What are the latest developments and open problems in Ito calculus?

[johnqwertyful]

I've been learning Ito calculus and it seems so underdeveloped. What's new? What are the open problems? Who's working on it? What are the big developments? Anything?

 

[Simon Bridge]

I've been learning Ito calculus and it seems so underdeveloped.

In what way? Please be specific.

What's new? What are the open problems? Who's working on it? What are the big developments? Anything?

It's the extension of regular calculus to stochastic processes.
But that should have been explained in the introductory materials you are using to learn from.
http://quantum.phys.cmu.edu/QIP/ito_calculus.pdf

 

[johnqwertyful]

In what way? Please be specific.

It's the extension of regular calculus to stochastic processes.
But that should have been explained in the introductory materials you are using to learn from.
http://quantum.phys.cmu.edu/QIP/ito_calculus.pdf

I learned what SDEs are, Feynman Kac, everything in introductory materials. What else is there? What's beyond SDEs, Ito's Lemma, Feynman Kac? What are the big open problems?

 

[Simon Bridge]

That's still a bit vague - but I think I see what you are getting at.
Have a look at: http://arxiv.org/pdf/math/0409277.pdf
.. p94 has a brief discussion of Wigner Gaussian matrices and points out that relaxing the "gaussian" requirement would be a tempting generalization. How to do this, would be an "open problem" for the field.

Each chapter has more examples.
That help?

 

[blue_raver22]

Ito calculus is a powerful mathematical tool used to model stochastic processes, which are systems that involve randomness or uncertainty. It has been widely used in various fields such as finance, physics, and engineering. While it may seem underdeveloped to some, there have been significant advancements in Ito calculus in recent years.

One of the major developments in Ito calculus is the extension to multivariate processes. This allows for a more accurate and comprehensive modeling of complex systems with multiple variables. Additionally, there have been advancements in the understanding and application of Ito calculus in non-equilibrium systems, which have important implications in fields such as statistical physics and biology.

Some open problems in Ito calculus include the development of more efficient numerical methods for solving stochastic differential equations and the exploration of its applications in machine learning and artificial intelligence. Researchers are also working on extending Ito calculus to include jump processes, which are commonly observed in financial markets and other real-world systems.

Some notable researchers working on Ito calculus include Professors Shige Peng, Jean Jacod, and Marc Yor. They have made significant contributions to the theory and applications of Ito calculus and continue to push the boundaries of this field.

In conclusion, while it may seem underdeveloped at first glance, there have been significant advancements in Ito calculus in recent years. Its applications continue to expand and there are many exciting open problems waiting to be solved. With the contributions of dedicated researchers, Ito calculus will continue to evolve and play a crucial role in understanding and predicting complex systems.