Girsanov Transformation Research Articles

The Onsager-Machlup Action Functional for Degenerate McKean-Vlasov Stochastic Differential Equations

The purpose of this paper is to investigate the existence of the Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations. To this end, we first derive Onsager-Machlup action functional for degenerate McKean-Vlasov stochastic differential equations with constant diffusion in a broad set of norms by Girsanov transformation, some conditioned exponential inequalities and It $\mathrm{\hat{o}}$ formulas for distribution dependent functional. Then an example is given to illustrate our results.

The Dynamics of the Hubbard Model Through Stochastic Calculus and Girsanov Transformation

As a typical quantum many body problem, we consider the time evolution of density matrix elements in the Bose-Hubbard model. For an arbitrary initial state, these quantities can be obtained from an SDE or stochastic differential equation system. To this SDE system, a Girsanov transformation can be applied. This has the effect that all the information from the initial state moves into the drift part, into the mean field part, of the transformed system. In the large N limit with g=UN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g=UN$$\\end{document} fixed, the diffusive part of the transformed system vanishes and as a result, the exact quantum dynamics is given by an ODE system which turns out to be the time dependent discrete Gross Pitaevskii equation. For the two site Bose-Hubbard model, the GP equation reduces to the mathematical pendulum and the difference of expected number of particles at the two lattice sites is equal to the velocity of that pendulum which is either oscillatory or it can have rollovers which then corresponds to the self trapping or insulating phase. As a by-product, we also find an equivalence of the mathematical pendulum with a quartic double well potential. Collapse and revivals are a more subtle phenomenom, in order to see these the diffusive part of the SDE system or quantum corrections have to be taken into account. This can be done with an approximation and collapse and revivals can be reproduced, numerically and also through an analytic calculation. Since expectation values of Fresnel or Wiener diffusion processes, we write the density matrix elements exactly in this way, can be obtained from parabolic second order PDEs, we also obtain various exact PDE representations. The paper has been written with the goal to come up with an efficient calculation scheme for quantum many body systems and as such the formalism is generic and applies to arbitrary dimension, arbitrary hopping matrices and, with suitable adjustments, to fermionic models.

Fluctuation theorem as a special case of Girsanov theorem

Stochastic thermodynamics is an important development in the direction of finding general thermodynamic principles for non-equilibrium systems. We believe stochastic thermodynamics has the potential to benefit from the measure-theoretic framework of stochastic differential equations (SDEs). Toward this, in this work, we show that fluctuation theorem (FT) is a special case of the Girsanov theorem, which is an important result in the theory of SDEs. We report that by employing Girsanov transformation of measures between the forward and the reversed dynamics of a general class of Langevin dynamic systems, we arrive at the integral fluctuation relation. Following the same approach, we derive the FT also for the overdamped case. Our derivation is applicable to both transient and steady state conditions and can also incorporate diffusion coefficients varying as a function of state and time, e.g. in the context of multiplicative noise. We expect that the proposed method will be an easy route towards deriving the FT irrespective of the complexity and non-linearity of the system.

Stochastic evolutionary p-Laplace equation: Large deviation principles and transportation cost inequality

We establish a large deviation principle for the strong solution of the evolutionary p-Laplace equation driven by small multiplicative Brownian noise, where the weak convergence approach plays a key role. Moreover, by using the Girsanov transformation along with the L1-contraction approach, we show the quadratic transportation cost inequality for the strong solution to the underlying problem.

The Onsager–Machlup action functional for degenerate stochastic differential equations in a class of norms

In this note, we show that the existence of the Onsager–Machlup action functional for degenerate stochastic differential equations in a large class of norms. The main ingredient of the proof is Girsanov transformation and some classical conditioned exponential inequalities.

Stochastic Data‐Driven Parameterization of Unresolved Eddy Effects in a Baroclinic Quasi‐Geostrophic Model

AbstractIn this work, a stochastic representation based on a physical transport principle is proposed to account for mesoscale eddy effects on the large‐scale oceanic circulation. This stochastic framework arises from a decomposition of the Lagrangian velocity into a smooth‐in‐time component and a highly oscillating noise term. One important characteristic of this random model, without any external forcing and damping, is that it conserves the total energy of the resolved flow for any realization. The proposed stochastic formulation is successfully implemented in a well established multi‐layered quasi‐geostrophic dynamical core. The empirical spatial correlation of the unresolved noise is calibrated from the eddy‐resolving simulation data. In particular, a stationary correction drift can be introduced in the noise through Girsanov transformation. This non‐intuitive term appears to be important in reproducing on a coarse mesh the eastward jet of the wind‐driven double‐gyre circulation. In addition, a projection method has been proposed to constrain the noise to act along the iso‐surfaces of the vertical stratification. The resulting noise enables us to improve the intrinsic low‐frequency variability of the large‐scale current.

