Lie Symmetry of Ito Stochastic Differential Equation Driven by Poisson Process

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1 American Review of Mathematics Statistics June 016, Vol. 4, No. 1, pp ISSN: (Print), (Online) Copyright The Author(s).All Rights Reserved. Published by American Research Institute for Policy Development DOI: /arms.v4n1a3 URL: Lie Symmetry of Ito Stochastic Differential Equation Driven by Poisson Process Aminu M. Nass 1 & E. Fredericks Abstract This work leads to an understing of the rom time change formulae for Poisson driven process in the context of Lie point symmetries without having to consult much of the intense Itocalculus theory needed to formally derive it. We apply a form invariance methodology to derive the formula apply it to a few examples. Key words: Lie symmetries; Poisson process; stochastic differential equation. 1. Introduction Lie symmetry theory of deterministic differential equations is well understood in literature [16, 17, 18, 19, 0] can used for many important applications in the context of differential equations. For instance, for determination of group-invariant solutions, solving the first order differential equation, reducing the order of higher ODE, reducing the number of variables of partial differential equations finding conservation laws. In contrast to the deterministic differential equation, only a few attempts have been made to extend Lie group theory to the stochastic differential equation. It is worth noticing that the theory is still developing. Gaeta Quintero [6] made the first approach to extend Lie symmetry of differential equations to Ito stochastic ordinary differential equations by which they consider a small class of transformations, i.e., fiber preserving transformations x = θ (t, x, ε ), t = θ (t, ε ). The method has been used to study the relationship between symmetries of stochastic systems to the symmetries of their corresponding Fokker-Planck equation. This is a restricted transformation that can only work to a fiber-preserving class of transformations which is a small sub-class of all possible transformations. The second attempted [3, 4, 5, 8, 10, 15] succeed in applying symmetry transformations that include all the dependent variables in the transformation x = θ (t, x, ε ), t = θ (t, x, ε ). 1 Department of Mathematics Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch 7701, South Africa. nssami001@myuct.ac.za Department of Mathematics Applied Mathematics, University of Cape Town, Private Bag X1, Rondebosch 7701, South Africa. ebrahim.fredericks@uct.ac.za

2 18 American Review of Mathematics Statistics, Vol. 4(1), June 016 This approach has been used to study the symmetry of scalar stochastic ordinary differential equations of first order [4] which reconciled the works of Meleshko S. V., Srihirun B. S. Schultz E. [8] Wafo Soh F.M. Mahomed [10]. Furthermore, the formal method for finding Lie Point symmetries of scalar Ito stochastic differential equations of the first order driven by the Wiener process was also discussed by E. Fredericks [3] with intention to correct reconcile the finding of Srihirun Schultz [8]. To the best of our knowledge in literature, all the methods above were applied only to the Ito stochastic differential equations driven by Wiener processes [3-1]. In this paper we extend the Lie symmetry methods to the class of Itostochastic differential equations driven by a Poisson process by implementing a more generalized Ito formula following the methodology of G. Gaeta [6] E Fredericks F. M. Mahomed [3]. We consider an Ito stochastic differential equation driven by Poisson processes; dx (t) = f t, X(t) dt + J t, X(t) dn(t) (1.1) with initial condition X(0) = x. So, equation (1.1) can be written in integral form as X (t) = x + f s, X(s) ds + J s, X(s) dn(s). (1.) Where f t, X(t) J t, X(t) are n 1 dimensional drift vector coefficients Poisson diffusion coefficient respectively, which are assumed to satisfy Ikeda Watanabe conditions for the uniqueness existence of the solution of (1.1)while dn(t) is the infinitesimal increment of the Poisson Process [1, 13, 14]. Symmetries of (1.1)are analysed by considering an infinitesimal generator H = τ(t, x) t + ξ (t, x). (1.3) The determining equations for Ito stochastic differential equations (SDE) driven by Poisson processes (1.1) are derived using Itocalculus are found to be non-stochastic. Starting with an arbitrary function Ft, X(t)which is once differential with respect to the spatial coordinate x differentiable once with respect to temporal variablet, the Ito Poisson diffusion process for Ft, X(t) of (1.1) exists [1, ] is df t, X(t) = F t + f F dt + F (t, X(t) + J t, X(t) F t, X(t) dn(t). (1.4) The Einstein summation convention is assumed throughout this paper. Let Γ () t, X(t) = F t + f F (1.5) Γ () t, X(t) = F (t, X(t) + J t, X(t) F t, X(t) (1.6) Therefore (1.4) can be written as; df t, X(t) = Γ () t, X(t)dt + Γ () t, X(t)dN(t). (1.7)

