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Vladikavkaz Mathematical Journal 2020, Volume 22, Issue 4, P. 68-86
УДК 519.642
DOI 10.46698/n8076-2608-1378-r
NEW NUMERICAL METHOD FOR SOLVING NONLINEAR STOCHASTIC INTEGRAL EQUATIONS
R. Zeghdane1
1 Department of Mathematics, Faculty of Mathematics and Informatics, University of Bordj-Bou-Arreridj, El-Anasser 34030, Bordj-Bou-Arreridj, Algeria E-mail: rebihae@yahoo.fr
Abstract. The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.
Key words: Chebyshev cardinal functions, stochastic operational matrix, Brownian motion, Ito integral, collocation method, numerical solution.
Mathematical Subject Classification (2010): 45G10, 65R20.
For citation: Zeghdane, R. New Numerical Method for Solving Nonlinear Stochastic Integral Equations, Vladikavkaz Math. J., 2020, vol. 22, no. 4, pp. 68-86. DOI: 10.46698/n8076-2608-1378-r.
1. Introduction
In recent years, the cardinal functions have been finding an important role in numerical analysis [1]. Both mathematicians and physicists have devoted considerable effort to find robust and stable analytical and numerical methods for solving stochastic differential equations, Adomian method [2], implicit Taylor methods [3, 4] and recently the operational matrices ofintegration for orthogonal polynomials Legendre wavelets, Chebyshev polynomials, etc. [5-20]. Several analytical and numerical methods have been proposed for solving various types of stochastic problems with the classical Brownian motion [10, 12, 14, 21-23]. Noting that finding the exact solutions for most of these equations is hard, therefore, we have to apply approximate numerical methods to obtain numerical solutions. This motivates our interest to propose an efficient and accurate computational method for solving stochastic integral equations. In [24] M. H. Heydari & al. used Chebyshev cardinal wavelets and their application in solving nonlinear stochastic differential equations with fractional Brownian motion. M. H. Heydari obtained a new method based on the Chebyshev cardinal functions for variable-order fractional optimal control problems [25]. An effective
© 2020 Zeghdane, R.
direct method to determine the numerical solution of Volterra-Fredholm integro-differential equations based on Chebyshev cardinal functions and deterministic operational matrices was also found in [26]. The aim of this paper is to use cardinal Chebyshev functions to solve nonlinear stochastic integral equations:
t t X (t) = Xo + y ki(t,s)[X (s)]p ds + y k2(t,s)[X (s)]q dB(s), (1)
0 0 under the initial condition X(0) = X0, where X(t) is an unknown process, the function k1 (t, s), k2(t, s) are defined on the square 0 ^ t, s ^ 1, X0 is a random variable, B(s) is a Brownian motion and p, q € N. After, we apply cardinal Chebyshev functions to SDE in the following general form
t t X(t) = X0 + y a(s,X(s)) ds + y b(s, X(s)) dB(s), (2)
00
where a(s,X(s,w)), b(s,X(s,w)) for s,t € [0,1] are known stochastic processes defined on the same filtered probability space (Q, F, Ft, P) with natural filtration Ft, X0 is the known random variable with E|X0|2 < +to and X(t) is unknown stochastic process which should be computed. The second integral in (2) is the Ito integral. Furthermore, all Lebesgue's and Ito integrals in (1) and (2) are well defined. The organization of this paper is as follows. In the second section, we give some preliminaries of stochastic calculus. We introduce Chebyshev cardinal functions and operational matrix of integration in Section 3. In Sections 4 and 5 we describe the numerical procedure of the numerical solution of the proposed problem. Convergence analysis of the method will be investigated in Section 6. To show the effectness of the numerical technique, some numerical examples are illustrated in Section 7. Finally, a brief conclusion is drawn on Section 8.
2. Preliminaries
Definition 1. Let V = V(S, T) be the class of functions g(t, w) : [0, to) —> R such that:
1) the function g(t, w) be B x F measurable, where B is the Borel a-algebra of R+;
2) the function g(t,w) is Ft-adapted (measurable);
3) E[/Jg2(t,w)dt] < to.
Lemma 1 (Ito isometry). For each X(t,w) € V(S,T), we have
E ( J X(s, w) dB(s) J = E ( J X2(s, w) ds J. 00 Lemma 2 (the Gronwall inequality). Let a, ft : [t0,T] —> R be integrable with
t
0 ^ a(t) ^ ft(t) + L J a(s) ds (3)
to
for t € [t0, T], where L> 0. Then
t
0 < a(t) < ft(t) + lJ eL(t-s)ft(s) ds, t € [t0,T]. (4)
to
3. Chebyshev Cardinal Functions
In this section, to construct the so called Chebyshev cardinal functions for the set of orthogonal Chebyshev polynomials TN(x), we will use the Taylor expansion of TN+i(x) in neighborhood the j-th root of TN+1(x), which gives
Tn+i(x) ~ Tn+1 (xj) + Tn+i,x(x - Xj) + o(x - Xj)2.