Importance sampling for reliability assessment of dynamic systems under general random process excitation

This paper develops a reliability assessment method for dynamic systems subjected to a general random process excitation. Safety assessment using direct Monte Carlo simulation is computationally expensive, particularly when estimating low probabilities of failure. The Girsanov transformation-based reliability assessment method is a computationally efficient approach intended for dynamic systems driven by Gaussian white noise, and this approach can be extended to random process inputs that can be represented as transformations of Gaussian white noise. In practice, dynamic systems may be subjected to inputs that may be better modeled as non-Gaussian and/or non-stationary random processes, which are not easily transformable to Gaussian white noise. We propose a computationally efficient scheme, based on importance sampling, which can be implemented directly on a general class of random processes — both Gaussian and non-Gaussian, and stationary and non-stationary. We demonstrate that this approach is in fact equivalent to Girsanov transformation when the uncertain inputs are Gaussian white noise processes. The proposed approach is demonstrated on a linear dynamic system driven by Gaussian white noise and Brownian bridge processes, a multi-physics aero-thermo-elastic model of a flexible panel subjected to hypersonic flow, and a nonlinear building frame subjected to non-stationary non-Gaussian random process excitation.

Change of Measure Enhanced Near-Exact Euler–Maruyama Scheme for the Solution to Nonlinear Stochastic Dynamical Systems

The present study utilizes the Girsanov transformation based framework for solving a nonlinear stochastic dynamical system in an efficient way in comparison to other available approximate methods. In this approach, a rejection sampling is formulated to evaluate the Radon-Nikodym derivative arising from the change of measure due to Girsanov transformation. The rejection sampling is applied on the Euler Maruyama approximated sample paths which draw exact paths independent of the diffusion dynamics of the underlying dynamical system. The efficacy of the proposed framework is ensured using more accurate numerical as well as exact nonlinear methods. Finally, nonlinear applied test problems are considered to confirm the theoretical results. The test problems demonstrates that the proposed exact formulation of the Euler-Maruyama provides an almost exact approximation to both the displacement and velocity states of a second order non-linear dynamical system.

On path independence of Girsanov transformation for stochastic differential equations

In this paper, we introduce results obtained over the past ten years on the path-independenceof the Girsanov transformation for stochastic differential equations, which was originated in 1990s from an important market efficiency property in mathematical finance studies. We end up the paper with a new result characterizing the solution of stochastic differential equations using nonlinear partial differential equations.

Supports for degenerate stochastic differential equations with jumps and applications

In the paper, we are concerned with degenerate stochastic differential equations with jumps. We first establish two theorems about supports for the solution laws of the degenerate stochastic differential equations, under different (sufficient) conditions. We then apply one of our results to a class of degenerate stochastic evolution equations (that is, stochastic differential equations in infinite dimensions) with jumps to obtain a characterisation of path-independence for the densities of their Girsanov transformations.

Transition pathways for a class of high dimensional stochastic dynamical systems with Lévy noise.

This work is devoted to deriving the Onsager-Machlup action functional for a class of stochastic differential equations with (non-Gaussian) Lévy process as well as Brownian motion in high dimensions. This is achieved by applying the Girsanov transformation for probability measures and then by a path representation. The Poincaré lemma is essential to handle such a path representation problem in high dimensions. We provide a sufficient condition on the vector field such that this path representation holds in high dimensions. Moreover, this Onsager-Machlup action functional may be considered as the integral of a Lagrangian. Finally, by a variational principle, we investigate the most probable transition pathways analytically and numerically.

Weak existence and uniqueness for McKean–Vlasov SDEs with common noise

This paper concerns the McKean–Vlasov stochastic differential equation (SDE) with common noise. An appropriate definition of a weak solution to such an equation is developed. The importance of the notion of compatibility in this definition is highlighted by a demonstration of its role in connecting weak solutions to McKean–Vlasov SDEs with common noise and solutions to corresponding stochastic partial differential equations (SPDEs). By keeping track of the dependence structure between all components in a sequence of approximating processes, a compactness argument is employed to prove the existence of a weak solution assuming boundedness and joint continuity of the coefficients (allowing for degenerate diffusions). Weak uniqueness is established when the private (idiosyncratic) noise’s diffusion coefficient is nondegenerate and the drift is regular in the total variation distance. This seems sharp when one considers using finite-dimensional noise to regularise an infinite dimensional problem. The proof relies on a suitably tailored cost function in the Monge–Kantorovich problem and representation of weak solutions via Girsanov transformations.

A BSDE approach to stochastic linear quadratic control problem

AbstractIn this article, we study a kind of linear quadratic optimal control problem driven by forward–backward stochastic differential equations (FBSDEs in short) with deterministic coefficients. The cost functional is defined by the solution of FBSDEs. By means of the Girsanov transformation, the original issue is turned equivalently into the classical LQ problem. By functional analysis approach, some necessary and sufficient conditions for the existence of optimal controls have been obtained. Moreover, we investigate the relationship between two groups of first‐order and second‐order adjoint equations. A new stochastic Riccati equation is derived, which leads to the state feedback form of optimal control. By introducing a new Hamiltonian function with an exponential factor, we establish the stochastic maximum principle to deal with the stochastic linear quadratic problem for forward–backward stochastic system with nonconvex control domain using first‐order adjoint equation. An illustrative example is given as well.

