Financial markets are highly uncertain environments where the evolution of asset prices exhibits random fluctuations over time, in particular due to complex and unpredictable market mechanisms. In this regard, stochastic analysis, which is at the intersection between the theory of probability and functional analysis, plays a fundamental role in financial modeling. 

Being aware of the non-specialist yet mathematically strong nature of the audience, I divide my talk into two major parts. The first part is mostly introductory, where I first give/recall elementary but indispensable notions of probability and stochastic calculus, then I illustrate the fundamentals of mathematical finance. I mention that throughout this part, I put the emphasis on the modeling aspects, most notably the extensive application of stochastic processes. 

In the second part, I present the topic of my doctoral research, i.e. Branching processes and multiple term structure modeling. I start by defining the multiple term structure framework and providing a construction of Continuous-state Branching processes with Immigration (CBI), which constitute a sophisticated class of stochastic processes. I carry on with examples of how CBI processes can be exploited for the modeling of financial markets where multiple term structures typically coexist. Finally, I propose some avenues of further developments.