New method detects data distribution changes and retraining is advised.
problem Detecting when data distribution changes for retraining prediction algorithms.
method Exchangeable martingales and conformal prediction.
result Guaranteed validity of the method, with efficiency explored.
The paper reviews methods for testing randomness and exchangeability in sequential data.
problem Testing the randomness and exchangeability of sequential data.
method Conformal martingales for online testing of randomness and exchangeability.
result Valid and efficiency results for various conformal martingale-based procedures.
Study detects concept shift in online data using martingales.
problem Detecting concept shift in online datasets.
method Exchangeable martingales and conformal prediction techniques.
result Decomposes concept shift into detectable components.
We consider the problem of quickest change-point detection in data streams. Classical change-point detection procedures, such as CUSUM, Shiryaev-Roberts and Posterior Probability statistics, are optimal only if the change-point model is known, which is an unrealistic assumption in typical applied problems. Instead we p…
A new method detects distribution shifts faster than existing CTMs.
problem Detecting distribution shifts in data streams with contamination issues.
method Uses a fixed reference dataset to compare each new sample, avoiding contamination.
result Detects distribution shifts faster and more reliably than standard CTMs.
Adaptive monitoring for AI systems detects and diagnoses shifts in data distribution.
problem Continuous monitoring of AI systems to detect and address unsafe behavior.
method Weighted-conformal martingales (WCTMs) for online monitoring of AI systems.
result Improved performance over state-of-the-art baselines on real-world datasets.
A new family of conformal test martingales based on Legendre polynomials for online exchangeability testing.
problem Detecting variance, skewness, and higher-order deviations from uniformity in online data.
method A family of conformal test martingales based on shifted Legendre polynomials.
result The Variational Legendre Jumper reduces exponential scaling to linear time with minimal loss in power.
We study strict local martingales via h-transforms, a method which first appeared in Delbaen-Schachermayer. We show that strict local martingales arise whenever there is a consistent family of change of measures where the two measures are not equivalent to one another. Several old and new strict local martingales are i…
Conformal Test Martingales can be 'blind' to significant changes in data distribution.
problem The converse of exchangeability does not hold, leading to potential blindness of CTMs.
method Explicit construction of A-cryptic change-point using bivariate Gaussian distributions. result CTMs can be perfectly cryptic to a significant change in marginal means.
Improved bounds for proximal gradient algorithms with computational errors.
problem Analyzing convergence of proximal gradient algorithms with inaccuracies.
method Deriving new tighter deterministic and probabilistic bounds for convex composite problems.
result Probabilistic bounds are more robust and accurate for algorithm verification and performance guarantees.
New algorithm selects robust martingale for optimal stopping problems.
problem Optimal stopping problems in stochastic processes.
method Randomized dual martingale minimization algorithm.
result Efficiently selects Doob martingale as close as possible.
Develops a martingale expansion for stochastic volatility models.
problem Approximating marginal distributions of stochastic volatility models.
method Martingale expansion framework for continuous stochastic volatility models.
result First-order perturbation expansions for small volatility-of-volatility and fast mean-reversion models.
Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.
problem Tackles the martingale Schrödinger bridge in arbitrary dimensions.
method Identifies continuous-time counterpart and relates to variational problems.
result Continuous martingale Schrödinger bridge coincides with Föllmer martingale in irreducible case.
M-FISHER detects and adapts to streaming data shifts with statistical validity and stability.
problem Detecting and adapting to distributional shifts in streaming data.
method Constructs an exponential martingale from non-conformity scores and applies Ville's inequality for detection. Fisher-preconditioned updates for adaptation.
result Establishes M-FISHER as a principled approach for robust, anytime-valid detection and geometrically stable adaptation.
Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.
problem Modeling continuous martingales with prescribed initial and terminal distributions.
method Developed geometric Bass martingales and established their properties.
result Explicit bijection and representation of geometric Bass martingales.
