A new simulation method for Volterra processes improves convergence for rough kernels.
problem Simulating Volterra processes with singular kernels.
method iVi (integrated Volterra implicit) scheme based on Inverse Gaussian distribution.
result The iVi scheme achieves weak convergence with few time steps, especially for rough kernels.
This research improves deep neural networks for parameter identification and prediction in stochastic Volterra integral equations.
problem Parameter identification and prediction in Volterra integral equations driven by Gaussian noise.
method Improved deep neural networks framework that incorporates inter-output relationships into the loss function.
result The framework enhances parameter estimation accuracy and provides accurate solutions for modeling stochastic systems.
Paper introduces cubature method for stochastic Volterra equations.
problem Solving stochastic Volterra integral equations efficiently.
method Derive stochastic Taylor expansion, introduce cubature measure.
result Cubature method is more efficient than Euler scheme under certain conditions.
Study small-time CLTs for stochastic Volterra equations with various kernels.
problem Understanding the behavior of stochastic Volterra equations with different kernels.
method Proved convergence of finite-dimensional distributions, functional CLT, and limit theorems for smooth transformations.
result Derived asymptotic pricing formulae for digital calls in rough volatility models.
Study proves optimal controls for stochastic Volterra equations with singular kernels.
problem Existence of optimal controls for stochastic Volterra equations with singular kernels.
method Sufficient conditions based on integrability and growth hypotheses.
result Existence of optimal relaxed and strict controls under classical convexity assumptions.
The paper analyzes the stationarity of stochastic Volterra integral equations and introduces fake stationary regimes.
problem Analyzing the stationarity of non-Markovian dynamical systems described by SVIEs.
method Investigates the properties of SVIE solutions, focusing on stationarity over finite and long time horizons, and introduces a deterministic stabilizer to induce a fake stationary regime.
result SVIEs do not exhibit a strong stationary regime unless the kernel is constant or degenerate, but a fake stationary regime can be achieved with a deterministic stabilizer.
We provide existence, uniqueness and stability results for affine stochastic Volterra equations with L1-kernels and jumps. Such equations arise as scaling limits of branching processes in population genetics and self-exciting Hawkes processes in mathematical finance. The strategy we adopt for the existence part is b…
This paper formulates and studies a stochastic maximum principle for forward-backward stochastic Volterra integral equations (FBSVIEs in short), while the control area is assumed to be convex. Then a linear quadratic (LQ in short) problem for backward stochastic Volterra integral equations (BSVIEs in short) is present …
This work studies nonnegativity-preserving kernels for stochastic equations and their applications.
problem Nonnegativity preservation in stochastic Volterra equations and related processes.
method Characterization and application of completely monotone kernels; approximation schemes for weak error.
result Positive linear combinations of decaying exponentials can be used for second-order approximation schemes.
Paper develops methods for solving complex stochastic equations using Malliavin calculus.
problem Existence, uniqueness, and regularity of solutions to BSVIEs.
method Malliavin calculus for tackling diagonal processes and nonlinear dependence.
result Developed well-posedness results for BSVIEs, including probabilistic interpretation of PDEs and portfolio optimization.
Paper solves stock loan pricing with finite maturity using integral equations.
problem Valuation of margin-call stock loans with finite maturities.
method Fourier Sine transform and Volterra integral equation approach.
result Integral representation of margin-call stock loan value.
Volterra square-root process boundary behavior and martingale measures
problem Boundary behavior of the Volterra square-root process
method Comparison principles for Volterra integral equations and generalized Riemann-Liouville fractional equations
result Finiteness of negative p-moments and atom at the boundary for rough kernels Researchers find the optimal exercise time for American options using a specific type of diffusion process.
problem Finding the optimal time to exercise American options with a time-dependent Ornstein-Uhlenbeck process.
method Optimal stopping problem, probabilistic arguments, non-linear Volterra-type integral equation, Picard iteration algorithm.
result They derive a non-linear Volterra-type integral equation and prove the exercise boundary's Lipschitz continuity and differentiability almost everywhere.
Deep-learning method solves BSVIEs and coupled systems.
problem High-dimensional, time-inconsistent stochastic control problems.
method Trains a neural network to approximate solution fields directly.
result Non-asymptotic error bound and scalable performance.
New PFPPs based on rank-dependent utility for better performance control.
problem Improving performance prediction in systems with short-term control.
method Introduces rank-dependent PFPPs, solves integral equations via Volterra theory.
result Existence of rank-dependent PFPPs under specific market conditions.
New method for pricing American options in time-dependent models, improving accuracy and efficiency.
problem Pricing American options in time-dependent models with improved accuracy and efficiency.
method Semi-analytical pricing using a nonlinear Volterra integral equation and numerical methods.
result Improved accuracy and efficiency in pricing American options compared to forward finite difference solvers.
