Symmetries of 𝑛th-Order Approximate Stochastic Ordinary Differential Equations

Abstract

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 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.

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 an 𝑛th-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 for 𝑛th-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

An 𝑛th-order Itô process has the following vector form: 𝑑𝐗^(𝑛−1)(𝑡)=𝐟𝑡,𝐗(𝑡),𝐗̇(𝑡),…,𝐗^(𝑛−1)(𝑡)𝑑𝑡+𝐆𝑡,𝐗(𝑡),𝐗̇(𝑡),…,𝐗^(𝑛−1)(𝑡)𝑑𝐖(𝑡),𝑑𝑋_𝑗^(𝑘)(𝑡)=𝑋_𝑗^(𝑘+1)(𝑡)𝑑𝑡,𝑋_𝑗⁽⁰⁾(𝑡)=𝑋_𝑗(𝑡) (2.1)(2.2)(2.3) 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),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 ()𝜕𝐻=𝜏𝑥,𝑡𝜕𝑡+𝜉_𝑗()𝜕𝑥,𝑡𝜕𝑥_𝑗, (2.4) 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 𝐱^(𝑛−1)=𝑒^{𝜃𝐻[𝑛−1]}𝑥^(𝑛−1),𝐱^(𝑘)=𝑒^{𝜃𝐻[𝑛−1]}𝑥^(𝑘)=𝑒^{𝜃𝐻[𝑘]}𝑥^(𝑘),𝑘<𝑛−1, (2.5) where 𝐻^[𝑛−1]=𝐻^(𝑛−2)+𝜉_𝑗^[𝑛−1]𝜕𝜕𝑥_𝑗^(𝑛−1)𝐻,𝑛≥1,_[0]=𝐻. (2.6) Applying Itô’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]_)𝑗𝑡,𝒳^(𝑟−1)(𝑡)𝑑𝑡+𝐺^𝑙_{(𝜉[𝑟])𝑗}𝑡,𝒳^(𝑟−1)(𝑡)𝑑𝑊_𝑙(𝑡), (2.7) where 𝑓_(𝜉^[𝑟]_)𝑗𝑡,𝒳^(𝑟−1)(𝑡)=𝜕𝜉_𝑗^[𝛾]𝜕𝑡+𝑓_𝑖𝜕𝜉_𝑗^[𝛾]𝜕𝑥_𝑖^(𝑛−1)+12^𝑀_𝑘=1𝐺^𝑘_𝑖𝐺^𝑘_𝑝𝜕²𝜉_𝑗^[𝛾]𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑝^(𝑛−1)+^𝑛−2_𝛼=0𝑥_𝑝^(𝛼+1)𝜕𝜉_𝑗^[𝑟]𝜕𝑥_𝑝^(𝛼),where𝐺𝑛≥2;_{𝑙𝜉[𝑟]𝑗}𝑡,𝒳^(𝑟−1)(𝑡)=𝜕𝜉_𝑗^[𝛾]𝜕𝑥_𝑖^(𝑛−1)𝐺^𝑙_𝑖𝑟,foreach𝑗rangingfrom1𝑡𝑜𝑁. (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 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 𝑑𝜏=𝑓_(𝜏)(𝐗(𝑡,𝜔),𝑡)𝑑𝑡+𝐺^𝑙_(𝜏)(𝐗(𝑡,𝜔),𝑡)𝑑𝑊_𝑙(𝐗(𝑡,𝜔),𝑡), (2.9) where 𝑓_(𝜏)(𝐗(𝑡,𝜔),𝑡)=𝜕𝜏𝜕𝑡+𝑓_𝑖𝜕𝜏𝜕𝑥_𝑖^(𝑛−1)+12^𝑀_𝑘=1𝐺^𝑘_𝑖𝐺^𝑘_𝑝𝜕²𝜏𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑝^(𝑛−1)+^𝑛−2_𝛼=0𝑥_𝑝^(𝛼+1)𝜕𝜏𝜕𝑥_𝑝^(𝛼), (2.10) which