prove ito's formula

This paper addresses the $L^p$-hedging problem ($p geq 1$) for financial derivatives by introducing a novel $L^p$-Wiener–Itô chaos expansion framework based on iterative Stratonovich integrals. Unlike classical orthogonal expansions, this approach applies to general exponentially integrable functionals driven by continuous semimartingales, relaxing orthogonality constraints to enhance approximation universality. The integrands are parameterized via stochastic neural networks and optimized using $L^p$ approximation theory and numerical training, enabling arbitrary-precision $L^p$-norm approximation of any $p$-integrable derivative. When $p = 2$, the method recovers minimum-variance hedging as a special case. The resulting hedging strategy admits closed-form expressions, facilitating real-time, low-overhead, high-accuracy approximate dynamic replication. The key contribution is the first extension of chaos decomposition to non-orthogonal, exponentially integrable continuous semimartingale settings—unifying theoretical optimality with computational tractability.

Pricing European put options under stochastic volatility models with time-varying parameters remains analytically intractable. Method: We derive an explicit asymptotic pricing formula using Malliavin calculus, expressing the option price as a mixed expansion around the Black–Scholes solution. Each term is computed analytically, and remainder terms are rigorously controlled. Contribution/Results: This yields the first asymptotic approximation with an explicit, verifiable error bound valid under general time-varying parameterizations. In the piecewise-constant parameter case, the expansion reduces to a closed-form solution, substantially accelerating calibration. Numerical experiments under the Stochastic Verhulst model confirm uniformly bounded approximation errors meeting practical accuracy requirements. Our framework unifies and extends the analytical tractability of classical volatility models, providing both a novel theoretical tool and an efficient algorithm for fast pricing and parameter calibration under complex stochastic volatility dynamics.

This paper addresses pricing and quadratic hedging of European and path-dependent options under a class of stochastic volatility models driven by infinite linear combinations of time-delayed signatures of Brownian motion. Methodologically, it introduces the first framework incorporating time-delayed signatures into volatility modeling, unifying classical models—including Heston, Stein–Stein, and Bergomi—as well as their path-dependent variants. The core contribution lies in constructing a Riccati equation system on an infinite-dimensional tensor algebra, coupled with joint characteristic functional derivation and numerical Fourier inversion, enabling analytic characterization and efficient computation of generalized volatility structures. Numerical experiments demonstrate that the method achieves both high accuracy and computational efficiency for complex path-dependent options, markedly improving dynamic hedging performance.

Existing generalization bounds for heavy-tailed stochastic optimization either rely on intractable information-theoretic quantities or yield only expectation-based guarantees. To address this, this paper establishes the first *computable, dimension-friendly, high-probability generalization bound* for heavy-tailed stochastic differential equation (SDE) optimizers. Methodologically, we introduce a novel entropy flow analysis framework grounded in the fractional-order Fokker–Planck equation, unifying heavy-tailed SDE theory with fractional PDE techniques. Our analysis reveals a structural-phase transition phenomenon: the impact of heavy tails on generalization—beneficial or detrimental—is governed by the underlying problem geometry. The resulting bound is fully computable, contains no unmeasurable terms, and exhibits improved dimension dependence compared to prior work. Extensive experiments across multiple models and datasets empirically validate the theoretical insights.

Modeling strongly path-dependent financial derivatives—such as exotic options and variance instruments—remains challenging due to the non-Markovian nature of path-dependent functionals. Method: This paper introduces the “occupied process” framework, augmenting the original process $X$ with its occupation measure flow $O$ to form a Markovian lifted system $(O,X)$. It defines the novel “occupation derivative”, unifying functional Itô calculus and mean-field derivatives, and recasts a broad class of path-dependent PDEs as parabolic equations in the occupation measure time variable. Contribution/Results: The framework enables an Itô calculus tailored to path occupation-time functionals and extends the Feynman–Kac formula accordingly. It yields closed-form solutions to local-time-driven optimal stopping problems, with direct applications to corridor variance swap pricing and path-dependent volatility modeling. By bridging stochastic analysis, mean-field theory, and financial mathematics, this work substantially expands both the theoretical foundations and practical applicability of path-dependent stochastic modeling.

This work addresses the absence of rigorous formalization of Itô integration and Itô’s formula in existing proof assistants, particularly the lack of machine-verified treatment of these constructs as martingale processes. Building upon Lean 4, Mathlib, and the BrownianMotion library, we develop an L²-theoretic Itô calculus on a bounded interval [0,T] by constructing the Itô integral via Hilbert space isometry, establishing it as an L²-continuous martingale, and proving Itô’s formula for C³ functions with an explicit remainder bound. To our knowledge, this is the first machine-checked verification of Itô’s formula in any proof assistant and the first formalization of the Itô integral as a martingale-valued process. A single structural identity uniformly yields adaptivity, the martingale property, contraction bounds, and both forms of Itô isometry. The entire development comprises approximately 7,200 lines of sorry-free code across 22 modules, with all main theorems validated under classical axioms.

This study addresses the challenge of computing sensitivities (Greeks) for complex path-dependent derivatives within Malliavin calculus, where explicit and tractable expressions are typically unavailable. Focusing on random variables given by finite linear combinations of signatures of time-augmented Brownian motion, the paper establishes, for the first time, purely algebraic formulations of the Malliavin derivative, Clark–Ocone representation, Ornstein–Uhlenbeck semigroup and its generator, and integration-by-parts formulas. By synergistically combining Malliavin variational methods, stochastic analysis, and the algebraic structure of path signatures, this approach overcomes traditional computational bottlenecks and enables efficient Greek computation under signature-based volatility models. Numerical experiments further demonstrate the performance differences among various Malliavin weights, validating the efficacy of the proposed framework.

This study addresses the absence of a forward stochastic partial differential equation (SPDE) framework suitable for pricing path-dependent derivatives within existing local stochastic volatility models. By leveraging Wentzell theory, the authors introduce the Itô–Wentzell formula into the Dupire equation for the first time, deriving a conditional forward SPDE. They further integrate this with the Musiela parametrization to formulate a dynamic model for rolling-maturity vanilla options. Additionally, a density-weighted Rao–Blackwell estimator is proposed to calibrate the leverage function with high accuracy and computational efficiency. This work not only establishes a novel modeling framework for path-dependent products under local stochastic volatility but also significantly enhances both the precision and speed of leverage function estimation.

This work addresses the failure of the classical Bismut–Elworthy–Li (BEL) formula for stochastic Volterra processes due to their path-dependent nature. To overcome this, the authors develop a novel integration-by-parts (IBP) formula based on Riemann–Liouville fractional derivatives, which interpolates between the standard chain rule and the BEL formula. Their analysis reveals a counterintuitive smoothing effect: the rougher the noise—characterized by a Hurst parameter $H \in (0,1/2)$—the smoother the resulting expectation functional. Specifically, directional differentiability along constant directions is guaranteed whenever the test function’s Hölder exponent satisfies $\beta > 2H$. The framework is further extended to establish first- and second-order BEL formulas for square-integrable directions in additive noise settings, with applications to forward and rough volatility models that clarify the trade-off between the regularity of the test function and the existence of directional derivatives.

研究通过路径空间信息流，使用变分恒等式方法解决非负鞅在随机时间点的精确控制问题，并分析了多种几何形态下的舍弃松弛。