Good papers on stochastic differential equations with applications in finance
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:
Fractional Brownian Motion in Finance (2003)
Fractional Brownian Motion and applications to financial modelling (2011)
A note on the use of fractional Brownian motion for financial modeling (2013)
To get a feel for recent research on this topic, here are some arXiv contributions from the last year or so:
Modeling the price of Bitcoin with geometric fractional Brownian motion
Pricing European option with the short rate under Subdiffusive fractional Brownian motion regime
Option Pricing Models Driven by the Space-Time Fractional Diffusion: Series Representation and Applications
The evaluation of geometric Asian power options under time changed mixed fractional Brownian motion
Series representation of the pricing formula for the European option driven by space-time fractional diffusion
Hedging in fractional Black-Scholes model with transaction costs
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).
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.