Study N-player and mean-field games in Itô-diffusion markets with competitive or homophilous interactions.
problem Optimal portfolio choice in a common market with N interacting players. method Analyzes N-player and mean-field games in incomplete and complete markets with CARA utilities and random risk tolerances. result Derives explicit or closed-form solutions for equilibrium processes and game values.
The paper develops a method for stochastic differential equations on manifolds using Schwartz morphisms and diffusion generators.
problem Representing stochastic differential equations on smooth manifolds.
method Using Schwartz morphisms and diffusion generators to construct SDEs on manifolds.
result An extended Ito formula for SDEs on manifolds.
Itô processes are the most common form of continuous semimartingales, and include diffusion processes. This paper is concerned with the nonparametric regression relationship between two such Itô processes. We are interested in the quadratic variation (integrated volatility) of the residual in this regression, over a un…
This paper analyzes sampling from heavy-tailed distributions using discretized Itô diffusions.
problem Sampling from heavy-tailed distributions with finite variance.
method Mean-square analysis of discretized Itô diffusions with weighted Poincaré inequalities.
result Explicit iteration complexity for obtaining samples close to target distributions in Wasserstein-2 metric.
Projects Markovian processes from Itô semimartingales with jumps.
problem Modeling Itô semimartingales with jumps using Markovian projections.
method Construct Markovian projections for Itô semimartingales with jumps using non-local FPKEs.
result Markovian projections match the marginal laws of the original process.
Extending Itô's formula to non-smooth functions is important both in theory and applications. One of the fairly general extensions of the formula, known as Meyer-Itô, applies to one dimensional semimartingales and convex functions. There are also satisfactory generalizations of Itô's formula for diffusion processes whe…
New method uses diffusions to measure sample quality in multivariate targets.
problem Measuring convergence to multivariate continuous targets.
method Ito diffusions and explicit multivariate Stein factor bounds.
result Established near-linear relationship between diffusion Stein discrepancies and Wasserstein distances.
We analyze a new Markov chain model for better sampling and optimization.
problem Developing a new Markov chain model for improved sampling and optimization.
method We introduce a new class of Ito chains with arbitrary noise and inexact drift/diffusion coefficients, proving a bound in W2-distance. result Our analysis provides improved or first results for various applications like SGLD, sampling, and boosting.
The paper analyzes performance criteria for competing fund managers in Ito-diffusion markets.
problem Analyzing performance of competing fund managers in Ito-diffusion markets.
method Developed forward relative performance criteria and forward Nash equilibrium for passive and competitive cases.
result Extended performance criteria for investment problems in Ito-diffusion markets.
Study optimal portfolio selection in a complex market with jumps and regime shifts.
problem Optimal portfolio selection in a market with jumps and regime shifts.
method Modeling a market with Lévy processes and regime switching, using various securities to complete the market, solving the portfolio selection problem for power and logarithmic utilities.
result Conditions for asymptotic-arbitrage-free market and solutions for optimal portfolio selection.
Neural networks can approximate complex stochastic equations well.
problem Approximating general stochastic differential equations.
method Identified neural network classes approximating continuous functions.
result Neural stochastic differential equations can approximate general stochastic differential equations arbitrarily well.
Neural Jump ODEs model Itô processes without adversarial training.
problem Generating samples from Itô processes with irregular data.
method Neural Jump ODEs framework for drift and diffusion approximation.
result NJODEs can recover true parameters of Itô processes in the limit.
Using results from our companion article [arXiv:1112.4824v2] on a Schauder approach to existence of solutions to a degenerate-parabolic partial differential equation, we solve three intertwined problems, motivated by probability theory and mathematical finance, concerning degenerate diffusion processes. We show that th…
Modeling financial volatility using quantum mechanics principles.
problem Capturing high volatility and spikes in financial asset prices.
method Agent-based model linking quantum mechanical jumps to socio-economic behavior.
result Model dynamics converge to Itô-diffusion price processes in large market limits.
Framework learns surrogates for molecular dynamics across multiple time-scales.
problem Stable molecular dynamics simulations require small time-steps, but long-time-scale moments need repeated simulations.
method Implicit Transfer Operator Learning with denoising diffusion probabilistic models and SE(3) equivariant architecture.
result Models can generate self-consistent stochastic dynamics across multiple time-scales.
