# Symmetries ofnth-Order Approximate Stochastic Ordinary Differential Equations

Authors: E. Fredericks; F. M. Mahomed
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

Symmetries ofnth-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 to model nature (e.g., earthquakes) or for testing the safety and reliability of models in construction engineering when looking at the impact of random perturbations.


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## Body

## 1. 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 Itô 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.Wafo Soh and Mahomed [5] gave an algorithm to obtain Lie point symmetries for both first- and nth-order SODEs. We briefly review their work and follow up with an extension from point symmetries to generalized symmetries.The first section begins with the transformations of the spatial, temporal, and Wiener variables for annth-order Itô process. These transformations have the same properties as stated in Fredericks and Mahomed [6].Using the Itô formula in conjunction with the infinitesimal transformations which preserve form invariance, we derive conditions fornth-ordered SODEs that ensure the recovery of invariance preserving finite transformations from the infinitesimal ones. This has not been done in the past.This is followed up with the development of recursive relations needed for finding the prolonged spatial infinitesimals in a SODE context by using the concept of form invariance. This differs from the methodology used by [5], where the recursive relation defined was predefined from an ODE context. As a result we also derive a conditioning on these prolonged spatial infinitesimal variables. We further derive a conditioning on the diffusion coefficient of the temporal generalised symmetry τ, which is similar to that of Ünal [4]. We conclude the article with an introduction of operators which generalize the determining equations for any order SODE that is adaptable to both point and generalized symmetries.The symmetries of high-ordered multidimensional stochastic ordinary differential equations (SODEs) are found using form invariance arguments on both the instantaneous drift and diffusion properties of the SODEs. We then apply this work to a generalized approximation analysis algorithm. The determining equations of SODEs are derived in an Itö calculus context. We also use the random time change formula used by Wafo Soh and Mahomed [5] to transform the Wiener processes.


