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.
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.
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.
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.
We consider a market model where there are two levels of information. The public information generated by the financial assets, and a larger flow of information that contains additional knowledge about a random time. This random time can represent many economic and financial settings, such as the default time of a firm…
We study the strong predictable representation property in filtrations initially enlarged with a random variable L. We prove that the strong predictable representation property can always be transferred to the enlarged filtration as long as the classical density hypothesis of Jacod (1985) holds. This generalizes the ex…
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.
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.
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.
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…
Let ψ be a multi-dimensional random variable. We show that the set of probability measures Q such that the Q-martingale StQ=EQ[ψ∣Ft] has the Martingale Representation Property (MRP) is either empty or dense in L∞-…
The problem of completeness of the forward rate based bond market model driven by a Lévy process under the physical measure is examined. The incompleteness of market in the case when the Lévy measure has a density function is shown. The required elements of the theory of stochastic integration over the compensated jump…
A machine learning model manages portfolio risk in high dimensions.
problem Managing risk in high-dimensional financial portfolios.
method A supervised learning approach using replicating martingales and polynomial/neural network bases.
result The model outperforms naive Monte Carlo and least-squares Monte Carlo methods.
Study optimal transport with backward martingale constraints in financial markets.
problem Optimal transport in financial markets with insider trading constraints.
method Maximal monotone set and minimal cost approach.
result Sharp conditions for uniqueness and representation of optimal transport plans.
We investigate aspects of semimartingale decompositions, approximation and the martingale representation for multidimensional correlated Markov processes. A new interpretation of the dependence among processes is given using the martingale approach. We show that it is possible to represent, in both continuous and discr…
In this paper we study mean-variance hedging under the G-expectation framework. Our analysis is carried out by exploiting the G-martingale representation theorem and the related probabilistic tools, in a contin- uous financial market with two assets, where the discounted risky one is modeled as a symmetric G-martingale…
Study derives new equation for reserves in non-monotone information scenarios.
problem Modeling reserves in situations where information is not always increasing.
method Infinitesimal approach to derive generalized stochastic Thiele equation.
result New equation allows for information discarding and solves open problems.
The dual representation of the martingale optimal transport problem in the Skorokhod space of multi dimensional cadlag processes is proved. The dual is a minimization problem with constraints involving stochastic integrals and is similar to the Kantorovich dual of the standard optimal transport problem. The constraints…
New method for non-arbitrage pricing in risky assets.
problem Non-arbitrage pricing in markets with non-negative risky assets.
method Constructing martingale measures and proving optional decomposition theorem.
result Deriving fair prices for European option contracts.
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…
Extends Clark-Ocone theorem to non-Malliavin differentiable random variables using Ito's formula.
problem Extending Clark-Ocone theorem to non-Malliavin differentiable random variables.
method Uses Ito's formula instead of Malliavin calculus.
result Explicit representation of locally risk-minimizing strategies for digital options in Levy models.
We consider dynamic sublinear expectations (i.e., time-consistent coherent risk measures) whose scenario sets consist of singular measures corresponding to a general form of volatility uncertainty. We derive a càdlàg nonlinear martingale which is also the value process of a superhedging problem. The superhedging strate…
DeepMartingale uses deep learning to solve complex optimal stopping problems efficiently.
problem Optimal stopping problems in high-dimensional continuous-time models.
method Leverages martingale representation and deep learning to directly optimize over parameterized martingales.
result DeepMartingale can approximate the true value function to any desired accuracy with neural networks of manageable size.
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.
Recurrent neural networks' hidden state can be reconstructed from its past, providing a theoretical framework for stability and tracking.
problem Hidden-state stability in RNNs
method Backward coherence analysis
result Almost-sure convergence, rates under mixing, interpretable limiting representation, finite pathwise stopping times, and theoretical framework for time-uniform confidence sequences.
We consider filtration consistent nonlinear expectations in probability spaces satisfying only the usual conditions and separability. Under a domination assumption, we demonstrate that these nonlinear expectations can be expressed as the solutions to Backward Stochastic Differential Equations with Lipschitz continuous …
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.
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…
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…
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.
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.
Paper analyzes mortality risk minimization with and without securitization.
problem Risk minimization in equity-linked mortality contracts with arbitrary death time.
method Optional martingale representation and enlarged filtration to consider death uncertainty.
result Quantifies the effect of mortality uncertainty on risk-minimizing strategies.
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.
The paper studies martingales and super-martingales under a convex set of measures.
problem Understanding martingales and super-martingales in a convex set of equivalent measures.
method Introduced local regular super-martingales and proved necessary and sufficient conditions for their regularity.
result Generalized Doob's decomposition theorem for super-martingales under a convex set of measures.
Simulates risk-neutral markets using neural spline flows.
problem Creating realistic risk-neutral market simulations.
method Developed a low-dimensional martingale representation and used neural spline flows for sampling.
result The calibrated simulator is closest to historical data with respect to Kullback-Leibler divergence.
We apply a quadratic hedging scheme developed by Foellmer, Schweizer, and Sondermann to European contingent products whose underlying asset is modeled using a GARCH process and show that local risk-minimizing strategies with respect to the physical measure do exist, even though an associated minimal martingale measure …
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.
We study the properties of nonlinear Backward Stochastic Differential Equations (BSDEs) driven by a Brownian motion and a martingale measure associated with a default jump with intensity process (λt). We give a priori estimates for these equations and prove comparison and strict comparison theorems. These results ar…
Study shows conditions for local martingales in SDEs with stochastic volatility.
problem Conditions for local martingales in stochastic differential equations with stochastic volatility.
method Examine sufficient conditions for components of SDEs to be strict local martingales or martingales.
result Components of SDEs can be strict local martingales or martingales under certain conditions.
We study a continuous-time financial market with continuous price processes under model uncertainty, modeled via a family P of possible physical measures. A robust notion NA1(P) of no-arbitrage of the first kind is introduced; it postulates that a nonnegative, nonvanishing claim cannot …
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.
The equivalence between multiportfolio time consistency of a dynamic multivariate risk measure and a supermartingale property is proven. Furthermore, the dual variables under which this set-valued supermartingale is a martingale are characterized as the worst-case dual variables in the dual representation of the risk m…
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.
This paper optimizes credit portfolios considering contagion risk and partial information.
problem Optimizing credit portfolios in a market with contagion risk and partial information.
method Formulated a stochastic control problem under partial observations, connected to a quadratic BSDE with jumps.
result Existence and uniqueness of solution to the BSDE, leading to optimization results.
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.