New method defines symmetries for diffusion processes.
problem Symmetries of diffusion processes.
method Geometric reformulation of martingale problem on manifolds.
result Natural definition of Lie symmetries for diffusion processes.
New method infers and samples point processes from latent diffusion.
problem Modeling point processes with latent diffusion.
method Itô's excursion theory for inference and sampling.
result Proposes a new method to infer and sample point processes.
We use probabilistic methods to study classical solutions for systems of interacting semilinear parabolic partial differential equations. In a modeling framework for a financial market with interacting Ito and point processes, such PDEs are shown to provide a natural description for the solution of hedging and valuatio…
Proposes overnight volatility model for better market dynamics.
problem Lack of high-frequency data during close-to-open period.
method Itô diffusion model with weighted least squares estimation.
result Developed and validated overnight volatility model.
The continuous-time random walk (CTRW) is a pure-jump stochastic process with several applications in physics, but also in insurance, finance and economics. A definition is given for a class of stochastic integrals driven by a CTRW, that includes the Ito and Stratonovich cases. An uncoupled CTRW with zero-mean jumps is…
Formula for option pricing in a stochastic volatility model with jumps.
problem Developing a formula for European option pricing in a complex stochastic volatility model.
method Fractional integral of a diffusion process, martingale representation, and Itô calculus for processes with jumps.
result A first-order approximation formula for option prices.
The paper introduces a new volatility model for state heterogeneous financial markets using high-frequency data.
problem State heterogeneity in financial volatility processes.
method Developed a state heterogeneous GARCH-Ito (SG-Ito) model based on continuous Ito diffusion process.
result Empirical studies reveal various state heterogeneities in S&P 500 index volatility.
Develops pathwise analysis for log-optimal portfolios using rough paths theory.
problem Analyzing stability and approximation of log-optimal portfolios.
method Pathwise approach based on càdlàg rough paths theory.
result Establishes pathwise stability and error estimates for log-optimal portfolios.
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 solves the inversion problem for jump processes using Markovian projections.
problem Calibrating jump-diffusion models with both local and stochastic features.
method Inverting Markovian projections for pure jump processes.
result Constructs calibrated local stochastic intensity (LSI) models for credit risk applications.
This work analyzes discrete diffusion models using stochastic integrals, providing error bounds and insights.
problem Error analysis for discrete diffusion models remains less understood.
method Proposes a comprehensive framework based on Lévy-type stochastic integrals.
result Obtains the first error bound for the τ-leaping scheme in KL divergence. We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pat…
Efficient diffusion model for symmetric manifolds reduces training and computation costs.
problem Heat kernel computations for manifold diffusion models are computationally expensive and infeasible.
method Spatially-varying covariance diffusion model, efficient objective derived via Ito's Lemma.
result Our model reduces training time and arithmetic operations by orders of magnitude.
The present paper introduces a jump-diffusion extension of the classical diffusion default intensity model by means of subordination in the sense of Bochner. We start from the bi-variate process (X,D) of a diffusion state variable X driving default intensity and a default indicator process D and time change it wi…
We derive a higher-order expansion for rough volatility models.
problem Characterizing and estimating rough volatility models.
method Higher-order asymptotic expansion of characteristic functions.
result Distinct roles of rough and jump dynamics in volatility.
The usual derivation of the Fokker-Planck partial differential eqn. assumes the Chapman-Kolmogorov equation for a Markov process. Starting instead with an Ito stochastic differential equation we argue that finitely many states of memory are allowed in Kolmogorov's two pdes, K1 (the backward time pde) and K2 (the Fokker…
Paper develops approximation and statistical theory for signature-based path regression.
problem Understanding how fast signatures approximate continuous path functionals.
method Develops \(L^2\) approximation rate for smooth functionals of Itô diffusions and establishes consistency of statistical learning procedures.
result Signature-based methods improve prediction over handcrafted features in various real-data applications.