## 2. Review of Wafo Soh and Mahomed [5] for nth-Order SODEs

Annth-order Itô process has the following vector form:
(2.1)dX(n-1)(t)=f(t,X(t),X˙(t),…,X(n-1)(t))dt+G(t,X(t),X˙(t),…,X(n-1)(t))dW(t),(2.2)dXj(k)(t)=Xj(k+1)(t)dt,(2.3)Xj(0)(t)=Xj(t)
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×M-matrix valued function and W(t) is an M-dimensional standard Wiener process. From here onwards we denote {X(t),X˙(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
(2.4)H=τ(x,t)∂∂t+ξj(x,t)∂∂xj,
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
(2.5)x¯(n-1)=eθH[n-1]x(n-1),x¯(k)=eθH[n-1]x(k)=eθH[k]x(k),k<n-1,
where
(2.6)H[n-1]=H(n-2)+ξj[n-1]∂∂xj(n-1),n≥1,H[0]=H.
Applying Itô’s formula on an arbitrarily ordered prolongation of a spatial infinitesimal, ξj[r](t,𝒳(r-1)(t)), gives
(2.7)dξj[r](t,𝒳(r-1)(t))=f(ξ[r])j(t,𝒳(r-1)(t))dt+G(ξ[r])jl(t,𝒳(r-1)(t))dWl(t),
where
(2.8)f(ξ[r])j(t,𝒳(r-1)(t))=∂ξj[γ]∂t+fi∂ξj[γ]∂xi(n-1)+12∑k=1M‍GikGpk∂2ξj[γ]∂xi(n-1)∂xp(n-1)+∑α=0n-2‍xp(α+1)∂ξj[r]∂xp(α),wheren≥2;G(ξ[r])jl(t,𝒳(r-1)(t))=∂ξj[γ]∂xi(n-1)Gilr,foreachjrangingfrom1toN.
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 Itô 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 Itô’s formula for the temporal variable, τ(x,t), gives
(2.9)dτ=f(τ)(X(t,ω),t)dt+G(τ)l(X(t,ω),t)dWl(X(t,ω),t),
where
(2.10)f(τ)(X(t,ω),t)=∂τ∂t+fi∂τ∂xi(n-1)+12∑k=1M‍GikGpk∂2τ∂xi(n-1)∂xp(n-1)+∑α=0n-2‍xp(α+1)∂τ∂xp(α),
which reduces to the total derivative, since the temporal infinitesimal is a point transformation
(2.11)f(τ)(X(t,ω),t)=D(τ),
where the total derivative is defined as
(2.12)D=∂∂t+∑α=0n-2‍xp(α+1)∂∂xp(α).
The diffusion coefficient of the temporal infinitesimal is given by
(2.13)G(τ)l(X(t,ω),t)=Gil∂τ∂xi(n-1)
reduces to zero as well because of the fact that we are dealing with point transformations, that is,
(2.14)G(τ)l(X(t,ω),t)=0.
The drift and diffusion coefficients of the (n-1)th-order spatial derivative are, respectively, transformed as
(2.15)fj(𝒳¯(n-1)(t¯),t¯)=fj(𝒳(n-1)(t),t)+θH[n-1](fj(𝒳(n-1)(t),t))+𝒪(θ2),(2.16)Gjl(𝒳¯(n-1)(t¯),t¯)=Gjl(𝒳(n-1)(t),t)+θH[n-1](Gjl(𝒳(n-1)(t),t))+𝒪(θ2).
The Itô formula of the finite time index transformation is
(2.17)dt¯=Γ(eθH[n-1](t))dt+Yl(eθH[n-1](t))dWl,
which Wafo Soh and Mahomed [5] simply write as
(2.18)dt¯=dt(1+θD(τ))+𝒪(θ2),
since the temporal infinitesimal is a point transformation. We also have that the transformed time index should keep invariance in the following probabilistic way:
(2.19)𝔼ℚ¯[dt¯(t,ω)]=dt¯(t,ω).
This requires
(2.20)Yl(eθH[n-1](t))=0,l=1,M,
which is automatically satisfied since τ is point transformation. Condition (2.19) also forces
(2.21)D(eθH[n-1](t))=Constant,
which is overlooked in [5]. Thus the finite transformation of the Wiener process is
(2.22)dW¯l(t¯,ω)=D(eθH[n-1](t))dWl(t,ω),
