Probability Theory

Malliavin Calculus

Цели урока

  • Understand the Malliavin derivative as differentiation of Brownian path functionals
  • Master the space D^{1,2} and the Malliavin-Sobolev norm
  • Analyze the divergence operator and its connection to the Skorokhod integral
  • Apply the integration by parts formula to compute option Greeks

Предварительные знания

  • Brownian motion and the Ito stochastic integral
  • Ito formula and stochastic differential equations
  • Hilbert spaces and operators
  • Wiener chaos expansion
  • Stochastic Finance and the Black-Scholes Formula

How does one compute the sensitivity of an exotic option to parameters when the payoff is an indicator of a barrier crossing? Malliavin transfers the derivative through integration by parts on Wiener space.

  • **Greeks computation:** delta and gamma of barrier options are computed via Malliavin IBP in Monte Carlo (Goldman Sachs, JPMorgan)
  • **Hypoelliptic operators:** Hormander's theorem is proven by a probabilistic method via non-degeneracy of the Malliavin matrix
  • **Diffusion models:** the score function in Stable Diffusion and DDPM is connected to the Malliavin derivative of the reverse process
  • **Stochastic control:** optimality criteria use variational methods on Wiener space

The Malliavin Derivative

In 1976 Paul Malliavin solved a problem that had stymied analysts: a purely probabilistic proof of Hormander's hypoellipticity theorem. His tool: differentiating a random variable along the Brownian path. Today the Malliavin derivative underpins the score functions of diffusion models in Stable Diffusion and DDPM.

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The score function nabla log p_t(x) in diffusion models (Stable Diffusion, DDPM) is expressed through the Malliavin derivative of the reverse process. This gives a rigorous mathematical justification of denoising score matching.

Along what direction is the Malliavin derivative of a random functional F taken?

Integration by Parts Formula

The power of Malliavin calculus lies in its integration by parts formula on Wiener space. It transfers a derivative with respect to a random variable X onto a weight functional, sidestepping non-differentiability of the integrand. Critical for computing Greeks of barrier and binary options at Goldman Sachs and JPMorgan: delta and gamma become accessible via Monte Carlo even for non-smooth payoffs.

For a barrier option, the payoff f(S_T)*1_{max S < B} is not differentiable in the classical sense. Malliavin IBP provides an explicit weight functional allowing one to compute Greeks via standard Monte Carlo without bias.

Why is the Malliavin IBP formula useful for computing Greeks?

Skorokhod Integral and Divergence

Anatoly Skorokhod in 1975 extended the Ito integral to non-adapted processes. The Malliavin divergence operator delta coincides with the Skorokhod integral: an extension that allows integration of processes 'looking into the future' of the Brownian path. This is needed in problems with reverse diffusion - precisely the reverse process in diffusion generative models.

Malliavin calculus unites analysis and probability

The Malliavin derivative transfers differential calculus to the infinite-dimensional Wiener space.

  • Stochastic calculus — The divergence operator delta extends the Ito integral to non-adapted processes via Skorokhod
  • Functional analysis — Sobolev-Watanabe spaces D^{k,p} on Wiener space are the analog of classical Sobolev spaces
  • Diffusion models — The score function in Stable Diffusion and DDPM is expressed through the Malliavin derivative of the reverse process
  • Infinite-dimensional probability — Wiener space is the first example of an infinite-dimensional probability space with a differential structure

What is the key difference between the Skorokhod and Ito integrals?

Malliavin Calculus: Bridge Between Analysis and Probability

The Malliavin derivative introduces differential structure into the infinite-dimensional Wiener space, unifying stochastic calculus, functional analysis, and modern deep learning.

  • Stochastic calculus — The divergence operator delta extends the Ito integral to non-adapted processes via the Skorokhod construction
  • Functional analysis — Sobolev-Watanabe spaces D^{k,p} on Wiener space are the infinite-dimensional analog of classical Sobolev spaces
  • Diffusion generative models — The score function nabla log p_t in Stable Diffusion and DDPM is expressed through the Malliavin derivative of the reverse process
  • Mathematical finance — The Malliavin IBP formula enables unbiased Monte Carlo computation of Greeks for barrier and binary options

Итоги

  • **Malliavin derivative** D_t F differentiates the functional F along the Brownian path in the direction of 1_{[0,t]} from the Cameron-Martin space
  • **Space D^{1,2}** is the closure of smooth functionals under the norm ||F||^2_{1,2} = E[F^2] + E integral (D_t F)^2 dt; the infinite-dimensional analog of Sobolev space W^{1,2}
  • **Divergence operator** delta = D* is adjoint to D in L^2; coincides with the Skorokhod integral, extending the Ito integral to non-adapted processes
  • **IBP formula** E[partial_i f(X) G] = E[f(X) delta(G (C_X^{-1} DX)_i)] transfers differentiation from a non-smooth payoff to a weight functional, enabling Greeks of barrier options via Monte Carlo
  • **Malliavin matrix** C_X controls smoothness of the law of X: non-degeneracy guarantees absolute continuity (Hormander's theorem)
  • **Score function** in diffusion models coincides with the Malliavin derivative of the reverse process, providing the mathematical foundation of denoising score matching
Malliavin Calculus