reduces to the total derivative, since the temporal infinitesimal is a point transformation 𝑓_(𝜏)(𝐗(𝑡,𝜔),𝑡)=𝐷(𝜏), (2.11) where the total derivative is defined as 𝜕𝐷=+𝜕𝑡^𝑛−2_𝛼=0𝑥_𝑝^(𝛼+1)𝜕𝜕𝑥_𝑝^(𝛼). (2.12) The diffusion coefficient of the temporal infinitesimal is given by 𝐺^𝑙_(𝜏)(𝐗(𝑡,𝜔),𝑡)=𝐺^𝑙_𝑖𝜕𝜏𝜕𝑥_𝑖^(𝑛−1) (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 𝑓_𝑗𝒳^(𝑛−1)𝑡,𝑡=𝑓_𝑗𝒳^(𝑛−1)(𝑡),𝑡+𝜃𝐻^[𝑛−1]𝑓_𝑗𝒳^(𝑛−1)(𝑡),𝑡+𝒪𝜃²,𝐺^𝑙_𝑗𝒳^(𝑛−1)𝑡,𝑡=𝐺^𝑙_𝑗𝒳^(𝑛−1)(𝑡),𝑡+𝜃𝐻^[𝑛−1]𝐺^𝑙_𝑗𝒳^(𝑛−1)(𝑡),𝑡+𝒪𝜃². (2.15)(2.16) The Itô formula of the finite time index transformation is 𝑑𝑡=Γ𝑒^{𝜃𝐻[𝑛−1]}(𝑡)𝑑𝑡+𝑌^𝑙𝑒^{𝜃𝐻[𝑛−1]}(𝑡)𝑑𝑊_𝑙, (2.17) which Wafo Soh and Mahomed [5] simply write as 𝑑𝑡=𝑑𝑡(1+𝜃𝐷(𝜏))+𝒪𝜃², (2.18) since the temporal infinitesimal is a point transformation. We also have that the transformed time index should keep invariance in the following probabilistic way: 𝔼_ℚ𝑑𝑡(𝑡,𝜔)=𝑑𝑡(𝑡,𝜔). (2.19) This requires 𝑌^𝑙𝑒^{𝜃𝐻[𝑛−1]}(𝑡)=0,𝑙=1,𝑀, (2.20) which is automatically satisfied since 𝜏 is point transformation. Condition (2.19) also forces 𝐷𝑒^{𝜃𝐻[𝑛−1]}(𝑡)=Constant, (2.21) which is overlooked in [5]. Thus the finite transformation of the Wiener process is 𝑑𝑊_𝑙𝑡,𝜔=𝐷𝑒^{𝜃𝐻[𝑛−1]}(𝑡)𝑑𝑊_𝑙(𝑡,𝜔), (2.22) which Wafo Soh and Mahomed [5] simplify as 𝑑𝑊_𝑙𝑡,𝜔=𝑑𝑊_𝑙()𝜖𝑡,𝜔1+2𝐷(𝜏)+𝒪𝜃², (2.23) 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 𝑑𝑋_𝑗^(𝑛−1)𝑡=𝑑𝑋_𝑗^(𝑛−1)(𝑡)+𝜃𝑓_(𝜉^[𝑛−1]_)𝑗𝑑𝑡+𝐺^𝑙_{𝜉[𝑛−1]𝑗}𝑑𝑊_𝑙(𝑡)+𝒪𝜃². (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 𝑑𝑋_𝑗^(𝑛−1)𝑡=𝑓_𝑗𝑡,𝒳^(𝑛−1)𝑡,𝜔𝑑𝑡+𝐺^𝑙_𝑗𝑡,𝒳^(𝑛−1)𝑡,𝜔𝑑𝑊_𝑙𝑡, (2.25) 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) gives 𝐟𝑡,𝒳^(𝑛−1)(𝑡)𝑑𝑡=𝐟𝑡,𝒳^(𝑛−1)(𝑡)+𝜃𝐷(𝜏)+𝐻^[𝑛−1]𝐟𝑡,𝒳^(𝑛−1)(𝑡)+^∞_𝑘=2𝜃^𝑘𝐷(𝜏)𝑘!