which Wafo Soh and Mahomed [5] simplify as
(2.23)dW¯l(t¯,ω)=dWl(t,ω)(1+ϵ2D(τ))+𝒪(θ2),
where [5] used a generalized binomial expansion of the square-root of the derivative of the transformed time index with respect to time. The Itô SODE associated with Lie point nth-ordered spatial transformation is
(2.24)dX¯j(n-1)(t¯)=dXj(n-1)(t)+θ(f(ξ[n-1])jdt+G(ξ[n-1])jldWl(t))+𝒪(θ2).
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
(2.25)dX¯j(n-1)(t¯)=fj(t¯,𝒳¯(n-1)(t¯,ω))dt¯+Gjl(t¯,𝒳¯(n-1)(t¯,ω))dW¯l(t¯),
where we denote {X-(t¯),X˙-(t¯),…,X-(r-1)(t¯)} by 𝒳¯(r-1)(t¯) for an arbitrary r∈ℕ.Expanding the drift componentfj(t¯,𝒳¯(n-1)(t¯,ω))dt¯ of (2.25) using (2.15) and (2.18) gives
(2.26)f(t,𝒳(n-1)(t))dt¯={+∑j=0k-2‍(kk-j)Hj(f(t,𝒳(n-1)(t)))(D(Hk-j(t))-[D(τ)]k-j))f(t,𝒳(n-1)(t))+θ(D(τ)+H[n-1])f(t,𝒳(n-1)(t))+∑k=2∞‍θkk!(+∑j=0k-2‍(kk-j)Hj(f(t,𝒳(n-1)(t)))(D(Hk-j(t))-[D(τ)]k-j))(D(τ)+H[n-1])kf(t,𝒳(n-1)(t))+∑j=0k-2‍(kk-j)Hj(f(t,𝒳(n-1)(t)))×(D(Hk-j(t))-[D(τ)]k-j)+∑j=0k-2‍(kk-j)Hj(f(t,𝒳(n-1)(t)))(D(Hk-j(t))-[D(τ)]k-j)))+∑j=0k-2‍(kk-j)Hj(f(t,𝒳(n-1)(t)))(D(Hk-j(t))-[D(τ)]k-j))}dt.
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
(2.27)eθD(τ)=D(eθH[n-1](t)),
which is satisfied as a result of condition (2.21). Whence the finite transformation of the Wiener process becomes
(2.28)dW¯l(t¯,ω)=eθD(τ)/2dWl(t,ω).
The diffusion component of (2.25) can easily be expanded with the utility of (2.16) and (2.23)
(2.29)Gjl(t,𝒳(n-1)(t))dW¯l={∑k=2∞‍θkk!(D(τ)2+H[n-1])kGjl(t,𝒳(n-1)(t))+θ(D(τ)2+H[n-1])Gjl(t,𝒳(n-1)(t))+∑k=2∞‍θkk!(D(τ)2+H[n-1])kGjl(t,𝒳(n-1)(t))}dWl.
This allows us to make a comparison with the Itô 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)f(ξ[n-1])j=(D(τ)+H[n-1])fj(𝒳(n-1)(t),t),G(ξ[n-1])jl=(D(τ)2+H[n-1])Gjl(𝒳¯(n-1)(t¯),t¯).
Constructing the prolonged variables was carried out in [5] by using preexisting recursive relations based on the Lie point theory for ODEs, that is,
(2.31)ξj[k]=Dξj[k-1]-xj(k)Dτ,ξj[0]=ξj,
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,
(2.32)dX¯j(k)(t¯)=X¯(k+1)(t¯)dt¯,
which after expanding yields the following θ-ordered relations:
(2.33)dX¯j(k)(t¯)=X(k+1)dt+θ(ξj[k+1]+xj(k+1)D(τ))dt+𝒪(θ2),(2.34)Γ(ξ[k])=(Γ(τ)+H[k+1])Xi(k+1),(2.35)Gij∂ξ[k]∂xi(n-1)=0,fork<n-1
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 Itô SODE associated with the transformation of the kth-ordered spatial transformation, that is,
(2.36)dX¯j(k)(t¯)=dXj(k)(t)+θ(f(ξ[k])jdt+G(ξ[k])jldWl(t))+𝒪(ϵ2),
which reduces to
(2.37)dX¯j(k)(t¯)=dXj(k)(t)+θD(ξj[k])dt+𝒪(θ2)
as a result of the fact that the lower ordered prolongation infinitesimals ξj[k], 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)D(ξj[k])=ξj[k+1]+xj(k+1)D(τ).
We now adapt the relations (2.30), (2.31), and (2.35) to an approximate SODE.


