An elementary introduction to the discrete Malliavin calculus

抄録

Malliavin calculus is a powerful mathematical framework in stochastic analysis, yet its continuous-time theory presents a steep learning curve for students due to its reliance on advanced functional analysis and infinite-dimensional measure theory. This paper provides a strictly elementary, self-contained tutorial on the core concepts of Malliavin calculus by shifting the framework to a discrete-time symmetric random walk. By introducing the discrete Malliavin derivative as a simple difference operator and utilizing a fundamental orthogonal decomposition, we circumvent the heavy analytical machinery typically required. We present elementary algebraic proofs for the three pillars of the theory: the Clark--Ocone formula, the chaos expansion (discrete Stroock formula), and the integration-by-parts formula. This exposition is intended as a pedagogical stepping stone, revealing the beautiful algebraic simplicity underlying Malliavin calculus before readers tackle the infinite-dimensional continuous setting.