This unit is very much like MTH3251 Financial Mathematics, but with more of a focus on stochastic calculus and underlying theorems. I guess it can be seen as a self-contained course in stochastic calculus alongside a bit of financial maths if you have some basics in calculus and probability. In particular, we look at:
1)Properties of Brownian motion (the most basic continuous time stochastic process which are random variables that move through time)
2)Ito calculus calculus incorporating Brownian motion (Itos formula underlies most, if not all studies in stochastic calculus which is taught here)
3)Stochastic differential equations diffusion processes (applications of Ito processes in ODEs to make SDEs and PDEs in particular, we look at conditions for which strong solutions exist for a particular SDE, and how to find them solutions are stochastic processes that have an explicit form that incorporate Brownian motion as well, we also talk a bit about martingales in more detail, which is relevant for a lot of theoretical work)
4)Weak solutions to SDEs there are two ways of estimating weak solutions when strong solutions are not available, change of time, and change of probability measure, for which we investigate them here. The crux of it is just to define a new Brownian motion that makes the SDE solvable explicitly.
5)Applications to finance we go through this very briefly, mainly discussion on applying the above concepts on asset pricing theorems and option pricing. These are all quite basic, but interested students can extend these concepts from these basics to complex methods such as more complex derivative pricing, and modelling. (Applications are taught in MTH5520 interest rate modelling where we look at stochastic calculus applied in bond markets).
I would recommend this unit compared to MTH3251, although the undergrad unit helps a lot with making this unit bearable. This unit goes much deeper into stochastic processes in continuous time and can be very interesting once you know where and why the mathematics is used in real-life applications something the maths department doesnt do very well tbh. The unit that is a straightforward extension to this is MTH5520 Interest Rate Modelling, which uses stochastic calculus as well, and I find that doing MTH5210 makes the content there more bearable too, compared to a lot of students who only did MTH3251. The concepts stick easier, and its much more interesting that way.