## 3. Symmetries ofnth Order Multidimensional Approximate Stochastic Ordinary Differential Equations

We now consider the following:(3.1)dX(n-1)(t¯)=f(t,X(t),X˙(t),…,X(n-1)(t),Rμ)dt+⋯+G(t,X(t),X˙(t),…,X(n-1)(t),Rν)dW(t),dXi(k)(t)=Xi(k+1)dt,Xi(0)(t)=Xi(t)
for k=0,1,…,n-2. 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:
(3.2)f(t,x(t),x˙(t),…,x(n-1)(t),Rμ)=ϵrμfr(t,x˙(t),…,x(n-1)(t)),
where the repeated index r runs from 0 to Rμ, where Rμ is the largest positive integer such that μRμ<2ρ and
(3.3)G(t,x(t),x˙(t),…,x(n-1)(t),Rν)=ϵrνGr(t,x(t),x˙(t),…,x(n-1)(t)),
where the repeated index r runs from 0 to Rν; Rν is the largest positive integer such that νRν<2ρ. The order of accuracy to which we choose to work is ρ.The spatial and temporal variables of our infinitesimal generator(3.4)H=τ(t,x,ρ)∂∂t+ξj(t,x,ρ)∂∂xj,
are defined as
(3.5)τ(t,x,ρ)=ϵrτr(t,x),ξ(t,x,ρ)=ϵrξr(t,x).
The repeated index runs from 0 to ρ, since throughout this paper we will be working on O(ϵρ). Using Itô’s formula on the (n-1)th-prolongation of the spatial, we get
(3.6)dξj[n-1]=(∂ξj[n-1]∂t+fi∂ξj[n-1]∂xi(n-1)+12∑s=1M‍GisGks∂2ξj[n-1]∂xi(n-1)∂xk(n-1)+∑α=0n-2‍xk(α+1)∂ξj[n-1]∂xk(α))dt+⋯+∂ξj[n-1]∂xi(n-1)GikdWk(t)=(fqi∂ξjl[n-1]∂xi(n-1)ϵl+μq(12∑s=1M‍Gi0sGk0s∂2ξjl[n-1]∂xi(n-1)∂xk(n-1)+f0i∂ξjl[n-1]∂xi+∂ξjl[n-1]∂t+∑α=0n-2‍xk(α+1)∂ξjl[n-1]∂xk(α))ϵl+⋯+12∑s=1M‍GipsGkps∂2ξjl[n-1]∂xi(n-1)∂xk(n-1)ϵl+2pν+∑s=1M‍GirsGkps∂2ξjl[n-1]∂xi(n-1)∂xk(n-1)ϵl+ν(r+p)+⋯+fqi∂ξjl[n-1]∂xi(n-1)ϵl+μq)dt+⋯+Gips∂ξjl[n-1]∂xs(n-1)ϵl+νpdW(i)
and on the temporal infinitesimal
(3.7)dτ=D(τ)dt=ϵl(∂τl∂t+∑α=0n-2‍xk(α+1)∂τl∂xk(n-1))dt
with
(3.8)Gisp∂τl∂xs(n-1)ϵl+νp=0,
which is automatically satisfied since τ is point symmetry. The repeated indices r, p, q, and l run from 0 to Rν-1, Rν, Rμ, and ρ, respectively in our repeated index summation convention; r<p. Thus, by substitution, we get
(3.9)dX¯(n-1)=dX(n-1)+⋯+θ(∂ξ[n-1]∂t+fi∂ξ[n-1]∂xi(n-1)+12∑k=1M‍GikGjk∂2ξ[n-1]∂xi(n-1)∂xj(n-1)+∑α=0n-2‍xj(α+1)∂ξ[n-1]∂xj(α))dt+θGji∂ξ[n-1]∂xi(n-1)dWt(j)+𝒪(θ2)dt¯=dt+θD(τ)dt+𝒪(θ2),dW¯t¯=dWt(l)(1+θ2D(τ))+𝒪(θ2).
The transformation of f and G under our prolongated infinitesimal generator H[β] is
(3.10)fi(t¯,𝒳^(n-1))=fi(t,𝒳(n-1))+θH[β]fi(t,𝒳(n-1))+𝒪(θ2)=fi+θ(τ(t,X˙(t))∂fi∂t+ξj[β](t,𝒳(n-1))∂fi∂xj(β))+𝒪(θ2)=ϵqfiq+θξjl[β]∂fiq∂xj(β)ϵl+μq+θτl∂fiq∂tϵl+μq+𝒪(θ2)=ϵqfiq+θϵl+μq(ξjl[β]∂fiq∂xj(β)+τl∂fiq∂t)+𝒪(θ2)Gki(t¯,𝒳^(n-1))=Gki(t,𝒳(n-1))+θH[β]Gki(t,𝒳(n-1))+𝒪(θ2)=Gki+θ(τ(t,X˙(t))∂Gki∂t+ξj[β](t,𝒳(n-1))∂Gki∂xj(β))+𝒪(θ2)=ϵpGkpi+θϵνp+l(ξjl[β]∂Gkpi∂xj(β)+τl∂Gkpi∂t)+𝒪(θ2),
where {X,X˙,…,X(n-1)} is represented by 𝒳(n-1) and the transformed set {X¯,X˙¯,…,X(n-1)¯} is represented by 𝒳^(n-1). The repeated indices q, p, l, and n run from 0 to Rμ, Rν, ρ, and n-1, respectively.


### 3.1. Operators

Thus the determining equations (2.30), (2.31), and (2.35) become
ϵμq+l(fmq(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμjΨj(τl))+Hβl(fmq))+⋯-ϵl(Γ0(ξml[n-1])+ϵν(r+p)℧rp(ξml[n-1])+ϵ2νpΥp(ξml[n-1])+ϵμqΨq(ξml[n-1]))=0,(3.12)ϵl+νp(Ypn-1k(ξml[n-1])Hβl(Gkpm)+12Gkpm×(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμqΨq(τl))-Yn-1pk(ξml[n-1]))=0,(3.13)ϵl+νpYn-1pi(τl)=0,(3.14)ϵl+νpYn-1pi(ξil[k])=0,k<n-1,(3.15)ϵl(Γ0(ξml[k])+ϵν(r+p)℧rp(ξml[k])+ϵ2νpΥp(ξml[k])+ϵμqΨq(ξml[k]))=ϵl(xi(k+1)(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμjΨj(τl))+ξil[k+1]),
respectively, where
(3.16)Γ0=12∑s=1M‍Gi0sGk0s∂2∂xi(n-1)∂xk(n-1)+fi0∂∂xi(n-1)+∂∂t+⋯+∑α=0n-1-1‍xk(α+1)∂∂xk(n-1),(3.17)℧rp=∑s=1M‍GirsGkps∂2∂xi(n-1)∂xk(n-1),(0≤r<p≤Rν),(3.18)Υp=12∑s=1M‍GipsGkps∂2∂xi(n-1)∂xk(n-1),(0<p≤Rν),(3.19)Ψq=fiq∂∂xi(n-1),(0<q,j≤Rμ),(3.20)Yn-1pi=Gipk∂∂xk(n-1),(0≤p≤Rν),(3.21)Hβl=τl∂∂t+ξjl[β]∂∂xj(β),(0≤β≤n-1).
Note that we cannot cancel out the terms ϵl and ϵl+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.We now apply our generalized methodology for finding approximate symmetries to the Itô system considered in [3]. Our application should be consistent with the determining equations found in Ibragimov et al. [3].


