Open research questions in Stochastic processes and financial applications

What the literature leaves open

• In particular: Whether the DTE-mismatch construction produces a stable, observable proxy for across different market regimes has not been tested on real data.

  Engineering the Mean-Reversion Delay in Dispersion Trading: A Market-Implied Finite-Dimensional Proxy for High-Dimensional Correlation Manifolds · 2026 · DOI

• 370 Asian Journal of Science, Technology, Engineering, and Art Rishav Jha, Kameshwar Sahani, Suresh Kumar Sahani, Ravi Kumar Raj, Dilip Kumar Sah Future work may explore adaptive methods for systems of SDEs, implicit adaptive schemes for stiff problems, and the application of adaptive methods to stochastic partial differential equations.

  Adaptive Time Stepping Numerical Schemes for Stochastic Differential Equations · 2026 · DOI

• While this work focused on compact manifolds, future research will explore extensions to non-compact settings, different classes of singular SPDEs, and the 14 deeper topological implications of the manifold’s structure on stochastic dynam- ics and universality.

  Universal Scaling Limits and Global Well-Posedness of Singular Stochastic Partial Differential Equations on Riemannian Manifolds · 2026 · DOI

• Future work could explore non-compact manifolds, where different techniques, perhaps involving weighted Sobolev spaces or suitable asymptotic conditions, would be required. Future research could explore the application of these techniques to specific physical models like stochastic quantization of gauge theories on curved spacetimes or the study of stochastic inflation models in cosmology, where both singularity and geometry are paramount.

  Universal Scaling Limits and Global Well-Posedness of Singular Stochastic Partial Differential Equations on Riemannian Manifolds · 2026 · DOI

• The proof that the integral in equation (4.4) equals zero relies on the non-decreasing/non-increasing structure of h on [0, 1/2] and [-1/2, 0] specific to one-dimensional domains. The extension of this key regularity argument to mean field games with common noise in higher dimensions requires developing new techniques beyond the current interval decomposition method.

  Intrinsic Regularization by Noise for 1d Mean Field Games · 2026 · DOI

• The paper develops the theory for one-dimensional mean field games on the circle S with periodic boundary conditions. The extension to higher-dimensional domains or mean field games on non-compact manifolds, where the regularity of the set {y : Xs(y) = Xs(x)} may differ fundamentally, has not been explored.

  Intrinsic Regularization by Noise for 1d Mean Field Games · 2026 · DOI

• The discrete time scheme presented in equations (4.8)-(4.9) using rearrangement operations has not been analyzed for convergence rates or numerical stability as the mesh size h varies. The relationship between the discrete approximation error and the regularization effect from noise in equation (4.5) needs explicit quantification.

  Intrinsic Regularization by Noise for 1d Mean Field Games · 2026 · DOI

• The paper establishes the connection between the noise-regularized mean field game formulation and standard mean field games under displacement monotonicity conditions (equation 4.6), but the extension to mean field games violating displacement monotonicity or satisfying only Lasry-Lions monotonicity remains unaddressed. The construction of the value field V through the master equation for such non-monotone cases requires further investigation.