3 Aminu M. Nas & E. Frederick 19 Using the Ito multiplication properties of Poisson processes [1, ] dn(t) dn(t) = dn(t), dn(t) dt = 0 dt dt = 0 And application of infinitesimal transformations the determining equations for (SDE) with Poisson processes are derived are non-stochastic. The main result can be summarised as Theorem 1.1: The Ito stochastic differential equation driven by Poisson processes dx (t) = f t, X(t) dt + J t, X(t) dn(t) (1.8) Where f t, X(t) J t, X(t) are the n 1 dimensional drift vector coefficient the Poisson diffusion coefficient, with infinitesimal generator H = τ(t, x) t + ξ (t, x) (1.9) Has the following determining equations; with additional conditions, f Γ () + λj Γ () + Hf Γ () t, X(t) = 0, (1.10) J Γ () + HJ Γ () t, X(t) = 0 (1.11) Γ () t, X(t) = 0, Γ () t, X(t) = c. (1.1) Where the operators Γt, X(t) Γ t, X(t)are defined as in (1.5) (1.6), λ > 0 is called the intensity of the jump process or jump rate. 1. Lie Group Transformation Consider a one parameter group of transformations of the time index t the spatial variable x respectively, with the infinitesimals t = θ (t, x, ε ), x = θ (t, x, ε ) θ ε = τ(t, x), θ ε = ξ(t, x) Satisfying the following initial conditions at ε = 0 t = t, x = x. A one parameter Lie group of infinitesimal transformations is therefore t = t + ετ(t, x) + O(ε) (.1)

4 0 American Review of Mathematics Statistics, Vol. 4(1), June 016 And X(t) = X(t) + ε ξ(t, x) + O(ε) (.) Where ε is the parameter of the group, with the corresponding generator of the Lie algebra of the form H = τ(t, x) t + ξ (t, x). follows The differential point transformations of the spatial, temporal the Poisson process variables are as dt = dt + ε dτ(t, x) + O(ε), (.3) dx(t) = dx(t) + ε dξ(t, x) + O(ε) (.4) dnt = dn(t) + ε dτ λ dt + dn(t) + O(ε). (.5) dt Using the Itoformula(1.7), we have the spatial temporal infinitesimals in Itoforms as dξ = Γ () dt + Γ () dτ = Γ () dt + Γ () dn(t) (.6) dn(t) (.7) where Γ (), Γ (), Γ () Γ () are the drift diffusion coefficients of the spatial temporal infinitesimals, respectively defined using the operators (1.5) (1.6). By substitution of the infinitesimal of spatial (.6) temporal variables (.7) in (.3), (.4) (.5), also using the Ito multiplication properties we proceed to get the group transformations of temporal, spatial jump variables in Ito forms dt = dt + ε Γ () dt + Γ () dn(t) + O(ε), (.8) dx = dx + ε Γ () dt + Γ () dnt = dn(t) + ε Γ () dt + Γ () dt dn(t) + O(ε) (.9) dn(t) λ dt + dn(t) + O(ε). (.10) Expending the Ito infinitesimal of the jump variable (.10) by utilising the Poisson process differential multiplication properties we get dnt = dn(t) + ε λ Γ () dt + Γ () 1.1 Invariance form of the Spatial Process dn(t) + Γ () dn(t) + Γ () dn(t) + O(ε). (.11) dt To ensures the recovery of the finite transformations from the infinitesimal transformation, we need to transform dx (t) into

