arXiv research

We present an elementary treatment of the Optional Decomposition Theorem for continuous semimartingales and general filtrations. This treatment does not assume the existence of equivalent local martingale measure(s), only that of strictly positive local martingale deflator(s).

We give an elementary proof of the celebrated Bichteler-Dellacherie Theorem which states that the class of stochastic processes $S$ allowing for a useful integration theory consists precisely of those processes which can be written in the form $S=M+A$, where $M$ is a local martingale and $A$ is a finite variation proce…

We extend Kyle's model to include stochastic liquidity and multiple assets.

Novel framework for risk-sensitive reinforcement learning using martingale decomposition.

The paper studies batch decompositions of random datasets with probabilistic similarity constraints.

In the paper, the martingales and super-martingales relative to a regular set of measures are systematically studied. The notion of local regular super-martingale relative to a set of equivalent measures is introduced and the necessary and sufficient conditions of the local regularity of it in the discrete case are fou…

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…

We analyse the structure of local martingale deflators projected on smaller filtrations. In a general continuous-path setting, we show that the local martingale part in the multiplicative Doob-Meyer decomposition of projected local martingale deflators are themselves local martingale deflators in the smaller informatio…

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…

We are concerned with a new type of supermartingale decomposition in the Max-Plus algebra, which essentially consists in expressing any supermartingale of class $(\mathcal{D})$ as a conditional expectation of some running supremum process. As an application, we show how the Max-Plus supermartingale decomposition allows…

No arbitrage in financial markets with special semimartingales.

Numerical observations on martingale couplings are confirmed under certain conditions.

The paper provides concentration inequalities for Markov chain variance estimators.

Paper defines saddle points in asymmetric Dynkin games using martingale theory.

In this paper, we study a type of reflected BSDE with a constraint and introduce a new kind of nonlinear expectation via BSDE with a constraint and prove the Doob-Meyer decomposition with respect to the super(sub)martingale introduced by this nonlinear expectation. We then apply the results to the pricing of American o…

Let $\mathbb{F}\subset \mathbb{G}$ be two filtrations and $S$ be a $\mathbb{F}$ semimartingale possessing a $\mathbb{F}$ local martingale deflator. Consider $τ$ a $\mathbb{G}$ stopping time. We study the problem whether $S^{τ-}$ or $S^τ$ can have $\mathbb{G}$ local martingale deflators. A suitable theoretical framework…

The paper studies reward concentration in MDPs, covering asymptotic and non-asymptotic settings.

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…

Paper studies pricing and hedging of nonreplicable insurance contracts using benchmark-neutral approach.

A new method uses deep learning for optimal stopping problems.

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measur…

New method for non-arbitrage pricing in risky assets.

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 …

This a free translation with additional explanations of {\em Processus à Accroissement Independants Chapitre I: La Décomposition de Paul Lévy}, by J.L. Bretagnolle, in {\em Ecole d'Eté de Probabilités}, Lecture Notes in Mathematics 307, Springer 1973. The Lévy-Khintchine representation of infinitely divisible distribut…

Given a Markovian Brownian martingale $Z$, we build a process $X$ which is a martingale in its own filtration and satisfies $X_1 = Z_1$. We call $X$ a dynamic bridge, because its terminal value $Z_1$ is not known in advance. We compute explicitly its semimartingale decomposition under both its own filtration $\cF^X$ an…

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…

New method for uncertainty analysis in TabPFN, a state-of-the-art tabular transformer.

In this paper, we introduce a numeraire-free and original probability based framework for financial markets. We reformulate or characterize fair markets, the optional decomposition theorem, superhedging, attainable claims and complete markets in terms of martingale deflators, present a recent result of Kramkov and Scha…

Empirical studies indicate the existence of long range dependence in the volatility of the underlying asset. This feature can be captured by modeling its return and volatility using functions of a stationary fractional Ornstein--Uhlenbeck (fOU) process with Hurst index $H \in (\frac{1}{2}, 1)$. In this paper, we analyz…

We consider the fundamental theorem of asset pricing (FTAP) and hedging prices of options under non-dominated model uncertainty and portfolio constrains in discrete time. We first show that no arbitrage holds if and only if there exists some family of probability measures such that any admissible portfolio value proces…

Formula for option pricing in a stochastic volatility model with jumps.

The paper develops new inequalities for Markov chain sums, linking them to mixing time.

In this work, we introduce a Monte Carlo method for the dynamic hedging of general European-type contingent claims in a multidimensional Brownian arbitrage-free market. Based on bounded variation martingale approximations for Galtchouk-Kunita-Watanabe decompositions, we propose a feasible and constructive methodology w…

Study uses G-BSDEs to decompose pricing kernels under robust G-expectation.

Signature volatility models are analyzed for existence, arbitrage, completeness, and hedging-error decomposition.

The results on the mean-variance hedging problem in Gouriéroux, Laurent and Pham (1998), Rheinländer and Schweizer (1997) and Arai (2005) are extended to discontinuous semimartingale models. When the numéraire method is used, we only assume the Radon-Nikodym derivative of the variance-optimal signed martingale measure …

New algorithm selects robust martingale for optimal stopping problems.

Develops a martingale expansion for stochastic volatility models.

Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.

Improved pricing method for illiquid assets using Lambert function.

Develops a framework for optimal investment in assets with different liquidity constraints.

Geometric Bass martingales linked to Brownian motion and geometric Brownian motion.

Existence proved for $q$-Bass martingales with specific marginals.

Study finds optimal martingale coupling between two distributions with minimal entropy.

Note on failure of Martingale Wasserstein Inequality in higher dimensions.

The paper studies projections of asset prices under equivalent martingale measures.

Extends optimal transport to dynamic and martingale settings.

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 $σ(M_t,v_t),$ with $v_t$ being a stochastic volatility term.