Information-Theoretic Matrix Inequalities and Diffusion Processes on Unimodular Lie Groups

Chirikjian, Gregory S.

doi:10.1007/978-3-030-02520-5_9

Cited by 3 publications

(12 citation statements)

References 69 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…where the square brackets reinforce that we are dealing with a matrix. This is not to be confused with the 'right' and 'left' Jacobian determinants that arise in the volume forms of the group, which would be the determinants of the matrices in (16) and (17), respectively. Note that the 'right' Jacobian has the g −1 appearing on the right whereas the 'left' Jacobian has the term on the left.…”

Section: Preliminary Definitionsmentioning

confidence: 99%

“…For non-unimodular groups, there exist distinct left-invariant and right-invariant Haar measures [35,36]. Using (16) and (17), the left and right Jacobians J L and J R of A f f + (1) expressed as matrices are,…”

Section: Option Price Evolution With Coupled Assets As a Diffusion Onmentioning

confidence: 99%

“…Diffusion equations on unimodular Lie groups, such as (24) and (28) can be solved approximately using mean and covariance propagation techniques developed in [6][7][8][9][10][11][12]16]. This technique would not be applicable directly on the affine group in the solution of Equation (34) since the group is not unimodular; instead, the later sections will show how the technique can be modified by converting (34) to a diffusion over the cotangent bundle of the affine group, which is unimodular.…”

Section: Mean and Covariance Propagation For Unimodular Lie Groupsmentioning

confidence: 99%

“…It is possible to obtain expressions for Σ(t) and µ(t) without explicitly solving for u(g, t) using the concepts of mean and covariance propagation developed in [6][7][8][9][10][11][12]16] by using the relation in (38). For X, Y, Z ∈ G, we have the Baker-Campbell-Hausdorff (BCH) formula:…”

Section: Consider a General Diffusion Equation For U(g T) On G In Tementioning

confidence: 99%

“…The benefit of reframing differential equations this way would be that variable-coefficient PDEs of N variables in R N may reduce to constant-coefficient PDEs on the Lie group G of dimension of at least N, which can be analysed using techniques developed in [6][7][8][9][10][11][12]. These applications motivated the development of mean and covariance propagation techniques to approximate the solution of diffusion equations in the special Euclidean group SE(N) and more generally, in unimodular groups [16,17]. Other numerical techniques to solve differential equations on Lie groups, through a generalisation of Euler and Runge-Kutta schemes have been developed in [18][19][20].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Jayaraman

Campolo

Chirikjian

2020

Entropy

Self Cite

View full text Add to dashboard Cite

The Black-Scholes partial differential equation (PDE) from mathematical finance has been analysed extensively and it is well known that the equation can be reduced to a heat equation on Euclidean space by a logarithmic transformation of variables. However, an alternative interpretation is proposed in this paper by reframing the PDE as evolving on a Lie group. This equation can be transformed into a diffusion process and solved using mean and covariance propagation techniques developed previously in the context of solving Fokker–Planck equations on Lie groups. An extension of the Black-Scholes theory with coupled asset dynamics produces a diffusion equation on the affine group, which is not a unimodular group. In this paper, we show that the cotangent bundle of a Lie group endowed with a semidirect product group operation, constructed in this paper for the case of groups with trivial centers, is always unimodular and considering PDEs as diffusion processes on the unimodular cotangent bundle group allows a direct application of previously developed mean and covariance propagation techniques, thereby offering an alternative means of solution of the PDEs. Ultimately these results, provided here in the context of PDEs in mathematical finance may be applied to PDEs arising in a variety of different fields and inform new methods of solution.

show abstract

Section: Preliminary Definitionsmentioning

confidence: 99%

Section: Option Price Evolution With Coupled Assets As a Diffusion Onmentioning

confidence: 99%

Section: Mean and Covariance Propagation For Unimodular Lie Groupsmentioning

confidence: 99%

Section: Consider a General Diffusion Equation For U(g T) On G In Tementioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Jayaraman

Campolo

Chirikjian

2020

Entropy

Self Cite

View full text Add to dashboard Cite

show abstract

Rate of Entropy Production in Stochastic Mechanical Systems

Chirikjian

2021

Entropy

Self Cite

View full text Add to dashboard Cite

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

show abstract

Rate of Entropy Production in Stochastic Mechanical Systems

Chirikjian

2021

Preprint

Self Cite

View full text Add to dashboard Cite

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker-Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

show abstract

Information-Theoretic Matrix Inequalities and Diffusion Processes on Unimodular Lie Groups

Cited by 3 publications

References 69 publications

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Black-Scholes Theory and Diffusion Processes on the Cotangent Bundle of the Affine Group

Rate of Entropy Production in Stochastic Mechanical Systems

Rate of Entropy Production in Stochastic Mechanical Systems

Contact Info

Product

Resources

About