Overview and History
Paul Malliavin's stochastic calculus of variations extends the calculus of variations from functions to stochastic processes. In particular, it allows the computation of derivatives of random variables.
Malliavin invented his calculus to provide a stochastic proof that Hörmander's condition implies the existence of a density for the solution of a stochastic differential equation; Hörmander's original proof was based on the theory of partial differential equations. His calculus enabled Malliavin to prove regularity bounds for the solution's density. The calculus has been applied to stochastic partial differential equations.
Read more about this topic: Malliavin Calculus
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