Since the first term in the right hand side vanishes, we can define the cardinal function of degree N in [-1,1] as follows [1, 27]
Cj(x) = Tn,+1^]-T, x € [-1,1], (5)
Tn+1,x (Xj )(x - Xj )
where the subscript x denotes x differentiation and xj are the zeros of TN+1(x) defined by
Xj = cos ((2{~ 1)7FV j = l,...,N + l, (6)
J V 2Ar + 2 /
with the kronecker property
Cj (xi) = Sji = s i if j i [0, if j = i.
3.1. Function approximation. To obtain cardinal functions in the interval [0,1], we change the variable t = then the shifted Chebyshev cardinal functions are defined on the interval [0,1] as follows:
C*(i)= Ci(2t - 1), i = 1,..., N + 1. (7)
Remark 1. The shifted Chebyshev cardinal functions are orthogonal with respect to the weight function w*(t) = w(2t — 1) on [0,1], where w(t) = j and we have
1
(C*(Í),C*(Í)> = J C*{t)C*{t)w*{t)dt = 2{n7[+ Sij. (8)
0
Theorem 1. Any function g(t) mean square integrable on [0,1] can expanded by element of shifted cardinal Chebyshev function as follow
N +1
g(t) = ¿ UjC*(t) = UT$n(t), (9)
j=1
where
xj + 1
are the shifted points of xj,
U = (U1,U2,...,UN+1)T, (t) = (Cld...,CN+1)T.
< If g(t) = EJl+i1 UC*(t), then
N +1
N+1
g(ti) = X) UjCJ'(ti) = E Uj ji-
i=l
i=l
Then ui = g(ti). >
Theorem 2. Any function g(t, s) mean square integrable on [0,1] x [0,1] can be approximated by cardinal functions as follow
N +1 N +1
= E E 'Sj)C*(i)C;(i) = $N(t)TKi$N(s), j = 1 i=1
where K1(iJ) = g(ij, Sj) and j, Sj are the corresponding shifted points of Xj. < The proof proceeds in a similar way as the proof of Theorem 1. > 3.2. Deterministic and stochastic operational matrices. Let
$N (t) = {C*1'C*2'...'C*N+1)T.
(10)
Lemma 3. We have
j $n(s) ds = P-1Q$n(t)
(11)
where the (N + 1) x (N + 1) matrix P is called the transform matrix (or Vandermonde's matrix) and is given by
/ 1 1. .1
tl t2 . - tN+l
P = t2 tl t2 . t2 . . t2 - bN+l
J.n -l J.n-l t2 . tN-l - bN+l
tN \ tl tN . t2 . - tN+l /
and
Q =
( tl t2 . - tN+l^
t2 2 2 ,2 lJV+l 2
,N-1 N-l ,N-1 2 N-l • ,N-1 lJV+l W-l
tN N ,N 2 ,N lJV+l
,N+1 1 ,N+1 2 iV+1 • ,N+1 rJV+l • w+i/
< Let ^j(i) = tl 1 for i = 1,... N + 1, by expanding ^(t) in (N + 1) terms of the shifted Chebyshev cardinal functions, we obtain
N+1
^i(t) = E ^(tj)Cj*(t), i = 1, 2,..., N + 1. j=1
Then
^ "i(t) \ "02 (t)
V^N +l(t)/
P
/ Ci(t) ^
C|(t)
VCN+i(t)/
= P $N (t).