Incomplete financial markets are considered, defined by a multi-dimensional non-homogeneous diffusion process, being the direct sum of an Itô process (the price process), and another non-homogeneous diffusion process (the exogenous process, representing exogenous stochastic sources). The drift and the diffusion matrix …
In mathematical Finance calculating the Greeks by Malliavin weights has proved to be a numerically satisfactory procedure for finite-dimensional Itô-diffusions. The existence of Malliavin weights relies on absolute continuity of laws of the projected diffusion process and a sufficiently regular density. In this article…
Method learns radial basis function distributions from samples.
problem Learning radial basis function distributions from training samples.
method Projected particle Langevin optimization method with distributionally robust optimization.
result Empirical measure of Langevin particles converges to a reflected Itô diffusion-drift process.
We consider optimal investment problems for a diffusion market model with non-observable random drifts that evolve as an Ito's process. Admissible strategies do not use direct observations of the market parameters, but rather use historical stock prices. For a non-linear problem with a general performance criterion, th…
Paper solves MV portfolio selection in jump-diffusion models with no-shorting constraint.
problem Mean-variance portfolio selection in jump-diffusion model with no-shorting constraint.
method Reduces problem to LQ control and finding a maximal point of a function, constructs viscosity solution.
result Explicit viscosity solution to Hamilton-Jacobi-Bellman equation, optimal controls derived.
Counterexample shows Ito integrand needn't be locally square integrable.
problem Ito integrand's square integrability condition is not always met.
method Provided a counterexample to Ito's Lemma's integrability condition.
result Ito integrand needn't be locally square integrable.
Derives conditions for no arbitrage in financial markets with stochastic or diffusion models.
problem Existence and absence of arbitrage in financial markets with stochastic or diffusion models.
method Integral tests, martingale and strict local martingale properties of stochastic exponentials, Markov switching models.
result Conditions for the existence of minimal martingale measure and its preservation under Markov switching.
We present a methodology for obtaining explicit solutions to infinite time horizon optimal stopping problems involving general, one-dimensional, Itô diffusions, payoff functions that need not be smooth and state-dependent discounting. This is done within a framework based on dynamic programming techniques employing var…
This research explores neural SDEs as deep latent Gaussian models in the diffusion limit.
problem Deep latent Gaussian models with time-inhomogeneous Markov chains and Gaussian perturbations.
method Develops variational inference for neural SDEs using stochastic automatic differentiation in Wiener space.
result The limiting latent object is an Itô diffusion process governed by neural nets.
Motivated by marginals-mimicking results for Itô processes via SDEs and by their applications to volatility modeling in finance, we discuss the weak convergence of the law of a hypoelliptic diffusions conditioned to belong to a target affine subspace at final time, namely L(Zt∣Yt=y) if $X_{\cdot}=(Y_\cd…
Estimates neural drift for stochastic equations, improving inference on noisy data.
problem Estimating drift in stochastic differential equations with neural networks.
method Non-parametric estimation using ReLU neural networks, enforcing theoretical bounds.
result Practical method for inference on noisy and rough functional data.
New algorithm improves sampling efficiency for complex systems.
problem Sampling with Markov chain Monte Carlo methods.
method Discretizing smooth Itô diffusions with stochastic Runge-Kutta.
result Sampling iterates converge to target distribution faster.
We recently showed that the S&P500 stock market index is well described by Tsallis non-extensive statistics and nonlinear Fokker-Planck time evolution. We argued that these results should be applicable to a broad range of markets and exchanges where anomalous diffusion and `heavy' tails of the distribution are present.…
Investor optimizes portfolio under dynamic risk preferences.
problem Optimizing investment under uncertain future risk attitudes.
method Developed a general equilibrium framework and solved for subgame-perfect equilibrium policies.
result Equilibrium policies include a novel hedging component to counteract anticipated risk aversion changes.
New method distinguishes stochastic from deterministic signals using excursion counts.
problem Distinguishing between stochastic and deterministic signals in discrete time series.
method Excursion and crossing theorems for continuous semimartingales, comparing empirical excursion counts to theoretical expectation.
result A robust data-driven diffusion test that classifies signals based on log-log slope deviation.
Developed a machine-checked Itô calculus for Brownian motion.
problem Formal verification of Itô calculus for Brownian motion.
method Machine-checked formalization in Lean over Mathlib.
result First machine-checked constructions of the Itô integral and Itô's formula.