Counterfactual analyses with graphical models based on local independence

Counterfactual distributions are important ingredients for policy analysis and decomposition analysis in empirical economics. In this article we develop modeling and inference tools for counterfactual distributions based on regression methods. The counterfactual scenarios that we consider consist of ceteris paribus changes in either the distribution of covariates related to the outcome of interest or the conditional distribution of the outcome given covariates. For either of these scenarios we derive joint functional central limit theorems and bootstrap validity results for regression-based estimators of the status quo and counterfactual outcome distributions. These results allow us to construct simultaneous confidence sets for function-valued effects of the counterfactual changes, including the effects on the entire distribution and quantile functions of the outcome as well as on related functionals. These confidence sets can be used to test functional hypotheses such as no-effect, positive effect, or stochastic dominance. Our theory applies to general counterfactual changes and covers the main regression methods including classical, quantile, duration, and distribution regressions. We illustrate the results with an empirical application to wage decompositions using data for the United States. As a part of developing the main results, we introduce distribution regression as a comprehensive and flexible tool for modeling and estimating the \textit{entire} conditional distribution. We show that distribution regression encompasses the Cox duration regression and represents a useful alternative to quantile regression. We establish functional central limit theorems and bootstrap validity results for the empirical distribution regression process and various related functionals.

The purpose of this chapter is to overview elements of the theory of stochastic differential equations, based on Wiener processes, for use in the subsequent chapters. This theory was founded by K. Itô in 1942 (Itô, Zenkoku Shijo Sugaku Danwakai 244:1352–1400, 1942; Itô, On stochastic differential equations. Memoirs of the American Mathematical Society, vol 4. AMS, New York City, 1951). His aim was to construct Markov processes, governed by Kolmogorov’s differential equations via Wiener processes, and to analyze their sample paths. After that, stochastic differential equations have been used to describe dynamical processes in random environments of various fields. Here we consider stochastic differential equations with random coefficients, because we aim at studying stochastic control problems.The chapter is organized as follows. Section 1.1 is preliminaries. The basic definitions and results on stochastic processes are collected for later use. Stochastic differential equations and stochastic analysis will be introduced in Sect. 1.2. Section 1.3 deals with asset pricing problems as an application of previous results.KeywordsBackward Stochastic Differential EquationsAsset Pricing ProblemsConstructing Markov ProcessesRandom CoefficientsBurkholder Davis Gundy InequalityThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preface Picture of Raphael Hoegh-Krohn Bibliography of Raphael Hoegh-Krohn 1. On the scientific work of Raphael Hoegh-Krohn Part I. Stochastic Analysis 2. Some Euclidean integer-valued random fields with Markov properties 3. A Beurling-Deny type structure theorem for Dirichlet forms on general state spaces 4. Log-concavity of radial Schroedinger wave functions and convergence of planetesimal diffusions 5. Dirichlet forms, diffusion processes and spectral dimensions for nested fractals 6. Large deviations for weak solutions of stochastic differential equations 7. On a class of probabilistic integrodifferential equations 8. Operator integrals and martingale integrals with general parameter 9. Wick multiplication and Ito-Skorohod stochastic differential equations 10. Multiscale analysis in random dynamics 11. A limiting distribution connected with fractional parts of linear forms 12. Rapport sur les representations chaotiques 13. Markov properties of solutions of stochastic partial differential equations in a finite volume 14. On stochastic evolution equations with non-homogeneous boundary conditions 15. Grey noise 16. Existence of invariant measures for diffusion process with infinite dimensional state space Part II. Infinite Dimensional Groups 17. On nonlinear equations associated with Lie algebras of diffeomorphism groups of two-dimensional manifolds 18. Fields of left-invariant standard Brownian motion processes on a smooth bundle of compact-simple Lie groups 19. Unitary highest weight representations of gauge groups Part III. Operator Algebras 20. Some non-commutative orbifolds 21. Ergodic actions of non-abelian compact groups 22. Positive projections onto Jordan algebras and their enveloping von Neumann algebras Part IV. Nonlinear Analysis and Applications 23. How many singularities can there be in an energy minimizing map from the ball to the sphere? 24. Front tracking for petroleum reservoirs 25. Quasi-periodic, finite-gap solutions of the modified Korteweg-de Vries equations 26. A new representation of soliton solutions of the Kadomtsev-Petviashvili equation 27. On scalar conservation laws in one dimension.