#### 3.1.1. Example 1

For their approximate stochastic ordinary differential equations,n-1=0, μ=1, ν=1/2, Rμ=1, Rν=1, and ρ=1. Thus the diffusion coefficient G, which was taken to be constant, and the drift f appeared as follows in the Itô system:
(3.22)dx=(f0+ϵf1)dt+ϵGdWt,
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)-12∑s=1M‍Gk1sGi1sϵ(∂2ξj0∂xi∂xk+ϵ∂2ξj1∂xi∂xk)+(ξi0+ϵξi1)(∂fj0∂xi+ϵ∂fj1∂xi)-fi0(∂ξj0∂xi+ϵ∂ξj1∂xi)-ϵfi1(∂ξj0∂xi+ϵ∂ξj1∂xi)+(τ0+ϵτ1)(∂fj0∂t+ϵ∂fj1∂t)-∂ξj0∂t-ϵ∂ξj1∂t+(fj0+ϵfj1)×(12ϵ∑s=1M‍Gk1sGi1s(∂2τ0∂xi∂xk+ϵ∂2τ1∂xi∂xk)+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0,ϵ(ξi0+ϵξi1)∂Gk1j∂xi-ϵGk1i(∂ξj0∂xi+ϵ∂ξj1∂xi))+ϵ(τ0+ϵτ1)∂Gk1j∂t+12ϵGk1j×(12ϵ(∂2τ0∂xi∂xl+ϵ∂2τ1∂xi∂xl)∑s=1M‍Gi1sGl1s+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0.
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
(3.24)-∂ξj0∂t-fi0∂ξj0∂xi+ξi0∂fj0∂xi+τ0∂fj0∂t+fj0fi0∂τ0∂xi+fj0∂τ0∂t=0,
which we get by comparing coefficients with no ϵ’s
(3.25)fj0fi0∂τ1∂xi-∂ξj1∂xifi0+fj0fi1∂τ0∂xi+fj1fi0∂τ0∂xi+12∂2τ0∂xi∂xl∑s=1M‍Gi1sGl1sfj0+∂τ1∂tfj0+ξi1∂fj0∂xi+ξi0∂fj1∂xi-fi1∂ξj0∂xi-12∂2ξj0∂xi∂xl∑s=1M‍Gi1sGl1s+∂fj0∂tτ1+∂fj1∂tτ0-∂ξj1∂t+∂τ0∂tfj1=0,
which all share the same coefficient ϵ. In a similar fashion, we get the following for ϵ and ϵ, respectively
(3.26)12Gk1j(∂τ0∂t+fi0∂τ0∂xi)-Gk1i∂ξj0∂xi=0,(3.27)-Gk1i∂ξj1∂xi+12Gk1j(∂τ1∂t+fi0∂τ1∂xi)+12Gk1jfi1∂τ0∂xi+Gk1j4∑s=1M‍Gi1sGl1s∂2τ0∂xi∂xl=0.
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.


#### 3.1.2. Example 2

We consider(3.28)dX˙=-ω2Xdt+σdW+ϵXdW.
By applying the condition (2.35), we have that
(3.29)ξ=ξ(t,x)
and thus the prolongation formula (2.34) becomes
(3.30)ξ[1]=D(ξ)-x˙D(τ),
where D is the total time derivative operator. Our determining equations at ϵ0 are
(3.31)-ω2xΓ0(τ0)+H0(-ω2x)=Γ0(ξ0[1]),H0(G0)+12G0Γ0(τ0)=Y0(ξ0[1]).
The determining equations at ϵ are
(3.32)-ω2xΓ0(τ1)+H1(-ω2x)-ω2xΥ1(τ0)=Γ0(ξ1[1])+Υ1(ξ0),(3.33)H1(G0)+12G0Γ0(τ1)=Y0(ξ1[1]).
At ϵ1/2 the determining equations are
(3.34)H0(G1)+12G1Γ0(τ0)+12G0℧10(τ0)=Y1(ξ0[1])
and the final determining equation at ϵ3/2 is
(3.35)12G0℧12(τ0)+12G1Γ0(τ1)=Y1(ξ1[1]).
Equation (3.31) for the infinitesimals at the zeroth echelon, that is, τ0 and ξ0, have been solved earlier in the oscillating-spring mass example
(3.36)τ0=C0,ξ0=C1cos(ωt)+C2sin(ωt).
Whence, (3.34) and (3.35) force
(3.37)ξ0=0,τ1=C3.
From (3.32), we get
(3.38)-ω2ξ1=D2(ξ1),
which solves as
(3.39)ξ1=C4cos(ωt)+C5sin(ωt).
Therefore we have
(3.40)ξ=ϵ(C4cos(ωt)+C5sin(ωt)),τ=C0+ϵC3.