+𝐻^{[𝑛−1]𝑘}𝐟𝑡,𝒳^(𝑛−1)(𝑡)+^𝑘−2_𝑗=0𝑘𝐻𝑘−𝑗^𝑗𝐟𝑡,𝒳^(𝑛−1)(𝑡)×𝐷𝐻^𝑘−𝑗(𝑡)−[𝐷(𝜏)]^𝑘−𝑗𝑑𝑡. (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 𝑒^{𝜃𝐷(𝜏)}=𝐷𝑒^{𝜃𝐻[𝑛−1]}(𝑡), (2.27) which is satisfied as a result of condition (2.21). Whence the finite transformation of the Wiener process becomes 𝑑𝑊_𝑙𝑡,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) 𝐺^𝑙_𝑗𝑡,𝒳^(𝑛−1)(𝑡)𝑑𝑊_𝑙=𝐺^𝑙_𝑗𝑡,𝒳^(𝑛−1)(𝑡)𝐷(𝜏)+𝜃2+𝐻^[𝑛−1]𝐺^𝑙_𝑗𝑡,𝒳^(𝑛−1)(𝑡)+^∞_𝑘=2𝜃^𝑘𝑘!𝐷(𝜏)2+𝐻^{[𝑛−1]𝑘}𝐺^𝑙_𝑗𝑡,𝒳^(𝑛−1)(𝑡)𝑑𝑊_𝑙. (2.29) 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, 𝑓_(𝜉^[𝑛−1]_)𝑗=𝐷(𝜏)+𝐻^[𝑛−1]𝑓_𝑗𝒳^(𝑛−1)(𝑡),𝐺,𝑡_{𝑙(𝜉[𝑛−1])𝑗}=𝐷(𝜏)2+𝐻^[𝑛−1]𝐺^𝑙_𝑗𝒳^(𝑛−1)𝑡,𝑡. (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, 𝜉_𝑗^[𝑘]=𝐷𝜉_𝑗^[𝑘−1]−𝑥_𝑗^(𝑘)𝐷_𝜏,𝜉_𝑗^[0]=𝜉_𝑗, (2.31) 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, 𝑑𝑋_𝑗^(𝑘)𝑡=𝑋^(𝑘+1)𝑡𝑑𝑡, (2.32) which after expanding yields the following 𝜃-ordered relations: 𝑑𝑋_𝑗^(𝑘)𝑡=𝑋^(𝑘+1)𝑑𝑡+𝜃𝜉_𝑗^[𝑘+1]+𝑥_𝑗^(𝑘+1)𝐷(𝜏)𝑑𝑡+𝒪𝜃²,Γ𝜉^[𝑘]=Γ(𝜏)+𝐻^[𝑘+1]𝑋_𝑖^(𝑘+1),𝐺^𝑗_𝑖𝜕𝜉^[𝑘]𝜕𝑥_𝑖^(𝑛−1)=0,for𝑘<𝑛−1 (2.33)(2.34)(2.35) 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, 𝑑𝑋_𝑗^(𝑘)𝑡=𝑑𝑋_𝑗^(𝑘)(𝑡)+𝜃𝑓_(𝜉^[𝑘]_)𝑗𝑑𝑡+𝐺^𝑙_{𝜉[𝑘]𝑗}𝑑𝑊_𝑙(𝑡)+𝒪𝜖², (2.36) which reduces to 𝑑𝑋_𝑗^(𝑘)𝑡=𝑑𝑋_𝑗^(𝑘)(𝑡)+𝜃𝐷𝜉_𝑗^[𝑘]𝑑𝑡+𝒪𝜃² (2.37) as a result of the fact that the lower ordered prolongation infinitesimals 𝜉_𝑗^[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 𝐷𝜉_𝑗^[𝑘]=𝜉_𝑗^[𝑘+1]+𝑥_𝑗^(𝑘+1)𝐷(𝜏). (2.38) We now adapt the relations (2.30), (2.31), and (2.35) to an approximate SODE.

