Hypercomplexification of SDEs Exact integration of some systems of stochastic differential equations through hypercomplexification

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by  Celestin Wafo Soh

Exact integration of some systems of stochastic
differential equations through hypercomplexification
7 C. Wafo Soh1,2 and F. M. Mahomed2
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1 Department of Mathematics and Statistical Sciences, Jackson State University
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2 JSU Box 17610, 1400 JR Lynch Street, Jackson, MS 39217, USA

n E-mail: Celestin.Wafo [email protected]
a
2DST-NRF Centre of Excellence in Mathematical and Statistical Sciences,
J
1 School of Computer Science and Applied Mathematics, University of the
2 Witwatersrand, Johannesburg, Wits 2050, South Africa

] E-mail: [email protected]
R
P
Abstract. We leverage commutative hypercomplex analysis to find closed-form
.
h
solutions of some systems of stochastic differential equations. Specifically, we obtain
t
a necessary and sufficient conditions under which a system of stochastic differential
m
equations can be transformed into a scalar one involving processes valued in a
[ commutative hypercomplex. In the event the targeted scalar stochastic differential

1 equation is solved by quadratures, we recover the solution of the original system
v by projecting the solution of the scalar stochastic differential equation along the
9
units of the underlying commutative hypercomplex. The conversion of a system
5
9 of stochastic differential equations involving real-valued processes into a scalar one
6 written in terms of hypercomplex-valued processes is termed hypercomplexification.
0
Bothhypercomplexificationanditsreversearemediatedbytheanalyticityofstochatic
.
1 differential equations data. They may be iterated in order to generate higher-
0
dimensionalintegrablesystemsofstochasticdifferentialequationsandsolvethem. We
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1 showcase the utility of hypercomplexification by treating several examples including
: linear, and linearizable systems of stochastic differential equations and stochastic
v
i Lotka-Volterra systems. Although we consider only random systems driven by white
X
noises, hypercomplexification is fundamentally algebraic, and it readily extends to
r
stochastic systems involving other types of disturbances.
a
PACS numbers: 02.50.Fz, 02.30.Fn, 02.10.De, 0.2.30.lk
Hypercomplexification of SDEs 2
1. Introduction
Many phenomena occur in environments governed by chance. The evolution of such
phenomena is often described by stochastic differential equations (SDEs). We may
roughly distinguish SDEs according to the type of noise that drives them. In this paper,
we are primarily concerned with SDEs driven by the so-called white noise, which are
pervasive in physics, finance, engineering and biology [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].
Finding closed-form solutions of SDEs i.e. solutions expressible in terms of
Lebesgue integrals and stochastic integrals of data is a formidable task. Even for scalar
SDEs, there is not a general procedure for finding exact solutions comparable to their
deterministic counterpart. Perhaps the largest class of scalar SDEs that is integrable
by quadratures is comprised of scalar linear SDEs. Thus the strategy often adopted
for finding exact solutions of a nonlinear scalar SDE is to ascertain whether it can be
invertibly mapped to a linear SDE. Such an approach was pioneered by Gard [12] and
extended by various researchers [13, 14, 15, 16, 17, 18].
