Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 263570, 15 pages
doi:10.1155/2012/263570
Research Article
Symmetries of nth-Order Approximate Stochastic
Ordinary Differential Equations
E. Fredericks1 and F. M. Mahomed2
1
Department of Mathematics and Applied Mathematics, University of Cape Town, Room 310.1,
Rondebosch 7700, South Africa
2
Centre for Differential Equations, Continuum Mechanics and Applications, School of Computational and
Applied Mathematics, University of the Witwatersrand, Johannesburg 2050, South Africa
Correspondence should be addressed to F. M. Mahomed, fazal.mahomed@wits.ac.za
Received 26 September 2012; Accepted 9 November 2012
Academic Editor: Asghar Qadir
Copyright q 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.
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 Ft, 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 Xt, ω which
drives the arbitrary function FXt, 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.
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Journal of Applied Mathematics
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:
dXn−1 t f t, Xt, Ẋt, . . . , Xn−1 t dt G t, Xt, Ẋt, . . . , Xn−1 t dWt,
k
k1
dXj t Xj
0
tdt,
Xj t Xj 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 × Mmatrix valued function and Wt is an M-dimensional standard Wiener process. From here
onwards we denote {Xt, Ẋt, . . . , Xn−1 t} by Xn−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 Oθ. As a result, the recoverability of the finite transformations, which
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3
keep invariance, from the infinitesimal ones is not verified. The symmetry operator H of
point symmetry is
H τx, t
∂
∂
ξj x, t
,
∂t
∂xj
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 −
1th-ordered spatial derivative is
xn−1 eθH
xk eθH
n−1
n−1
xn−1 ,
xk
2.5
k
eθH xk ,
k < n − 1,
where
∂
n−1
H n−1 H n−2 ξj
n−1
∂xj
,
n ≥ 1,
2.6
H 0 H.
Applying Itô’s formula on an arbitrarily ordered prolongation of a spatial infinitesimal,
ξj r t, Xr−1 t, gives
dξj r t, Xr−1 t fξr j t, Xr−1 t dt Glξr j t, Xr−1 t dWl t,
2.7
where
M
∂ξj γ
∂ξj γ
∂2 ξj γ
1
fξr j t, Xr−1 t fi n−1 Gki Gkp n−1 n−1
∂t
2 k1
∂x
∂x
∂xp
i
n−2
α1
xp
α0
∂ξj γ l
Gl ξr j t, Xr−1 t G r,
n−1 i
∂xi
i
∂ξj
r
α
∂xp
,
where n ≥ 2;
2.8
for each j ranging from 1 to N.
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
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Journal of Applied Mathematics
indices l and k run from 1 to M. Similarly, the Itô’s formula for the temporal variable, τx, t,
gives
dτ fτ Xt, ω, tdt Glτ Xt, ω, tdWl Xt, ω, t,
2.9
where
fτ Xt, ω, t M
n−2
∂τ
∂2 τ
1
∂τ
α1 ∂τ
fi n−1 Gki Gkp n−1 n−1 xp
,
α
∂t
2
∂xi
∂xi
∂xp
∂xp
α0
k1
2.10
which reduces to the total derivative, since the temporal infinitesimal is a point transformation
fτ Xt, ω, t Dτ,
2.11
where the total derivative is defined as
D
n−2
∂ α1 ∂
xp
.
α
∂t α0
∂xp
2.12
The diffusion coefficient of the temporal infinitesimal is given by
Glτ Xt, ω, t Gli
∂τ
n−1
∂xi
2.13
reduces to zero as well because of the fact that we are dealing with point transformations,
that is,
Glτ Xt, ω, t 0.
2.14
The drift and diffusion coefficients of the n − 1th-order spatial derivative are, respectively,
transformed as
n−1 t , t fj Xn−1 t, t θH n−1 fj Xn−1 t, t O θ2 ,
fj X
2.15
n−1 Glj X
t , t Glj Xn−1 t, t θH n−1 Glj Xn−1 t, t O θ2 .
2.16
The Itô formula of the finite time index transformation is
n−1
n−1
dt Γ eθH t dt Y l eθH t dWl ,
2.17
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5
which Wafo Soh and Mahomed 5 simply write as
dt dt1 θDτ O θ2 ,
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:
EQ dtt, ω dtt, ω.