This article is devoted to the existence of strong solutions to stochastic differential equations (SDEs). Compared with Ito's theory, we relax the assumptions on the volatility term and replace the global Lipschitz continuity condition with a local Lipschitz continuity condition and a Hoelder continuity condition. In particular, our general SDE covers the Cox–Ingersoll–Ross SDE as a special case. We note that the general weak existence theory presumably extends to our general SDE (although the explicit time dependence of the drift term and the volatility term might require some extra considerations). However, avoiding weak existence theory we prove the existence of a strong solution directly using a priori estimates (the so-called energy estimates) derived from the SDE. The benefit of this approach is that the argument only requires some basic knowledge about stochastic and functional analysis. Moreover, the underlying principle has developed to become one of the cornerstones of the modern theory of partial differential equations (PDEs). In this sense, the general goal of this article is not just to establish the existence of a strong solution to the SDE under consideration but rather to introduce a new principle in the context of SDEs that has already proven to be successful in the context of PDEs.

Stochastic differential equations (SDEs) are an important class of mathematical models for areas such as physics and finance. Usually the model outputs are in the form of statistics of the dependent variables, generated from many solutions of the SDE using different samples of the random variables. Challenges in using existing conventional digital computer architectures for solving SDEs include: rapidly generating the random input variables for the SDE solutions, and having to use numerical integration to solve the differential equations. Recent work by our group has explored using hybrid analog-digital computing to solve differential equations. In the hybrid computing model, we solve differential equations by encoding variables as continuous values, which evolve in continuous time. In this paper we review the prior work, and study using the architecture, in conjunction with analog noise, to solve a canonical SDE, the Black-Scholes SDE.

Digital currency markets exhibit extreme volatility, heavy tails, and bursty order-flow that complicate surveillance and heighten manipulation risk. This systematic review synthesizes the state of the art in stochastic differential equation (SDE)–based approaches for detection and monitoring of abusive practices in cryptocurrencies. Following PRISMA 2020 and PICOS, an Elsevier-indexed search (2015–2025) yielded 273 records; after screening and eligibility assessment, 20 primary studies (2018–2025) were included. The literature clusters into six methodological families: (i) stochastic volatility and jump–diffusion models for heavy-tailed returns and volatility smiles; (ii) Hawkes and related point-process models for clustered order arrivals, spoofing, pump-and-dump, and wash trading; (iii) Markov and regime-switching diffusions that delineate latent “fair” versus manipulated regimes; (iv) hybrid SDE–machine learning frameworks for high-frequency prediction; (v) change-point and sequential detection methods grounded in likelihood ratios and optimal stopping; and (vi) meta-studies consolidating performance trends. Across studies, Hawkes-type intensities consistently outperform Poisson and threshold baselines in event detection; regime-switching models align with known market breaks; and hybrid neural–SDE systems achieve the strongest forecasting but at reduced interpretability. We formalize a taxonomy linking model structure to surveillance objective, and we delineate a practical trade-off between transparency (classical SDEs) and predictive accuracy (hybrid models). The review highlights open needs in explainable hybrid designs, reproducible datasets, and real-time deploy ability for exchanges and regulators. By connecting applied probability, financial engineering, and control, the paper clarifies how SDE frameworks can underpin robust, auditable market-integrity tools for digital assets.

Stochastic differential equations (SDEs), which models uncertain phenomena as the time evolution of random variables, are exploited in various fields of natural and social sciences such as finance. Since SDEs rarely admit analytical solutions and must usually be solved numerically with huge classical-computational resources in practical applications, there is strong motivation to use quantum computation to accelerate the calculation. Here, we propose a quantum-classical hybrid algorithm that solves SDEs based on variational quantum simulation (VQS). We first approximate the target SDE by a trinomial tree structure with discretization and then formulate it as the time-evolution of a quantum state embedding the probability distributions of the SDE variables. We embed the probability distribution directly in the amplitudes of the quantum state while the previous studies did the square-root of the probability distribution in the amplitudes. Our embedding enables us to construct simple quantum circuits that simulate the time-evolution of the state for general SDEs. We also develop a scheme to compute the expectation values of the SDE variables and discuss whether our scheme can achieve quantum speed-up for the expectation-value evaluations of the SDE variables. Finally, we numerically validate our algorithm by simulating several types of stochastic processes. Our proposal provides a new direction for simulating SDEs on quantum computers.