Existence proved for q-Bass martingales with specific marginals.
problem Constructing martingales with prescribed marginals close to a reference measure.
method Geometric analysis of parametrized convex polygonal chains.
result Existence and uniqueness of q-Bass martingales with finitely supported initial marginals. Study finds optimal martingale coupling between two distributions with minimal entropy.
problem Finding the optimal martingale coupling between two distributions with minimal relative entropy.
method Solving a dual problem to find the log-density of the optimal coupling, which represents the marginal and martingale constraints.
result The log-density of the optimal coupling is given by a triplet of real functions representing the marginal and martingale constraints.
In the paper, the martingales and super-martingales relative to a convex set of equivalent measures are systematically studied. The notion of local regular super-martingale relative to a convex set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the disc…
Note on failure of Martingale Wasserstein Inequality in higher dimensions.
problem Analyzing failure of Martingale Wasserstein Inequality in higher dimensions.
method Checking failure in dimension d≥2 and proving a stronger inequality in all dimensions.
result A stronger Maximal Martingale Wasserstein Inequality holds in all dimensions.
The paper studies projections of asset prices under equivalent martingale measures.
problem Understanding the impact of information on asset price bubbles and arbitrage opportunities.
method Analyzes optional projections of local martingales into a smaller filtration under equivalent martingale measures.
result Provides general results and specific examples like inverse Bessel process and stochastic volatility models.
This paper introduces an arbitrage-free conic martingale model for credit risk.
problem The lack of an arbitrage-free conic martingale model for credit risk.
method Developed an arbitrage-free conic martingale called Φ-martingale.
result The Φ-martingale model satisfies the immersion property and is suitable for practical applications in credit risk.
Extends optimal transport to dynamic and martingale settings.
problem Dynamic and martingale relaxation of optimal transport problems.
method Extends Benamou-Brenier formula to weak optimal transport and introduces barycentric optimal transport.
result Relates barycentric optimal transport to martingale Benamou-Brenier formula.
We exhibit sufficient conditions such that components of a multidimensional SDE giving rise to a local martingale M are strict local martingales or martingales. We assume that the equations have diffusion coefficients of the form σ(Mt,vt), with vt being a stochastic volatility term.
Dual martingales improve primal optimal stopping problem efficiency.
problem Optimal stopping problem in the primal formulation.
method Investigation of dual martingales to improve primal methods.
result Accurate dual martingale approximations reduce primal problem variance.
A strict local martingale is a local martingale which is not a martingale. There are few explicit examples of "naturally occurring" strict local martingales with jumps available in the literature. The purpose of this paper is to provide such examples, and to illustrate how they might arise via filtration shrinkage, a p…
In this paper, we obtain stability results for martingale representations in a very general framework. More specifically, we consider a sequence of martingales each adapted to its own filtration, and a sequence of random variables measurable with respect to those filtrations. We assume that the terminal values of the m…
In this paper we introduce the concept of conic martingales}. This class refers to stochastic processes having the martingale property, but that evolve within given (possibly time-dependent) boundaries. We first review some results about the martingale property of solution to driftless stochastic differential equations…
Authors prove the existence of a martingale measure in credit risk models.
problem Existence of an equivalent martingale measure in hazard process models of credit risk.
method By identifying a no-arbitrage condition, the authors construct a measure that turns discounted stock and bond prices into martingales.
result The existence of a martingale measure is demonstrated in credit risk models.
A concept of martingale-fair index of return, consistent with Arbitrage Free Pricing Theory, is introduced. An explicit formula for the average rate of return of a group of investment/pension funds in a discrete time stochastic model is derived and several properties of this index are shown. In particular, it is proven…
New method finds closest martingale to Brownian motion.
problem Finding optimal martingale interpolating marginals.
method Martingale Sinkhorn algorithm, iterative scheme.
result Algorithm yields Bass potential in arbitrary dimension.
We compute and discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of Ornstein-Uhlenbe…
Dynamic reinsurance aims to minimize surplus risk using martingale transport.
problem Minimizing surplus risk in dynamic reinsurance.
method Martingale optimal transport techniques.
result A tractable solution analogous to the Bass martingale is found.