## 3.1. Operators

Thus the determining equations (2.30), (2.31), and (2.35) become
ϵμq+l(fmq(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμjΨj(τl))+Hβl(fmq))+⋯-ϵl(Γ0(ξml[n-1])+ϵν(r+p)℧rp(ξml[n-1])+ϵ2νpΥp(ξml[n-1])+ϵμqΨq(ξml[n-1]))=0,(3.12)ϵl+νp(Ypn-1k(ξml[n-1])Hβl(Gkpm)+12Gkpm×(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμqΨq(τl))-Yn-1pk(ξml[n-1]))=0,(3.13)ϵl+νpYn-1pi(τl)=0,(3.14)ϵl+νpYn-1pi(ξil[k])=0,k<n-1,(3.15)ϵl(Γ0(ξml[k])+ϵν(r+p)℧rp(ξml[k])+ϵ2νpΥp(ξml[k])+ϵμqΨq(ξml[k]))=ϵl(xi(k+1)(Γ0(τl)+ϵν(r+p)℧rp(τl)+ϵ2νpΥp(τl)+ϵμjΨj(τl))+ξil[k+1]),
respectively, where
(3.16)Γ0=12∑s=1M‍Gi0sGk0s∂2∂xi(n-1)∂xk(n-1)+fi0∂∂xi(n-1)+∂∂t+⋯+∑α=0n-1-1‍xk(α+1)∂∂xk(n-1),(3.17)℧rp=∑s=1M‍GirsGkps∂2∂xi(n-1)∂xk(n-1),(0≤r<p≤Rν),(3.18)Υp=12∑s=1M‍GipsGkps∂2∂xi(n-1)∂xk(n-1),(0<p≤Rν),(3.19)Ψq=fiq∂∂xi(n-1),(0<q,j≤Rμ),(3.20)Yn-1pi=Gipk∂∂xk(n-1),(0≤p≤Rν),(3.21)Hβl=τl∂∂t+ξjl[β]∂∂xj(β),(0≤β≤n-1).
Note that we cannot cancel out the terms ϵl and ϵl+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.We now apply our generalized methodology for finding approximate symmetries to the Itô system considered in [3]. Our application should be consistent with the determining equations found in Ibragimov et al. [3].


### 3.1.1. Example 1

For their approximate stochastic ordinary differential equations,n-1=0, μ=1, ν=1/2, Rμ=1, Rν=1, and ρ=1. Thus the diffusion coefficient G, which was taken to be constant, and the drift f appeared as follows in the Itô system:
(3.22)dx=(f0+ϵf1)dt+ϵGdWt,
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)-12∑s=1M‍Gk1sGi1sϵ(∂2ξj0∂xi∂xk+ϵ∂2ξj1∂xi∂xk)+(ξi0+ϵξi1)(∂fj0∂xi+ϵ∂fj1∂xi)-fi0(∂ξj0∂xi+ϵ∂ξj1∂xi)-ϵfi1(∂ξj0∂xi+ϵ∂ξj1∂xi)+(τ0+ϵτ1)(∂fj0∂t+ϵ∂fj1∂t)-∂ξj0∂t-ϵ∂ξj1∂t+(fj0+ϵfj1)×(12ϵ∑s=1M‍Gk1sGi1s(∂2τ0∂xi∂xk+ϵ∂2τ1∂xi∂xk)+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0,ϵ(ξi0+ϵξi1)∂Gk1j∂xi-ϵGk1i(∂ξj0∂xi+ϵ∂ξj1∂xi))+ϵ(τ0+ϵτ1)∂Gk1j∂t+12ϵGk1j×(12ϵ(∂2τ0∂xi∂xl+ϵ∂2τ1∂xi∂xl)∑s=1M‍Gi1sGl1s+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0.
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
(3.24)-∂ξj0∂t-fi0∂ξj0∂xi+ξi0∂fj0∂xi+τ0∂fj0∂t+fj0fi0∂τ0∂xi+fj0∂τ0∂t=0,
which we get by comparing coefficients with no ϵ’s
(3.25)fj0fi0∂τ1∂xi-∂ξj1∂xifi0+fj0fi1∂τ0∂xi+fj1fi0∂τ0∂xi+12∂2τ0∂xi∂xl∑s=1M‍Gi1sGl1sfj0+∂τ1∂tfj0+ξi1∂fj0∂xi+ξi0∂fj1∂xi-fi1∂ξj0∂xi-12∂2ξj0∂xi∂xl∑s=1M‍Gi1sGl1s+∂fj0∂tτ1+∂fj1∂tτ0-∂ξj1∂t+∂τ0∂tfj1=0,
which all share the same coefficient ϵ. In a similar fashion, we get the following for ϵ and ϵ, respectively
(3.26)12Gk1j(∂τ0∂t+fi0∂τ0∂xi)-Gk1i∂ξj0∂xi=0,(3.27)-Gk1i∂ξj1∂xi+12Gk1j(∂τ1∂t+fi0∂τ1∂xi)+12Gk1jfi1∂τ0∂xi+Gk1j4∑s=1M‍Gi1sGl1s∂2τ0∂xi∂xl=0.
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.


