arXiv subjects

Christian Bender

Publications and source records attributed to Christian Bender.

At least 19 recordsLinked to original sources

arXiv2026

Optional Stopping for Superhedging Supermartingales

Superhedging supermartingales, introduced by the authors in previous work, are non-probabilistic processes defined via subadditive outer integrals that carry a purely financial interpretation in terms of superhedging cost. Building on the Leinert-K\"onig theory of non-lattice integration, the present paper establishes several results that are classical in probability theory but whose non-probabilistic proofs require fundamentally new arguments: (i) a tower inequality for the conditional outer integral \overline{\sigma}_j applied at stopping times, reducing to equality when the integrand is conditionally integrable; (ii) three versions of Doob's optional stopping theorem, organised by the class of supermartingale and the range of the stopping times; and (iii) Dubins' upcrossing inequality in both finite- and infinite-time horizons. A key structural result, property (K)-a.e., identifies conditions under which the two superhedging operators \overline{\sigma}_j and \overline{I}_j coincide on non-negative functions, extending the scope of all preceding results to the positive operator \overline{I}_j. None of the proofs invoke classical measure-theoretic tools; in particular, (classical) integrability and measurability are not assumed. The analogues of classical stochastic results acquire a purely financial interpretation and, in this way, gain depth and generality by providing a context that is independent of any a priori probabilistic structure.

math.PR↗

arXiv2024

Limit Theorems for the Dynamical Foundation of the Fractional Brownian Motion and Related Models of Anomalous Diffusion with Random Diffusion Coefficient and Time-Dependent Random Hurst parameter

Anomalous diffusion is an established phenomenon but still a theoretical challenge in non-equilibrium statistical mechanics. Physical models are built incrementally, and the most recent and most general family is based on the fractional Brownian motion (fBm) with a random diffusion coefficient (superstatistical fBm) together with a time-dependent random Hurst parameter. We provide here a dynamical foundation for such general family of models. We consider a dynamical system describing the motion of a test-particle surrounded by $N$ Brownian particles with different masses. This dynamic is governed by underdamped Langevin equations. Physical principles of conservation of momentum and energy are met. We prove that, in the limit $N\to\infty$, the test-particle diffuses in time according to a quite general (non-Markovian) Gaussian process whose covariance function is determined by the distribution of the masses of the surround-particles. In particular, with proper choices of the distribution of the masses of the surround-particles, we obtain fBm together with a number of other special cases of interest in modelling anomalous diffusion including time-dependent anomalous exponent. Furthermore, when the ensemble heterogeneity of the surround-particles embodying the environment becomes non-uniform and joins with the individual inhomogeneity of the test-particles, we show that, in the limit $N\to\infty$, the test-particle diffuses in time according to a quite general conditionally Gaussian process that can be calibrated into a fBm with random diffusion coefficient and random time-dependent Hurst parameter. We conclude our study by reporting the generalised Kolmogorov--Fokker--Planck equations associated to these highly general processes.

math.PR↗

arXiv2024

On the grid-sampling limit SDE

In our recent work [3] we introduced the grid-sampling SDE as a proxy for modeling exploration in continuous-time reinforcement learning. In this note, we provide further motivation for the use of this SDE and discuss its wellposedness in the presence of jumps.

stat.ML↗

arXiv2024

A random measure approach to reinforcement learning in continuous time

We present a random measure approach for modeling exploration, i.e., the execution of measure-valued controls, in continuous-time reinforcement learning (RL) with controlled diffusion and jumps. First, we consider the case when sampling the randomized control in continuous time takes place on a discrete-time grid and reformulate the resulting stochastic differential equation (SDE) as an equation driven by suitable random measures. The construction of these random measures makes use of the Brownian motion and the Poisson random measure (which are the sources of noise in the original model dynamics) as well as the additional random variables, which are sampled on the grid for the control execution. Then, we prove a limit theorem for these random measures as the mesh-size of the sampling grid goes to zero, which leads to the grid-sampling limit SDE that is jointly driven by white noise random measures and a Poisson random measure. We also argue that the grid-sampling limit SDE can substitute the exploratory SDE and the sample SDE of the recent continuous-time RL literature, i.e., it can be applied for the theoretical analysis of exploratory control problems and for the derivation of learning algorithms.

cs.LG↗

arXiv2023

Entropy-Regularized Mean-Variance Portfolio Optimization with Jumps

Motivated by the trade-off between exploitation and exploration in reinforcement learning, we study a continuous-time entropy-regularized mean variance portfolio selection problem in the presence of jumps. We propose an exploratory SDE for the wealth process associated with multiple risky assets which exhibit L\'evy jumps. In contrast to the existing literature, we study the limiting behavior of the natural discrete-time formulation of the wealth process associated to a randomized control in order to derive the continuous-time dynamics. We then show that an optimal distributional control of the continuous-time entropy-regularized exploratory mean-variance problem is Gaussian. The respective optimal wealth process solves a linear SDE whose representation is explicitly obtained.

