Symmetries of n th-Order Approximate Stochastic Ordinary Differential Equations

Symmetries of nth-order approximate stochastic ordinary differential equations SODEs are studied. The determining equations of these SODEs are derived in an Itô calculus context. These determining equations are not stochastic in nature. SODEs are normally used tomodel nature e.g., earthquakes or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.


Introduction
The modelling power of a SODE has been applied to many diverse fields of research, from the modelling of turbulent diffusion to neuronal activity in the brain.Models such as these are often influenced by more than one Wiener process.In these models, we assume that the Wiener processes are independent of one another.As a result of this increase in the number of Wiener processes affecting the model, the form of the It ô formula is slightly different to the one used in Fredericks and Mahomed 1, 2 .The Itô formula is able to relate an arbitrary sufficiently smooth function F t, x of time and space to a particular SODE, of which it is a solution.This formula, however, needs the SODE of the spatial random process X t, ω which drives the arbitrary function F X t, ω , ω .The application of an SODE to an approximate analysis algorithm has been done by Ibragimov et al. 3 for scalar SODEs of first order.We extend this work for higher dimensions and order.We derive a similar conditioning on the temporal infinitesimal τ as had been performed by Ünal 4 and Fredericks and Mahomed 2 .We introduce operators to write the determining equations in a neater form.

Review of Wafo Soh and Mahomed [5] for nth-Order SODEs
An nth-order It ô process has the following vector form: f t, X t , Ẋ t , . . ., X n−1 t dt G t, X t , Ẋ t , . . ., X n−1 t dW t , 2.1 for k 0, 1, . . ., n − 2. Since the instantaneous mean, f, is an N-vector valued function, the index j runs from one to N, that is, j 1, . . ., N. The diffusion coefficent G is an N × Mmatrix valued function and W t is an M-dimensional standard Wiener process.From here onwards we denote {X t , Ẋ t , . . ., X n−1 t } by X n−1 t .The context of this processes is that both the instantaneous drift and diffusion coefficients are Lipschitz continuous with respect to the right norm.A good example of the type of norm used for this is given by 7 in their seventh chapter.
The Lie point transformation methodology used by Wafo Soh and Mahomed 5 does all calculations to O θ .As a result, the recoverability of the finite transformations, which keep invariance, from the infinitesimal ones is not verified.The symmetry operator H of point symmetry is where there is summation j 1, N. However since we are dealing with an nth-ordered SODE, prolongation formulation is necessary.In the Banach space, the transformation for the n − 1 th-ordered spatial derivative is where

2.6
Applying It ô's formula on an arbitrarily ordered prolongation of a spatial infinitesimal, ξ j r t, X r−1 t , gives dξ j r t, X r−1 t f ξ r j t, X r−1 t dt G l ξ r j t, X r−1 t dW l t , 2.7 where , where n ≥ 2; i r, for each j ranging from 1 to N.

2.8
If the summation operator runs from a nonnegative value, for example, 0, to a negative one; that is, −1, the outcome of the entire summation is set to zero.With this convention, we are able to recover the It ô formula for first-order SODEs.Due to the repeated index summation convention, the spatial indices i and p both run from 1 to N in the summation; the Wiener indices l and k run from 1 to M. Similarly, the It ô's formula for the temporal variable, τ x, t , gives dτ f τ X t, ω , t dt G l τ X t, ω , t dW l X t, ω , t , 2.9 where 2.10 which reduces to the total derivative, since the temporal infinitesimal is a point transformation where the total derivative is defined as

2.12
The diffusion coefficient of the temporal infinitesimal is given by

2.13
reduces to zero as well because of the fact that we are dealing with point transformations, that is, G l τ X t, ω , t 0.

2.14
The drift and diffusion coefficients of the n − 1 th-order spatial derivative are, respectively, transformed as

2.16
The It ô formula of the finite time index transformation is dt Γ e θH n−1 t dt Y l e θH n−1 t dW l , 2.17 which Wafo Soh and Mahomed 5 simply write as since the temporal infinitesimal is a point transformation.We also have that the transformed time index should keep invariance in the following probabilistic way:

2.24
Wafo Soh and Mahomed 5 make the assumption that only the system of nth order SODEs, 2.1 , remains invariant under the symmetry operator 2.4 , which implies that where we denote {X t , Ẋ t , . . ., X r−1 t } by X r−1 t for an arbitrary r ∈ N.
Expanding the drift component f j t, X n−1 t, ω dt of 2.25 using 2.15 and 2.18

2.26
In order for the finite transformations to keep invariance, we require a higher ordered θ-terms to be solely dependent on the O 1 and O θ terms, this forces the condition which is satisfied as a result of condition 2.21 .Whence the finite transformation of the Wiener process becomes dW l t, ω e θD τ /2 dW l t, ω .