### 3.1.2. Example 2

We consider(3.28)dX˙=-ω2Xdt+σdW+ϵXdW.
By applying the condition (2.35), we have that
(3.29)ξ=ξ(t,x)
and thus the prolongation formula (2.34) becomes
(3.30)ξ[1]=D(ξ)-x˙D(τ),
where D is the total time derivative operator. Our determining equations at ϵ0 are
(3.31)-ω2xΓ0(τ0)+H0(-ω2x)=Γ0(ξ0[1]),H0(G0)+12G0Γ0(τ0)=Y0(ξ0[1]).
The determining equations at ϵ are
(3.32)-ω2xΓ0(τ1)+H1(-ω2x)-ω2xΥ1(τ0)=Γ0(ξ1[1])+Υ1(ξ0),(3.33)H1(G0)+12G0Γ0(τ1)=Y0(ξ1[1]).
At ϵ1/2 the determining equations are
(3.34)H0(G1)+12G1Γ0(τ0)+12G0℧10(τ0)=Y1(ξ0[1])
and the final determining equation at ϵ3/2 is
(3.35)12G0℧12(τ0)+12G1Γ0(τ1)=Y1(ξ1[1]).
Equation (3.31) for the infinitesimals at the zeroth echelon, that is, τ0 and ξ0, have been solved earlier in the oscillating-spring mass example
(3.36)τ0=C0,ξ0=C1cos(ωt)+C2sin(ωt).
Whence, (3.34) and (3.35) force
(3.37)ξ0=0,τ1=C3.
From (3.32), we get
(3.38)-ω2ξ1=D2(ξ1),
which solves as
(3.39)ξ1=C4cos(ωt)+C5sin(ωt).
Therefore we have
(3.40)ξ=ϵ(C4cos(ωt)+C5sin(ωt)),τ=C0+ϵC3.


