Stochastic Calculus as Operator Factorization An Operator-Covariant Derivative and Unified Representation
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914311612399616 |
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| author | Fontes, Ramiro |
| author_facet | Fontes, Ramiro |
| contents | We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = δ_X Π_X D_X F, valid for F_T^X-measurable F in L^2(Ω) when the driving process X has the representation property. For a square-integrable process X with stochastic integral δ_X, we define the operator-covariant derivative D_X := δ_X* as the Hilbert space adjoint of δ_X. Combined with predictable projection Π_X, this yields a unified Clark-Ocone representation. The operator D_X F is defined as an adjoint for all F in L^2(Ω), without differentiability assumptions; the representation holds when X has the predictable representation property, and reduces to the Galtchouk-Kunita-Watanabe projection when it does not. The framework requires no reproducing kernel Hilbert space or Cameron-Martin structure, and applies to non-Gaussian processes. We work out concrete examples including Brownian motion, general continuous martingales, and compensated Poisson processes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_09976 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stochastic Calculus as Operator Factorization An Operator-Covariant Derivative and Unified Representation Fontes, Ramiro Probability Primary 60H07, Secondary 60H05, 60G15 We present a unified operator-theoretic framework for stochastic calculus based on the factorization (Id - E)F = δ_X Π_X D_X F, valid for F_T^X-measurable F in L^2(Ω) when the driving process X has the representation property. For a square-integrable process X with stochastic integral δ_X, we define the operator-covariant derivative D_X := δ_X* as the Hilbert space adjoint of δ_X. Combined with predictable projection Π_X, this yields a unified Clark-Ocone representation. The operator D_X F is defined as an adjoint for all F in L^2(Ω), without differentiability assumptions; the representation holds when X has the predictable representation property, and reduces to the Galtchouk-Kunita-Watanabe projection when it does not. The framework requires no reproducing kernel Hilbert space or Cameron-Martin structure, and applies to non-Gaussian processes. We work out concrete examples including Brownian motion, general continuous martingales, and compensated Poisson processes. |
| title | Stochastic Calculus as Operator Factorization An Operator-Covariant Derivative and Unified Representation |
| topic | Probability Primary 60H07, Secondary 60H05, 60G15 |
| url | https://arxiv.org/abs/2601.09976 |