On Lie Group Classification of a Scalar Stochastic Differential Equation

Abstract

Lie point symmetry group classification of a scalar stochastic differential equation (SDE) with one-dimensional Brownian motion is presented. First we prove that the admitted symmetry group is at most three-dimensional. Then the classification is carried out with the help of Lie algebra realizations by vector fields.

Random Lie-point symmetries

“…As we have mentioned, these generators were already obtained in [4], [8] and [9]. Let us now compute the random symmetries of the stochastic ODE (4.1).…”

“…Suppose f has a primitive, namely, F(t). This equation was considered by Kozlov in [8] and [9]. The author obtains its symmetries and shows that this equation represents all the scalar stochastic equations that possess a 2−dimensional Lie group of standard symmetries.…”

“…On the other hand, the connection between symmetry methods and stochastic differential equations is more recent. This connection started with Misawa ( [11], 1994) and Albeverio and Fei ( [1], 1995), and it was followed by many other authors until today ( [4], [5], [7], [8], [9], [10], [15], [17], [18]). …”

Group Quantization of Quadratic Hamiltonians in Finance

“…The classical limit → 0 corresponds to the zero volatility limit. 17 ε G equals the W Sp(2, R) cocycle in ( 16) plus the coboundary generated by (px)/2 (see section 2.4).…”

“…veloped in in physics is often based in the formal similarities between the Black-Scholes equation and the quantum mechanical Schrodinger equation. These formal similarities have been explored both from the point of view of Lie algebra invariance ( [12], [53], [36], [17] ) and global symmetries of the Black-Scholes Hamiltonian operator( [20], [23], [22], [21]).…”