t
Since the matrix P is invertible, $N(t) = P 1^N(t), where
^n (t) =
^ 01 (t) \ 02 (t)
V0n+1(t)7
Hence
t t $N (s) ds = f P-1^N(s) ds = P-1 f ^N(s) ds = P-1
t
tN+l
\n+T/
Now, let gi{t) = t, i = 1,2,..., N + 1, we have = J]^1 gi(tj)C*(t) = Q$N(t). Then
J $N(s) ds = P-1Q$N(t). >
Lemma 4. Assume $N(t) = (C, C|,... ,CN+^T and U = (u1,u2,... ,«N+1)T. Then
$n (t)$N (t)U = U $n (t),
(12)
where U = diag[u1,«2,...,«N+1]. < We have
$n(t)$N(t)U *
( C{(t)Ci(t) Ci(t)C|(t) ... Ci(t)CN+1(t) ^ C| (t)Ci (t) C| (t)C| (t) ... C| (t)CN+1(t)
VCN+1(t)Ci(t) cn+1(t)C|(t) ... cn+1(t)CN+1(t)/
/ «1 \
«2
V«N +1/
and expanding Ci(t)Cj(t), i,j = 1,2,...,N + 1, by the elements of Chebyshev cardinal functions, we get
N+1 N+1
Ci(t)Cj(t) * Ci(tfc)Cj(tk)Ck(t) * 5ifcjCfc(t). fc=1 fc=1
From this we conclude
$n (t)$N (t)U *
(C\ (t) 0 0 C2* (t)
0
/ «1 \
= U $N (t). >
or
. 0 «2
V 0 0 ... CN+1(t)/ V«N+1/
Lemma 5 [26]. If we consider X(t) * UT$N(t), then for every p € N, we have
[X(t)]P * Up$N(t) * UT(^)P-1$N(t),
[X(t)]p * [«1, «2, . . . , «N+1] ^N(t),
(13)
(14)
where U = diag(«1,«2,... ,«n+1).
t
2
t
2
t
3.3. Stochastic operational matrices of integration. In this subsection, we give stochastic operational matrix of integration with respect to Brownian motion we have
t t t J (s) dB(s) = y P(s) dB(s) = P-1 f ^N(s) dB(s)
0 0
' t t t /dB(s),/sdB(s), /sNdB(.)
Lo o o
we apply Ito formula, we get
/ t \
' JdB(s) X o
J sdB (s) o
} s2 dB(s) 0
,/sN dB(s) \o
= B(i)^N(t) -
/ B(s) ds 0 Jt
2 / sB(s) ds
N/sN-1B(s) ds o
= An (t) = (ai)i=o,....
N,
where
a, = t*B(t) - i si-1B(s) ds, i = 0,..., N.
For the integral J^ s* 1B(s) ds, we can use Simpson rule as follow
s^Bis) ds~l (&-lB{<d) B(1)+ t^Bit)
i = 1,2,
so
^ = mt) -1\ (4 (01 \ Q) + f-^w
i = 1, 2,
a, = B (t) for i = 0.
Also we approximate B(t) and £>(|) for 0 ^ t ^ 1 by 13(0.5) and 13(0.25), then we obtain
P-1An (t)
/B (0.5)
P
1
0
0 |.B(0.5) — |.B(0.25) 0
0 0
1
t t2
(l-f)i?(0.5)-3jUi3(0.25)/Viwy
0
t
t
0
Then where
P-1An (t) = P(t) = P-iAsP (t) = Ps$N (t)
(15)
/B (0.5)
As =
0 f 5(0.5) - | 5(0.25) 0
(l-f)5(0.5)-3^5(0.25)/
and Ps = P MsP is (N + 1) x (N + 1) stochastic operational matrix. Finally,
(t) dB(t) ~ Ps$N(t).
(16)
4. Numerical Method for Solving Stochastic Integral Equation (1)
In section, we describe numerical technique for solving stochastic integral equation (1), first we approximate the functions k1(t,s), k2(t,s) and X(t) by elements of the basis C*, i = 1, 2, . . . , N + 1, as follow
X(t) * UT$N(t), k1(t,s) * $N(t)TK1$N(s), k2(t,s) * $N(t)K2$N(s). (17)
Then, we approximate the integrals J^ k1(t, s)[X(s)]pds and f^ k2(t, s)[X(s)]qdB(s), we obtain
t t t
ykl(t,s)[X(s)]pds (t)T(s)$N(s)TUpds ~ (t)TKi J(s)$N(s)TUpds
0
~ (t)TKiU
(s) ds
(18)
~ (t)TKiUpP-1Q$N(t),
where Up = diag(u1, u2,..., uN+1) and Up are the coefficients of Xp(t) in the basis (t). Let Uq be the coefficients of Xq(t) in the basis (t). Then we have
k2(t,s)[X(s)]qdB(s) ~ / (t)TK2$N(s)$N(s)TUqdB(s)
(s) dB(s)
(19)
~ (t)T (s)$N (s)' Uq dB (s) ~ (t)T K2Uq
0
~ (t)TK2UqPs$N(t). We replace equations (17), (18) and (19) in equation (1), we get
UT(t) - Xo - (t)TKlUpP-1Q$n(t) - (t)TK2UqPs$N(t) = 0. (20)
To solve equation (20), we have three methods.
0
0
t
t
t
t
t
t
1. First, by collacting equation (20) in (N + 1) points tj, j = 1,2,... + 1, shifted points
of xj, we obtain
UT(tj) - Xo - (tj)TKlUpP(tj) - (tj)TK2Uq(tj) =0,
(21)
j = 1,2, + 1.