  Intrinsic Regularization by Noise for 1d Mean Field Games · 2026 · DOI

• The high-probability bounds in Theorem 3.3 decay as O(√ln K/√K), but the tightness of these bounds and whether this rate is optimal for the stochastic proximal linearized ADMM applied to sparse distribution control constrained by random elliptic equations is not discussed or compared against lower bounds.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The paper recommends ̺ ∈ [1.3, 1.6] for robust convergence based on numerical experiments, but provides no theoretical justification or guidelines for selecting optimal ̺ values as functions of problem parameters (regularization coefficients α, β, mesh size h, and iteration count K) in the stochastic proximal linearized ADMM framework.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The benchmark solution in Example 1 is computed by averaging results over 50 independent repetitions of the stochastic algorithm, but no formal comparison is provided against deterministic solvers (e.g., interior point methods, second-order methods) or other stochastic optimization algorithms for sparse PDE-constrained control problems.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The finite element discretization using linear elements (mesh sizes h = 2⁻⁵ to 2⁻⁷) is tested only on a 2D square domain. Scalability of the stochastic proximal linearized ADMM to higher-dimensional problems (3D), non-convex domains, and the interaction between discretization error and stochastic approximation error in these settings has not been investigated.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The paper provides high-probability convergence bounds (Theorem 3.3) but does not characterize how the constant C₃ scales with problem dimensionality or how the convergence rate degrades as the domain Ω increases in complexity beyond the tested [0,1]² domain. The relationship between mesh size h, domain geometry, and the constants in the probability bounds needs explicit analysis.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The paper demonstrates the stochastic proximal linearized ADMM method only on uniformly distributed random diffusion coefficients on [0.5, 1.5] in the elliptic constraint equation. The algorithm's convergence behavior and high-probability bounds under other random distributions (log-normal, Gaussian, non-uniform) and under different types of uncertainty in the PDE constraint remain unexplored.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The relaxation factor ̺ exhibits numerical instability as it approaches the theoretical upper bound of 2 in the stochastic proximal linearized ADMM algorithm, compromising the positive definiteness of the matrix operator H. The specific threshold values and conditions under which this instability manifests across different regularization parameters (α, β) and mesh resolutions require systematic characterization.

  Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · 2026 · DOI

• The continuity proof of X(·) in Lemma D.4 requires that the free boundary point xₙₖᵢ satisfies xₙₖᵢ ≥ a for some a > 0 (equation 81), but the relationship between this lower bound a and the problem parameters (initial capital, constraint parameters, time horizon) is not explicitly determined or analyzed.

  Optimal Dividend Payout with Path-dependent Drawdown Constraint · 2026 · DOI

• Lemma D.3 establishes an estimate (80) for the dependence of v(x,c) on the constraint level c with constant B, but the paper does not provide a constructive method or bound for computing B in terms of the problem parameters (r, γ, b, T). This limits the practical applicability of the estimate.

  Optimal Dividend Payout with Path-dependent Drawdown Constraint · 2026 · DOI

• The paper establishes uniqueness of solutions for problem (9) and related equations, but does not characterize the rate of convergence of the approximating sequence {vnk} to the limit function v in the W²ₚ norm. The dependence of this convergence rate on the discretization parameter Δc* is not quantified.

  Optimal Dividend Payout with Path-dependent Drawdown Constraint · 2026 · DOI

• The proofs rely on establishing properties of the value function v(x,c) and free boundary X(c) in the theoretical setting, but the paper does not investigate the sensitivity of the optimal dividend payout policy to perturbations in the constraint parameters (b, T, r, γ). How robust is the switching boundary X(c) to small changes in the drawdown constraint specification?

  Optimal Dividend Payout with Path-dependent Drawdown Constraint · 2026 · DOI

• The paper establishes continuity and boundedness of the switching boundary X(·) through Lemmas D.2-D.5, but does not address numerical computation of X(c) for practical implementation. The specific discretization scheme, convergence rate of the finite difference approximations used in Section C, and computational complexity of determining X(c) across the interval [0, c̄) remain unspecified.

  Optimal Dividend Payout with Path-dependent Drawdown Constraint · 2026 · DOI

• The paper mentions in Remark 4.5 that Theorem 3.1 can apply to fourth-order parabolic SPDEs with Lipschitz continuous drift/diffusion; explicit derivation of convergence rates for such higher-order equations and identification of the critical spatial regularity parameter θ analogous to the second-order case is not provided.

  OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · 2026 · DOI

• Theorem 4.2 applies to Q-Wiener processes satisfying Assumption 2.1 with θ ∈ (0, 2); the extension to rougher colored noise regimes with θ ≥ 2 or to non-Gaussian driving noise (e.g., Lévy processes) requires developing corresponding regularity theory and time-stepping error bounds.

  OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · 2026 · DOI

• The paper establishes convergence under classical conditions (4.9) for trace-class noise with parameters F and G satisfying specific growth rates; validation on benchmark SPDEs with nonstandard covariance structures (e.g., space-fractional operators, Matérn-type covariances) would test whether the θ/d spatial rate is universal.

  OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · 2026 · DOI

• Remark 4.1 notes that the approach in Lemma 3.1 yields only (1/4 − ǫ)-Hölder continuity when X₀ ∈ Lᵖ(Ω; Ḣᵝ) with β ∈ (0, 1/2]; the precise characterization of which eigenmodes and integration techniques optimize this Hölder exponent for different parabolic SPDE types remains unresolved.

  OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · 2026 · DOI

• The temporal convergence rate θ/2 ∧ γ in Theorem 4.2 requires the restrictive assumption γ ≤ 1/2 on the Hölder regularity parameter; investigation of whether this constraint can be relaxed for SPDEs with specific drift/diffusion structures (e.g., gradient-type nonlinearities) would determine if sharper temporal rates are achievable.

  OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · 2026 · DOI

Most-cited papers in Stochastic processes and financial applications

• Modeling and forecasting realized volatility with the fractional Ornstein–Uhlenbeck process · Journal of Econometrics · 2021 · 66 citations
• Interest rates forecasting: Between Hull and White and the CIR#—How to make a single‐factor model work · Journal of Forecasting · 2021 · 30 citations
• Parametric estimation for a parabolic linear SPDE model based on discrete observations · Journal of Statistical Planning and Inference · 2020 · 24 citations
• The Chinese equity index options market · Emerging Markets Review · 2020 · 15 citations
• On solvability and optimal controls for impulsive stochastic integrodifferential varying-coefficient model · Automatika · 2024 · 10 citations
• Fractional Brownian Motions · Acta Physica Polonica B · 2020 · 9 citations
• Semiflow selection and Markov selection theorems · Topological Methods in Nonlinear Analysis · 2020 · 9 citations
• Stochastic Approximation: From Statistical Origin to Big-Data, Multidisciplinary Applications · Statistical Science · 2021 · 5 citations
• Asymptotic autonomy of bi-spatial attractors for stochastic retarded Navier-Stokes equations · Topological Methods in Nonlinear Analysis · 2021 · 5 citations
• Some examples of solutions to an inverse problem for the first-passage place of a jump-diffusion process · Control and Cybernetics · 2022 · 4 citations

Most recent work

• Strong convergence of a fully discrete finite element approximation of non-Lipschitz SPDEs with multiplicative noise · Applied Mathematics Letters · 2026
• Rates of convergence of finite element approximations of second-order mean field games with nondifferentiable Hamiltonians · Mathematics of Computation · 2026
• McKean-Vlasov SPDEs Driven by Poisson Random Measure: Well-Posedness and Large Deviation Principle · Applied Mathematics & Optimization · 2026
• Optimal Dividend Payout with Path-dependent Drawdown Constraint · Applied Mathematics & Optimization · 2026
• Short-Time Behavior of the At-the-Money Implied Volatility for the Jump-Diffusion Stochastic Volatility Bachelier Model · SIAM Journal on Financial Mathematics · 2026
• Nonregular McKean–Vlasov Equations and Calibration Problem in Local Stochastic Volatility Models · Mathematics of Operations Research · 2026
• OptimalStrongConvergenceRateofSpectralGalerkin Exponential Euler Scheme for Parabolic SPDEs · CSIAM Transactions on Applied Mathematics · 2026
• Utility Maximization Under Endogenous Pricing · Mathematics of Operations Research · 2026
• Stochastic Proximal Linearized ADMM for Sparse Distribution Control Problem Constrained by Random Elliptic Equation · Journal of Computational Mathematics · 2026
• Infinite Horizon Mean-Field Linear-Quadratic Optimal Control Problems with Switching and Indefinite-Weighted Costs · Applied Mathematics & Optimization · 2026

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