3. Symmetries of 𝑛th Order Multidimensional Approximate Stochastic Ordinary Differential Equations

We now consider the following: 𝑑𝐗^(𝑛−1)𝑡=𝐟𝑡,𝐗(𝑡),𝐗̇(𝑡),…,𝐗^(𝑛−1)(𝑡),𝑅_𝜇𝑑𝑡+⋯+𝐆𝑡,𝐗(𝑡),𝐗̇(𝑡),…,𝐗^(𝑛−1)(𝑡),𝑅_𝜈𝑑𝐖(𝑡),𝑑𝑋_𝑖^(𝑘)(𝑡)=𝑋_𝑖^(𝑘+1)𝑋𝑑𝑡,_𝑖(0)(𝑡)=𝑋_𝑖(𝑡) (3.1) 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: 𝐟𝑡,𝐱(𝑡),𝐱̇(𝑡),…,𝐱^(𝑛−1)(𝑡),𝑅_𝜇=𝜖^𝑟𝜇𝐟^𝑟𝑡,𝐱̇(𝑡),…,𝐱^(𝑛−1)(𝑡), (3.2) 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 𝐆𝑡,𝐱(𝑡),𝐱̇(𝑡),…,𝐱^(𝑛−1)(𝑡),𝑅_𝜈=𝜖^𝑟𝜈𝐆^𝑟𝑡,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 𝜕𝐻=𝜏𝑡,𝐱,𝜌𝜕𝑡+𝜉_𝑗𝜕𝑡,𝐱,𝜌𝜕𝑥_𝑗, (3.4) are defined as 𝜏𝑡,𝐱,𝜌=𝜖^𝑟𝜏^𝑟(),𝜉𝑡,𝐱𝑡,𝐱,𝜌=𝜖^𝑟𝜉^𝑟(𝑡,𝐱). (3.5) The repeated index runs from 0 to 𝜌, since throughout this paper we will be working on 𝑂(𝜖^𝜌). Using Itô’s formula on the (𝑛−1)th-prolongation of the spatial, we get 𝑑𝜉_𝑗^[𝑛−1]=𝜕𝜉_𝑗^[𝑛−1]𝜕𝑡+𝑓_𝑖𝜕𝜉_𝑗^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)+12^𝑀_𝑠=1𝐺^𝑠_𝑖𝐺^𝑠_𝑘𝜕²𝜉_𝑗^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1)+^𝑛−2_𝛼=0𝑥_𝑘^(𝛼+1)𝜕𝜉_𝑗^[𝑛−1]𝜕𝑥_𝑘^(𝛼)𝑑𝑡+⋯+𝜕𝜉_𝑗^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)𝐺^𝑘_𝑖𝑑𝑊_𝑘(𝑡)=12^𝑀_𝑠=1𝐺_𝑖^0𝑠𝐺_𝑘^0𝑠𝜕²𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1)+𝑓⁰_𝑖𝜕𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑖+𝜕𝜉_𝑗^{𝑙[𝑛−1]}+𝜕𝑡^𝑛−2_𝛼=0𝑥_𝑘^(𝛼+1)𝜕𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑘^(𝛼)𝜖^𝑙1+⋯+2^𝑀_𝑠=1𝐺_𝑖^𝑝𝑠𝐺_𝑘^𝑝𝑠𝜕²𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1)𝜖^{𝑙+2𝑝𝜈}+^𝑀_𝑠=1𝐺