2.19
This requires
n−1
Y l eθH t 0,
l 1, M,
2.20
which is automatically satisfied since τ is point transformation. Condition 2.19 also forces
n−1
D eθH t Constant,
2.21
which is overlooked in 5. Thus the finite transformation of the Wiener process is
dW l t, ω D eθH n−1 t dWl t, ω,
2.22
which Wafo Soh and Mahomed 5 simplify as
dW l t, ω dWl t, ω 1 Dτ O θ2 ,
2
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 nthordered spatial transformation is
n−1 t
dX j
n−1
dXj
t
l
θ fξn−1 j dt G ξn−1 j dWl t O θ2 .
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
n−1 t
dX j
fj t, X
where we denote {Xt, Ẋt, . . . , X
n−1 r−1
t, ω
dt Glj
r−1
t} by X
t, X
n−1 t, ω
dW l t ,
t for an arbitrary r ∈ N.
2.25
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Journal of Applied Mathematics
Expanding the drift component fj t, X
n−1
t, ωdt of 2.25 using 2.15 and 2.18
gives
⎧
⎨
f t, Xn−1 t dt f t, Xn−1 t θ Dτ H n−1 f t, Xn−1 t
⎩
∞ k
θ
k2
k!
⎛
k ⎝ Dτ H n−1 f t, Xn−1 t
k−2 k
H j f t, Xn−1 t
k−j
j0
2.26
⎞⎫
⎬
× D H k−j t − Dτk−j ⎠ dt.
⎭
In order for the finite transformations to keep invariance, we require a higher ordered θ-terms
to be solely dependent on the O1 and Oθ terms, this forces the condition
n−1
eθDτ D eθH t ,
2.27
which is satisfied as a result of condition 2.21. Whence the finite transformation of the
Wiener process becomes
dW l t, ω eθDτ/2 dWl t, ω.
2.28
The diffusion component of 2.25 can easily be expanded with the utility of 2.16 and 2.23
Glj
t, X
n−1
t dW l Dτ
H n−1 Glj t, Xn−1 t
Glj t, Xn−1 t θ
2
k ∞ k
θ Dτ
n−1
l
n−1
H
Gj t, X
t dWl .
k!
2
k2
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,
fξn−1 j Dτ H n−1 fj Xn−1 t, t ,
Glξn−1 j
n−1 Dτ
n−1
l
H
t ,t .
Gj X
2
2.30
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7
Constructing the prolonged variables was carried out in 5 by using preexisting recursive
relations based on the Lie point theory for ODEs, that is,
k
ξj k Dξj k−1 − xj Dτ ,
ξj 0 ξj ,
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,
k dX j
k1 t X
t dt,
2.32
which after expanding yields the following θ-ordered relations:
k dX j
k1
t X k1 dt θ ξj k1 xj
Dτ dt O θ2 ,
k1
Γ ξk Γτ H k1 Xi
,
j
Gi
∂ξk
n−1
∂xi
0,
2.33
2.34
for k < n − 1
2.35
a new condition, which is not mentioned in 5, which is automatically satisfied since the
terms ξk , where k < n − 1, are not functions of xn−1 . In conjunction with this, we have the
Itô SODE associated with the transformation of the kth-ordered spatial transformation, that
is,
k dX j
k
t dXj t θ fξk j dt Gl ξk j dWl t O 2 ,
2.36
which reduces to
k dX j
k
k
t dXj t θD ξj dt O θ2
2.37
k
as a result of the fact that the lower ordered prolongation infinitesimals ξj , are not a function
of xn−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
k
k1
ξj k1 xj
D ξj
Dτ.
We now adapt the relations 2.30, 2.31, and 2.35 to an approximate SODE.