We have (tj) = e^, where e^ denotes the column of ordre j of identity matrix I of order N + 1. Then we obtain a nonlinear system included N + 1 unknowns (ul,u2,... +l)T and N + 1 equations, Newton method can be used to obtain accurate solution of nonlinear systems.
2. Here, we approximate (tj)TKlU7pP(tj) and (tj)TK2Uq(tj) as follow Lemma 6. We have
(t)TKlUpP(t) ^ Mi$N(t), (22)
(t)TK2U(t) ^ M2$N(t), (23)
where_Ml and M2 areJN + 1) row vectors including elements equal to the diagonal entries of KlUpP-lQ and K2UqPs respectibely.
< It is easy to proof identity (22) and (23). >
We replace (22) and (23) in equation (20), we get
UT(t) - Xo - Mi$n(t) - M2$w(t) = 0. (24)
Hence
[UT - Ao - Mi - M2]$w(t)=0, (25)
where A0 is (N + 1) row vector including elements equal to X0. The obtained system (25) is a nonlinear system with N + 1 unknowns (ul, u2,..., uN+l)T.
3. We can use orthogonality condition.
5. Solving Stochastic Integral Equation (2)
We approximate equation (2) as follows:
zi(t) = a(t,X(t)), z2(t) = b(t,X(t)), t € [0,1]. (26)
By using equation (2) and (26), we have
zi(t) = aft, X0 + / zi(s) ds + /z2(s)dB(s)^,
V 0 0 J
z2(t) = &(t, Xo + / zi(s) ds + / z2(s)dB(s^ . 00
(27)
By expanding zl(t) and z2(t) by elements of cardinal functions, we get
zi(t) = UT (t), z2(t) = U2T (t). (28)
By substituting equation (28) in (27), we obtain
t t
I r ttTP,
zi(t) = a t, X0 + / Uf (s) ds + / U2T(s) dB(s)
V 0 0
z2(t) = b(t, X0 + / Uf(s) ds + / U2T(s) dB(s)) ,
V 0 0 /
which is equivalent to
zi(t) = aft, X0 + Uf / (s) ds + U2T / (s) dB(s)) ,
< V 0 0 )
z2(t) = b(t, X0 + Uf (s) ds + U2T (s) dB(s)) . 00
By using equation (11) and (16), we get
iuf(t) = a(t, X0 + UiTP(t) + U2T(t)), \uf(t) = b(t, X0 + UfP(t) + U2T(t)).
We collocate (29) at shifted points tj, j = 1,2,... N + 1, and we arrive at
rUfeN = a(tj, X0 + UfP-lQeN + UfPsef), \ufeN = b(tj, X0 + Uf P-lQeN + Uf Pj),
(29)
(30)
(31)
(32)
where e^ denotes the column of ordre j of identity matrix I of order N + 1. The system (32) can be solved for the unknown Ul and U2 with Matlab software packages or by the Newton's iterative method. By determining Ul and U2, we can determine the approximate solution of X (t) as follow
Xn (x) = X0 + Uf P(t) + Uf (t). (33)
6. Convergence Analysis
In this section, we investigate the convergence and error analysis of the proposed method in the Sobolev space.
Definition 2 [28]. The Sobolev space Hm(a, b) is defined as follow:
Hm(a,b) = {u € LW(a, b), u(j)(t) € L^(a,b), j = 0,1,... ,m}, (34)
where w be a weight function and m ^ 0 be an integer.
Remark 2. The Sobolev space Hm(a, b) is endowed with the following weighted inner product
m b
<u(t),v(i)>mw = £ / u(jVjMi) dt. (35)
^=1 a
The space Hm(a, b) is a Hilbert space with the following norm
lu(t)llHm(a,6) = (a,6)J . (36)
i=1
Lemma 7 [28]. Let
1 N+l
u € H™(-1,1), w(t) = —== and Ijmu = Y^ UjCj(t)
v 1 — x2
be the Chebyshev interpolant of u(t) Then, the truncated error u — INu satisfies
|u - 1NU|LW(-1,1) < E I^Hl* (-1,1)) , (37)
- j=min(m,N)
where is a positive constant independent of N and dependent on m. Moreover, in the maximum norm, it yields
(m, \ è
j=min(m,N ) /
where Cm is a positive constant independent of N and dependent on m, and ||u||Loo(-1,1) = sup-1<t<1 |u(t)|. Theorem 3. Let
N+1
u € (0,1), w *(t) = w(2t - 1) and INu = ^ UjC*(t), Uj = u(tj)
j=1
be the Chebyshev interpolant of u(t). Then, the truncated error u - INu satisfies
( m í^\2j \ ^
Y: (2) 11^11^(0,1) ' (39)
\j=min(m,N) V 7 /
where is a positive constant independent of N and dependent on m. Moreover, in the maximum norm, it yields
'1
- j=min(m,N)
1« " ^IL^i) £ 2 11^11^(0,1) ' (4°)
1
2j \ 2
where is a positive constant independent of N and dependent on m, and ||u||Lk>(0,1) = supo <t<1|u(t)|.
