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Symmetries of th-Order Approximate Stochastic Ordinary Differential Equations
Symmetries of th-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.
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 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 which drives the arbitrary function . The application of an SODE to an approximate analysis algorithm has been done by Ibragimov et al.  for scalar SODEs of first order. We extend this work for higher dimensions and order. We derive a similar conditioning on the temporal infinitesimal as had been performed by Ünal  and Fredericks and Mahomed . We introduce operators to write the determining equations in a neater form.
Wafo Soh and Mahomed  gave an algorithm to obtain Lie point symmetries for both first- and th-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 th-order Itô process. These transformations have the same properties as stated in Fredericks and Mahomed .
Using the Itô formula in conjunction with the infinitesimal transformations which preserve form invariance, we derive conditions for th-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 , 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 , which is similar to that of Ünal . 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  to transform the Wiener processes.
2. Review of Wafo Soh and Mahomed  for th-Order SODEs
An th-order Itô process has the following vector form: for . Since the instantaneous mean, , is an -vector valued function, the index runs from one to , that is, . The diffusion coefficent is an -matrix valued function and is an -dimensional standard Wiener process. From here onwards we denote by . 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  in their seventh chapter.
The Lie point transformation methodology used by Wafo Soh and Mahomed  does all calculations to . As a result, the recoverability of the finite transformations, which keep invariance, from the infinitesimal ones is not verified. The symmetry operator of point symmetry is where there is summation . However since we are dealing with an th-ordered SODE, prolongation formulation is necessary. In the Banach space, the transformation for the th-ordered spatial derivative is where Applying Itô’s formula on an arbitrarily ordered prolongation of a spatial infinitesimal, , gives where If the summation operator runs from a nonnegative value, for example, 0, to a negative one; that is, , 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 and both run from to in the summation; the Wiener indices and run from to . Similarly, the Itô’s formula for the temporal variable, , gives where which reduces to the total derivative, since the temporal infinitesimal is a point transformation where the total derivative is defined as The diffusion coefficient of the temporal infinitesimal is given by reduces to zero as well because of the fact that we are dealing with point transformations, that is, The drift and diffusion coefficients of the th-order spatial derivative are, respectively, transformed as The Itô formula of the finite time index transformation is which Wafo Soh and Mahomed  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: This requires which is automatically satisfied since is point transformation. Condition (2.19) also forces which is overlooked in . Thus the finite transformation of the Wiener process is which Wafo Soh and Mahomed  simplify as where  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 th-ordered spatial transformation is Wafo Soh and Mahomed  make the assumption that only the system of th order SODEs, (2.1), remains invariant under the symmetry operator (2.4), which implies that where we denote by for an arbitrary .
Expanding the drift component of (2.25) using (2.15) and (2.18) gives In order for the finite transformations to keep invariance, we require a higher ordered -terms to be solely dependent on the and terms, this forces the condition which is satisfied as a result of condition (2.21). Whence the finite transformation of the Wiener process becomes The diffusion component of (2.25) can easily be expanded with the utility of (2.16) and (2.23) This allows us to make a comparison with the Itô SODE associated with the th-ordered spatial transformation (2.24), which furnishes the determining equations used by Wafo Soh and Mahomed , that is, Constructing the prolonged variables was carried out in  by using preexisting recursive relations based on the Lie point theory for ODEs, that is, for . The sketch of the methodology used for Lie point symmetries for th-ordered SODEs by Wafo Soh and Mahomed  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 -ordered relations: a new condition, which is not mentioned in , which is automatically satisfied since the terms , where , are not functions of . In conjunction with this, we have the Itô SODE associated with the transformation of the th-ordered spatial transformation, that is, which reduces to as a result of the fact that the lower ordered prolongation infinitesimals , are not a function of for . Thus the recursive relations, defined by Wafo Soh and Mahomed  from an ODE context, are easily derived using a form invariance philosophy, namely We now adapt the relations (2.30), (2.31), and (2.35) to an approximate SODE.