_𝑖^𝑟𝑠𝐺_𝑘^𝑝𝑠𝜕²𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1)𝜖^{𝑙+𝜈(𝑟+𝑝)}+⋯+𝑓^𝑞_𝑖𝜕𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑖^(𝑛−1)𝜖^{𝑙+𝜇𝑞}𝑑𝑡+⋯+𝐺_𝑖^𝑝𝑠𝜕𝜉_𝑗^{𝑙[𝑛−1]}𝜕𝑥_𝑠^(𝑛−1)𝜖^{𝑙+𝜈𝑝}𝑑𝑊^(𝑖) (3.6) and on the temporal infinitesimal 𝑑𝜏=𝐷(𝜏)𝑑𝑡=𝜖^𝑙𝜕𝜏^𝑙+𝜕𝑡^𝑛−2_𝛼=0𝑥_𝑘^(𝛼+1)𝜕𝜏^𝑙𝜕𝑥_𝑘^(𝑛−1)𝑑𝑡 (3.7) with 𝐺^𝑝𝑠_𝑖𝜕𝜏^𝑙𝜕𝑥_𝑠^(𝑛−1)𝜖^{𝑙+𝜈𝑝}=0, (3.8) 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 𝑑𝑋^(𝑛−1)=𝑑𝑋^(𝑛−1)+⋯+𝜃𝜕𝜉^[𝑛−1]𝜕𝑡+𝑓_𝑖𝜕𝜉^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)+12^𝑀_𝑘=1𝐺^𝑘_𝑖𝐺^𝑘_𝑗𝜕²𝜉^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑗^(𝑛−1)+^𝑛−2_𝛼=0𝑥_𝑗^(𝛼+1)𝜕𝜉^[𝑛−1]𝜕𝑥_𝑗^(𝛼)𝑑𝑡+𝜃𝐺^𝑖_𝑗𝜕𝜉^[𝑛−1]𝜕𝑥_𝑖^(𝑛−1)𝑑𝑊_𝑡^(𝑗)+𝒪𝜃²𝑑𝑡=𝑑𝑡+𝜃𝐷(𝜏)𝑑𝑡+𝒪𝜃²,𝑑𝑊_𝑡=𝑑𝑊_𝑡^(𝑙)𝜃1+2𝐷(𝜏)+𝒪𝜃². (3.9) The transformation of 𝐟 and 𝐆 under our prolongated infinitesimal generator 𝐻^[𝛽] is 𝑓_𝑖𝑡,𝒳^(𝑛−1)=𝑓_𝑖𝑡,𝒳^(𝑛−1)+𝜃𝐻^[𝛽]𝑓_𝑖𝑡,𝒳^(𝑛−1)+𝒪𝜃²=𝑓_𝑖+𝜃𝜏𝑡,𝐗̇(𝑡)𝜕𝑓_𝑖𝜕𝑡+𝜉_𝑗^[𝛽]𝑡,𝒳^(𝑛−1)𝜕𝑓_𝑖𝜕𝑥_𝑗^(𝛽)+𝒪𝜃²=𝜖^𝑞𝑓^𝑞_𝑖+𝜃𝜉^𝑙_𝑗^[𝛽]𝜕𝑓^𝑞_𝑖𝜕𝑥_𝑗^(𝛽)𝜖^{𝑙+𝜇𝑞}+𝜃𝜏^𝑙𝜕𝑓^𝑞_𝑖𝜖𝜕𝑡^{𝑙+𝜇𝑞}+𝒪𝜃²=𝜖^𝑞𝑓^𝑞_𝑖+𝜃𝜖^{𝑙+𝜇𝑞}𝜉^𝑙_𝑗^[𝛽]𝜕𝑓^𝑞_𝑖𝜕𝑥_𝑗^(𝛽)+𝜏^𝑙𝜕𝑓^𝑞_𝑖𝜃𝜕𝑡+𝒪²𝐺^𝑖_𝑘𝑡,𝒳^(𝑛−1)=𝐺^𝑖_𝑘𝑡,𝒳^(𝑛−1)+𝜃𝐻^[𝛽]𝐺^𝑖_𝑘𝑡,𝒳^(𝑛−1)+𝒪𝜃²=𝐺^𝑖_𝑘+𝜃𝜏𝑡,𝐗̇(𝑡)𝜕𝐺^𝑖_𝑘𝜕𝑡+𝜉_𝑗^[𝛽]𝑡,𝒳^(𝑛−1)𝜕𝐺^𝑖_𝑘𝜕𝑥_𝑗^(𝛽)+𝒪𝜃²=𝜖^𝑝𝐺_𝑘^𝑝𝑖+𝜃𝜖^{𝜈𝑝+𝑙}𝜉^𝑙_𝑗^[𝛽]𝜕𝐺_𝑘^𝑝𝑖𝜕𝑥_𝑗^(𝛽)+𝜏^𝑙𝜕𝐺_𝑘^𝑝𝑖𝜃𝜕𝑡+𝒪², (3.10) where {𝑋,𝑋̇,…,𝑋^(𝑛−1)} is represented by 𝒳^(𝑛−1) and the transformed set {𝑋,𝑋̇,…,𝑋^(𝑛−1)} is represented by 𝒳^(𝑛−1). The repeated indices 𝑞, 𝑝, 𝑙, and 𝑛 run from 0 to 𝑅_𝜇, 𝑅_𝜈, 𝜌, and 𝑛−1, respectively.