Study on fake stationary Volterra Heston model for non-stationary processes.
problem Non-stationary nature of true Volterra equations.
method Weak notion of stationarity (fake stationary regime) for inhomogeneous affine Stochastic Volterra equations.
result Existence of limiting distributions in the long run, which may depend on initial state.
Pathwise uniqueness shown for specific stochastic equations.
problem Stochastic Volterra equations with singular kernels and Hölder coefficients.
method Established pathwise uniqueness through Hölder continuity of coefficients.
result Pathwise uniqueness and existence of unique strong solutions.
Large deviation principles for multivariate stochastic volatility models.
problem Understanding the behavior of log-processes in multivariate stochastic volatility models.
method Establishing a comprehensive sample path large deviation principle for log-processes.
result Asymptotic formulas for first exit times and barrier option prices derived from the LDP.
Study on non-negative solutions for stochastic Volterra equations with jumps.
problem Existence and uniqueness of non-negative solutions for stochastic Volterra equations with jumps and non-Lipschitz coefficients.
method Developed a nonnegative approximation approach and used Yamada--Watanabe approximation technique for convergence proof.
result Established conditions for strong existence and pathwise uniqueness of non-negative solutions.
The study analyzes prediction errors in systems with memory kernels, providing bounds and stability results.
problem Prediction errors in stochastic dynamical systems with memory kernels.
method Analysis of generalized Langevin equations (GLEs) with Volterra equations, integrating synchronized noise coupling and weighted norms.
result Prediction discrepancies decay at a rate determined by the memory kernel's decay, quantitatively bounded by kernel estimation errors.
Volterra signature provides a clear, interpretable feature for history-dependent systems.
problem Learning from non-Markovian time series with implicit memory mechanisms.
method Develops Volterra signature as a tensor algebra representation weighted by a temporal kernel, proving injectivity and universal approximation.
result Volterra signature leads to linear functionals and universal approximation, improving dynamic learning tasks.
Neural SVEs model complex systems with memory, outperforming traditional methods.
problem Modeling systems with memory effects and irregular behavior.
method Introducing neural stochastic Volterra equations as a physics-inspired architecture.
result Neural SVEs outperform neural SDEs and DeepONets in various applications.
New financial model with sandwiched volatility for option pricing.
problem Developing a new financial model for option pricing.
method Introducing a new model with stochastic volatility driven by a Gaussian Volterra process, ensuring the solution is sandwiched between two arbitrary Hölder continuous functions.
result Developed an algorithm for pricing options with discontinuous payoffs using Malliavin calculus.
The study examines stability of Hamiltonian Poisson integrators on both integrable and non-integrable systems.
problem Investigating stability properties of Hamiltonian Poisson integrators.
method Examples of Lotka-Volterra dynamics and numerical investigations of a non-integrable system are used.
result The existence of a modified Hamiltonian is crucial for the stability of Hamiltonian Poisson integrators.
Study approximates rough stochastic volatility models using diffusion processes.
problem High computational cost in simulating rough stochastic volatility models.
method Approximates stochastic Volterra equations with an N-dimensional diffusion process.
result Approximations converge strongly with superpolynomial rate in N.
New IBP formulae for rough stochastic Volterra processes.
problem Deriving IBP formulae for path-dependent stochastic Volterra processes.
method Developed a new fractional IBP formula that interpolates between standard and Bismut-Elworthy-Li formulae.
result For rough noise, the expectation is differentiable along constant directions under certain Hölder continuity conditions.
New method for pricing barrier options in time-dependent λ-SABR model.
problem Pricing barrier options in the time-dependent λ-SABR model.
method Modified integral transform method and Fourier-Bessel series solution.
result Semi-analytical solution for barrier options in λ-SABR model.
We establish an explicit expression for the conditional Laplace transform of the integrated Volterra Wishart process in terms of a certain resolvent of the covariance function. The core ingredient is the derivation of the conditional Laplace transform of general Gaussian processes in terms of Fredholm's determinant and…
Model for high-frequency trading with rough volatility.
problem High-frequency trading dynamics and rough volatility modeling.
method Stochastic partial differential equation (SPDE) with rough volatility driven by a Hawkes process.
result The volatility path of the SPDE is rougher than that driven by a standard Brownian motion.
Investor optimizes investment timing with future knowledge, overcoming transaction costs.
problem Optimal investment timing with future peeking, constrained by transaction costs.
method Solves control problem with infinite-dimensional memory using Gaussian Volterra integral equations.
result Explicit solution to optimal investment problem in Bachelier setting.
The paper derives formulas for pricing geometric Asian options in the Volterra-Heston model.
problem Pricing geometric Asian options in the Volterra-Heston model.
method Derives semi-closed formulas using Fourier transforms and Riccati-Volterra equations.
result Derives formulas for pricing geometric Asian options with fixed and floating strikes.
