scholarly journals VARIETIES OF SIGNATURE TENSORS

2019 ◽  
Vol 7 ◽  
Author(s):  
CARLOS AMÉNDOLA ◽  
PETER FRIZ ◽  
BERND STURMFELS
Keyword(s):  
Brownian Motion ◽  
Algebraic Geometry ◽  
Lie Groups ◽  
Stochastic Analysis ◽  
Piecewise Linear ◽  
Nilpotent Lie Groups ◽  
Rough Paths ◽  
Iterated Integrals ◽  
Parametric Curve

The signature of a parametric curve is a sequence of tensors whose entries are iterated integrals. This construction is central to the theory of rough paths in stochastic analysis. It is examined here through the lens of algebraic geometry. We introduce varieties of signature tensors for both deterministic paths and random paths. For the former, we focus on piecewise linear paths, on polynomial paths, and on varieties derived from free nilpotent Lie groups. For the latter, we focus on Brownian motion and its mixtures.

scholarly journals Random transformations and invariance of semimartingales on Lie groups

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Sergio Albeverio ◽  
Francesco C. De Vecchi ◽  
Paola Morando ◽  
Stefania Ugolini
Keyword(s):  
Brownian Motion ◽  
Lie Groups ◽  
Stochastic Analysis ◽  
Invariance Properties ◽  
Explicit Condition ◽  
Stochastic Characteristics ◽  
Necessary And Sufficient

Abstract Invariance properties of semimartingales on Lie groups under a family of random transformations are defined and investigated, generalizing the random rotations of the Brownian motion. A necessary and sufficient explicit condition characterizing semimartingales with this kind of invariance is given in terms of their stochastic characteristics. Non-trivial examples of symmetric semimartingales are provided and applications of this concept to stochastic analysis are discussed.

scholarly journals Sampling and interpolation on some nilpotent Lie groups

2016 ◽  
Vol 28 (2) ◽  
Author(s):  
Vignon Oussa
Keyword(s):  
Lie Groups ◽  
Nilpotent Lie Groups

AbstractLet

Approximations of boundary crossing probabilities for a Brownian motion

1999 ◽  
Vol 36 (4) ◽  
pp. 1019-1030 ◽  
Author(s):  
Alex Novikov ◽  
Volf Frishling ◽  
Nino Kordzakhia
Keyword(s):  
Brownian Motion ◽  
Power Series ◽  
Numerical Integration ◽  
Piecewise Linear ◽  
Boundary Crossing ◽  
Square Root ◽  
Girsanov Transformation ◽  
Piecewise Linear Approximations ◽  
Infinite Power ◽  
Crossing Probabilities

Using the Girsanov transformation we derive estimates for the accuracy of piecewise approximations for one-sided and two-sided boundary crossing probabilities. We demonstrate that piecewise linear approximations can be calculated using repeated numerical integration. As an illustrative example we consider the case of one-sided and two-sided square-root boundaries for which we also present analytical representations in a form of infinite power series.

scholarly journals An invariant for unitary representations of nilpotent Lie groups.

1987 ◽  
Vol 34 (1) ◽  
pp. 23-30 ◽  
Author(s):  
C. Benson ◽  
G. Ratcliff
Keyword(s):  
Lie Groups ◽  
Unitary Representations ◽  
Nilpotent Lie Groups
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