3. Symmetries of th Order Multidimensional Approximate Stochastic Ordinary Differential Equations
We now consider the following: for . The function is an approximate drift, which is an vector-valued function, . is an matrix-valued function approximating diffusion and is an -dimensional Wiener process. Here and are defined as follows: where the repeated index runs from to , where is the largest positive integer such that and where the repeated index runs from to ; is the largest positive integer such that . The order of accuracy to which we choose to work is .
The spatial and temporal variables of our infinitesimal generator are defined as The repeated index runs from to , since throughout this paper we will be working on . Using Itô’s formula on the th-prolongation of the spatial, we get and on the temporal infinitesimal with which is automatically satisfied since is point symmetry. The repeated indices , , , and run from to , , , and , respectively in our repeated index summation convention; . Thus, by substitution, we get The transformation of and under our prolongated infinitesimal generator is where is represented by and the transformed set is represented by . The repeated indices , , , and run from to , , , and , respectively.
Thus the determining equations (2.30), (2.31), and (2.35) become respectively, where Note that we cannot cancel out the terms and 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 . Our application should be consistent with the determining equations found in Ibragimov et al. .
3.1.1. Example 1
For their approximate stochastic ordinary differential equations, , , , , , and . Thus the diffusion coefficient , which was taken to be constant, and the drift appeared as follows in the Itô system: where the drift is a vector and the constant diffusion coefficient is a matrix with dimension . The determining equations are Now since we are working to order , we get the following groups of determining equations which are exactly what Ibragimov et al.  get which we get by comparing coefficients with no ’s which all share the same coefficient . In a similar fashion, we get the following for and , respectively Notice that we used (2.21) and the fact that was constant to simplify the above.
Remark 3.1. Our application is consistent with the findings of  for this example.
3.1.2. Example 2
We consider By applying the condition (2.35), we have that and thus the prolongation formula (2.34) becomes where is the total time derivative operator. Our determining equations at are The determining equations at are At the determining equations are and the final determining equation at is Equation (3.31) for the infinitesimals at the zeroth echelon, that is, and , have been solved earlier in the oscillating-spring mass example Whence, (3.34) and (3.35) force From (3.32), we get which solves as Therefore we have
4. Concluding Comments
Lie group analysis for th-ordered Itô SODEs was first pursued in Wafo Soh and Mahomed . 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  for one-dimensional SODEs.
In this more general approximate approach to higher order SODE, we derive the same conditioning as Ünal  did without recourse to the Itô’s multiplication table for the transformed variables. Our results are consistent with that of  in the first-order case. However, we have a generalization to th-order SODEs. We also applied our method to an example taken from  as well as another.
- E. Fredericks and F. M. Mahomed, “A formal approach for handling Lie point symmetries of scalar first-order Itô stochastic ordinary differential equations,” Journal of Nonlinear Mathematical Physics, vol. 15, supplement 1, pp. 44–59, 2008.
- E. Fredericks and F. M. Mahomed, “An alternative “W-symmetries” approach to Lie point symmetries of scalar first-order itô stochastic ordinary differential equations,” In press.
- N. H. Ibragimov, G. Ünal, and C. Jogréus, “Group analysis of stochastic differential systems: approximate symmetries and conservation laws,” ALGA, vol. 1, pp. 95–126, 2004.
- G. Ünal, “Symmetries of Itô and Stratonovich dynamical systems and their conserved quantities,” Nonlinear Dynamics, vol. 32, no. 4, pp. 417–426, 2003.
- C. Wafo Soh and F. M. Mahomed, “Integration of stochastic ordinary differential equations from a symmetry standpoint,” Journal of Physics A, vol. 34, no. 1, pp. 177–192, 2001.
- E. Fredericks and F. M. Mahomed, “Symmetries of first-order stochastic ordinary differential equations revisited,” Mathematical Methods in the Applied Sciences, vol. 30, no. 16, pp. 2013–2025, 2007.
- Z. Brzeźniak and T. Zastawniak, Basic Stochastic Processes, Springer, 2002.
Copyright © 2012 E. Fredericks and F. M. Mahomed. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.