5 Aminu M. Nas & E. Frederick 1 dx (t) = f t, X(t) dt + J t, X(t) dn(t) (.1) where the transformed drift component f t, X(t) jump component J t, X(t) using the infinitesimal generator respectively are H = τ(t, x) t + ξ (t, x), f t, X(t) = f + ε Hf t, X(t) = f t, X(t) + ε τ f t + ξ f t, X(t) (.13) J t, X(t) = J + ε HJ t, X(t) = J t, X(t) + ε τ J t + ξ J t, X(t) (.14) 1. Poisson Invariance Properties We apply the invariance to the moments of the Poisson process to ensure it remains invariant under the group transformations, viz the instantaneous mean variance of the Poisson process which are: E [dn(t)] = λ dt (.15) E [dn(t) dn(t)] = λ dt. (.16) The invariance of the instantaneous mean of the transformed Poisson process under new measure Q is E dn(t) = λ dt (.17) Exping (.17) using the Ito forms of jump (.8) temporal group transformations (.11) we get Γ () t, X(t) = 0 (.18) Next, we apply the invariance form to instantaneous variance of the transformed Poisson process measure (.16) from which using (.11) we have E dn(t) dn(t) = λ dt (.19) Thus, using (.18) the Itotemporal group transformation (.8) we have derived the following generalized rom time change formula t = Γ () (s)ds (.0) With

6 American Review of Mathematics Statistics, Vol. 4(1), June 016 Γ () t, X(t) = constant = c (.1) Using the probabilistic invariance property of the transformed time index differential, i.e., E dt = dt. (.) Finally, we can conclude from (.18) the temporal infinitesimal τ(t, x)does not depend on x, therefore τ(t, x) = τ(t). Definition.1The infinitesimal transformations (.3) (.4) i.e., t = t + ετ(t, x) + O(ε), X = X + εξ(t, x) + O(ε)(.3) Are called Lie symmetry transformations of (1.1) if they leave the Ito stochastic differential equation (1.1) dx (t) = f t, X(t) dt + J t, X(t) dn(t)(.4) And the infinitesimal moments for the differential Poisson process i.e., E [dn(t)] = λ dt, E [dn(t) dn(t)] = λ dt E [dt] = dt(.5) Invariant. Where λ > 0 ε is the jump intensity group parameter respectively.. Determining Equations In this section, we will derive the determining equations for the admitted symmetries of (1.1). The intention is to transform dx (t) into dx (t) = f t, X(t) dt + J t, Xt dn(t) (3.1) Substituting the transformed drift coefficient (.13), Poisson vector coefficients(.14), Itoforms of temporal (.8) Poisson group transformation (.11) into (3.1) we get dx t = dx (t) + ε f Γ () t, X(t) + λj Γ ()t, X(t) + Hf dt + Γ () t, X(t) + J Γ ()t, X(t) + HJ dn(t) (3.) Therefore, by comparing transformed stochastic differential equation (3.) the Ito form of the spatial group transformation (.9) we have the following determining equations f Γ () + λj Γ () + Hf Γ () t, X(t) = 0, (3.3) J Γ () + HJ Γ () t, X(t) = 0. (3.4)

