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

Jayaraman, Amitesh S.; Campolo, Domenico; Chirikjian, Gregory S.

doi:10.3390/e22040455

Cited by 7 publications

(3 citation statements)

References 36 publications

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“…From the above, and the definition of conjugate momenta,

Therefore, the two phase spaces have volume elements that are related as:

The determinant of the upper-triangular block matrix in the above equation is equal to 1, illustrating the invariance:

The key to this result is how

transforms in ( 40 ). A similar result holds in the Lie group setting wherein the cotangent bundle of a Lie group can be endowed with an operation making it unimodular even when the underlying group is not [ 45 ]. This is analogous to why ( 4 ) requires the metric tensor weighting and is coordinate dependent and ( 41 ) is not.…”

Section: Classical Statistical Mechanics As Stochastic Mechanicsmentioning

confidence: 81%

Rate of Entropy Production in Stochastic Mechanical Systems

Chirikjian

2021

Entropy

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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.

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“…From the above, and the definition of conjugate momenta,

Therefore, the two phase spaces have volume elements that are related as:

The determinant of the upper-triangular block matrix in the above equation is equal to 1, illustrating the invariance:

The key to this result is how

Section: Classical Statistical Mechanics As Stochastic Mechanicsmentioning

confidence: 81%

Rate of Entropy Production in Stochastic Mechanical Systems

Chirikjian

2021

Entropy

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show abstract

“…The key to this result is how p transforms in (40). A similar result holds in the Lie group setting wherein the cotangent bundle of a Lie group can be endowed with an operation making it unimodular 2 even when the underlying group is not [48]. This is analogous to why (4) requires the metric tensor weighting and is coordinate dependent and (41) is not.…”

Section: Properties Of Phase Spacementioning

confidence: 81%

Rate of Entropy Production in Stochastic Mechanical Systems

Chirikjian

2021

Preprint

Self Cite

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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.

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“…Jayaraman, et-al. [21] who transformed this equation into a diffusion equation and solved it by using mean and covariance propagation techniques which were developed previously in the context of solving Fokker-Planck equations on Lie groups.…”

Section: Introductionmentioning

confidence: 99%