Scholes model, prices follow geometric Brownian motion.
The Ito integral is one of the major components in stochastic calculus. It is defined as the integral
\begin{equation*}
\int HdX
\end{equation*}
where $X$ is a semimartingale & $H$ is a locally bounded predictable process. Note that we cannot use the generalized Riemann-Stieltjes integral because strong bounded variation is assumed & Brownian motion is not of bounded variation.
Stochastic analysis is looking at the interplay between analysis & probability.
Examples of research topics include linear & nonlinear SPDEs, forward-backward SDEs, rough path theory, asymptotic behaviour of stochastic processes, filtering, sequential monte carlo methods, particle approximations, & statistical methods for stochastic processes.
Something I am interested in is how probabilistic techniques can be used to settle problems in in harmonic analysis, such as proving the $L^p$ boundedness of the Riesz transform. For $f\in C^1_K:$
\begin{equation*}
||R_jf||_p\leq c||f||_p,~1<p<\infty
\end{equation*}
for some constant $c.$
A technique is to give a probabilistic interpretation of the Riesz transform before proving $L^p$ boundedness:
\begin{equation*}
R_jf(x)=c\lim_{s\to\infty}\mathbb{E}^{(0,s)}_{x}\int^{\tau}_{0}A\bigtriangledown u(Z_r)\cdot
\end{equation*}
where $u$ is the harmonic extension of $f,~f\in C^{\infty}_K.$ Here $A$ is the $(d+1)\times (d+1)$ matrix with $A_{ik}$ zero unless $i=d+1,~k=j,$ in which case it is one. Here, $Z_t$ is Brownian motion in $\mathbb{R}^2.$
I hope this is okay. I am happy to expand on any points in more detail if you would like.