Throughout this work, we shall consider systems of SDEs of the form
dX = a(t,X )dt+b(t,X )dW , (1)
t t t t
where X (ω) = X(t)(ω) = X(t,ω) = [X (t,ω)]T, a(t,X ) = [a (t,X )]T, b(t,X ) =
t 1:n t 1:n t t
[b (t,X )] , W (ω) = W(t,ω) = [W (t,ω)]T, ω is the stochastic variable, the
ij t 1≤i,j≤n t 1:n
W ’s are independent one-dimensional standard Wiener processes, and Eq. (1) is
i
taken in Itˆo’s sense. Given an n-dimensional commutative hypercomplex (i.e. a
finite-dimensional vector space equipped with a multiplication which is associative,
commutative,distributivewithrespecttovectoradditionandcompatiblewiththevector
space field multiplication), with units e ,e ,...,e , we want to find out the necessary
1 2 n
and sufficient conditions on a and b such that under the substitution
Z (ω) = X (t,ω)e +X (t,ω)e +···+X (t,ω)e , (2)
t 1 1 2 2 n n
the system (1) is transformed into an integrable scalar SDE of the form
dZ = f(t,Z )dt+g(t,Z )dW , (3)
t t t t
with
W = W (t,ω)e +W (t,ω)e +···+W (t,ω)e . (4)
t 1 1 2 2 n n
Intheschemeoutlinedabove, Eq. (3)willbecalledthebase SDEandthetransformation
of Eq. (1) into Eq. (3) via the replacement (2) will be termed hypercomplexification.
Note that after solving the base SDE i.e. Eq. (3), the components of the solution of
Eq. (1) are obtained by projecting Z along the units e to e .
t 1 n
The remainder of this paper is organized as follows. In Section 2, we introduce
hypercomplex numbers and provide a succinct introduction to hyprcomplex analysis.
Also, we show how hypecomplex analysis may be employed for seeking exact solutions
of systems of SDEs. Section 3 is devoted to the uncovering of some systems of linear
SDEs which can be solved in closed-form. Section 4 is dedicated to the problem of
Hypercomplexification of SDEs 3
linearizing a system of SDEs through hypercomplexification. In Section 5, we explain
how one can generate integrable stochastic Lotka-Volterra systems starting from the
scalar one. Finally, in Section 6, we summarize our findings and we elaborate on some
problems we could not solve.
2. Hypercomplex analysis and hypercomplexification of stochastic
differential equations
Complex numbers and quaternions are particular instances of hypercomplexes. The
general theory of hypercomplexes was developed and completed in the nineteen
century. In this section, we provide a concise yet operational account of the theory
of hypercomplexes with an emphasis on commutative ones that are employed through
this work. For an in-depth treatment of hypercomplexes, we refer the reader to the
works [19, 20, 21, 22].
2.1. Hypercomplex numbers
A hypercomplex H is a finite-dimensional vector space equipped with a multiplication
which is associative, distributive with respect to vector addition, compatible with the
multiplication of the vector space field, and admits an identity that we will sometimes
abusively denote 1 when there is no risk of confusion. In this section, we shall denote
the field of the vector space F. In practice, we shall commonly take F as R or C.
When the multiplication of a hypercomplex is commutative, the hypercomplex is called
a commutative hypercomplex. For all the applications envisaged in this paper, we solely
use commutative hypercomplexes.
Now, suppose that we have fixed a basis of a hypercomplex H, {e ,e ,...,e }, so
1 2 n
that
H = {z = x e +x e +...+x e | (x ,x ,...,x ) ∈ Fn}. (5)
1 1 2 2 n n 1 2 n
The elements of a basis of H are called the units of the hypercomplex. Note that we
can always select one of the units to be the identity of the hypercomplex. Owing to the
closure property of the multiplication of H which states that the product of two generic
element of H must belong to H, we must have e e ∈ H for all i,j = 1 : n. Thus,
i j
n
(cid:88)
e e = γ e , for all i,j = 1 : n, (6)
i j ijk k
k=1
where the γ ’s belong to F. The γ ’s are the structure constants of the hypercomplex
ijk ijk
H associated with the units e ,e ,...,e . It can be shown that the multiplication of H
1 2 n
is commutative if and only if
γ = γ , for all i,j,k = 1 : n. (7)
ijk jik
Hypercomplexification of SDEs 4
It can be verified that the multiplication of H is associative if and only if (e e )e =
i j k
e (e e ), for all i,j,k = 1 : n, which translates into
i j k
n n
(cid:88) (cid:88)
γ γ = γ γ , for all i,j,k,t = 1 : n. (8)
ijs skt ist jks
s=1 s=1
The element (cid:15) = (cid:80)n (cid:15) e is the multiplicative identity of H if and only if (cid:15)e = e and
i=1 i i i i
e (cid:15) = e for all i = 1 : n. That is
i i
n n
(cid:88) (cid:88)
(cid:15) γ = (cid:15) γ = δ for all i,j = 1 : n, (9)
k ikj k kij ij
k=1 k=1
where δ is Kronecker’s symbol. In practice, a hypercomplex is specified by providing
ij
th th
its multiplication table i.e. a table in which the entry located on the i row and j
column, is e e . For commutative hypercomplexes, such a table is symmetric which
i j
respect to its main diagonal.
