Explicit conditions for martingale property of financial models.
problem Well-definedness of semimartingale Libor models.
method Express conditions in terms of semimartingale characteristics.
result Allows careful discussion of Libor model construction.
Develops a multivariate aggregation property for unbiased risk premium estimation.
problem Estimating unbiased risk premia from high-frequency returns.
method Introduces a general multivariate aggregation property for multivariate martingales and log martingales.
result Defines realised third and fourth moments for unbiased risk premium measurement.
Study on martingale property and moment explosions in signature volatility models.
problem Analyzing the martingale property and moment explosions in signature volatility models.
method Fine analysis of the explosion time of a signature stochastic differential equation.
result The price process is a true martingale if and only if the order of the linear form is odd and a correlation parameter is negative.
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.
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.
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.
The paper introduces conic martingales within boundaries and provides a method to construct them.
problem Developing martingale processes within specified boundaries.
method Review and construction of martingale solutions to driftless SDEs, focusing on [0,1]. result An analytically tractable martingale with separable coefficient is identified.
New analysis shows LLMs don't follow Bayesian inference in ICL.
problem Does in-context learning in LLMs follow Bayesian inference?
method Analyzes ICL through the martingale property, a requirement for Bayesian inference.
result Violations of the martingale property show LLMs don't follow Bayesian inference.
Lions and Musiela (2007) give sufficient conditions to verify when a stochastic exponential of a continuous local martingale is a martingale or a uniformly integrable martingale. Blei and Engelbert (2009) and Mijatović and Urusov (2012c) give necessary and sufficient conditions in the case of perfect correlation (ρ=1).…
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.
Stability proved for martingale and weak transport problems.
problem Stability of martingale and weak optimal transport problems.
method Established stability through unconventional topology considering temporal structure of martingales.
result Proved stability of martingale and weak transport 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 …
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…
Proves existence and uniqueness of SDE solutions with Lipschitz coefficients driven by continuous martingales.
problem Existence and uniqueness of solutions for SDEs with Lipschitz coefficients.
method Picard's iterative procedure and model-free Burkholder-Davis-Gundy inequality.
result Existence and uniqueness of solutions for SDEs with Lipschitz coefficients driven by continuous, model-free martingales.
Study shows stock price is a martingale if volatility's driving Brownian motion is negatively correlated with the stock.
problem Determining the martingale property of stock prices in fractional stochastic volatility models.
method Analyzed a class of fractional stochastic volatility models, including the rough Bergomi model, focusing on the correlation between stock and volatility.
result The stock price is a true martingale if and only if the correlation between the driving Brownian motions of the stock and the volatility is nonpositive.
The main result of this paper that a martingale evolution can be chosen for Libor such that all the Libor interest rates have a common market measure; the drift is fixed such that each Libor has the martingale property. Libor is described using a field theory model, and a common measure is seen to be emerge naturally f…
The paper proves stability of martingale representations in a broad context.
problem Stability of martingale representations in a general framework.
method Extensive use of martingale theory and convergence properties.
result Each component of the martingale representation converges under Skorokhod topology.
We study the class of Azéma-Yor processes defined from a general semimartingale with a continuous running maximum process. We show that they arise as unique strong solutions of the Bachelier stochastic differential equation which we prove is equivalent to the drawdown equation. Solutions of the latter have the drawdown…
Optimal transport with scalar martingales defined over multiple periods.
problem Finding optimal transport plans with specific properties over multiple time periods.
method Introducing left-monotone transports and characterizing them through various properties.
result Left-monotone transports are unique under certain conditions and have specific order properties.
The strong predictable representation property is proven for filtrations with a random variable under a density hypothesis.
problem Proving the strong predictable representation property in filtrations with a random variable.
method Using the density hypothesis of Jacod (1985), the strong predictable representation property is transferred to the enlarged filtration.
result The strong predictable representation property can always be transferred to the enlarged filtration under the density hypothesis.
Extends martingale theory to non-monotone information in jump processes.
problem Non-monotone information dynamics in financial and insurance applications.
method Develops a general theory of martingale representations for non-monotone filtrations.
result Introduces a symmetric counterpart to martingale representations that quantifies information loss.
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.
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…
Study on implied volatility in strict local martingale models, showing how to detect price bubbles.
problem Detecting price bubbles in financial models with strict local martingale behavior.
method Asymptotic expansion and duality method based on absolutely continuous measure change.
result Strict local martingale property can be determined from the asymptotic expansion of implied volatility.
Study shows measures with MRP are dense in probability space.
problem Endogenous completeness in financial economics.
method Analytic fields and probability measures over open sets.
result Set of measures with MRP is either empty or dense.
We present simple new examples of pure-jump strict local martingales. The examples are constructed as exponentials of self-exciting affine Markov processes. We characterize the strict local martingale property of these processes by an integral criterion and by non-uniqueness of an associated ordinary differential equat…
The study examines the supports of extremal martingale measures with given marginals in a two-period setting.
problem Investigating the supports of extremal martingale measures with pre-specified marginals in a two-period setting.
method Established equivalence between extremality and denseness in L1(Q), provided combinatorial sufficient conditions for weak exact predictable representation property (WEP), and studied the relation between cycles and extremality. result Developed necessary and sufficient conditions for the weak exact predictable representation property (WEP) in terms of 2-net and deadlock for finite support of the first marginal. 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.