2.38
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Journal of Applied Mathematics
3. Symmetries of nth Order Multidimensional Approximate Stochastic
Ordinary Differential Equations
We now consider the following:
dXn−1 t f t, Xt, Ẋt, . . . , Xn−1 t, Rμ dt · · ·G t, Xt, Ẋt, . . . , Xn−1 t, Rν dWt,
k
k1
dXi t Xi
dt,
0
Xi t Xi 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
Wt is an M-dimensional Wiener process. Here f and G are defined as follows:
f t, xt, ẋt, . . . , xn−1 t, Rμ rμ fr t, ẋt, . . . , xn−1 t ,
3.2
where the repeated index r runs from 0 to Rμ , where Rμ is the largest positive integer such
that μRμ < 2ρ and
G t, xt, ẋt, . . . , xn−1 t, Rν rν Gr t, xt, ẋt, . . . , xn−1 t ,
3.3
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
∂
∂
ξj t, x, ρ
,
H τ t, x, ρ
∂t
∂xj
3.4
τ t, x, ρ r τ r t, x,
ξ t, x, ρ r ξr t, x.
3.5
are defined as
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9
The repeated index runs from 0 to ρ, since throughout this paper we will be working on
Oρ . Using Itô’s formula on the n − 1th-prolongation of the spatial, we get
⎛
n−1
dξj
⎞
n−1
2 n−1
M
n−2
∂ξj n−1
∂ξj n−1 1 ∂
ξ
∂ξ
j
j
α1
⎠dt
fi n−1 ⎝
Gs Gs
x
α
∂t
2 s1 i k ∂xn−1 ∂xn−1 α0 k
∂x
∂x
i
··· ⎛
∂ξj n−1
n−1
∂xi
k
k
Gki dWk t
⎛
M
⎜ 1
0s 0s
⎝
⎜
⎝ 2 Gi Gk
s1
i
ln−1
∂2 ξj
n−1
∂xi
0
n−1
∂xk
ln−1
fi
∂ξj
∂xi
ln−1
∂ξj
∂t
n−2
α0
ln−1
∂ξj
α1
xk
ln−1
α
∂xk
⎞
⎠l
3.6
ln−1
M
M
∂2 ξj
∂2 ξj
1
ps ps
l2pν
rs ps
··· Gi Gk
G
G
lνrp
i
k
n−1
n−1
n−1
n−1
2 s1
∂x
∂x
∂x
∂x
s1
i
i
k
⎞
ln−1
q ∂ξ
j
· · · f i n−1 lμq ⎠dt
∂xi
··· k
ln−1
∂ξj
ps
Gi
lνp dW i
n−1
∂xs
and on the temporal infinitesimal
dτ Dτdt
⎛
⎞
n−2
l
l
∂τ
∂τ
α1
⎠dt
l⎝
x
n−1
∂t α0 k
∂x
3.7
k
with
p
Gsi
∂τ l
n−1
∂xs
lνp 0,
3.8
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
⎛
dX
n−1
dX n−1 · · · θ⎝
M
∂ξn−1 1 ∂2 ξn−1
∂ξn−1
fi n−1 Gki Gkj n−1 n−1
∂t
2 k1
∂x
∂x
∂x
i
⎞
i
j
n−2
n−1
n−1
j
α1 ∂ξ
⎠dt θGi ∂ξ
xj
dWt O θ2
j
α
n−1
∂xj
∂xi
α0
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Journal of Applied Mathematics
dt dt θDτdt O θ2 ,
dW t l
dWt
θ
1 Dτ O θ2 .
2
3.9
The transformation of f and G under our prolongated infinitesimal generator H β is
n−1 fi t, Xn−1 θH β fi t, Xn−1 O θ2
fi t, X
⎞
∂fi
∂fi
β
⎠ O θ2
ξj t, Xn−1
fi θ⎝τ t, Ẋt
β
∂t
∂x
⎛
j
q
q fi
q
q fi
∂fi lμq
θτ
O θ2
∂t
q
∂fi
q
β
θξjl
β
θlμq ⎝ξjl
β
∂xj
lμq
l
⎛
q
∂fi
β
∂xj
⎞
q
∂fi
⎠ O θ2
τ
∂t
l
3.10
n−1 Gi t, Xn−1 θH β Gi t, Xn−1 O θ2
Gik t, X
k
k
⎛
⎞
i
∂Gi
∂G
β
k
k ⎠
Gik θ⎝τ t, Ẋt
O
θ2
ξj t, Xn−1
β
∂t
∂x
j
⎛
p
pi
Gk
β
θνpl ⎝ξjl
pi
∂Gk
β
∂xj
pi
τ
l
∂Gk
∂t
⎞
⎠ O θ2 ,
where {X, Ẋ, . . . , X n−1 } is represented by Xn−1 and the transformed set {X, Ẋ, . . . , X n−1 } is
n−1 . The repeated indices q, p, l, and n run from 0 to Rμ , Rν , ρ, and n − 1,
represented by X
respectively.