The classification of lower-dimensional hypercomplexes was done by several
researchers [19, 22, 23, 24, 25, 26] using often different or complementary techniques. In
our recent paper [27], we have summarized from the work of Study [24] the classification
of commutative hypercomplexes up to dimension four. We shall use the nomenclature
of [27] for lower-dimensional commutative hypercomplexes.
There are three nonequivalent classes of two-dimensional commutative hypecom-
plexes over R that can be treated at once through the consideration of the so-called
generalized complex numbers [28, 29]. The set of general hypercomplex numbers is de-
noted C and its units may be chosen as e = 1, e = i with its multiplication table
p 1 2
given by: e2 = 1, e2 = p, e e = e e = e , where p is a real number. Put another way
1 2 1 2 2 1 2
C = {x+iy|x,y ∈ R} and the algebra is done in C like in R under the proviso that
p p
i2 = p.
AccordingtoStudy’s[24]classification, thereareuptoequivalencetransformations,
five three-dimensional and twelve four-dimesional commutative hypercomplexes over R.
Note that some high-dimensional hypercomplexes may be generated through direct sum
or direct product [22, 27] of lower-dimensional ones.
2.2. A primer on hypercomplex analysis
The investigation of analytic functions of hypercomplex variables was initiated by
Scheffers [30, 31] for commutative hypercomplexes. His works was extended in several
directions by Ketchum [32] who in particular generalized all important theorems of
the classical complex analysis to functions of hypercomplex variables in commutative
hypercomplexes. Our aim in this subsection is to provide a brief account of this theory.
We shall restrict ourselves to situations that suit our need in this paper. For an in-dept
treatment of commutative hypercomplex analysis, we refer the reader to the seminal
papers [30, 31, 32].
Throughout this subsection, H will be a commutative hypercomplex over real
numbers, with units e ,e ,...,e , and corresponding structure constants γ , i,j,k =
1 2 n ijk
Hypercomplexification of SDEs 5
1 : n. We denote the multiplicative identity of H by (cid:15) = (cid:80)n (cid:15) e . We topologize H
i=1 i i
trough the Euclidian norm defined for z = (cid:80)n x e by ||z|| = (cid:112)(cid:80)n x2.
i=1 i i i=1 i
A function of a hypercomplex variable f(z) = (cid:80)n f (x )e is differentiable at
i=1 i 1:n i
z ∈ H if there is u ∈ H such that
0
lim ||f(z +h)−f(z)−uh|| = 0. (10)
||h||→0
We call u the derivative of f at z and we denote it by f(cid:48)(z ).
0 0
The function f is monogenic in a region (open connected set) U of H if it is
differentiable at every point in U and the f are analytic on U.
