Study shows Euler scheme converges at strong order 1/2 for Cox-Ingersoll-Ross process.
problem Analyzing convergence of Euler approximations for Cox-Ingersoll-Ross process.
method Full truncation Euler scheme applied to Cox-Ingersoll-Ross process.
result Strong order 1/2 convergence in L^p of scheme to exact solution.
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.
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.
Paper addresses Heston model under violated Feller condition, deriving new change of measure conditions.
problem Investigates Heston model under Feller condition violation.
method Derives sufficient conditions for equivalent martingale measure and true martingale stock price process.
result New conditions for change of measure and martingale properties in Heston model are established.
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 study the solution to Kolmogorov-Feller equation and by using it provide pricing formulas of well known some options under jump-diffusion model.
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…
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…
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.
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 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. Model shows how adding liquidity in banking system can lead to defaults.
problem Systemic risk and interbank lending in banking systems.
method Simple model using Feller diffusions and Markov Nash equilibrium.
result Adding liquidity leads to defaults in banking system.
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.
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.
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.
The paper analyzes Euler approximations for complex volatility models with strong convergence rates.
problem Analyzing strong convergence rates for Euler approximations in stochastic path-dependent volatility models.
method Proposes a Monte Carlo simulation scheme combining log-Euler and truncation/Euler-Maruyama schemes.
result Establishes strong convergence rate of 1/2 for the approximation process up to a critical time.
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. 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.
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…
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.
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…
Valuing FF contracts in time-dependent models
problem Valuing American options and Flexible Forwards contracts
method Recursive Riccati solution and Volterra equation
result FF contracts priced faster than traditional methods
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 …
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
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.
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.
New Upsilon invariants rule out stable equivalence of knot complexes.
problem Stable equivalence of knot complexes and its invariants.
method Secondary Upsilon invariants defined by Kim and Livingston.
result Relations between Upsilon invariants do not extend to stable equivalence.
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.…
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…
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.
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.
Derives Black-Scholes model using central limit theorem.
problem Deriving the Black-Scholes formula for European options.
method Central limit theorem argument for undergraduate understanding.
result Elementary derivation accessible to undergraduates.
The Fano ratio offers a more diversified portfolio than the Sharpe ratio in optimization.
problem Optimizing portfolio performance using the Sharpe ratio.
method Introducing and optimizing the Fano ratio as an alternative to the Sharpe ratio.
result Optimizing for the Fano ratio leads to more diversified and less skewed portfolios.
New method to decompose portfolio performance ratios.
problem Understanding the drivers of portfolio performance ratios.
method Using Euler's theorem, decomposes performance ratios into modified ratios.
result Derives condition for new asset to improve portfolio performance.
Omega ratio is shown to be equivalent to Sharpe ratio under certain distributional assumptions.
problem Comparing Omega ratio to Sharpe ratio as performance indicators.
method Computation and analysis of Omega ratio for normal distribution and proof for elliptic distributions.
result Omega ratio is equivalent to Sharpe ratio for returns with elliptic distributions.
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.
FORE evaluates occupancy ratios without requiring Bellman completeness.
problem Offline reinforcement learning occupancy ratio estimation.
method Fitted occupancy-ratio evaluation (FORE) using adjoint Bellman recursion.
result FORE achieves convergence in KL without Bellman completeness.
Hydrodynamics principles applied to finance, solving complex market models.
problem Understanding financial market dynamics using physics principles.
method Unified mathematical framework using Kelvin waves to solve differential and pseudo-differential equations.
result Solved various financial models including volatility and variance swaps.
Optimal option portfolios under Sharpe Ratio maximization with skew-elliptical t-distributed returns
problem Optimal option portfolios under Sharpe Ratio maximization
method Formulation for explicit portfolio weights
result Different optimal portfolios for Sharpe Ratio and return-to-Value-at-Risk (VaR) ratio
The paper proposes an asset allocation strategy using the Sortino ratio for better performance.
problem Traditional asset allocation methods like the Sharpe ratio do not penalize negative returns adequately.
method The Sortino ratio is used to maximize asset allocation, penalizing only negative return variances.
result The Sortino ratio-based strategy outperforms traditional methods like the Kelly criterion.