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I recently completed reading the book "Stochastic Differential Equations" by Bernt Oksendal which is the first time ever I was exposed to the topic. Now I am interested in pursuing research ( Ph.D.) SDEs and its applications in finance and I would like some help finding some recent papers related to or useful when doing research. I have already looked at some papers on MathSciNet by the same author but I would much appreciate if anyone can suggest some journals or papers/ articles that are relevant and useful in the current times. Thank you in advance!

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  • $\begingroup$ Not a paper but one of the main (``must-have") references is the Karatzas and Shreve textbook. $\endgroup$ – user102087 Aug 11 '18 at 2:21
  • $\begingroup$ Yes I have been referring that book while reading Oksendal's book @WaitakereCity $\endgroup$ – Heisenberg Aug 11 '18 at 2:41
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    $\begingroup$ I mean it's an entire field. What exactly are you interested in? SDEs with applications in finance is a massive field. I don't do finance but I do work in stochastic analysis. Something that is really popular right now are rough volatility models. I've heard something about Heston model being pretty standard now. Just some phrases to look for. $\endgroup$ – user69208 Aug 11 '18 at 2:46
  • $\begingroup$ @ZacharySelk Yes, I understand it is a massive field, which is the reason why I posted this here so that I can get some good suggestions and get exposed to some interesting subfields. I will read about what you mentioned. Thank you! $\endgroup$ – Heisenberg Aug 11 '18 at 3:08
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    $\begingroup$ Start with Stephen Shreve's books (Stochastic Calculus for Finance I and II), and also Martingale Methods in Financial Modelling by Marek Musiela and Marek Rutkowski, and then follow up with papers therein. These are basic texts, and now somewhat dated. Look into Levy processes - they are more general than Brownian motion, and include jump processes and fat tail distributions. Current research is moving towards machine learning. $\endgroup$ – zab Aug 13 '18 at 3:50
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As indicated in the comments, the field is very wide, but I understand from the comment of the OP to zab's answer that there is a specific interest in the more narrow subtopic of applications of fractional Brownian motion to quantitative finance. Here are some overviews:

To get a feel for recent research on this topic, here are some arXiv contributions from the last year or so:

The 2013 paper referred to above notes that the application of fractional Brownian motion to financial modeling still has several unsolved problems of a foundational nature, so this might a fruitful area of research for someone entering the field (it seems a less mature topic than others).

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For basic theory: Stephen Shreve's books (Stochastic Calculus for Finance I and II) and Martingale Methods in Financial Modelling by Marek Musiela and Marek Rutkowski. Also have a look at Oksendal's book on Jump Diffusions.

For numerical treatment of SDEs: Numerical Solution of Stochastic Differential Equations by Platen and Kloeden

For generalization of stochastic calculus to Lévy processes: Lévy Processes and Stochastic Calculus by David Applebaum

One thing to keep in mind is that SDEs and related technology was built around mathematically "nice" objects like Brownian motion, Markov processes and martingales, just because in these cases, theoretical calculations could be done by hand. In my view, these objects are too nice for the real world. For example, the Markov property almost never holds in human systems - people have memory. Martingales are non-anticipating processes, but to a model that cannot account for things like insider information, trader intuition, self-fulfilling herd behaviour etc, financial markets may look like anticipating systems. With recent leaps in computational power (cloud computing, super-computers for hire), brute force computational methods are becoming more important. I'd keep an eye on non-parametric black-box models like neural networks too.

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  • $\begingroup$ @zab What about fractional Brownian motion? It must be more useful for real-world applications $\endgroup$ – Heisenberg Aug 13 '18 at 21:51
  • $\begingroup$ I used these books during my Master's too. Thanks, @zab! $\endgroup$ – Dendi Suhubdy Aug 14 '18 at 16:52

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