i
Now assume that f is monogenic in a region U. Then, on this region, we must have
df(z) = f(cid:48)(z)dz. (11)
Substitute f(cid:48)(z) = (cid:80)n f(cid:48)(z)e , dz = (cid:80)n dx e into (11) to obtain
i=1 i i i=1 i i
n
(cid:88) ∂f (cid:88)
idx = γ f(cid:48) dx . (12)
∂x j kji k j
j
j=1 j,k
Since Eq. (12) must hold true for all increment dx , j = 1 : n, we must have
j
n
∂f (cid:88)
i = γ f(cid:48) for all i,k = 1 : n. (13)
∂x jki j
k
j=1
Multiply both sides of Eq. (13) by (cid:15) (the kth component of the multiplicative identity)
k
and sum both sides from k = 1 to k = 1. Then employ the identity (9) on the left-hand
side to arrive at
n
(cid:88) ∂f
f(cid:48) = (cid:15) i for all i = 1 : n. (14)
i k ∂x
k
k=1
Substituting Eq. (14) into Eq. (13) yields
n
∂f (cid:88) ∂f
i k
= (cid:15) γ · (15)
(cid:96) jki
∂x x
k (cid:96)
j,(cid:96)=1
Note that the system (15) generalizes the classical Cauchy-Riemann equations. We shall
call it Scheffers’ equations owing to the following theorem proved by Scheffers [30, 31].
Theorem 2.1 The function f(z) = (cid:80)n f (x )e , where z = (cid:80)n x e ∈ H, and H
i=1 i 1:n i i=1 i i
is a commutative hypercomplex with structure constants γ , i,j,k = 1 : n, is monogenic
ijk
on a region U if and only if the f ’s are continuously infinitely differentiable on U and
i
the system (15) is satisfied on U.
Afunctionofahypercomplexvariablez isanalyticatapointz ifitcanbeexpressed
0
as the sum of a power series in (z−z ) in the vicinity of z . Such a function is analytic in
0 0
a region U if it is analytic at every point in U. It can be verified that analytic functions
are monogenic. In fact, Scheffers [30, 31] proved that monogenic functions are analytic.
Thus, in the sequel, we shall use monogenic and analytic as synonymous when dealing
with functions of hypercomplex variables.
Hypercomplexification of SDEs 6
Several classical analytic functions have their hypercomplex counterparts which are
also analytic. The machinery commonly employed to achieve the hypercomplexification
of these functions includes series expansion, analytic continuation or integral
representation. In what follows, we shall illustrate this point by constructing the
exponential functions on C and A4 [27] through series expansion.
p 3
Let z = x+iy ∈ C . We define the exponential of z as follows [27, 28, 29]:
p
ez = exeiy
(cid:88)∞ (iy)k
= ex
k!
k=1
(cid:40) (cid:41)
(cid:88)∞ (iy)2k (cid:88)∞ (iy)2k+1
= ex +
(2k)! (2k +1)!
k=1 k=1
(cid:40) (cid:41)
(cid:88)∞ pky2k (cid:88)∞ pky2k+1
= ex +i [i2 = p]
(2k)! (2k +1)!
k=1 k=1
= ex[cos (y)+isin (y)], (16)
p p
where we have set
(cid:88)∞ pky2k (cid:88)∞ pky2k+1
cos (y) = and sin (y) = · (17)
p p
(2k)! (2k +1)!
k=1 k=1
By noticing that eiye−iy = 1, we arrive at the generalized Pythagorean identity
cos (y)−p sin (y) = 1. (18)
p p
Let us now turn our attention to the commutative hypercomplex A4 [27] whose
3
units are e = 1, e = i and e = j and the multiplication table is given by: i2 = j,
1 2 3
ij = 0 and j2 = 0. Let us express a generic element of A4 as z = t+xi+yj. So, we
3
may define ez as
ez = eteixejy
(cid:88)∞ (ix)k (cid:88)∞ (jx)k
= et
k! k!
k=1 k=1
(cid:18) x2 (cid:19)
= et 1+xi+ j (1+jy)
2
(cid:20) (cid:18)x2 (cid:19) (cid:21)
= et 1+xi+ +y j . (19)
2
Then,thenaturallogarithm,lnz,isobtainedbysolvingtheequationz = exp(a+bi+cj)
for a, b and c. The foregoing equation yields the system
t = ea, (20)
x = eab, (21)
(cid:18)b2 (cid:19)
y = ea +c . (22)
2
Hypercomplexification of SDEs 7
Thesolutionofthesystem(20)-(22)isgivenby: a = lnt, b = x/t, andc = y/t−x2/(2t2).