3.1. Operators

Thus the determining equations (2.30), (2.31), and (2.35) become 𝜖^{𝜇𝑞+𝑙}𝑓^𝑞_𝑚Γ⁰𝜏^𝑙+𝜖^{𝜈(𝑟+𝑝)}℧_𝑟𝑝𝜏^𝑙+𝜖^2𝜈𝑝Υ_𝑝𝜏^𝑙+𝜖^𝜇𝑗Ψ_𝑗𝜏^𝑙+𝐻^𝑙_𝛽𝑓^𝑞_𝑚+⋯−𝜖^𝑙Γ⁰𝜉_𝑚^{𝑙[𝑛−1]}+𝜖^{𝜈(𝑟+𝑝)}℧_𝑟𝑝𝜉_𝑚^{𝑙[𝑛−1]}+𝜖^2𝜈𝑝Υ_𝑝𝜉_𝑚^{𝑙[𝑛−1]}+𝜖^𝜇𝑞Ψ_𝑞𝜉_𝑚^{𝑙[𝑛−1]}𝜖=0,_{𝑙+𝜈𝑝}𝐻^𝑙_𝛽𝐺_𝑘^𝑝𝑚+12𝐺_𝑘^𝑝𝑚×Γ⁰𝜏^𝑙+𝜖^{𝜈(𝑟+𝑝)}℧_𝑟𝑝𝜏^𝑙+𝜖^2𝜈𝑝Υ_𝑝𝜏^𝑙+𝜖^𝜇𝑞Ψ_𝑞𝜏^𝑙−𝑌^𝑝𝑘_𝑛−1𝜉_𝑚^{𝑙[𝑛−1]}=0,𝜖^{𝑙+𝜈𝑝}𝑌^𝑝𝑖_𝑛−1𝜏^𝑙=0,𝜖^{𝑙+𝜈𝑝}𝑌^𝑝𝑖_𝑛−1𝜉_𝑖^𝑙[𝑘]=0,𝑘<𝑛−1,𝜖^𝑙Γ⁰𝜉_𝑚^𝑙[𝑘]+𝜖^{𝜈(𝑟+𝑝)}℧_𝑟𝑝𝜉_𝑚^𝑙[𝑘]+𝜖^2𝜈𝑝Υ_𝑝𝜉_𝑚^𝑙[𝑘]+𝜖^𝜇𝑞Ψ_𝑞𝜉_𝑚^𝑙[𝑘]=𝜖^𝑙𝑥_𝑖^(𝑘+1)Γ⁰𝜏^𝑙+𝜖^{𝜈(𝑟+𝑝)}℧_𝑟𝑝𝜏^𝑙+𝜖^2𝜈𝑝Υ_𝑝𝜏^𝑙+𝜖^𝜇𝑗Ψ_𝑗𝜏^𝑙+𝜉_𝑖^𝑙[𝑘+1], (3.12)(3.13)(3.14)(3.15) respectively, where Γ⁰=12^𝑀_𝑠=1𝐺_𝑖^0𝑠𝐺_𝑘^0𝑠𝜕²𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1)+𝑓⁰_𝑖𝜕𝜕𝑥_𝑖^(𝑛−1)+𝜕𝜕𝑡+⋯+^𝑛−1−1_𝛼=0𝑥_𝑘^(𝛼+1)𝜕𝜕𝑥_𝑘^(𝑛−1),℧_𝑟𝑝=^𝑀_𝑠=1𝐺_𝑖^𝑟𝑠𝐺_𝑘^𝑝𝑠𝜕²𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1),0≤𝑟<𝑝≤𝑅_𝜈,Υ_𝑝=12^𝑀_𝑠=1𝐺_𝑖^𝑝𝑠𝐺_𝑘^𝑝𝑠𝜕²𝜕𝑥_𝑖^(𝑛−1)𝜕𝑥_𝑘^(𝑛−1),0<𝑝≤𝑅_𝜈,Ψ_𝑞=𝑓^𝑞_𝑖𝜕𝜕𝑥_𝑖^(𝑛−1),0<𝑞,𝑗≤𝑅_𝜇,𝑌^𝑝𝑖_𝑛−1=𝐺_𝑖^𝑝𝑘𝜕𝜕𝑥_𝑘^(𝑛−1),0≤𝑝≤𝑅_𝜈,𝐻^𝑙_𝛽=𝜏^𝑙𝜕𝜕𝑡+𝜉_𝑗^𝑙[𝛽]𝜕𝜕𝑥_𝑗^(𝛽),0≤𝛽≤𝑛−1. (3.16)(3.17)(3.18)(3.19)(3.20)(3.21) Note that we cannot cancel out the terms 𝜖^𝑙 and 𝜖^{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.