On the validity of the Girsanov transformation method for sensitivity analysis of stochastic chemical reaction networks

We investigate the validity of the Girsanov Transformation (GT) method for parametric sensitivity analysis of stochastic models of chemical reaction networks. The validity depends on the likelihood ratio process being a martingale and the commutation of a certain derivative with expectation. We derive some exponential integrability conditions which imply both these requirements. We provide further conditions in terms of a reaction network that imply these exponential integrability conditions.

Selfsimilarity of diffusions’ first passage times

Considering a general diffusion process that runs over the non-negative half-line, this paper addresses the first-passage time (FPT) to the origin: the time it takes the process to get from an arbitrary fixed positive level to the level zero. Inspired by the special features of Brownian motion, three types of FPT selfsimilarity are introduced: (i) stochastic, which holds in ‘real space’; (ii) Laplace, which holds in ‘Laplace space’; and (iii) joint, which is the combination of the stochastic and Laplace types. Analysis establishes that the three types of FPT selfsimilarity yield, respectively and universally, the following FPT distributions: inverse-gamma; inverse-Gauss; and Levy–Smirnov. Moreover, the analysis explicitly pinpoints the classes of diffusion processes that produce the three types of selfsimilar FPTs. Shifting from general diffusion dynamics to Langevin dynamics, it is shown that the three classes collapse, respectively, to the following specific processes: diffusion in a logarithmic potential; Brownian motion with drift; and Brownian motion. Also, the effect of the Girsanov transformation on the three types of selfsimilar FPTs is investigated, as well as the effect of initiating the diffusion process from its steady-state level (rather than from a fixed positive level). This paper presents a novel approach to the exploration of first-passage times.

Transportation inequalities for doubly perturbed stochastic differential equations with Markovian switching

<abstract><p>In this paper, we focus on a class of doubly perturbed stochastic differential equations with Markovian switching. Using the Girsanov transformation argument we establish the quadratic transportation inequalities for the law of the solution of those equations with Markovian switching under the $ d_2 $ metric and the uniform metric $ d_{\infty} $.</p></abstract>

On path-independent Girsanov transform

This article provides a unified and more direct approach to path-independent Girsanov transformation for (possibly time-inhomogeneous) Markov processes and their corresponding parabolic partial differential equations. The approach is applicable to a large class of Markov processes, including diffusion processes determined by stochastic differential equations, and Markov processes with discontinuous sample paths. In particular, we apply it to reflected diffusions, diffusions with jumps and a class of piecewise deterministic Markov processes that are Markov. The approach of this paper is also applicable to a class of time-inhomogeneous Markov processes that have jumps only at integer times, where the path-independent Girsanov transform have to be given by discontinuous but piecewise differentiable functions. In the last section of this paper, we illustrate this by a simplified stochastic model from insurance mathematics.

A robust Kalman–Bucy filtering problem

A generalized Kalman–Bucy model under model uncertainty and a corresponding robust problem are studied in this paper. We find that this robust problem is equivalent to an estimated problem under a sublinear operator. By Girsanov transformation and the minimax theorem, we prove that this problem can be reformulated as a classical Kalman–Bucy filtering problem under a new probability measure. The equation which governs the optimal estimator is obtained. Moreover, the optimal estimator can be decomposed into the classical optimal estimator and a term related to the model uncertainty parameter under some condition.

Space-distribution PDEs for path independent additive functionals of McKean–Vlasov SDEs

Let [Formula: see text] be the space of probability measures on [Formula: see text] with finite second moment. The path independence of additive functionals of McKean–Vlasov SDEs is characterized by PDEs on the product space [Formula: see text] equipped with the usual derivative in space variable and Lions’ derivative in distribution. These PDEs are solved by using probabilistic arguments developed from Ref. 2. As a consequence, the path independence of Girsanov transformations is identified with nonlinear PDEs on [Formula: see text] whose solutions are given by probabilistic arguments as well. In particular, the corresponding results on the Girsanov transformation killing the drift term derived earlier for the classical SDEs are recovered as special situations.

Exact simulation of first exit times for one-dimensional diffusion processes

The simulation of exit times for diffusion processes is a challenging task since it concerns many applications in different fields like mathematical finance, neuroscience, reliability… The usual procedure is to use discretization schemes which unfortunately introduce some error in the target distribution. Our aim is to present a new algorithm which simulates exactly the exit time for one-dimensional diffusions. This acceptance-rejection algorithm requires to simulate exactly the exit time of the Brownian motion on one side and the Brownian position at a given time, constrained not to have exit before, on the other side. Crucial tools in this study are the Girsanov transformation, the convergent series method for the simulation of random variables and the classical rejection sampling. The efficiency of the method is described through theoretical results and numerical examples.