Thus, the natural logarithm on A4 is defined as follows:
3
x (cid:18)y x2 (cid:19)
lnz = lnt+ i+ − j. (23)
t t 2t2
From Eqs. (19) and (23), we infer that for any real number m,
(cid:20) mx (cid:18)my m(m−1)x2(cid:19) (cid:21)
zm = exp(mlnz) = tm 1+ i+ + j . (24)
t t 2t2
In fact we can even define zz2 for z ,z ∈ A4 by noticing that zz2 = exp(z lnz ) and
1 1 2 3 1 2 1
using the formulas (19) and (23).
From the formula (24), we deduce that
(cid:88)∞ z2m (cid:88)∞ t2m (cid:88)∞ t2m−1x
cos(z) = (−1)m = (−1)m +i (−1)m
(2m)! (2m)! (2m−1)!
m=0 m=0 m=1
(cid:88)∞ (−1)m (cid:18) x2(cid:19)
+j 2myt2m−1 +2m(2m−1)t2m−2
(2m)! 2
m=0
(cid:18) x2cost(cid:19)
= cost−(xsint)i− ysint+ j. (25)
2
Likewise, we find that
(cid:18) x2sint(cid:19)
sin(z) = sint+(xcost)i+ ycost− j. (26)
2
The reader may verify that amazingly cos2z + sin2z = 1 for all z ∈ A4 such that the
3
latter equation makes sense.
2.3. Hypercomplexification of stochastic differential equations
It is hard to find the general exact solution of a scalar SDE. Usually, exact solutions
of scalar ODEs are found through clever use of the stochastic chain rule (Itoˆ’s lemma)
to map these SDEs to integrable SDEs. Arguably, the largest class of scalar integrable
SDEsistheclassmadeoflinearSDEs. Thusthemainstrategyforfindingexactsolutions
of a scalar SDE consists in looking for invertible change of both the dependent and
dependentvariablesthatconverttheunderlyingSDEintoalinearone. Suchanapproach
was pioneer by Gard [12] and extended by several researchers [13, 14, 15, 16, 17, 18].
For systems of SDEs, the situation is even more complicated than for scalar
SDE. Indeed one has to deal with the coupling of the equations and the fact that in
general, linear system of SDEs are not exactly integrable. So even if one contemplates
transforming a system of SDE to a linear system of SDEs, the target system may not
be exactly integrable.
Our proposal for the search of integrable systems of SDEs consists in ascertaining
whetherbyappropriatehypercomplexsubstitutionsonecantransformsthesesystemsto
scalar SDEs in which the involved process are valued in commutative hypercomplexes.
In the event one can transform a system of SDEs to a scalar SDE and the scalar
Hypercomplexification of SDEs 8
SDE is solvable by quadratures, one recovers the solution of the original system by
projecting the solution of the target scalar SDE along the units of the commutative
hypercomplex: The exact solution of the scalar SDE will be expressed in terms
of Lebesgue and stochastic integrals involving data and elementary functions whose
hypercomplex counterpart can be generated as discussed in the previous subsection. In
this scheme, we shall call the target scalar SDE the base equation [33]. The conversion
of a systems of SDEs in which the dependent variables and data are real-valued to a
scalar SDE which has hypercomplex-valued dependent variable and data, will be termed
hypercomplexification.