< The proof proceeds in a same manner as the one of Theorem (5.4) in [24]. >
Theorem 4. Suppose X(t) € Hm(0,1) and XN(x) be the exact and approximate solutions of equation (2), respectively, furthermore, we suppose that
(H1) |a(t, X1 (t)) - a(t,X2(t))| + |b(t,X1 (t)) - b(t,X2(t))| < L|X1 - X2I (Lipschitz condition).
(H2) |a(t,X(t))| + |b(t,X(t))| < L(1 + |X|) (Linear growth condition), where t € [0,1], X1,X2 € R and Lj are positive constants for i = 1,2.
(H3) E|Xo|2 < TO. Then Xn(t) converges to X (t) in L2.
< Let ew (t) = X (t) — Xw (t) be an error function of approximate solution Xw (t) to the exact solution X(t),
t t X(t) — Xn(t) = J (zi(s) — Zi(s)) ds + J (Z2(s) — ¿2(s)) dB(s), (41)
a a
where Zj(t), i = 1,2, are given by zi(t) = a(t,X(t)), z2(t) = b(t,X(t)), also Zj(t), i = 1,2, is approximated form of zj(t) by schifted cardinal Chebyshev function
Zi(t) = appN (a(t, Xn (t)), ¿2(t) = appN (b(t,Xw (t)),
zf (t) = a(t, Xn (t)), z2N (t) = b(t, Xn t)), t t ew(t) = J (zi(s) — Zi(s)) ds + J (z2(s) — Z2(s)) dB(s),
E |ew(t)|2 = E
t
(zi(s) — Zi(s)) ds + J (z2(s) — Z2(s)) dB(s)
Using the inequality (a + b)2 ^ 2(a2 + b2), we get
00 2 ^ Oi^2 , T,2\
E |ew(t)|2 < 2E
t 2 t y (zi(s) — Zi(s)) ds +2E y (z2(s) — Z2(s)) dB(s)
by the Schwartz inequality and Ito isometry, we get
E|ew(t)|2 < 2E^ ^ |zi(s) — Zi(s)|2 dsj + 2E^ j |z2(s) — Z2(s)|2 ds j, consequently,
2E^ ^ |zi(s) — Zi(s)12 dsj < 4E^ y |zi(s) — zf (s)f dsj + 4E^ j |zf (s) — Zi(s)^ ds
2EÎ I |z2(s) — Z2(s)|2 ds] < 4EÎ I |z2(s) — zN(s)|2 ds] + 4EÎ I |z22(s) — Z2(s)|2 ds
By using Theorem 3, there exists aj (m, N), j = 1,2, such that
E||zf (s) — Zj(s)||2 < (aj(m,N))2, j = 1,2,
where
at(m,N ) = CmN-
m '1\2j
E
^ j=min(m,N)
VN )(j)
(o.i)
i = 1, 2.
2
2
t
t
t
Then
E|era(t)|2<4(ai(m,N) + a2(m,N))2 + 4Î ^E|zi(s) — z^(s)|2ds +JE |z2(s) — zn(s)|2dsj.
00
By using Lipschitz condition, we get
t
E|era(t)|2 < 4(ai(m,N) + a2(m,N))2 + 8^E|era(s)|2ds, (42)
0
hence by Gronwall inequality we obtain E|eN(t)|2 —> 0, as N —> oo. >
Remark 3. We can see that if m is sufficiently large than the error in Lemma (7) is sufficiently small.
7. Numerical Examples
To demonstrate the accuracy and effectiveness of the method proposed herein, we have applied it to several examples. These examples are solved in different references, so the numerical results obtained here can be compared with those of other numerical methods. In order to analyze the error of the method we introduce the absolute error, with M simulations
ew(t) = |X(t) - XN(t)|.
Example 1. Consider the deterministic Volterra integral equation of the kind as follows [29]:
t
- 8exp(21) + 6sin(i) + 3cos(i) + 5exp(-i)) - J (exp(s -t)+ sin(t - s)X(s)) ds,
0
where the exact solution is X (t) = exp(2t). The numerical results are summarized in Table 1.