Solving stochastic differential equations through genetic programming and automatic differentiation

Stochastic differential equations are a flexible way to model continuous probability distributions. The most popular differential equations are for non-stationary Lognormal, non-stationary Normal and stationary Ornstein-Uhlenbeck distributions. The probability densities are known for these distributions and the assumptions behind the differential equations are well understood. Unfortunately, the assumptions do not fit most situations. In economics and finance, prices and quantities are usually stationary and positive. The Lognormal and Normal distributions are nonstationary and the Normal and Ornstein-Uhlenbeck distributions allow negative prices and quantities. This study derives a stochastic differential equation that includes most of the classical probability distributions as special cases and greatly expands the number distributions that can be used in models of stochastic dynamic systems.

This paper focuses on a class of stochastic differential equations with mixed delay based on Lyapunov stability theory, Itô formula, stochastic analysis, and inequality technique. A sufficient condition for existence and uniqueness of the adapted solution to such systems is established by employing fixed point theorem. Some sufficient conditions of exponential stability and corollaries for such systems are obtained by using Lyapunov function. By utilizing Doob’s martingale inequality and Borel-Cantelli lemma, it is shown that the exponentially stable in the mean square of such systems implies the almost surely exponentially stable. In particular, our theoretical results show that if stochastic differential equation is exponentially stable and the time delay is sufficiently small, then the corresponding stochastic differential equation with mixed delay will remain exponentially stable. Moreover, time delay upper limit is solved by using our theoretical results when the system is exponentially stable, and they are more easily verified and applied in practice.

The theory of stochasic integrals and stochastic differential equations was established by K Ito [3, 4] (also see [2]). In past four decade years, Ito’s stochastic analysis has established for itself the central role in modern probability theory. Ito’s theory of stochastic differential equations has been one of the most important tools. However, Ito’s construction of stochastic integrals over Brownian motion possesses an essentially random characterization, and is meaningless for a single Brownian path. The Ito map obtained by solving Ito’s stochastic differential equations is nowhere continuous on the Wiener space.

In [Stochastic Analysis: 331–346, 1991, Annals of Probability 26: 267–307, 1998] Kurtz and Protter resp. Jacod and Protter specify the asymptotic error distribution of the Euler method for stochastic differential equations (SDEs) with smooth coefficients growing at most linearly. The required differentiability and linear growth of the coefficients rule out some popular SDEs as for instance the Cox–Ingersoll–Ross (CIR) model, the Heston model, or the stochastic Brusselator. In this article, we partially extend one of the fundamental results in [Jacod and Protter, Annals of Probability 26: 267–307, 1998], so that also the mentioned examples are covered. Moreover, we compare by means of simulations the asymptotic error distributions of the CIR model and the geometric Brownian motion with mean reversion.

Modeling real-time balancing power demands in wind power systems using stochastic differential equations

Likelihood ratios are used in computer simulation to estimate expectations with respect to one law from simulation of another. This importance sampling technique can be implemented with either the likelihood ratio at the end of the simulated time horizon or with a sequence of likelihood ratios at intermediate times. Since a likelihood ratio process is a martingale, the intermediate values are conditional expectations of the final value and their use introduces no bias. We provide conditions under which using conditional expectations in this way brings guaranteed variance reduction. We use stochastic orderings to get positive dependence between a process and its likelihood ratio, from which variance reduction follows. Our analysis supports the following rough statement: for increasing functionals of associated processes with monotone likelihood ratio, conditioning helps. Examples are drawn from recursively defined processes, Markov chains in discrete and continuous time, and processes with Poisson input.

This thesis is centered on the theory of stochastic processes and their applications in biological systems characterized by a complex environment. Three case studies have been modeled by the use of the three fundamental tools of stochastic processes: the master equation (ME), the stochastic differential equation (SDE) and the partial differential equation (PDE). The principal approach here applied to deal with complexity is the characterization of the system by means of probability distributions describing each a parameter of the model or the introduction of fractional order derivatives to include non-local and memory effects maintaining the linearity in the equations. In Chapter 1 we briefly review the theory of stochastic processes. In Chapter 2 we derive a birth-death process master equation to test if Long Interspersed Elements (LINEs) can be modeled according to the neutral theory of biodiversity. In Chapter 3 we derive a model of anomalous diffusion based on a Langevin approach in which anomalous behavior arise in the asymptotic intermediate state as a consequence of the heterogeneity of the system, from the superposition of Ornstein-Uhlenback processes. In Chapter 4 we propose an extension of the cable equation, used to describe anomalous diffusion phenomena as the signal conduction in spiny dendrites, by introducing a Caputo time fractional derivative.