In order to reify the above narrative, consider the following scalar SDE
dZ(t) = a(t,Z(t))dt+b(t,Z(t))dW(t), (27)
where Z(t) = (cid:80)n X (t)e is valued in a commutative hypercomplex over reals, H,
i=1 i i
with units e ,e ,...,e , W(t) = (cid:80)m W (t)e , and m ≤ n. In other sections, we
1 2 n i=1 i i
shall take m = n for the sake of simplicity. By letting a = (cid:80)m a (t,X (t))e
i=1 i 1:n i
and b = (cid:80)m b (t,X (t))e , and using the linear independent of the units of H, the
i=1 i 1:n i
equation (27) implies
n m
(cid:88)(cid:88)
dX (t) = a (t,X (t))dt+ γ b (t,X (t))dW (t). (28)
i i 1:n ijk j 1:n k
k=1 j=1
The scalar equation (27) is the base equation of the system (28). In order to reverse
the procedure i.e. start from the system (28) and arrive at Eq. (27), one must require
that the γ ’s be the structure constants of a commutative hypercomplex and that the
ijk
functions a and b be analytic in Z as functions valued in that hypercomplex. Thus,
these functions must satisfy Scheffers’ equations (15).
In practice, one generates Eq. (28) by using the classification of lower-dimensional
commutative hypercomplexes [24, 27] and a base equation one knows how to solve. We
shall illustrate this procedure in the remaining sections.
Note that the above procedure may be iterated in such a way that one may end
up with a high-dimensional integrable system of SDEs. Indeed, if the system (28) is
integrable, we can also hypercomplexify it using another hypercomplex, K, say. The
resultingsystemcouldbegeneratedfromthebaseequation(27)usingthedirectproduct
[27] of H by K, H ⊗ K. Thus stating from a scalar integrable SDE, we may build
a hierarchy of integrable systems of SDEs that may be nontrivial to solve without
employing the hypercomplexification procedure.
3. Integrable systems of linear stochastic differential equations
In this section, we characterize linear systems of SDEs that are integrable by
hypercomplexification. It is important to stress that although scalar linear SDEs are
integrable, systems of linear SDEs do not in general admit closed-form solutions [12].
Consider a scalar linear SDE
dZ(t) = (f (t)+f (t)Z(t))dt+[g (t)+g (t)Z(t)]dW(t) (29)
1 2 1 2
Hypercomplexification of SDEs 9
in which the functions f , f , g and g are deterministic functions and we have dropped
1 2 1 2
the stochastic variable as it is customary in stochastic calculus. Its closed-form solution
is [12]
(cid:26) (cid:90) t
Z(t) = E(t) Z(0)+ E−1(s)[f (s)−g (s)g (s)]ds
1 1 2
0
(cid:90) t (cid:27)
+ E−1(s)g (s)dW(s) , (30)
1
0
where
(cid:26)(cid:90) t(cid:20) 1 (cid:21) (cid:90) t (cid:27)
E(t) = exp f (s)− g2(s) ds+ g (s)dW(s) . (31)
2 2 2 2
0 0
Now, let us treat the scalar quantities as elements of a commutative hypercomplex H
with units e to e , and structure constants γ where i,j = 1 : n:
1 n ij
n
(cid:88)
Z(t) = X (t)e , (32)
i i
i=1
n
(cid:88)
f (t) = f (t)e , (33)
1 1i i
i=1
n
(cid:88)
f (t) = f (t)e , (34)
2 2i i
i=1
n
(cid:88)
g (t) = g (t)e , (35)
1 1i i
i=1
n
(cid:88)
g (t) = g (t)e , (36)
2 2i i
i=1
n
(cid:88)
dW(t) = dW (t)e . (37)
i i
i=1
Now, substitute Eqs. (32)-(37) into Eq. (29) to obtain after some algebraic
manipulations
(cid:32) (cid:33)
n
(cid:88)
dX (t) = f (t)+ γ f (t)X (t) dt
i 1i k(cid:96)i 2k (cid:96)
k,(cid:96)=1
(cid:32) (cid:33)
n n
(cid:88) (cid:88)
+ γ g (t)+ γ γ g (t)X (t) dW (t), (38)
k(cid:96)i 1k kpm m(cid:96)i 2k p (cid:96)
k,(cid:96)=1 m,p=1
for all i = 1 : n. Conversely, starting from the system (38) and provided that the
γ ’s appearing in it are structure constants of a commutative hypercomplex with basis
ijk
e ,e ,...,e , we can transform Eq. (38) into Eq.(29) via the substitution (32). We has
1 2 n
thus established the following theorem.