Table 1. The absolute errors obtained by the proposed method with different values of N for Example 1
t N = 4 N = 10 N=15
0 8.1011 E- 3 6.4010 E- 9 8.3377 E- 14
0.2 5.3252 E- 3 6.6734 E- 9 2.5157 E- 13
0.4 9.8748 E- 3 1.0813 E- 9 3.5527 E- 15
0.6 1.0258 E- 3 8.5579 E- 9 2.0872 E- 14
0.8 1.5953 E- 2 4.6935 E- 9 1.4264 E- 12
1 7.2225 E 2 1.1335 E 7 3.9968 E 13
Example 2. Consider the deterministic Volterra integral equation of the second kind as follows:
t
X (t) = cos(t) — J (t — s) cos(t — s)X (s) ds, 0
where the exact solution is X(t) = |(2cos\/3t + 1). The numerical results are shown in Fig. 1-2.
0.95
0.8
0.85
Approximate solution
+ Exact solution
X 1 1 t-
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Fig. 1. The graphs of exact and approximate solutions for N = 4 for Example 2.
1.05
0.95
0.85,
0.05
0.1
0.15
0 2
0.25
0.3
0.35
0.4
Fig. 2. The graphs of exact and approximate solutions for N = 2 for Example 2.
The proposed method, can be also applied to nonlinear deterministic Fredhoml integral equations.
Example 3. Consider the Fredholm integral equation of the second kind [30]
X(t) = exp ( 2 t+[±y\+ f exp - Q )s )X(s) ds,
(43)
with the exact solution X(t) = exp(2t). The computational results are compared with that obtained in [30] and are illustrated in Table 2.
Table 2. The absolute errors obtained by the proposed method with different values of N for Example 3
t CJl N = 6 N = 10 rn = 64 [30] rn = 128 [30]
0 1.0589 E- 4 7.6683 E- 6 6.2495 E- 11 5.6999 E—5 4.0000 E-5
0.2 8.3927 E- 5 7.8574 E- 6 4.6068 E- 11 1.2000 E-4 1.9999 E- 5
0.4 4.1799 E- 5 8.3148 E- 6 5.3258 E- 11 9.9992 E-5 3.0000 E- 5
0.6 4.4243 E- 5 8.7391 E- 6 5.5025 E- 11 4.5999 E-4 4.9999 E- 5
0.8 9.9533 E- 5 9.1235 E- 6 5.0805 E- 11 7.5999 E-4 2.9999 E - 5
1 1.4078 E- -4 9.8403 E- ■6 7.3655 E- 11 3.5000 E-4 4.9999 E-5
Example 4. Consider the deterministic Riccati differential equation
u'(t) + u2(t) - 1 = 0, u(0)=0.
(44)
The exact solution is given by u(t) = exp(2t)+i • ^he numerical results of this example are given in Table 3, and are compared with the results obtained in [31].
i
Table 3. The absolute errors obtained by the proposed method with two values of N for Example 4
t N = 6 (Present method) N =12 (Present method) m = 12 [31]
0.1 1.2775 E-6 1.6259 E-11 1.11 E-10
0.2 2.6439 E-6 2.5123 E-12 2.04 E-10
0.3 1.3688 E-7 2.3986 E-11 2.10 E-12
0.4 2.8560 E-6 2.2805 E-11 2.23 E-10
0.5 7.3035 E-7 1.3141 E-11 4.03 E-10
0.6 2.39994 E-6 2.3181 E-13 1.79 E-10
0.7 9.8334 E-7 1.1980 E-11 8.59 E-11
0.8 2.5757 E-6 1.4748 E-11 2.70 E-10
0.9 2.8394 E-7 5.0553 E-12 1.89 E-10
1.0 2.6817 E 6 2.3652 E 11 2.66 E 11
Example 5. Let us consider the problem
t t
X(t) = Xo + J a2 cos(X(s)) sin3(X(s)) ds - a J sin2(X(s)) dB(s), t € [0,1]. (45)
0 0
The exact solution is X(t) = arccot(aB(s) + cot(X0)). The computed errors for N = 5, a = 1/8 and X0 = n/32, X0 = 0.1, X0 = 0.01, X0 = 1 are summarized in Table 4.
