Functional Integration by Parts Formulae for Stochastic Volterra Processes

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Highly relevant to quantitative trading and derivatives pricing. The paper directly addresses computation of Greeks (sensitivities) under rough volatility models, which are the state-of-the-art in equity derivatives pricing. The IBP formulae provide exact, unbiased Monte Carlo estimators for pathwise sensitivities without finite-difference approximations, crucial for risk management. The forward variance curve sensitivity (a new 'Greek') is directly applicable to variance swap and VIX hedging. The rough Bergomi and rough Heston models covered are industry standards. However, the paper is purely theoretical with no numerical implementation or backtesting, limiting immediate practical deployment.

Extremely high complexity. Implementation would require: (1) numerical solution of stochastic Volterra equations with singular kernels, (2) computation of Riemann-Liouville fractional derivatives of conditional expectations, (3) Malliavin calculus operations in infinite-dimensional settings, (4) careful handling of the trade-off parameter α between test function and direction regularity, (5) Monte Carlo estimation of the derived weights involving stochastic integrals with power-law kernels. The theoretical framework involves fractional calculus, generalized Grönwall inequalities, Stone-Weierstrass on non-locally compact spaces, and stochastic Fubini theorems. Practical implementation for rough volatility Greeks would require significant numerical analysis expertise beyond the mathematical theory.

This is a pure theoretical mathematics paper with complete proofs provided in the main text and Appendix A. All results are rigorously derived from stated assumptions (Assumption Kernels and Assumption Coefficients). The mathematical framework is self-contained with definitions of all operators (Riemann-Liouville fractional derivatives, Malliavin calculus, weighted spaces). However, the extreme technical complexity of the proofs (fractional calculus, stochastic Fubini theorems, generalized Grönwall inequalities, Stone-Weierstrass on non-locally compact spaces) makes independent verification very challenging. No numerical experiments or code are provided.