Simplified derivation and simulation of Feller Diffusion.
problem Deriving the probability density function of Feller Diffusion.
method Fourier Transform and Method of Characteristics for derivation; simulation algorithms for validation.
result Confirmation of hitting time probabilities via simulation.
We study the solution to Kolmogorov-Feller equation and by using it provide pricing formulas of well known some options under jump-diffusion model.
We consider Feller mean-reverting square-root diffusion, which has been applied to model a wide variety of processes with linearly state-dependent diffusion, such as stochastic volatility and interest rates in finance, and neuronal and populations dynamics in natural sciences. We focus on the statistical mixing (or sup…
The asymptotic behavior of the heat kernel of a Riemannian manifold gives rise to the classical concepts of parabolicity, stochastic completeness (or conservative property) and Feller property (or C0-diffusion property). Both parabolicity and stochastic completeness have been the subject of a systematic study whic…
Constructs rank-based continuous semimartingales for financial markets.
problem Model financial markets using rank-based diffusions.
method Uses Dirichlet forms and Feller property to construct semimartingales.
result Establishes nonexistence of triple collisions and simplified rank process dynamics.
Transform drift of diffusions without knowing if measure change is a martingale.
problem Transforming drift in diffusions without knowing if measure change is a martingale.
method Characterize when measure change local martingale is a true martingale.
result Complete characterization of measure change local martingale being a true martingale.
A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient esti…
We solve continuous-time reinforcement learning using distributional Hamilton-Jacobi-Bellman equations.
problem Predicting the distribution of returns in continuous-time, stochastic environments.
method We derive a distributional Hamilton-Jacobi-Bellman equation for Itô diffusions and Feller-Dynkin processes, and propose an algorithm based on a JKO scheme.
result We propose an online control algorithm that can be used to approximately solve the distributional HJB equation.
Extends Feller theory to non-locally compact spaces for stochastic equations.
problem Stochastic partial differential equations and fractional processes.
method Extended Feller processes and proofs of folklore results.
result No condition of generalized Feller semigroups can be dropped.
Study on VIX options pricing in SABR model, showing infinite prices due to volatility explosion.
problem Infinite VIX futures and call prices due to volatility explosion in SABR model.
method Analyzing SABR model, showing vt as unique solution to diffusion process, proving explosion using Feller test, proposing capped volatility process. result VIX futures and call prices are infinite for any maturity due to volatility explosion, but capped volatility process mitigates this issue.
This paper discusses properties of a Doubly Stochastic Poisson Process (DSPP) where the intensity process belongs to a class of affine diffusions. For any intensity process from this class we derive an analytical expression for probability distribution functions of the corresponding DSPP. A specification of our results…
In this paper, the relevance of the Feller conditions in discrete time macro-finance term structure models is investigated. The Feller conditions are usually imposed on a continuous time multivariate square root process to ensure that the roots have nonnegative arguments. For a discrete time approximate model, the Fell…
Study simulates Heston-type local stochastic volatility model using particle method.
problem Simulate calibrated Heston-type local stochastic volatility model with non-standard coefficients.
method Monte Carlo particle method, Euler-Maruyama scheme, full truncation Euler scheme.
result Strong convergence of Euler-Maruyama scheme with rate 1/2 in time, up to a logarithmic factor.
We compute exact values respectively bounds of "distances" - in the sense of (transforms of) power divergences and relative entropy - between two discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration is arbitrarily Poisson-distributed (leading to arbitra…
When dealing with Heston's stochastic volatility model, the change of measure from the subjective measure P to the objective measure Q is usually investigated under the assumption that the Feller condition is satisfied. This paper closes this gap in the literature by deriving sufficient conditions for the existence of …
Based on ideas of Pigolla and Setti \cite{PS} we prove that immersed submanifolds with bounded mean curvature of Cartan-Hadamard manifolds are Feller. We also consider Riemannian submersions π:M→N with compact minimal fibers, and based on various criteria for parabolicity and stochastic completeness, see \c…
We propose a simple model of the banking system incorporating a game feature where the evolution of monetary reserve is modeled as a system of coupled Feller diffusions. The Markov Nash equilibrium generated through minimizing the linear quadratic cost subject to Cox-Ingersoll-Ross type processes creates liquidity and …
We build a sequence of empirical measures on the space D(R_+,R^d) of R^d-valued càdlàg functions on R_+ in order to approximate the law of a stationary R^d-valued Markov and Feller process (X_t). We obtain some general results of convergence of this sequence. Then, we apply them to Brownian diffusions and solutions to …
New method learns diffusion transition density for Bayesian inference.
problem Bayesian inference on diffusions with inaccessible boundaries.
method Neural Galerkin framework to solve FP equation with Dirac mass.
result Approximates likelihood function for efficient posterior sampling.