Table 4. The absolute errors obtained by the proposed method with different values of X0 for Example 5
t X0 = 0.01 X0 = tt/32 X0 = 0.1 X0 = 1
0 8.2145 E- 6 4.0132 E- 4 8.3099 E- 4 6.2593 E- 2
0.1 7.7400 E- 6 6.8875 E- 4 7.8514 E- 4 5.9772 E- 2
0.2 1.0725 E- 6 8.6983 E- 4 1.1750 E- 4 1.1500 E- 2
0.3 4.6979 E- 7 4.2429 E- 4 4.6663 E- 4 3.2472 E- 2
0.4 4.2535 E- 6 9.1225 E- 5 3.0996 E- 5 1.2364 E- 3
0.5 7.6467 E- 6 1.3240 E- 4 7.8170 E- 4 6.1292 E- 2
0.6 3.0515 E- 6 1.2116 E- 4 3.2278 E- 4 2.8464 E- 2
0.7 6.1677 E- 6 3.2922 E- 4 6.1092 E- 4 4.1968 E- 2
0.8 1.5208 E- 6 6.1442 E- 4 1.3615 E- 4 4.9793 E- 3
0.9 3.2037 E 6 9.8149 E 4 3.0564 E 4 1.7478 E 2
Example 6 (Stochastic Lotka-Volterra model). Lotka-Volterra model also known as the predator-prey equations, in deterministic subclasses, are well-known and have been an active area of research concerning ecological population modeling [32]. The logistic model is often represented as follow:
dX(t) = X(t) (61 - anX(t) - ai2Y(t))dt + ^X(t)dBi(t), dY(t) = Y(t) (62 - a2iX(t) - a22Y(t))dt + (t)dBi(t),
with initial conditions X(0) = X0, Y(0) = Y0, where a11, a12, a21, a22, 61, 62, a1 and are parameters. The application of the proposed method, gives the corresponding nonlinear system
UT = XT + 61UTP-1Q - anU2TP-1Q - a12UP-1Q + T, = Y0T + 62 VTP-1Q - a21VTUP-1Q - a22^2TP-1Q + TPB2,
where
X (t) = UT Qn (t), Y (t) = VtQn (t), X 2(t) = Uf Qn (t), Y 2(t) = V^Qn (t), F = diag[v1,v2,... ,^N+1], U = diag[«1,«2,...,«n+J,
F2 = « . . . v2+0T, U2 = (u2, u2, . . . «N+1)T,
with U = («1,«2,... ,«n+1), V = (v1,v2,...,vn+1). In this example, we take X (0) = 0.5, Y(0) = 1 and 61 = 20, B2 = -30, a11 = a22 = 0, a12 = a21 = 25 and a1 = = 1. We take M = 80 simulations for N = 8 and M = 30 for N = 5, we compute the means of X(t) and Y(t). The numerical results are shown in Fig. 3-4.
0 0.2 0.4 t 0.6 0.8 1
Fig. 3. Approximate solutions for M = 80 and N = 8 for Example 6.
0 0,2 0.4 t 0.6 0.8 1
Fig. 4. Approximate solutions for M = 30 and N = 5 for Example 6.
Example 7. Consider the following nonlinear stochastic Itô integral equation
t t X(t) = 1 + J X(t) ^ - X2(t)j dt + J 0.25X(t) d,B(t), t € [0,1],
0
exp(0.25B(t))
with the exact solution
X (t) =
'1 + 2/exp(0.5B(s)) ds
(46)
(47)
where X(t) is a stochastic process defined on the probability space (Q, F, P). The numerical results with M = 150 simulations are shown in Table 5 and are compared with the results obtained in [10].
Table 5. The absolute errors obtained by the proposed method with different values of N for Example 7
t N = 4: N = 8 N= 10 N = 4 [10] N = 8 [10] N = 10 [10]
0 1.6360 E- 3 4.0901 E- 2 1.4337 E- 1 8.17 E- 2 2.76 E- 2 9.29 E 2
0.1 3.4591 E 2 7.6948 E- 2 8.2714 E- 2 5.29 E- 2 2.51 E- 2 6.31 E 2
0.2 1.1814 E 1 6.9798 E- 2 1.3960 E- 3 2.89 E- 2 2.59 E- 2 3.86 E 2
0.3 9.468 E- 2 2.4183 E- 2 1.7747 E- 2 6.7 E- 3 3.06 E- 2 1.65 E 2
0.4 7.5338 E 2 9.7591 E- 3 8.4280 E- 3 1.59 E- 2 3.84 E- 2 4.3 E- 3
0.5 7.7120 E 2 6.4695 E- 3 5.9106 E- 2 4.12 E- 2 4.87 E- 2 2.41 E 2
0.6 6.1632 E 2 9.0339 E- 3 1.2380 E- 2 7.25 E- 2 6.08 E- 2 4.31 E 2
0.7 5.4542 E 2 1.0809 E- 1 4.8138 E- 2 1.141 E -1 7.42 E- 2 6.12 E 2
0.8 6.9447 E 2 6.1975 E- 2 4.8274 E- 2 1.714 E -1 8.89 E- 2 7.87 E 2
0.9 6.6438 E 2 2.3192 E 2 8.8528 E 2 2.512 E 1 1.055 E 1 9.55 E 2
Example 8 (the basic Black-Scholes model). Consider the following linear stochastic equation
dX(t) = XX(t)dt + ¡X(t)dW(t), X(0) = Xo, t € [0,1], (48) where the exact solution is given by
X{t) = exp ^A - ^t2) t + fJiWit)^.