3.1. Operators
Thus the determining equations 2.30, 2.31, and 2.35 become
q q μql fm Γ0 τ l νrp rp τ l 2νp Υp τ l μj Ψj τ l Hβl fm
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11
ln−1
ln−1
νrp rp ξm
· · · − l Γ0 ξm
ln−1
ln−1
2νp Υp ξm
μq Ψq ξm
0,
3.11
lνp
1 pm
pm
Hβl Gk Gk
2
pk
ln−1
0,
× Γ0 τ l νrp rp τ l 2νp Υp τ l μq Ψq τ l − Yn−1 ξm
pi
lνp Yn−1 τ l 0,
pi
lk
lνp Yn−1 ξi
0,
3.12
3.13
k < n − 1,
3.14
lk
lk
lk
lk
l Γ0 ξm νrp rp ξm 2νp Υp ξm μq Ψq ξm
k1
lk1
Γ0 τ l νrp rp τ l 2νp Υp τ l μj Ψj τ l ξi
,
l xi
3.15
respectively, where
Γ0 M
n−1−1
α1 ∂
∂2
∂
1
∂
0s
0
·
·
·
G0s
G
f
xk
,
i
n−1
n−1
2 s1 i k ∂xn−1 ∂xn−1
∂t
∂x
∂x
α0
i
rp i
k
M
ps
Grs
i Gk
s1
Υp ∂2
n−1
n−1
∂xi
∂xk
0 ≤ r < p ≤ Rν ,
,
M
∂2
1
ps ps
Gi Gk n−1 n−1 ,
2 s1
∂x
∂x
i
pi
n−1
∂xi
pk
Yn−1 Gi
Hβl τ l
0 < p ≤ Rν ,
3.17
3.18
k
∂
q
Ψq fi
3.16
k
0 < q, j ≤ Rμ ,
,
∂
n−1
∂xk
,
∂
lβ ∂
ξj
,
β
∂t
∂x
0 ≤ p ≤ Rν ,
0≤β ≤n−1 .
3.19
3.20
3.21
j
Note that we cannot cancel out the terms l and lνp 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.
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Journal of Applied Mathematics
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:
√
dx f0 f1 dt GdWt ,
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
⎞
⎞
⎛
⎞
⎛
⎛
M
∂fj0
∂ξj0
∂2 ξj0
∂2 ξj1
∂fj1
∂ξj1
1
⎠ ξ0 ξ1 ⎝
⎠ − f 0⎝
⎠
−
G1s G1s ⎝
i
i
i
2 s1 k i
∂xi ∂xk
∂xi ∂xk
∂xi
∂xi
∂xi
∂xi
⎛
−
fi1 ⎝
∂ξj0
∂xi
∂ξj1
∂xi
⎞
⎛
⎠ τ 0 τ 1 ⎝
∂fj0
∂t
∂fj1
∂t
⎞
⎠−
∂ξj0
∂t
−
∂ξj1
∂t
fj0 fj1
M
∂2 τ 0
∂τ 0
∂2 τ 1
∂τ 1
1 1s 1s
0
fi
G G
×
2 s1 k i ∂xi ∂xk
∂xi ∂xk
∂xi
∂xi
∂τ 0
∂τ 0
∂τ 1
∂τ 1
0,
fi1
∂xi
∂xi
∂t
∂t
⎞
⎛
∂G1j √
∂G1j 1 √ 1j
∂ξj0
∂ξj1
√ √ 0
k
k
⎠ τ 0 τ 1
⎝
ξi ξi1
− G1i
Gk
k
∂xi
∂xi
∂xi
∂t
2
×
M
∂τ 0
∂2 τ 0
∂2 τ 1
∂τ 1
1s 1s
0
G G fi
∂xi ∂xl
∂xi ∂xl s1 i l
∂xi
∂xi
∂τ 0
∂τ 0
∂τ 1
∂τ 1
1
0.