Investigates mean-variance portfolio selection in non-Markovian markets.
problem Continuous-time Markowitz mean-variance portfolio selection in fake stationary affine Volterra models.
method Stochastic factor solution to a Riccati BSDE, deriving explicit solutions as multi-dimensional Riccati-Volterra equations.
result Analytical closed-form expressions for optimal portfolio policies and mean-variance efficient frontier.
New algorithms compute Volterra signature efficiently for time series analysis.
problem Efficient computation of Volterra signature with matrix-valued kernels.
method Decomposed Chen-type convolution relation, introduced FFT-based and exact recursion algorithms.
result Efficient algorithms for Volterra signature computation with various complexities.
Derives semi-closed form prices for barrier options in the Hull-White model.
problem Calculating prices of barrier options in the Hull-White model with time-dependent parameters.
method Applies generalized integral transform and heat potentials to solve linear Volterra equations of the first kind.
result The method provides more efficient and accurate solutions compared to finite difference methods.
We present three models of stock price with time-dependent interest rate, dividend yield, and volatility, respectively, that allow for explicit forms of the optimal exercise boundary of the finite maturity American put option. The optimal exercise boundary satisfies the nonlinear integral equation of Volterra type. We …
Researchers compute Greeks for rough Volterra SV models using Malliavin calculus.
problem Computing Greeks under rough Volterra stochastic volatility models.
method Malliavin calculus techniques, extending integration by parts to non-square integrable functionals.
result Formulas for computing Greeks (Delta, Gamma, Rho, Vega) under various rough Volterra SV models.
The Volterra square-root process shows non-uniqueness of limiting distributions and regularity of its law.
problem Non-uniqueness of limiting distributions in the Volterra square-root process.
method Establishing existence of limiting distributions using integrability of the Volterra convolution kernel and exponential-affine transformation.
result The limiting distributions of the Volterra square-root process depend on the initial state and belong to weighted Besov spaces.
New framework analyzes SGD dynamics in large samples and dimensions.
problem Analyzing stochastic gradient descent in large-scale settings.
method Inspired by random matrix theory, new framework for fixed stepsize and finite sum settings.
result SGD dynamics become deterministic in the large sample and dimensional limit, governed by a Volterra integral equation.
Defines speculative bubbles in discrete-time models based on discounted stock price losing mass.
problem Characterizing speculative bubbles in discrete-time models.
method Introduces a new definition based on discounted stock price behavior and provides probabilistic characterizations.
result Speculative bubbles in discrete time are linked to solutions of a linear Volterra integral equation.
New theorem handles stochastic Volterra semimartingales.
problem Classic Fubini theorem restrictions for Volterra semimartingales.
method Introduced measure-valued stochastic integration.
result Proved new stochastic Fubini theorem.
This paper optimizes portfolio selection for multivariate affine and quadratic Volterra models with rough volatilities.
problem Optimizing portfolio selection for multivariate models with rough volatilities and stochastic correlations.
method Investigates continuous-time Markowitz mean-variance problem for multivariate affine and quadratic Volterra models using Riccati backward stochastic differential equations (BSDEs).
result Derives explicit solutions for BSDEs in affine Volterra models and new analytic formulae for quadratic models.
Study measures inequality in social-economic systems using Fokker-Planck equations and Lotka-Volterra dynamics.
problem Measuring inequality in oscillatory social-economic systems described by Fokker-Planck equations and Lotka-Volterra dynamics.
method Used Fokker-Planck equations and Lotka-Volterra dynamics to model inequality, focusing on coefficient of variation as a measure.
result Inequality initially tends to decrease in oscillatory systems, contrary to steady-state models.
Unified approach to stochastic Volterra systems' deviations.
problem Large and moderate deviations for stochastic Volterra systems.
method Weak convergence approach by Budhijara, Dupuis and Ellis.
result Unified treatment of deviations for a broad class of stochastic Volterra equations.
Study on Volterra Cox-Ingersoll-Ross process, proving asymptotic independence and ergodicity.
problem Analyzing the Volterra Cox-Ingersoll-Ross process and its properties.
method Fine asymptotic analysis of Volterra Riccati equation, affine transformation formula.
result Proves asymptotic independence and ergodicity of the process.
We study several optimal stopping problems that arise from trading a mean-reverting price spread over a finite horizon. Modeling the spread by the Ornstein-Uhlenbeck process, we analyze three different trading strategies: (i) the long-short strategy; (ii) the short-long strategy, and (iii) the chooser strategy, i.e. th…
We consider stochastic partial differential equations appearing as Markovian lifts of matrix valued (affine) Volterra type processes from the point of view of the generalized Feller property (see e.g., \cite{doetei:10}). We introduce in particular Volterra Wishart processes with fractional kernels and values in the con…
Paper solves Merton's portfolio problem in a non-Markovian, non-semimartingale model.
problem Merton's portfolio optimization in a fake stationary Volterra-Heston model.
method Stochastic factor solution to a Riccati BSDE, combined with martingale optimality principle.
result Derives semi-closed form optimal strategies and value function.