Without knowing you personally or the curriculum at your school,
it is not possible to give responsible advice on what you should
take next term. You need to discuss this with an adviser. You should be clear that you are currently planning on a career in industry rather than academia. Perhaps
ideas arising on this page could be included in that discussion.
Most beginning courses in stochastic processes include Markov chains
and some simple queueing processes. Usually an introductory course
in probability is a prerequisite. Material in a stochastic processes
course is used in other parts of applied probability modeling:
relaibility theory, sequential statistical analysis,
modern computational methods such as Markov Chain Monte Carlo (MCMC),
and many other topics (including those in the Comments).
If you are serious about an industrial career, I think you should study some basics of computer science and get acquainted with
a computer language. Among my colleagues, it is debated whether this is best done in classes or by self study. I suspect many people benefit by learning the basics in a class. The first computer
language you learn may soon be out of fashion, but basic principles
won't
Also, please read to explore the connections among pure mathematics, statistics,
probability, and computation.
These days one reads a lot about uses
'big data', 'data science', 'machine learning', and 'data mining'
in industry and national security. These are emerging interdisciplinary subjects. Standards for what methods and results are useful are still being established, but I think it is already clear
that some of the ideas will turn out to be very important.
The Ito calculus extends the methods of classical calculus to stochastic functions of random variables.
The Malliavin calculus extends the classical calculus of variations to stochastic functions. Just as the variational calculus allows considering derivatives in infinite dimensional function space, the Malliavin calculus extends stochastic analysis to infinite dimensional space.
In other words, I think the analogy between the Ito and Malliavin calculi is the same as that between the classical multivariable calculus and the variational calculus.
Check out this work by Han Zhang, which has an introductory description of both.
Best Answer
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.