math.OC↗

arXiv2023

A segment-wise dynamic programming algorithm for BSDEs

We introduce and analyze a family of linear least-squares Monte Carlo schemes for backward SDEs, which interpolate between the one-step dynamic programming scheme of Lemor, Warin, and Gobet (Bernoulli, 2006) and the multi-step dynamic programming scheme of Gobet and Turkedjiev (Mathematics of Computation, 2016). Our algorithm approximates conditional expectations over segments of the time grid. We discuss the optimal choice of the segment length depending on the `smoothness' of the problem and show that, in typical situations, the complexity can be reduced compared to the state-of-the-art multi-step dynamic programming scheme.

math.PR↗

arXiv2022

Subordination principle and Feynman-Kac formulae for generalized time-fractional evolution equations

We consider generalized time-fractional evolution equations of the form $$u(t)=u_0+\int_0^tk(t,s)Lu(s)ds$$ with a fairly general memory kernel $k$ and an operator $L$ being the generator of a strongly continuous semigroup. In particular, $L$ may be the generator $L_0$ of a Markov process $\xi$ on some state space $Q$, or $L:=L_0+b\nabla+V$ for a suitable potential $V$ and drift $b$, or $L$ generating subordinate semigroups or Schr\"{o}dinger type groups. This class of evolution equations includes in particular time- and space- fractional heat and Schr\"odinger type equations. We show that a subordination principle holds for such evolution equations and obtain Feynman-Kac formulae for solutions of these equations with the use of different stochastic processes, such as subordinate Markov processes and randomly scaled Gaussian processes. In particular, we obtain some Feynman-Kac formulae with generalized grey Brownian motion and other related self-similar processes with stationary increments.

math.PR↗

arXiv2021

Conditional Non-Lattice Integration, Pricing and Superhedging

Closely motivated by financial considerations, we develop an integration theory which is not classical i.e. it is not necessarily associated to a measure. The base space, denoted by $\mathcal{S}$ and called a trajectory space, substitutes the set $\Omega$ in probability theory and provides a fundamental structure via conditional subsets $\mathcal{S}_{(S,j)}$ that allows the definition of conditional integrals. The setting is a natural by-product of no arbitrage assumptions that are used to model financial markets and games of chance (in a discrete infinite time framework). The constructed conditional integrals can be interpreted as required investments, at the conditioning node, for hedging an integrable function, the latter characterized a.e. and in the limit as we increase the number of portfolios used. The integral is not classical due to the fact that the original vector space of portfolio payoffs is not a vector lattice. In contrast to a classical stochastic setting, where price processes are associated to conditional expectations (with respect to risk neutral measures), we uncover a theory where prices are naturally given by conditional non-lattice integrals. One could then study analogues of classical probabilistic notions in such non-classical setting, the paper stops after defining trajectorial martingales the study of which is deferred to future work.

math.PR↗

arXiv2021

Model-Free Finance and Non-Lattice Integration

Starting solely with a set of possible prices for a traded asset $S$ (in infinite discrete time) expressed in units of a numeraire, we explain how to construct a Daniell type of integral representing prices of integrable functions depending on the asset. Such functions include the values of simple dynamic portfolios obtained by trading with $S$ and the numeraire. The space of elementary integrable functions, i.e. the said portfolio values, is not a vector lattice. It then follows that the integral is not classical, i.e. it is not associated to a measure. The essential ingredient in constructing the integral is a weak version of the no-arbitrage condition but here expressed in terms of properties of the trajectory space. We also discuss the continuity conditions imposed by Leinert (Archiv der Mathematik, 1982) and K\"onig (Mathematische Annalen, 1982) in the abstract theory of non-lattice integration from a financial point of view and establish some connections between these continuity conditions and the existence of martingale measures

q-fin.MF↗

arXiv2021

Supporting Land Reuse of Former Open Pit Mining Sites using Text Classification and Active Learning

Open pit mines left many regions worldwide inhospitable or uninhabitable. To put these regions back into use, entire stretches of land must be renaturalized. For the sustainable subsequent use or transfer to a new primary use, many contaminated sites and soil information have to be permanently managed. In most cases, this information is available in the form of expert reports in unstructured data collections or file folders, which in the best case are digitized. Due to size and complexity of the data, it is difficult for a single person to have an overview of this data in order to be able to make reliable statements. This is one of the most important obstacles to the rapid transfer of these areas to after-use. An information-based approach to this issue supports fulfilling several Sustainable Development Goals regarding environment issues, health and climate action. We use a stack of Optical Character Recognition, Text Classification, Active Learning and Geographic Information System Visualization to effectively mine and visualize this information. Subsequently, we link the extracted information to geographic coordinates and visualize them using a Geographic Information System. Active Learning plays a vital role because our dataset provides no training data. In total, we process nine categories and actively learn their representation in our dataset. We evaluate the OCR, Active Learning and Text Classification separately to report the performance of the system. Active Learning and text classification results are twofold: Whereas our categories about restrictions work sufficient ($>$.85 F1), the seven topic-oriented categories were complicated for human coders and hence the results achieved mediocre evaluation scores ($<$.70 F1).