7 Aminu M. Nas & E. Frederick 3 The invariance of the instantaneous mean of the transformed differential Poisson process (.17) gives additional conditions i.e., from (.18) (.1)we get Γ () t, X(t) = 0 Γ () t, X(t) = c. (3.5) Equation (3.3) can be interpreted using the definition of first prolongation of an infinitesimal generator for non-stochastic ordinary differential equations as follows H [] [] = H + η. (3.6) Where x = dx dt = D x (3.7) η [] = D (ξ ) x D (τ) (3.8) = ξ t + x ξ x τ t + x τ (3.9) with total time derivative D defined as D = t + x + x + (3.10) Using the definition of first prolongation on x f at x = f, can be expressed as H [] ( x f ) = η [] H(f ). (3.11) Using (3.8) (3.11) equation (3.4) can be written as H [] ( x f ) λj Γ () τ t + f τ = 0. (3.1) Where the operators Γ () t, X(t), Γ () t, X(t) are defined in (1.5), (1.6) respectively, λ is called the jump rate or jump intensity of the Poisson process. Remark 3.1: The extra condition obtained from the invariance of the instantaneous mean of the transformed differential Poisson process (.17) forces the temporal infinitesimal τ(t, x) to be a function of the time variable only. This implies that we are now dealing with a fiber-preserving infinitesimal generator i.e., 3. Applications H = τ(t) t + ξ (t, x). (3.13) In this section, we are going to apply the derived determining equations of Poisson Ito stochastic differential equations obtained in the previous section to some Poisson process models to show how the determining equations can be used to find the admitted Lie point symmetries of each model.

8 4 American Review of Mathematics Statistics, Vol. 4(1), June 016 Example 4.1: Consider the Poisson SDE, linear in the state process X(t), with constant coefficients, dx(t) = X(t)u (t) dt + v (t) dn(t) (4.1) With initial condition X(t ) = x > 0, u (t) = is called the drift or deterministic coefficient v (t) = 1 is the jump amplitude coefficient of the jump term, with jump rateλ = λ. Using the determining equations (3.3) (3.4)respectively we have xγ () + λ x Γ () + ξ(t, x) Γ () t, X(t) = 0 (4.) x τ(t) t + λ x τ(t) ξ(t, x) ξ(t, x) + ξ(t, x) x = 0 (4.3) t t x Γ () + ξ(t, x) Γ () t, X(t) = 0 (4.4) x τ(t) + ξ(t, x + x) + ξ(t, x) = 0. (4.5) t Using (.18) (.1) we get the temporal infinitesimal as τ(t) = c t + c. (4.6) Substituting the temporal infinitesimal (4.6) in (4.3) (4.5) respectively gives c x(λ + 4) ξ(t, x) ξ(t, x) + ξ(t, x) x = 0(4.7) t c x + ξ(t, x) ξ(t, x) = 0. (4.8) Differentiating (4.7) with respect to x gives c (λ + 4) Differentiating (4.8) with respect to x gives ξ(t, x) + ξ(t, x) ξ(t, x) x ξ(t, x) t = 0. (4.9) c + ξ(t, x) ξ(t, x) = 0. (4.10) Differentiating (4.10) with respect to t gives ξ(t, x) t Equation (4.11) implies = ξ(t, x). (4.11) t

9 Aminu M. Nas & E. Frederick 5 ξ(t, x) t = ξ(t) t = df(t). (4.1) dt Solving the differential equation (4.1) we get ξ(t, x) = f(t)x + g(x). (4.13) By substituting (4.13) into (4.9) we get c (λ + 4) = df(t) + x d g(x) dt dx. (4.14) When differentiating (4.14) with respect to time we get d f(t) dt = 0. (4.15) Solving the ordinary differential equation (4.15) implies f(t) is linear in t i.e., f(t) = c t + c. (4.16) After substituting (4.16) into (4.13) we arrive at the spatial infinitesimal ξ(t, x) = (c t + c )x + g(x). (4.17) Substituting (4.17) into (4.14) results in c (λ + 4) Which implies that d g(x) dx = = c + x d g(x) dx, (4.18) ( ) c. (4.19) x Solving the differential equation (4.19) for g(x) finally gives g(x) = ( ) c (xln x x) + c x + c, (4.0) Therefore, using (4.0) the special infinitesimal (4.17) can be written as ξ(t, x) = (c t + c )x + ( ) However, substituting (4.1) in (4.8) we have c (xln x x) + c x + c. (4.1)

