N-SYMMETRY OF ITô STOCHASTIC DIFFERENTIAL EQUATION DRIVEN BY POISSON PROCESS

Lie point symmetry transformation of the class of Itô stochastic differential equation driven by Poisson Processes was successfully carried out. We consider symmetries involving not only spatial and time variables (t, x), but also the Poisson process term N(t). The result was achieved by following the invariance methodology of Lie point transformation and the use of Itô formula for Poisson stochastic differential equation without enforcing any conditions to the momenta of the stochastic process. AMS Subject Classification: 60H10, 76M60


Introduction
In [4] G. Gaeta extend his earlier work [6] to introduced W-symmetries by considering symmetries that involve both the spatial x, temporal variables t and the vector Wiener process w(t).However, G. Gaeta [4] enforced conditions that transformed Wiener process to be consistent with the original process in terms of its momenta, i.e., the instantaneous mean and variance of the transformed process are forced to be exactly the instantaneous mean and variance of the original Wiener process.
In this paper we extend [4] to the class of Itô stochastic differential equations driven by Poisson process dX(t, N ) = f (t, X(t, N ))dt + J(t, X(t, N ))dN (t). ( Where f (t, x) and J(t, x) are the n × 1 dimensional drift vector coefficient and Poisson diffusion coefficient respectively.Here dN (t) is the infinitesimal increment of the Poisson process.We assume that the coefficient functions satisfy the Ikeda Watanabe conditions for the existence and uniqueness of the solution of (1) [1,7].Lie symmetry of (1) were discussed by extending [5] to consider infinitesimal generator that include the group transformations of Poisson process N (t) i.e., we now consider infinitesimal involving not only the spatial and time variables (t, x), but also the Poisson process variable N(t) i.e., The determining equations for Itô stochastic differential equation driven by Poisson processes (1) are successfully derived in an Itô calculus context, and they are found to be deterministic even though they represent a stochastic process.The primary tools for finding admitted Lie point symmetry transformations for stochastic differential equation (SDEs) are Itô formula and the random change of time.
Theorem 1 (Itô Lemma for Poisson diffusion processm see [1,2]).Let Y (t) = F (t, X(t)), such that the function F (t, X(t)) is once continuously differentiable with respect to x and t.Let the X(t) process satisfy the Itô stochastic differential equation with finite jump of the form, X(0) = x 0 where f (t, X(t)) and J(t, X(t)) are bounded and integrable respectively.Then Einstein summation convention is assumed through out this paper, for the matter of convenient let introduce operators and rewrite Finally, using Itô multiplication properties of stochastic differential equation driven by Poisson diffusion process [1,2] and application of infinitesimal transformations the determining equations for the Poisson process stochastic differential equation ( 1) are derived.Precisely the following result was obtained.
Theorem 2. The Itô stochastic differential equation driven by Poisson processes where the coefficient functions f (t, X(t)) and J(t, X(t)) are n × 1 dimensional drift vector coefficient and Poisson diffusion coefficients, with infinitesimal generator has admitted the following determining equations; with additional conditions, Where the operators Γ(t, X(t, N )) and Γ * (t, X(t, N )) are defined as in ( 5) and (6), and λ > 0 is called the jump intensity.And the infinitesimals τ (t, x, N ), ξ(t, x, N ) and φ(t, x, N ) are called the admitted symmetries of (1) if and only if they satisfied the determining equations (11-15).

Lie Group Transformations
Consider a one parameter group of transformation with time index t, the spatial variable x and Poisson process variable N respectively, with the infinitesimals satisfying the following initial conditions at ǫ = 0 Where ǫ is the parameter of the group, hence the corresponding generator of the Lie algebra is of the form of The group transformations can be expressed in term of of the symmetry operator (17) as t = e ǫH (t) (18) x = e ǫH (x) and The and Where the operators Γ(t, x) and Γ * (t, x) have been defined in ( 5) and ( 6) respectively, these operators are in fact instantaneous and standard deviation of the temporal, spatial and Poisson infinitesimals τ , ξ and φ respectively.

Poisson Invariance Properties
Before deriving the determining equations we apply the invariant to the moments of the Poisson processes to make sure it remain invariance under the group transformations, viz the instantaneous mean and variance of the Poisson process which are: The invariance of the instantaneous mean of the transformed Poisson process under new measure using ( 21) and ( 22) equation ( 29) gives Now expending (30) using (27) gives Next we apply the invariant form to instantaneous variance of the transformed Poisson process measure i.e., which using ( 22) gives using the probabilistic invariance property of the transformed time index differential, i.e.
Therefore the generalised infinitesimal jump process is This is a generalised random time change formula for the Poisson process.

Invariance Form of the Spatial Process
To ensure the recovery of the finite transformations from the infinitesimal transformation, we need to transform dX(t) into where the transformed drift component using our generator (17) is and the transformed Poisson amplitude is Expanding the drift component (44) we have The following condition is necessary to ensure the recovery of the finite transformations from the infinitesimal transformation Condition ( 47) is to ensure that the higher order terms depend on the first term associated with O(ǫ).All the ordered terms contribute in the construction of the finite transformations, the zeroth and the first order terms contribute towards the construction of the infinitesimal transformations.This also forces the instantaneous drift coefficient of the temporal infinitesimal to be a constant.i.e.
Expanding the Poisson amplitude (45) gives if they leaves (51) and infinitesimal moments of the Poisson process i.e., invariant, where λ > 0 is the Poisson jump intensity or simply the jump rate.

Determining Equations
This section is devoted to finding the determining equations of the admitted Lie symmetries of the stochastic differential equation driven by Poisson processes (1).
The intention is to transform given SDE driven by Poisson processes (1) into This can be achieved by substituting ( 46) and ( 49) into (53) which gives Therefore, by comparing equation ( 54) and ( 23) we have the following determining equations and Equations ( 36) and (37) give and From (57) we have Equations ( 58) and (59) give To show the relationship between (55) and the first prolongation of ordinary differential equations we proceed by using the definition of first prolongation of an infinitesimal generator for non-stochastic ordinary differential equations and with total time derivative D t defined as using the definition of first prolongation on ( ẋ − f ) at ẋ = f , can be expressed as H [1] Using ( 66) and ( 63) equation ( 55) can be write as Where Γ(t, X(t, N )) and Γ * (t, X(t, N )) are defined using the operators ( 5) and ( 6) respectively, and λ > 0 is called the jump intensity.

Conclusion
Lie point symmetry transformations for the class of Itô stochastic differential equations driven by Poisson Processes was successfully carried out.We considered symmetries involving not only spatial x and time variables t, but also included the Poisson process term N (t).This was achieved by following the methodology of Lie point transformations [3,8] and the use of Itô formula for Poisson stochastic differential equations [1] without enforcing any conditions at the momenta of the stochastic processes.We ensured the instantaneous mean and variance of the Poisson stochastic processes remained invariant under the transformation (16).
Finally, the determining equations for the Itô stochastic differential equation driven by Poisson processes dX i (t, N ) = f i (t, X(t, N ))dt + J i (t, X(t, N ))dN (t) (152) were successfully derived and they were found to be non-stochastic even though they represent stochastic processes.Determining equations found were later applied to a few examples which, while simple, are non-trivial to find their correspondent admitted Lie symmetries.Classification of the given examples are presented in Table 1 below.

) Definition 3 .
The infinitesimals transformation t = e ǫH (t), x = e ǫH (x) and N = e ǫH (N ) (50) are called admitted Lie symmetry transformations of stochastic differential equations driven by Poisson processes

Table 1 :
Lie Group Classification