We study convergence properties of the full truncation Euler scheme for the Cox-Ingersoll-Ross process in the regime where the boundary point zero is inaccessible. Under some conditions on the model parameters (precisely, when the Feller ratio is greater than three), we establish the strong order 1/2 convergence in $L^…
The paper proves a Laplacian comparison theorem on weighted Riemannian manifolds and applies it to diffusion processes.
problem Analyzing diffusion processes on Riemannian manifolds with weighted metrics.
method Proving a Laplacian comparison theorem and applying it to various geometric and analytic properties of diffusion processes.
result Optimal conditions on m-Bakry-Émery Ricci tensor for various geometric and analytic properties to hold on weighted complete Riemannian manifolds. Positive braids have a signature bound by their Betti number.
problem Bounding the signature of positive braids.
method Using the first Betti number as a lower bound for the signature.
result The signature is bounded from below by one-quarter of the first Betti number.
New cobordisms found between certain quasipositive knots.
problem Existence of complex cobordisms between quasipositive knots.
method Construction of specific knots and cobordisms of specified genus.
result No complex cobordism exists between certain quasipositive knots.
This paper uses entropy to derive stock price dynamics and option valuation.
problem Deriving stock price dynamics and option valuation from information constraints.
method Develops an entropic inference framework to derive stochastic processes from information constraints, representing price changes through two channels: continuous and jump.
result The derived dynamics is the Merton jump diffusion, with Geometric Brownian Motion as the no jump limit.
The paper studies Lévy processes on compact manifolds, proving properties of their semigroups.
problem Analyzing Lévy processes on compact Riemannian manifolds.
method Proving properties of Feller semigroups and generators on Lp spaces. result The generator has a discrete spectrum of eigenvalues and the semigroup is trace-class when the process has a non-trivial Brownian part.
Study decay and compact support of solutions to certain nonlinear PDEs.
problem Decay and compact support properties of positive solutions to Δpu≥Λ(u) on manifolds. method Nonlinear PDE analysis, Feller property, integral Ricci curvature conditions.
result Characterization of stochastic completeness for the p-Laplacian. Develops high-order approximations for financial models, proving convergence and regularity.
problem Challenges in approximating and regularizing the Heston model due to its square root diffusion term.
method Random grid technique, Cox-Ingersoll-Ross (CIR) process, log-Heston process, PDE analysis.
result Achieves weak approximations of any order for smooth test functions in the Heston model, extending to log-Heston process.
Adaptive method improves numerical solution of Cox-Ingersoll-Ross model.
problem Approximating solutions to the Cox-Ingersoll-Ross model efficiently.
method Path-bounded timestepping with hybrid approach, including a backstop method.
result The adaptive method is strongly convergent, with strong error control.
Proves generalization bounds for SGD using Feller processes and Hausdorff dimension.
problem Characterizing generalization properties of SGD in deep learning.
method Proves generalization bounds for SGD under Feller process approximation, linking generalization error to the Hausdorff dimension of trajectories.
result Generalization error controlled by the Hausdorff dimension of trajectories, which is linked to the tail behavior of the driving process.
Paper tackles risk-sensitive impulse control for continuous-time processes.
problem Risk-sensitive impulse control for continuous-time Feller-Markov processes.
method Probabilistic approach to solve Bellman equation and construct optimal strategy.
result Optimal strategy approximated by dyadic impulse strategies.
Abstract reviews Markov processes with jumps on manifolds and Lie groups.
problem Analyzing Markov processes with jumps in geometric settings.
method Stochastic differential equations, Courrège theorem, invariant Markov processes.
result Developments in Lie groups and manifolds under various actions.