2.28
The diffusion component of 2.25 can easily be expanded with the utility of 2.16 and 2.23

2.29
This allows us to make a comparison with the It ô SODE associated with the nth-ordered spatial transformation 2.24 , which furnishes the determining equations used by Wafo Soh and Mahomed 5 , that is,

2.30
Constructing the prolonged variables was carried out in 5 by using preexisting recursive relations based on the Lie point theory for ODEs, that is, for k ≤ n.The sketch of the methodology used for Lie point symmetries for nth-ordered SODEs by Wafo Soh and Mahomed 5 ends here.However, it is possible to construct the recursive relations using form invariance arguments on the SODEs described in 2.2 , that is, which after expanding yields the following θ-ordered relations: a new condition, which is not mentioned in 5 , which is automatically satisfied since the terms ξ k , where k < n − 1, are not functions of x n−1 .In conjunction with this, we have the It ô SODE associated with the transformation of the kth-ordered spatial transformation, that is, as a result of the fact that the lower ordered prolongation infinitesimals ξ k j , are not a function of x n−1 for k < n − 1 .Thus the recursive relations, defined by Wafo Soh and Mahomed 5 from an ODE context, are easily derived using a form invariance philosophy, namely

2.38
We now adapt the relations 2.30 , 2.31 , and 2.35 to an approximate SODE.

Symmetries of nth Order Multidimensional Approximate Stochastic Ordinary Differential Equations
We now consider the following: The function f is an approximate drift, which is an N vector-valued function, i 1, . . ., N. G is an N × M matrix-valued function approximating diffusion and W t is an M-dimensional Wiener process.Here f and G are defined as follows: where the repeated index r runs from 0 to R μ , where R μ is the largest positive integer such that μR μ < 2ρ and G t, x t , ẋ t , . . ., x n−1 t , R ν rν G r t, x t , ẋ t , . . ., x n−1 t , 3.3 where the repeated index r runs from 0 to R ν ; R ν is the largest positive integer such that νR ν < 2ρ.The order of accuracy to which we choose to work is ρ.
The spatial and temporal variables of our infinitesimal generator are defined as τ t, x, ρ r τ r t, x , ξ t, x, ρ r ξ r t, x .

3.5
The repeated index runs from 0 to ρ, since throughout this paper we will be working on O ρ .Using Itô's formula on the n − 1 th-prolongation of the spatial, we get which is automatically satisfied since τ is point symmetry.The repeated indices r, p, q, and l run from 0 to R ν − 1, R ν , R μ , and ρ, respectively in our repeated index summation convention; r < p.Thus, by substitution, we get

3.9
The transformation of f and G under our prolongated infinitesimal generator H β is

Operators
Thus the determining equations 2.30 , 2.31 , and 2.35 become respectively, where

3.21
Note that we cannot cancel out the terms l and l νp in 3.11 and 3.12 , respectively, in order to simplify them.These terms are a part of the summation convention implied by the repeated indices.These terms contribute to the order of error as a result of this implication.

Journal of Applied Mathematics
We now apply our generalized methodology for finding approximate symmetries to the It ô system considered in 3 .Our application should be consistent with the determining equations found in Ibragimov et al. 3 .

Example 1
For their approximate stochastic ordinary differential equations, n − 1 0, μ 1, ν 1/2, R μ 1, R ν 1, and ρ 1.Thus the diffusion coefficient G, which was taken to be constant, and the drift f appeared as follows in the It ô system: where the drift is a N × 1 vector and the constant diffusion coefficient is a matrix with dimension N × M. The determining equations are

3.23
Now since we are working to order ρ, we get the following groups of determining equations which are exactly what Ibragimov et al. 3 get which we get by comparing coefficients with no 's 3.25 which all share the same coefficient .In a similar fashion, we get the following for √ and , respectively G 1s i G 1s l ∂ 2 τ 0 ∂x i ∂x l 0.

3.27
Notice that we used 2.21 and the fact that G was constant to simplify the above.
Remark 3.1.Our application is consistent with the findings of 3 for this example.

3.28
By applying the condition 2.35 , we have that ξ ξ t, x 3.29 and thus the prolongation formula 2.34 becomes where D is the total time derivative operator.Our determining equations at 0 are

3.31
The determining equations at are 32

3.33
At 1/2 the determining equations are and the final determining equation at 3/2 is

3.37
From 3.32 , we get