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What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven.
I have read online and some answers include:
Stefan's problem for American Options
The lack of an analog of Clark-Haussmann formula in deterministic calculus
Explicit formulas for replicating strategies
among others. Could you suggest some really nice mathematical (and not statistical) ideas?
If you want to address interesting problems that are interesting for financial mathematics, I do not believe you have the good list.
Pricing. For instance, most of explicit formulas for pricing that are not available yet will never be. In this direction, you should have a look at simulation techniques. See for instance Nonlinear Option Pricing. Interesting convergences remains to be proved.
Stochastic Calculus. In terms of extending the reach of stochastic calculus (like Malliavin did some years ago), you can have a look at the Ito-Tanaka Trick and its extensions. It gives deep clues about the regularity of stochastic functions. See for instance Stochastic regularization effects of semi-martingales on random functions.
Mean-Field Games. It you like PDE and / or probabilities, you should definitely read papers on MFG. They are great. Then have been mainly introduced by two great mathematicians (PL Lions and JM Lasry). See their seminal Mean Filed Game paper. If you want to read a "simple" paper (restricted on 1st orders PDE), have a look at Mean field games systems of first order. For a probabilistic viewpoint, see Probabilistic Analysis of Mean-Field Games.
Optimal Transport. For BSDE and transport big fans, martingale transport is your topic. Have a look at Complete Duality for Martingale Optimal Transport on the Line. Nizar Touzi wrote a lot of papers in the area.
Transaction Costs. Solving investment problems with transaction costs give birth to interesting situations. It is the less generic topic of my list, but I like it a lot. The bibliography of Asymptotic Lower Bounds for Optimal Tracking: a Linear Programming Approach will give you an good overview of papers.
More Details. If you want to find yourself interesting topics, you can have a look at the program of the Louis Bachelier seminar (a reference in math finance for reasearchers).