Theorem 3.1 A system of n linear SDEs that can be hypercomplexified provided
has to have the form (38), where the γk’s are structure constants of a commutative
ij
hypercomplex. The solution of the system (38) is obtained by projecting Eq.(30) along
the units of the underlying commutative hypercomplex.
Hypercomplexification of SDEs 10
Remark 3.1 Stochastic differential equations of the form
3 3
(cid:88) (cid:88)
du = − A (t)u dt+ B (t)dW , i = 1 : 3 (39)
i ij j ij j
j=1 j=1
arise in the study of the dispersion of particles in turbulent flows [34]. By employing
three-dimensional hypercomplexes, we may infer from the system (38) integrable forms
of Eq. (39).
Let us illustrate Theorem 3.1 by using C as our commutative hypercomplex. For
p
C , the system (38) becomes:
p
dX (t) = [f (t)+f (t)X (t)+pf (t)X (t)]dt
1 11 21 1 22 2
+[g (t)+g (t)X (t)+pg (t)X (t)]dW (t)
11 21 1 22 2 1
+p [g (t)+g (t)X (t)+g (t)X (t)]dW (t), (40)
12 22 1 21 2 2
dX (t) = [f (t)+f (t)X (t)+f (t)X (t)]dt
2 12 22 1 21 2
+[g (t)+g (t)X (t)+g (t)X (t)]dW (t)
12 22 1 21 2 1
+[g (t)+g (t)X (t)+pg (t)X (t)]dW (t). (41)
11 21 1 22 2 2
In order to express the solution of the system (40)-(41) in closed-form using Eq.(30), we
introduce the following notations
(cid:90) t(cid:20) 1 (cid:21)
a(t,τ) = f (s)− (g2 (s)+pg2 (s)) ds
21 2 21 22
τ
(cid:90) t (cid:90) t
+ g (s)dW (s)+p g (s)dW (s), (42)
21 1 22 2
τ τ
(cid:90) t
b(t,τ) = [f (s)−g (s)g (s)]ds
22 21 22
τ
(cid:90) t (cid:90) t
+ g (s)dW (s)+ g (s)dW (s). (43)
22 1 21 2
τ τ
Then, the general solution of the system (40)-(41) is given by
X (t) = ea(t,0)[X (0)cos [b(t,0)]+pX (0)sin [b(t,0)]
1 1 p 2 p
(cid:90) t
+ ea(t,s)[cos [b(t,s)]{f (s)−g (s)g (s)−pg (s)g (s)}
p 11 11 21 12 22
0
+psin [b(t,s)]{f (s)−g (s)g (s)−g (s)g (s)}]ds
p 21 12 21 11 22
(cid:90) t
+ ea(t,s)[cos [b(t,s)]g (s)+psin [b(t,s)]g (s)]dW (s)
p 11 p 12 1
0
(cid:90) t
+p ea(t,s)[cos [b(t,s)]g (s)+sin [b(t,s)]g (s)]dW (s), (44)
p 12 p 11 2
0
X (t) = ea(t,0)[X (0)sin [b(t,0)]+X (0)cos [b(t,0)]]
2 1 p 2 p
(cid:90) t
(cid:8)
+ ea(t,s)[sin [b(t,s)]{f (s)−g (s)g (s)−pg (s)g (s)}
p 11 11 21 12 22
0
+cos [b(t,s)]{f (s)−g (s)g (s)−g (s)g (s)}]}ds
p 21 12 21 11 22