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, 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 Itô system: 𝑑𝐱=𝐟⁰+𝜖𝐟¹𝑑𝑡+√𝜖𝐆𝑑𝐖_𝑡, (3.22) where the drift is a 𝑁×1 vector and the constant diffusion coefficient is a matrix with dimension 𝑁×𝑀. The determining equations are −12^𝑀_𝑠=1𝐺_𝑘^1𝑠𝐺_𝑖^1𝑠𝜖𝜕²𝜉⁰_𝑗𝜕𝑥_𝑖𝜕𝑥_𝑘𝜕+𝜖²𝜉¹_𝑗𝜕𝑥_𝑖𝜕𝑥_𝑘+𝜉⁰_𝑖+𝜖𝜉¹_𝑖𝜕𝑓⁰_𝑗𝜕𝑥_𝑖+𝜖𝜕𝑓¹_𝑗𝜕𝑥_𝑖−𝑓⁰_𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖+𝜖𝜕𝜉¹_𝑗𝜕𝑥_𝑖−𝜖𝑓¹_𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖+𝜖𝜕𝜉¹_𝑗𝜕𝑥_𝑖+𝜏⁰+𝜖𝜏¹𝜕𝑓⁰_𝑗𝜕𝑡+𝜖𝜕𝑓¹_𝑗−𝜕𝑡𝜕𝜉⁰_𝑗𝜕𝑡−𝜖𝜕𝜉¹_𝑗+𝑓𝜕𝑡^0𝑗+𝜖𝑓¹_𝑗×12𝜖^𝑀_𝑠=1𝐺_𝑘^1𝑠𝐺_𝑖^1𝑠𝜕²𝜏⁰𝜕𝑥_𝑖𝜕𝑥_𝑘𝜕+𝜖²𝜏¹𝜕𝑥_𝑖𝜕𝑥_𝑘+𝑓⁰_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝜖𝜕𝜏¹𝜕𝑥_𝑖+𝜖𝑓¹_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝜖𝜕𝜏¹𝜕𝑥_𝑖+𝜕𝜏⁰𝜕𝑡+𝜖𝜕𝜏¹√𝜕𝑡=0,𝜖𝜉⁰_𝑖+𝜖𝜉¹_𝑖𝜕𝐺_𝑘^1𝑗𝜕𝑥_𝑖−√𝜖𝐺_𝑘^1𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖+𝜖𝜕𝜉¹_𝑗𝜕𝑥_𝑖)+√𝜖𝜏⁰+𝜖𝜏¹𝜕𝐺_𝑘^1𝑗+1𝜕𝑡2√𝜖𝐺_𝑘^1𝑗×12𝜖𝜕²𝜏⁰𝜕𝑥_𝑖𝜕𝑥_𝑙𝜕+𝜖²𝜏¹𝜕𝑥_𝑖𝜕𝑥_𝑙^𝑀_𝑠=1𝐺_𝑖^1𝑠𝐺_𝑙^1𝑠+𝑓⁰_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝜖𝜕𝜏¹𝜕𝑥_𝑖+𝜖𝑓¹_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝜖𝜕𝜏¹𝜕𝑥_𝑖+𝜕𝜏⁰𝜕𝑡+𝜖𝜕𝜏¹𝜕𝑡=0. (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 −𝜕𝜉⁰_𝑗𝜕𝑡−𝑓⁰_𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖+𝜉⁰_𝑖𝜕𝑓⁰_𝑗𝜕𝑥_𝑖+𝜏⁰𝜕𝑓⁰_𝑗𝜕𝑡+𝑓⁰_𝑗𝑓⁰_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝑓⁰_𝑗𝜕𝜏⁰𝜕𝑡=0, (3.24) which we get by comparing coefficients with no 𝜖’s 