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I am physics student close to my masters and I intend to enter finance after my PhD (Investmentbanking, Hedge- / Quantfonds, Riskmanagement).

For that, I am aware that a PhD in theoretical physics would be preferable since it will increase my programming and modelling skills.

However, I think I lack a necessary mathematical background. Financial and insurance mathematics is based on stochastics. Now, I only had the undergraduate introductory stochastics course with the mathematicians as well as my basic knowledge of probability related stuff in theoretical physics (e.g. statistical mechanics).

But I did not hear pure probability theory / stochastic calculus, nor statistics.

I tried searching for books. First for financial mathematics, but most of them seem to be addressed to economics students and not to the typical mathematician / physicist working in finance.

Similiarly, while there are lots of statistics books out there, lots of them address psychologists or medicine students, while others address mathematicians.

Do I have to learn the "hardcore" version of the mathematicians for my desired job field or is the basic stuff for psychologists enough?

Do I have to learn stochastic calculus?

And, more importantly, do I have to work through these fields first before getting a book about say financial mathematics or is the undergraduate stochastics course enough to understand financial / insurance maths?

I would like to have some guidance in what to learn next and what kind of book to buy. I have ~4 years left to learn in my freetime.

Thanks!