arXiv research

Unified framework for PE and TD methods in continuous time and space.

Deviation inequalities for stochastic approximation methods.

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…

Novel bounds improve TD learning consistency in RL.

Efficiently computes robust option prices using multi-marginal martingale transport.

The study examines markets with multiple numéraires and finds equivalent martingale measures.

The martingale optimal transport aims to optimally transfer a probability measure to another along the class of martingales. This problem is mainly motivated by the robust superhedging of exotic derivatives in financial mathematics, which turns out to be the corresponding Kantorovich dual. In this paper we consider the…

We present a set of high-probability inequalities that control the concentration of weighted averages of multiple (possibly uncountably many) simultaneously evolving and interdependent martingales. Our results extend the PAC-Bayesian analysis in learning theory from the i.i.d. setting to martingales opening the way for…

Researchers created a continuous Markov martingale that mimics Brownian motion but lacks the strong Markov property.

New boundary condition for Black-Scholes equations in strict local martingale models.

A constrained informationally efficient market is defined to be one whose price process arises as the outcome of some equilibrium where agents face restrictions on trade. This paper investigates the case of short sale constraints, a setting which despite its simplicity, generates new insights. In particular, it is show…

Develops European power option pricing under correlated interest rate and asset processes.

Study a continuous portfolio optimization with a new CVaR-like constraint using martingale approach.

We provide non-asymptotic convergence rates of the Polyak-Ruppert averaged stochastic gradient descent (SGD) to a normal random vector for a class of twice-differentiable test functions. A crucial intermediate step is proving a non-asymptotic martingale central limit theorem (CLT), i.e., establishing the rates of conve…

Develops a new duality between entropy martingale optimal transport and nonlinear pricing-hedging.

New algorithm selects robust martingale for optimal stopping problems.

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 $\mathcal{S}$ of significant sets, which we call Arbitrage de la classe $\mathcal{S}$. The choice of $\mathcal{S}$ reflects into the int…

Develops a martingale expansion for stochastic volatility models.

Extends martingale Schrödinger bridge to arbitrary dimensions and characterizes it.

It is generally understood that a given one-dimensional diffusion may be transformed by Cameron-Martin-Girsanov measure change into another one-dimensional diffusion with the same volatility but a different drift. But to achieve this we have to know that the change-of-measure local martingale that we write down is a tr…

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.

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

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.

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

As a crucial problem in statistics is to decide whether additional variables are needed in a regression model. We propose a new multivariate test to investigate the conditional mean independence of Y given X conditioning on some known effect Z, i.e., E(Y|X, Z) = E(Y|Z). Assuming that E(Y|Z) and Z are linearly related, …

Extends optimal transport to dynamic and martingale settings.

We discuss the difference between locally risk-minimizing and delta hedging strategies for exponential Lévy models, where delta hedging strategies in this paper are defined under the minimal martingale measure. We give firstly model-independent upper estimations for the difference. In addition we show numerical example…

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…

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.

Dual martingales improve primal optimal stopping problem efficiency.

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…

Paper develops probabilistic bounds for a stochastic gradient algorithm in non-convex problems.

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…

Paper analyzes convergence of two time-scale stochastic approximation using martingale approach.

The paper develops a method for self-normalized inference in adaptive experiments.

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…

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…

Following closely the construction of the Schrodinger bridge, we build a new class of Stochastic Volatility Models exactly calibrated to market instruments such as for example Vanillas, options on realized variance or VIX options. These models differ strongly from the well-known local stochastic volatility models, in p…

New method finds closest martingale to Brownian motion.

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…

Conic martingales refer to Brownian martingales evolving between bounds. Among other potential applications, they have been suggested for the sake of modeling conditional survival probabilities under partial information, as usual in reduced-form models. Yet, conic martingale default models have a special feature; in co…

Maximal concentration bounds for stochastic approximation with heavy-tailed noise.

Dynamic reinsurance aims to minimize surplus risk using martingale transport.

The stochastic exponential $Z_t=\exp\{M_t-M_0-(1/2) <M,M>_t\}$ 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 $M_t=\int_0^t b(Y_u)\,dW_u$ and $Y$ is a one-dimensional diffusion drive…

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 …