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Background:

I am an undergraduate maths student and I am tasked with writing a 10 page essay on a mathematical topic outside of what we studied yet, so something new and perhaps more advanced. The whole point of the essay is to get a student to learn about something new on his/her own by consulting various books.

Since my future interest lies in mathematical finance and since I enjoy Partial Differential Equations; I thought about choosing a topic concerning PDE's in finance. I have taken ODE's and PDE's undergraduate courses.

Aim: I would like to explore Brownian Motion, Black-Scholes Equation and topics such as Option Pricing and if I am not wrong I would have to learn about Stochastic Partial Differential Equations. -How should I delve into this field? The issue I am having is that I do not really know which areas of maths I actually have to learn in order to explore the above. Book recommendations would be helpful.

Thank you

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  • $\begingroup$ I would think (I could be wrong) that as an undergraduate, you would have difficulty understanding and learning stochastic PDEs in a short space of time especially if you have only taken undergraduate ODEs and PDEs courses. $\endgroup$ – Mattos Nov 29 '17 at 15:09
  • $\begingroup$ @Mattos I understand that I cannot really learn stochastic PDE's thoroughly. Although what I am looking for is at least some part so I can discuss some topic(s) related to finance. Ordinary PDE's would also suffice but I am not sure if I can discuss them in financial topics without SPDE's... $\endgroup$ – MilTom Nov 29 '17 at 21:13
  • $\begingroup$ There's also HJB equations and optimal control theory, a quite "natural" link to finance $\endgroup$ – Shalop Aug 10 '18 at 1:12
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For books on SPDEs in general you can look at this question. There is also my faviorite book on SPDEs, namely PDE and Martingale Methods in Option Pricing by Pasucci. I think it is exactly the book you are looking for. It is carefully explaining everything you'll need later to investigate SPDEs in mathematical finance. The Black-Scholes model is thoroughly discussed and many more market models. There are also numerical methods.

Otherwise if you want to focus more on ODEs I'd recommend Stochastic Differential Equations by Oksendal. Here, also all concepts are introduced in mathematically rigoros sense and in the last chapter several applications in mathematical finance are discussed, e.g. in option pricing.

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