## 3.1.1. Example 1

For their approximate stochastic ordinary differential equations,n-1=0, μ=1, ν=1/2, Rμ=1, Rν=1, and ρ=1. Thus the diffusion coefficient G, which was taken to be constant, and the drift f appeared as follows in the Itô system:
(3.22)dx=(f0+ϵf1)dt+ϵGdWt,
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)-12∑s=1M‍Gk1sGi1sϵ(∂2ξj0∂xi∂xk+ϵ∂2ξj1∂xi∂xk)+(ξi0+ϵξi1)(∂fj0∂xi+ϵ∂fj1∂xi)-fi0(∂ξj0∂xi+ϵ∂ξj1∂xi)-ϵfi1(∂ξj0∂xi+ϵ∂ξj1∂xi)+(τ0+ϵτ1)(∂fj0∂t+ϵ∂fj1∂t)-∂ξj0∂t-ϵ∂ξj1∂t+(fj0+ϵfj1)×(12ϵ∑s=1M‍Gk1sGi1s(∂2τ0∂xi∂xk+ϵ∂2τ1∂xi∂xk)+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0,ϵ(ξi0+ϵξi1)∂Gk1j∂xi-ϵGk1i(∂ξj0∂xi+ϵ∂ξj1∂xi))+ϵ(τ0+ϵτ1)∂Gk1j∂t+12ϵGk1j×(12ϵ(∂2τ0∂xi∂xl+ϵ∂2τ1∂xi∂xl)∑s=1M‍Gi1sGl1s+fi0(∂τ0∂xi+ϵ∂τ1∂xi)+ϵfi1(∂τ0∂xi+ϵ∂τ1∂xi)+∂τ0∂t+ϵ∂τ1∂t)=0.
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
(3.24)-∂ξj0∂t-fi0∂ξj0∂xi+ξi0∂fj0∂xi+τ0∂fj0∂t+fj0fi0∂τ0∂xi+fj0∂τ0∂t=0,
which we get by comparing coefficients with no ϵ’s
(3.25)fj0fi0∂τ1∂xi-∂ξj1∂xifi0+fj0fi1∂τ0∂xi+fj1fi0∂τ0∂xi+12∂2τ0∂xi∂xl∑s=1M‍Gi1sGl1sfj0+∂τ1∂tfj0+ξi1∂fj0∂xi+ξi0∂fj1∂xi-fi1∂ξj0∂xi-12∂2ξj0∂xi∂xl∑s=1M‍Gi1sGl1s+∂fj0∂tτ1+∂fj1∂tτ0-∂ξj1∂t+∂τ0∂tfj1=0,
which all share the same coefficient ϵ. In a similar fashion, we get the following for ϵ and ϵ, respectively
(3.26)12Gk1j(∂τ0∂t+fi0∂τ0∂xi)-Gk1i∂ξj0∂xi=0,(3.27)-Gk1i∂ξj1∂xi+12Gk1j(∂τ1∂t+fi0∂τ1∂xi)+12Gk1jfi1∂τ0∂xi+Gk1j4∑s=1M‍Gi1sGl1s∂2τ0∂xi∂xl=0.
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.


## 3.1.2. Example 2

We consider(3.28)dX˙=-ω2Xdt+σdW+ϵXdW.
By applying the condition (2.35), we have that
(3.29)ξ=ξ(t,x)
and thus the prolongation formula (2.34) becomes
(3.30)ξ[1]=D(ξ)-x˙D(τ),
where D is the total time derivative operator. Our determining equations at ϵ0 are
(3.31)-ω2xΓ0(τ0)+H0(-ω2x)=Γ0(ξ0[1]),H0(G0)+12G0Γ0(τ0)=Y0(ξ0[1]).
The determining equations at ϵ are
(3.32)-ω2xΓ0(τ1)+H1(-ω2x)-ω2xΥ1(τ0)=Γ0(ξ1[1])+Υ1(ξ0),(3.33)H1(G0)+12G0Γ0(τ1)=Y0(ξ1[1]).
At ϵ1/2 the determining equations are
(3.34)H0(G1)+12G1Γ0(τ0)+12G0℧10(τ0)=Y1(ξ0[1])
and the final determining equation at ϵ3/2 is
(3.35)12G0℧12(τ0)+12G1Γ0(τ1)=Y1(ξ1[1]).
Equation (3.31) for the infinitesimals at the zeroth echelon, that is, τ0 and ξ0, have been solved earlier in the oscillating-spring mass example
(3.36)τ0=C0,ξ0=C1cos(ωt)+C2sin(ωt).
Whence, (3.34) and (3.35) force
(3.37)ξ0=0,τ1=C3.
From (3.32), we get
(3.38)-ω2ξ1=D2(ξ1),
which solves as
(3.39)ξ1=C4cos(ωt)+C5sin(ωt).
Therefore we have
(3.40)ξ=ϵ(C4cos(ωt)+C5sin(ωt)),τ=C0+ϵC3.


## 4. Concluding Comments

Lie group analysis fornth-ordered Itô SODEs was first pursued in Wafo Soh and Mahomed [5]. Though it had only been done for point symmetries, it has led to many interesting findings in this paper. We have shown that it is possible to derive the prolongation formulas by using the philosophy of form invariance.With the use of the philosophy that the properties of the Wiener processes should remain invariant under the Lie group transformations, we derive conditions on the temporal and lower level derivative spatial infinitesimals that are a generalization of the condition derived by Ünal [4] for one-dimensional SODEs.In this more general approximate approach to higher order SODE, we derive the same conditioning as Ünal [4] did without recourse to the Itô’s multiplication table for the transformed variables. Our results are consistent with that of [3] in the first-order case. However, we have a generalization to nth-order SODEs. We also applied our method to an example taken from [3] as well as another.



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*Source: 263570-2012-12-27.xml*