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.
Study optimal semistatic portfolios using martingale Schrödinger bridges.
problem Optimizing semistatic portfolios in a dynamic stock market.
method Minimizing entropy among calibrated martingale measures.
result Explicit solution for optimal semistatic portfolios exists.
This papers addresses the stock option pricing problem in a continuous time market model where there are two stochastic tradable assets, and one of them is selected as a numéraire. It is shown that the presence of arbitrarily small stochastic deviations in the evolution of the numéraire process causes significant chang…
Complete duality theory for martingale optimal transport on the line.
problem Optimal transport between two probability measures on the real line.
method Quasi-sure formulation of the dual problem, showing complete duality theory for general marginals and measurable reward functions.
result Absence of duality gap and existence of dual optimizers for general marginals and measurable reward functions.
The paper develops general, discrete, non-probabilistic market models and minmax price bounds leading to price intervals for European options. The approach provides the trajectory based analogue of martingale-like properties as well as a generalization that allows a limited notion of arbitrage in the market while still…
Efficient variance reduction for Markov chains using martingale representations.
problem Reducing variance in estimating additive functionals of Markov chains.
method A novel discrete time martingale representation approach for variance reduction.
result The proposed method achieves a lower cost-to-variance product than the naive approach.
Fast pricing of American-style options has been a difficult problem since it was first introduced to financial markets in 1970s, especially when the underlying stocks' prices follow some jump-diffusion processes. In this paper, we propose a new algorithm to generate tight upper bounds on the Bermudan option price witho…
This paper gives new concentration inequalities for the spectral norm of a wide class of matrix martingales in continuous time. These results extend previously established Freedman and Bernstein inequalities for series of random matrices to the class of continuous time processes. Our analysis relies on a new supermarti…
In the "positive interest" models of Flesaker-Hughston, the nominal discount bond system is determined by a one-parameter family of positive martingales. In the present paper we extend this analysis to include a variety of distributions for the martingale family, parameterised by a function that determines the behaviou…
A new stochastic volatility model with quadratic drift prevents moment explosions and preserves stock price martingale property.
problem Avoiding moment explosions and preserving stock price martingale property in stochastic volatility models.
method Introduces a one-factor stochastic volatility model with quadratic drift and a linear dispersion function, showing that the quadratic term is crucial.
result The model prevents moment explosions and preserves the martingale property of the stock price process.
Study uses viscosity solutions to solve control problems involving measure-valued martingales.
problem Stochastic control problems with measure-valued martingale state processes.
method Viscosity solution approach exploiting structural properties of MVM processes.
result Value function is the unique viscosity solution to the HJB equation.
New probabilistic approach to optimal transport using martingales.
problem Optimal transport between given distributions.
method Martingale formulation of the Benamou-Brenier problem.
result Unique solution mimics Brownian motion and provides time-consistent interpolations.
We study a novel pricing operator for complete, local martingale models. The new pricing operator guarantees put-call parity to hold for model prices and the value of a forward contract to match the buy-and-hold strategy, even if the underlying follows strict local martingale dynamics. More precisely, we discuss a chan…
We describe an abstract control-theoretic framework in which the validity of the dynamic programming principle can be established in continuous time by a verification of a small number of structural properties. As an application we treat several cases of interest, most notably the lower-hedging and utility-maximization…
This paper does not suppose a priori that the evolution of the price of a financial asset is a semimartingale. Since possible strategies of investors are self-financing, previous prices are forced to be finite quadratic variation processes. The non-arbitrage property is not excluded if the class A of admiss…
Strict local martingales may admit arbitrage opportunities with respect to the class of simple trading strategies. (Since there is no possibility of using doubling strategies in this framework, the losses are not assumed to be bounded from below.) We show that for a class of non-negative strict local martingales, the s…
We develop the fundamental theorem of asset pricing in a probability-free infinite-dimensional setup. We replace the usual assumption of a prior probability by a certain continuity property in the state variable. Probabilities enter then endogenously as full support martingale measures (instead of equivalent martingale…
In a model independent discrete time financial market, we discuss the richness of the family of martingale measures in relation to different notions of Arbitrage, generated by a class S of significant sets, which we call Arbitrage de la classe S. The choice of S reflects into the int…
Invariance times relate stopping times to martingale properties in probability theory.
problem Characterizing stopping times that preserve martingale properties.
method Analyzing filtrations and stopping times in probability spaces.
result Characterized invariance times in terms of Azéma supermartingales.
Investigates optimal consumption and investment strategies in non-Markovian markets with unbounded parameters.
problem Optimal consumption and investment strategies in non-Markovian markets with unbounded parameters.
method Martingale optimal principle and quadratic BSDEs with exponential moment.
result Establishes optimal strategies for consumption and investment.