10 6 American Review of Mathematics Statistics, Vol. 4(1), June 016 c x + (c t + c )x + Which can be simplified to get c x + c = ( ) = (c t + c )x + ( ) c (xln x x) + c x + c ( ) c (xln x x) + c x + c. (4.) c (xln 4 ). (4.3) Further comparison of the coefficients of powers of x in (4.3), gives x c = c ( ) x c = 0. Thus, the spatial infinitesimal (4.1) finally becomes ξ(t, x) = c ln 4 (λ + 4) tx + ln 16 So we have three symmetry generators corresponding to the infinitesimals xln x x + c ln 16 x + c x. (4.4) H = t t + ln 4 (λ + 4) tx + ln 16 xln x x ln 16, H = t, H = x. (4.5) The infinitesimal generators (4.5) give the following Lie bracket relations in Table 1 below H, H H H H H 0 H H ln 16 H H 0 0 H H ln Table 1: Commentator table for the Lie algebra generators (4.5) The commentator table shows that the infinitesimals generators (4.5) is closed under Lie bracket relations hence is a Lie algebra, where H is linear combination of H H given as H = α H + H with α = ln ln λ. (4.6) ln 16 Example 4.: Consider a Poisson driven stochastic differential equation

11 Aminu M. Nas & E. Frederick 7 dx(t) = kt dt + b dn(t) with b 0 (4.7) And initial condition X(0) = x. Using the determining equations (3.3) (3.4)respectively we have kt τ t τ b λ kt + τ t τ ξ ξ kt ktτ(t, x) = kt t (4.8) b τ τ kt = ξ(t, x + b) ξ(t, x). (4.9) t Using equation (.18) (.1) we get the temporal infinitesimal as τ(t) = c t + c. (4.30) Using temporal infinitesimal (4.30) in (4.8) (4.9) we respectively have b λ c kt kt(c t + c ) = ξ ξ kt t (4.31) And ξ(t, x + b) ξ(t, x) = b c. (4.3) Differentiating (4.31) (4.3) with respect to x respectively gives ξ t ξ kt = 0 (4.33) ξ(t, x + b) = Equation (4.34) implies ξ(t, x) = ξ(t) ξ(t, x). (4.34) = f(t). (4.35) Differentiating (4.35) with respect to x gives ξ = 0, (4.36) Solving the differential equation (4.36) we have

12 8 American Review of Mathematics Statistics, Vol. 4(1), June 016 ξ(t, x) = f(t)x + g(t). (4.37) Substituting (4.37) into (4.33) implies df(t) = 0. (4.38) dt Equation (4.38) implies f(t) is constant i.e., f(t) = c, (4.39) Therefore, using (4.39) (4.37) we have ξ(t, x) = c x + g(t). (4.40) Substituting (4.40) into (4.3) gives this relation c = c. (4.41) Using (4.40) (4.41), equation (4.31) gives b λ c 3kt ktc = dg(t) dt Solving the differential equation (4.4) gives kt c. (4.4) g(t) = c bλt 5kt 6 kt c + c. (4.43) Therefore, substituting (4.43) into (4.40) the spatial infinitesimal finally becomes ξ(t, x) = c bλt 5kt 6 + x kt c + c. (4.44) Finally, the Poisson diffusion model admitted three dimensional symmetry infinitesimal generators; H = t t + bλt 5kt 6 + x, H = t kt, H =. (4.45) With the corresponding Lie bracket relations of the generators (4.45) given in Table as H, H H H H H 0 H H H H 0 0 H H 0 0 Table : Commentator table for the Lie algebra generators (4.45)