cs.CL↗

arXiv2021

Stochastic solutions of generalized time-fractional evolution equations

We consider a general class of integro-differential evolution equations which includes the governing equation of the generalized grey Brownian motion and the time- and space-fractional heat equation. We present a general relation between the parameters of the equation and the distribution of the underlying stochastic processes, as well as discuss different classes of processes providing stochastic solutions of these equations. For a subclass of evolution equations, containing Saigo-Maeda generalized time-fractional operators, we determine the parameters of the corresponding processes explicitly. Moreover, we explain how self-similar stochastic solutions with stationary increments can be obtained via linear fractional L\'evy motion for suitable pseudo-differential operators in space.

math.PR↗

arXiv2019

`Regression Anytime' with Brute-Force SVD Truncation

We propose a new least-squares Monte Carlo algorithm for the approximation of conditional expectations in the presence of stochastic derivative weights. The algorithm can serve as a building block for solving dynamic programming equations, which arise, e.g., in non-linear option pricing problems or in probabilistic discretization schemes for fully non-linear parabolic partial differential equations. Our algorithm can be generically applied when the underlying dynamics stem from an Euler approximation to a stochastic differential equation. A built-in variance reduction ensures that the convergence in the number of samples to the true regression function takes place at an arbitrarily fast polynomial rate, if the problem under consideration is smooth enough.

math.ST↗

arXiv2018

Ito's Formula for Gaussian Processes with Stochastic Discontinuities

We introduce a Skorokhod type integral and prove an Ito formula for a wide class of Gaussian processes which may exhibit stochastic discontinuities. Our Ito formula unifies and extends the classical one for general (i.e., possibly discontinuous) Gaussian martingales in the sense of Ito integration and the one for stochastically continuous Gaussian non-martingales in the Skorokhod sense, which was first derived in Alos et al. (Ann. Probab. 29, 2001).

math.PR↗

arXiv2016

Pathwise Iteration for Backward SDEs

We introduce a novel numerical approach for a class of stochastic dynamic programs which arise as discretizations of backward stochastic differential equations or semi-linear partial differential equations. Solving such dynamic programs numerically requires the approximation of nested conditional expectations, i.e., iterated integrals of previous approximations. Our approach allows us to compute and iteratively improve upper and lower bounds on the true solution starting from an arbitrary and possibly crude input approximation. We demonstrate the benefits of our approach in a high dimensional financial application.

math.NA↗

arXiv2016

Discretizing Malliavin calculus

Suppose $B$ is a Brownian motion and $B^n$ is an approximating sequence of rescaled random walks on the same probability space converging to $B$ pointwise in probability. We provide necessary and sufficient conditions for weak and strong $L^2$-convergence of a discretized Malliavin derivative, a discrete Skorokhod integral, and discrete analogues of the Clark-Ocone derivative to their continuous counterparts. Moreover, given a sequence $(X^n)$ of random variables which admit a chaos decomposition in terms of discrete multiple Wiener integrals with respect to $B^n$, we derive necessary and sufficient conditions for strong $L^2$-convergence to a $σ(B)$-measurable random variable $X$ via convergence of the discrete chaos coefficients of $X^n$ to the continuous chaos coefficients of $X$. In the special case of binary noise, our results support the known formal analogies between Malliavin calculus on the Wiener space and Malliavin calculus on the Bernoulli space by rigorous $L^2$-convergence results.

math.PR↗

arXiv2016

A general non-existence result for linear BSDEs driven by Gaussian processes

In this paper, we study linear backward stochastic differential equations driven by a class of centered Gaussian non-martingales, including fractional Brownian motion with Hurst parameter $H\in (0,1)\setminus \{\frac12\}$. We show that, for every choice of deterministic coefficient functions, there is a square integrable terminal condition such that the equation has no solution.

math.PR↗

arXiv2015

A generalised Itō formula for Lévy-driven Volterra processes

We derive a generalised Itō formula for stochastic processes which are constructed by a convolution of a deterministic kernel with a centred Lévy process. This formula has a unifying character in the sense that it contains the classical Itō formula for Lévy processes as well as recent change-of-variable formulas for Gaussian processes such as fractional Brownian motion as special cases. Our result also covers fractional Lévy processes (with Mandelbrot-Van Ness kernel) and a wide class of related processes for which such a generalised Itō formula has not yet been available in the literature.

math.PR↗

arXiv2014

Maximal inequalities for fractional L\'evy and related processes

In this paper we study processes which are constructed by a convolution of a deterministic kernel with a martingale. A special emphasis is put on the case where the driving martingale is a centred L\'evy process, which covers the popular class of fractional L\'evy processes. As a main result we show that, under appropriate assumptions on the kernel and the martingale, the maximum process of the corresponding `convoluted martingale' is $p$-integrable and we derive maximal inequalities in terms of the kernel and of the moments of the driving martingale.

math.PR↗