𝑓⁰_𝑗𝑓⁰_𝑖𝜕𝜏¹𝜕𝑥_𝑖−𝜕𝜉¹_𝑗𝜕𝑥_𝑖𝑓⁰_𝑖+𝑓⁰_𝑗𝑓¹_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝑓¹_𝑗𝑓⁰_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+12𝜕²𝜏⁰𝜕𝑥_𝑖𝜕𝑥_𝑙^𝑀_𝑠=1𝐺_𝑖^1𝑠𝐺_𝑙^1𝑠𝑓⁰_𝑗+𝜕𝜏¹𝑓𝜕𝑡^0𝑗+𝜉¹_𝑖𝜕𝑓⁰_𝑗𝜕𝑥_𝑖+𝜉⁰_𝑖𝜕𝑓¹_𝑗𝜕𝑥_𝑖−𝑓¹_𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖−12𝜕²𝜉⁰_𝑗𝜕𝑥_𝑖𝜕𝑥_𝑙^𝑀_𝑠=1𝐺_𝑖^1𝑠𝐺_𝑙^1𝑠+𝜕𝑓⁰_𝑗𝜏𝜕𝑡¹+𝜕𝑓¹_𝑗𝜏𝜕𝑡⁰−𝜕𝜉¹_𝑗+𝜕𝑡𝜕𝜏⁰𝑓𝜕𝑡^1𝑗=0, (3.25) which all share the same coefficient 𝜖. In a similar fashion, we get the following for √𝜖 and 𝜖, respectively 12𝐺_𝑘^1𝑗𝜕𝜏⁰𝜕𝑡+𝑓⁰_𝑖𝜕𝜏⁰𝜕𝑥_𝑖−𝐺_𝑘^1𝑖𝜕𝜉⁰_𝑗𝜕𝑥_𝑖=0,−𝐺_𝑘^1𝑖𝜕𝜉¹_𝑗𝜕𝑥_𝑖+12𝐺_𝑘^1𝑗𝜕𝜏¹𝜕𝑡+𝑓⁰_𝑖𝜕𝜏¹𝜕𝑥_𝑖+12𝐺_𝑘^1𝑗𝑓¹_𝑖𝜕𝜏⁰𝜕𝑥_𝑖+𝐺_𝑘^1𝑗4^𝑀_𝑠=1𝐺_𝑖^1𝑠𝐺_𝑙^1𝑠𝜕²𝜏⁰𝜕𝑥_𝑖𝜕𝑥_𝑙=0. (3.26)(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.

3.1.2. Example 2

We consider 𝑑𝑋̇=−𝜔²𝑋𝑑𝑡+𝜎𝑑𝑊+√𝜖𝑋𝑑𝑊. (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 𝜉^[1]=𝐷(𝜉)−𝑥̇𝐷(𝜏), (3.30) where 𝐷 is the total time derivative operator. Our determining equations at 𝜖⁰ are −𝜔²𝑥Γ⁰𝜏⁰+𝐻⁰−𝜔²𝑥=Γ⁰𝜉^0[1],𝐻⁰𝐺⁰+12𝐺⁰Γ⁰𝜏⁰=𝑌⁰𝜉^0[1]. (3.31) The determining equations at 𝜖 are −𝜔²𝑥Γ⁰𝜏¹+𝐻¹−𝜔²𝑥−𝜔²𝑥Υ₁𝜏⁰=Γ⁰𝜉^1[1]+Υ₁𝜉⁰,𝐻¹𝐺⁰+12𝐺⁰Γ⁰𝜏¹=𝑌⁰𝜉^1[1]. (3.32)(3.33) At 𝜖^1/2 the determining equations are 𝐻⁰𝐺¹+12𝐺¹Γ⁰𝜏⁰+12𝐺⁰℧₁₀𝜏⁰=𝑌¹𝜉^0[1] (3.34) and the final determining equation at 𝜖^3/2 is 12𝐺⁰℧₁₂𝜏⁰+12𝐺¹Γ⁰𝜏¹=𝑌¹𝜉^1[1]. (3.35) 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 𝜏⁰=𝐶₀,𝜉⁰=𝐶₁cos(𝜔𝑡)+𝐶₂sin(𝜔𝑡). (3.36) Whence, (3.34) and (3.35) force 𝜉⁰𝜏=0,₁=𝐶₃. (3.37) From (3.32), we get −𝜔²𝜉¹=𝐷²𝜉¹, (3.38) which solves as 𝜉¹=𝐶₄cos(𝜔𝑡)+𝐶₅sin(𝜔𝑡). (3.39) Therefore we have 𝜉=𝜖𝐶₄cos(𝜔𝑡)+𝐶₅sin(𝜔𝑡),𝜏=𝐶₀+𝜖𝐶₃. (3.40)

4. Concluding Comments

Lie group analysis for 𝑛th-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.