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.
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.
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.
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.
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.
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. 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…
We consider a Poisson process η on a measurable space $(\BY,\mathcal{Y})$ equipped with a partial ordering, assumed to be strict almost everwhwere with respect to the intensity measure λ of η. We give a Clark-Ocone type formula providing an explicit representation of square integrable martingales (defined with re…
Dual representation of Kantorovich functional using martingale measures.
problem Representation of Kantorovich functional on Skorokhod space.
method Choquet capacity generated by martingale measures with constraints.
result Dual representation of Kantorovich functional.
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.
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.
Paper proves equivalence between risk measure consistency and supermartingale property.
problem Consistency of multivariate risk measures over time.
method Proves equivalence between time consistency and supermartingale property, characterizes dual variables.
result Characterizes dual variables under which supermartingale is a 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…
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.
The paper describes how martingales can be represented after a random time in financial models.
problem Representing martingales after a random event in financial markets.
method Explicit representation of G-local martingales in terms of F-local martingales and parameters of the random time.
result Comprehensive representation of G-local martingales, complementing previous work.
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 develop a new method to value securities with uncertain default or death times.
problem Valuation of securities with uncertain default or death times in markets with additional information.
method Expansion of filtration and martingale representation theorem to handle uncertainty and risk.
result Any martingale in the large filtration stopped at a random time can be decomposed into orthogonal local martingales.
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).…
Let Q and P be equivalent probability measures and let ψ be a J-dimensional vector of random variables such that dPdQ and ψ are defined in terms of a weak solution X to a d-dimensional stochastic differential equation. Motivated by the problem of \emph{endoge…
Study BSDEs with default jump, proving properties and pricing claims.
problem Properties and pricing of BSDEs with default jumps.
method Properties and comparison theorems for BSDEs driven by Brownian motion and martingale measure with default jump.
result Representation of BSDE solutions involving conditional expectation and adjoint exponential semi-martingale.
Study dynamic risk measures and performance indices using distortion functions.
problem Investigate time consistency of dynamic risk measures and performance indices generated by distortion functions.
method Analyze dynamic coherent risk measures (DCRMs) and dynamic weighted value at risk measures, proving their equivalence. Establish properties of families of DCRMs generated by distortion functions and define corresponding dynamic coherent acceptability indices (DCAIs). Examine time consistency of DCRMs and DCAIs.
result DCRM generated by distortion functions are sub-martingale time consistent but not super-martingale time consistent and not weakly acceptance time consistent.
Develops a method for solving optimal stopping problems with multiple exercise rights.
problem Optimal stopping with multiple exercise rights under model uncertainty.
method Pathwise duality approach based on robust martingale dual representation.
result Establishes upper and lower bounds that converge to the true solution.
In this paper we consider a class of BSDEs with drivers of quadratic growth, on a stochastic basis generated by continuous local martingales. We first derive the Markov property of a forward--backward system (FBSDE) if the generating martingale is a strong Markov process. Then we establish the differentiability of a FB…
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…
The article provides representations of exchange option prices under SVJD dynamics.
problem Modeling and pricing exchange options under stochastic volatility and jumps.
method Develops representations for European and American exchange options using SVJD dynamics and equivalent martingale measures.
result Derives integro-partial differential equations and representations for exchange option prices.
Paper tackles utility maximization with job-switching and retirement constraints.
problem Maximizing utility with job-switching and retirement constraints.
method Dual-martingale approach and double obstacle problem theory.
result Characterization of optimal job-switching strategy and wealth boundaries.
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…
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.
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.
The paper models asset prices using Wiener chaos expansions for efficient calibration to implied volatility surfaces.
problem Calibrating to implied volatility surfaces using flexible martingale models.
method Constructing an over-parameterized martingale model based on Wiener chaos expansions and conditional expectations.
result The method enables fast calibration to implied volatility surfaces and demonstrates flexibility through numerical experiments.
Study shows bond market incompleteness with Lévy noise.
problem Incompleteness of forward rate based bond market model driven by Lévy noise.
method Examined the incompleteness of the market when Lévy measure has a density function. Presented theory of stochastic integration and integral representation of local martingales.
result Proven incompleteness of the bond market model under Lévy noise.
Expected signatures map data streams to lower dimensions, improving ML performance.
problem Leveraging model-free embeddings for domain-agnostic machine learning.
method Expected signatures map data streams to lower dimensions, with convergence results bridging empirical and theoretical estimators.
result A modified expected signature estimator with lower mean squared error for martingale processes.
In this article we develop an explicit formula for pricing European options when the underlying stock price follows a non-linear stochastic differential delay equation (sdde). We believe that the proposed model is sufficiently flexible to fit real market data, and is yet simple enough to allow for a closed-form represe…
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…
Study minimizes risk with partial data and wealth constraints.
problem Minimizing risk with partial data and wealth constraints.
method New approach using martingale representation and Clark-Ocone representation.
result Explicit solutions provided for special cases.