New infinite-rank summand found in knot concordance group.
problem Finding an infinite-rank summand in the smooth knot concordance group.
method Using iterated satellite operations with the Mazur pattern.
result Existence of a topologically slice knot K whose iterated satellites span an infinite-rank summand. Paper calculates homology and intersection form of trisected 4-manifolds with boundary.
problem Calculating homology and intersection form of trisected 4-manifolds with boundary.
method Uses relative trisection diagrams to calculate homology and intersection form.
result Describes a representative of the second Stiefel-Whitney class using relative trisection diagrams.
Two Heegaard Floer knot complexes are called stably equivalent if an acyclic complex can be added to each complex to make them filtered chain homotopy equivalent. Hom showed that if two knots are concordant, then their knot complexes are stably equivalent. Invariants of stable equivalence include the concordance invari…
Study on PDEs in Heston model with unique solution and convergence proof.
problem Analyzing PDEs in the Heston model for financial applications.
method Regularity results, verification theorem, unique viscosity solution, convergence proof.
result Unique viscosity solution for wide initial and source data.
Formulas for tau and epsilon concordance invariants of braided satellite knots
problem tau and epsilon invariants of satellite knots
method tau and epsilon invariants of braided satellite knots
result tau and epsilon formulas for braided satellite knots
The Heston model stands out from the class of stochastic volatility (SV) models mainly for two reasons. Firstly, the process for the volatility is non-negative and mean-reverting, which is what we observe in the markets. Secondly, there exists a fast and easily implemented semi-analytical solution for European options.…
In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider t…
The paper introduces new processes for modeling multivariate volatility.
problem Developing new stochastic processes for multivariate volatility modeling.
method Introducing Volterra Wishart and Volterra pure jump processes with fractional kernels.
result Affine covariance processes for multivariate volatility modeling.
The Black-Scholes model (sometimes known as the Black-Scholes-Merton model) gives a theoretical estimate for the price of European options. The price evolution under this model is described by the Black-Scholes formula, one of the most well-known formulas in mathematical finance. For their discovery, Merton and Scholes…
Paper studies long-run risk optimization with dyadic impulses for unbounded processes.
problem Long-run risk optimization problem with unbounded and non-uniformly ergodic processes.
method Adapting weight norm approach, combining geometric drift and local minorization property.
result Existence of solution to Bellman equation for risk-averse parameters.
Automates learning of multivariate diffusions for generative models.
problem Lack of automated methods for choosing and optimizing diffusion processes in generative models.
method Develops a recipe to maximize likelihood without model-specific analysis, parameterizes diffusion for target noise, and optimizes the inference diffusion process.
result Automatic search over all linear diffusions for generative models.
Masking diffusion outperforms other discrete diffusion models by incorporating jump times into the model.
problem Improving the performance of discrete diffusion models.
method Conditioning on the jump schedule of discrete Markov processes.
result Schedule-conditioned discrete diffusion (SCUD) models outperform classical and masking diffusion models.
Diffusion-GAN uses diffusion to improve GAN training stability and realism.
problem Stability and realism issues in training GANs.
method Diffusion-GAN employs a forward diffusion chain to generate Gaussian-mixture distributed instance noise, with adaptive diffusion process and timestep-dependent discriminator.
result Diffusion-GAN produces more realistic images with higher stability and data efficiency.
Study shows how heat leaks from material sets in low diffusivity scenarios.
problem Understanding heat leakage from material sets in low diffusivity limits.
method Generalized leading-order asymptotics for time-dependent diffusion processes.
result Diffusive transport out of a material set is proportional to the surface area of the set boundary.
New blurring diffusion models bridge heat dissipation and denoising.
problem Developing a new generative modeling approach.
method Connecting blurring to Gaussian diffusion with non-isotropic noise.
result Proposed Blurring Diffusion Models offer the best of both Gaussian denoising and inverse heat dissipation.
A new financial model merges long-range dependence and leverage effects.
problem Challenges posed by financial markets' stylized facts.
method Develops a fractional and mixed-fractional CEV model using fractional calculus.
result Analytical valuation formula for European Call options and Greeks.
Study mass transport in low-diffusivity using Lagrangian coordinates.
problem Mass preserving transport of passive tracers in low-diffusivity limit.
method Lagrangian coordinates, time-averaged diffusion equation, weighted manifold structure.
result Leading order asymptotics extend to dominant nontrivial singular value in low-diffusivity limit.