13 Aminu M. Nas & E. Frederick 9 The Lie bracket relations in Table above show that the infinitesimal generator (4.45) satisfied Lie commutative relation properties hence forms a Lie algebra, where 4. Conclusion H = H H is the linear combination of H H. Lie Symmetry analysis for Ito stochastic differential equations driven the by Poisson processes was carried out, infinitesimals of the Poisson process dn(t) were derived using the moments invariance properties of the process. Determining equations were derived found to be deterministic even though they describe stochastic differential equation. Examples are given to show how the determining equations can be used to find the symmetries, symmetries admitted by (1.1) are found to be fiber-preserving symmetries. Finally, the Lie bracket relation was obtained which shows that all the infinitesimal generators found are closed under the Lie bracket hence they form a Lie algebra. Classification of the given examples is presented in Table 3. Group Dimension Basis Operators Equations 3 H = t t + ln 4 (λ + 4) tx ln 16 dx(t) = X(t) dt + dn(t) + xln x x ln 16, H = t, H = x. 3 H = t t + bλt 5kt 6 + x, H = t kt, H = dx(t) = kt dt + b dn(t), b 0 5. References Table 3: Lie Group Classification B. Floyd Hanson, Applied Stochastic Processes Control for Jump-Diffusions: Modelling, Analysis Computation, Society for Industrial Applied Mathematics (007). P. E. Kloeden E. Platen, Numerical Solution of Stochastic Differential Equations Springer, Applications of Mathematics 64 (010). E. Fredericks F. M Mahomed, Formal Approach for Hling Lie Point Symmetries of Scalar First-Order ItoStochastic Ordinary Differential Equations, Journal of Nonlinear Mathematical Physics Volume 15 (008), E. Fredericks F. M Mahomed, Symmetries of First-Order Stochastic Ordinary Differential Equations Revisited, Mathematical Methods in the Applied Sciences 30 (007), G. Unal, Symmetries of Ito Stratonovich dynamical systems their conserved quantities, Nonlinear Dynamics, 3 (003), G. Gaeta Q. R. Quintero, Lie-point symmetries stochastic differential equations. J Phys A: Math Gen; 3(1999), F.M. Mahomed, C. A. Pooe C. Wafo Soh, Fundamental solutions for zero-coupon bond pricing models, Nonlinear Dynamics, 36 (004), S. V. Meleshko, B. S. Srihirun E. Schultz, on the definition of an admitted Lie group for stochastic differential equations, Communications in Nonlinear Science Numerical Simulation 1 (8) (006),

14 30 American Review of Mathematics Statistics, Vol. 4(1), June 016 G. Gaeta, Symmetry of stochastic equations, Journal of Proceedings National Academy of Sciences Ukraine, 50 (004), Wafo Soh F. M. Mahomed, Integration of stochastic ordinary differential equations from a symmetry stpoint, Journal of Physics A: Mathematical General, 34 (001), G. Gaeta, Lie-point symmetries stochastic differential equations: II. J. Phys. A.: Math. Gen., 33 (000), G. Unal, Ismail Iyigunler C. K. Masood, Linearization of one-dimensional no autonomous jump-diffusion stochastic differential equations, Journal of Nonlinear Mathematical Physics, 14(007), Erinc Ozden G. Unal, Linearization of second order jump-diffusion stochastic differential equations, Int. J. Dynam. control of Nonlinear Mathematical Physics, 1 (013), G. U nal, A. Sanver I. Iyigunler, Exact linearization of one-dimensional jump-diffusion stochastic differential equations, Non-linear Dynamics, 51(1-) (008), 1-8. Roman Kozlov, On Lie Group Classification of a Scalar Stochastic Differential Equation Journal of Nonlinear Mathematical Physics, 18(sup1) (011), P. E. Hydon, Symmetry methods for differential equations. Cambridge University Press, Cambridge (000). N. H. Ibragimov, Elementary Lie group analysis ordinary differential equations. John Wiley Sons, Chichester (1999). Bluman George, Cheviakov Alexei Anco Stephen, Applications of Symmetry Methods to Partial Differential Equations, Applied Mathematical Sciences, Volume 168, Springer-Verlag New York (010). Bluman George Anco Stephen, Symmetry Integration Methods for Differential Equations, Applied Mathematical Sciences, Volume 154, Springer-Verlag New York (00). Aminu M. Nass, Symmetry Analysis Invariants Solutions of Laplace Equation on Surfaces of Revolution, Advances in Mathematics: Scientific Journal 3, no.1 (014), 3-31.