fi
∂xi
∂xi
∂t
∂t
1
2
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
−
∂ξj0
∂t
−
fi0
∂ξj0
∂xi
ξi0
∂fj0
∂xi
τ
0
∂fj0
∂t
fj0 fi0
∂τ 0
∂τ 0
0,
fj0
∂xi
∂t
3.24
Journal of Applied Mathematics
13
which we get by comparing coefficients with no ’s
∂τ
fj0 fi0
1
∂xi
−
∂ξj1
∂xi
− fi1
fi0
∂ξj0
∂xi
−
∂τ
fj0 fi1
0
∂xi
M
∂fj0
∂fj1
1 ∂2 τ 0 ∂τ 1 0
1s 1s 0
1
0
f ξi
G G f ξi
∂xi 2 ∂xi ∂xl s1 i l j
∂t j
∂xi
∂xi
∂τ
fj1 fi0
0
2 0 M
∂fj0
∂fj1
∂ξj1 ∂τ 0
1 ∂ ξj 1s
1
0
τ
τ
f 1 0,
G1s
G
−
2 ∂xi ∂xl s1 i l
∂t
∂t
∂t
∂t j
which all share the same coefficient . In a similar fashion, we get the following for
respectively
0
∂ξj0
1 1j ∂τ 0
0 ∂τ
− G1i
Gk
fi
0,
k
2
∂t
∂xi
∂xi
−G1i
k
∂ξj1
1 1j
G
∂xi 2 k
∂τ 1
∂τ 1
fi0
∂t
∂xi
1j M
2 0
1 1j ∂τ 0 Gk 1s ∂ τ
Gk fi1
G1s
0.
i Gl
2
∂xi
4 s1
∂xi ∂xl
3.25
√
and ,
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 −ω2 Xdt σdW √
XdW.
3.28
By applying the condition 2.35, we have that
ξ ξt, x
3.29
and thus the prolongation formula 2.34 becomes
ξ1 Dξ − ẋDτ,
3.30
where D is the total time derivative operator. Our determining equations at 0 are
−ω2 xΓ0 τ 0 H 0 −ω2 x Γ0 ξ01 ,
1
H 0 G0 G0 Γ0 τ 0 Y 0 ξ01 .
2
3.31
14
Journal of Applied Mathematics
The determining equations at are
−ω2 xΓ0 τ 1 H 1 −ω2 x − ω2 xΥ1 τ 0 Γ0 ξ11 Υ1 ξ0 ,
1
H 1 G0 G0 Γ0 τ 1 Y 0 ξ11 .
2
3.32
3.33
At 1/2 the determining equations are
1
1
H 0 G1 G1 Γ0 τ 0 G0 10 τ 0 Y 1 ξ01
2
2
3.34
and the final determining equation at 3/2 is
1
1 0
G 12 τ 0 G1 Γ0 τ 1 Y 1 ξ11 .
2
2
3.35
Equation 3.31 for the infinitesimals at the zeroth echelon, that is, τ 0 and ξ0 , have been solved
earlier in the oscillating-spring mass example
τ 0 C0 ,
ξ0 C1 cosωt C2 sinωt.
3.36
Whence, 3.34 and 3.35 force
ξ0 0,
τ 1 C3 .
3.37
From 3.32, we get
−ω2 ξ1 D2 ξ1 ,
3.38
ξ1 C4 cosωt C5 sinωt.
3.39
which solves as
Therefore we have
ξ C4 cosωt C5 sinωt,
τ C0 C3 .
3.40
Journal of Applied Mathematics
15
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.
References
1 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.
2 E. Fredericks and F. M. Mahomed, “An alternative “W-symmetries” approach to Lie point symmetries
of scalar first-order itô stochastic ordinary differentialequations,” In press.
3 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.
4 G. Ünal, “Symmetries of Itô and Stratonovich dynamical systems and their conserved quantities,”
Nonlinear Dynamics, vol. 32, no. 4, pp. 417–426, 2003.
5 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.
6 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.
7 Z. Brzeźniak and T. Zastawniak, Basic Stochastic Processes, Springer, 2002.
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