The stochastic exponential Zt=exp{Mt−M0−(1/2)<M,M>t} of a continuous local martingale M is itself a continuous local martingale. We give a necessary and sufficient condition for the process Z to be a true martingale in the case where Mt=∫0tb(Yu)dWu and Y is a one-dimensional diffusion drive…
Efficiently computes robust option prices using multi-marginal martingale transport.
problem Computing robust option prices under martingale constraints.
method Extending state space, sequential martingale structure, entropic regularisation.
result Fast computation of optimal solutions for large problems.
We propose procedures for testing whether stock price processes are martingales based on limit order type betting strategies. We first show that the null hypothesis of martingale property of a stock price process can be tested based on the capital process of a betting strategy. In particular with high frequency Markov …
We extend martingale transport results to weak martingale transport.
problem Applying martingale transport results to weak martingale transport.
method Change of numeraire for weak martingale transport.
result Established the correspondence between stretched Brownian motion and its geometric counterpart.
We discuss martingales, detrending data, and the efficient market hypothesis for stochastic processes x(t) with arbitrary diffusion coefficients D(x,t). Beginning with x-independent drift coefficients R(t) we show that Martingale stochastic processes generate uncorrelated, generally nonstationary increments. Generally,…
We derive integral tests for the existence and absence of arbitrage in a financial market with one risky asset which is either modeled as stochastic exponential of an Ito process or a positive diffusion with Markov switching. In particular, we derive conditions for the existence of the minimal martingale measure. We al…
A martingale \int H.dZ is defined as having Dimension k if H has rank k almost surely, almost all t. Dimension can be used as a geometric invariant to classify and study martingales. We also define general Brownian motions in higher dimensions.
Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.
problem Constructing a continuous Markov martingale with Brownian marginals that misses the strong Markov property.
method Developed a new approach to create a continuous Markov martingale that differs from Brownian motion in terms of the strong Markov property.
result A continuous Markov martingale with Brownian marginals that lacks the strong Markov property was successfully constructed.
We study martingale inequalities from an analytic point of view and show that a general martingale inequality can be reduced to a pair of deterministic inequalities in a small number of variables. More precisely, the optimal bound in the martingale inequality is determined by a fixed point of a simple nonlinear operato…
Extends specific relative entropy to multidimensional continuous martingales.
problem Mutual singularity of martingale laws in continuous time.
method Extension of specific relative entropy from one to multiple dimensions, including closed-form expressions for simple examples.
result Establishes that the lower bound on specific relative entropy from Gantert carries over to higher dimensions and is tight.
The study examines how market completeness is lost when filtering down the information set.
problem Loss of market completeness under filtration shrinkage.
method Bayesian filtering approach to analyze local martingale deflators and their projections.
result Projections of deflators in smaller filtrations are not sufficient to span all local martingale deflators.
The paper extends Strassen's theorem to include biased martingales for American options.
problem Existence of martingales for arbitrage-free prices of American options.
method Derives an extension of Strassen's theorem linking biased martingales to strengthened convex order.
result Characterizes the strengthened convex order through integrals with respect to compensated Poisson processes.
Numerical observations on martingale couplings are confirmed under certain conditions.
problem Understanding the validity of numerical observations on maximizers and minimizers of martingale couplings.
method Investigation of sufficient conditions and counterexamples for the property to hold.
result The non-decreasing property of martingale couplings is preserved for maximizers under specific conditions.
Study characterizes martingales on fiber bundles for harmonic map analysis.
problem Characterizing martingales on fiber bundles for harmonic maps.
method Investigated ablaP-martingales on principal fiber bundles P(M,G) with a projectable connection. result New characterizations of harmonic maps established.
The paper develops option pricing methods for bilateral Gamma stock models.
problem Developing accurate option pricing measures for bilateral Gamma stock models.
method Incorporates various mathematical techniques including Esscher transforms, minimal entropy martingale measures, and p-optimal martingale measures. result Illustrates the theory with a numerical example, providing practical application of the methods.
We generalize the classical Bochner formula for the heat flow on M to martingales on the path space PM, and develop a formalism to compute evolution equations for martingales on path space. We see that our Bochner formula on PM is related to two sided bounds on Ricci curvature in much the same manner that the classical…