The results obtained for X = -10, ¡i = 1, N = 5 and M = 100 simulations of this example are given in Table 6 and in Fig. 5-6.
Table 6. Computed errors for Example 8
t Xo = 0.001 Xo = 0.01 Xo = 1
0 6.0012 E- 5 2.1938 E- 3 1.2309 E- 1
0.1 7.1472 E- 4 2.5322 E- 3 9.5855 E- 2
0.2 8.3066 E- 4 1.5726 E- 3 2.8627 E- 2
0.3 7.0929 E- 4 6.0209 E- 4 4.0362 E- 2
0.4 4.8256 E- 4 4.1796 E- 4 1.8396 E- 2
0.5 2.3794 E- 4 3.8518 E- 4 2.7327 E- 2
0.6 5.5484 E- 5 4.4513 E- 4 3.7373 E- 2
0.7 2.6947 E- 5 5.3577 E- 4 2.9311 E- 2
0.8 1.8354 E- 6 5.6066 E- 4 5.1787 E- 3
0.9 8.6807 E- -5 5.4154 E- -4 2.0294 E- ■2
Fig. 5. Exact and approximate solutions for X0 = 0.01 for Example 8.
Fig. 6. Exact and approximate solutions for X0 = 0.001 for Example 8.
8. Conclusion
Some stochastic differential equations can be written as stochastic Volterra integral equations. There are many stochastic integral equations which can not be solved analytically. In recent decade, many researcher are trying to develop the numerical methods for solving stochastic integral equations. In this paper, we introduced the cardinal Chebyshev functions, then the deterministic and stochastic operational matrices of these orthogonal functions have been obtained. These matrices can be also used to solve linear and nonlinear differential equations. These cardinal functions was used and applied for solving linear and nonlinear Volterra integral equations. The convergence and error analysis of the proposed method were investigated. Finally, several examples were included to demonstrate the applicability of the presented approach, the method of Chebyshev cardinal functions proposed in this paper can be further expanded to solve systems of stochastic integro-and integral equations for futur studies.
Acknowledgments. The author wish to express their gratitude to the Editor and referees for their helpful comments, suggestions and careful reading of the paper which have helped to improve the quality of the paper.
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Received October 12, 2020 Rebiha Zeghdane
Department of Mathematics, Faculty of Mathematics and Informatics, University of Bordj-Bou-Arreridj, El-Anasser 34030, Bordj-Bou-Arreridj, Algeria, Associate Professor E-mail: rebihae@yahoo.fr
Владикавказский математический журнал 2020, Том 22, Выпуск 4, С. 68-86
НОВЫЙ ЧИСЛЕННЫЙ МЕТОД РЕШЕНИЯ НЕЛИНЕЙНЫХ СТОХАСТИЧЕСКИХ ИНТЕГРАЛЬНЫХ УРАВНЕНИЙ
Ребиха Зегдане1
1 Университет Бордж-Бу-Арреридж, Алжир, 34030, Бордж-Бу-Арреридж, Эль-Анасе E-mail: rebihae@yahoo.fr,
Аннотация. Цель статьи — применить кардинальные функции Чебышева к численному решению стохастических интегральных уравнений Вольтерра. Метод основан на разложении искомого приближенного решения по кардинальным функциями Чебышева. Для упомянутых базисных функций выводится новая операционная матрица интегрирования. Точнее, искомое решение разлагается в терминах кардинальных функций Чебышева с неизвестными коэффициентами. Подставляя указанное разложение в исходную задачу, операционная матрица сводит стохастическое интегральное уравнение к системе алгебраических уравнений. Исследованы сходимость и оценка погрешности в пространстве Соболева. Метод подвергнут численной оценке путем решения тестовых задач, взятых из литературы, с помощью которых демонстрируется вычислительная эффективность метода. С вычислительной точки зрения решение, полученное этим методом, отлично согласуется с решениями, полученными в других работах, и его эффективно использовать при решении различных задач.
Ключевые слова: кардинальные функции Чебышева, стохастическая операциональная матрица, броуновское движение, интеграл Ито, метод коллокации, численное решение. Mathematical Subject Classification (2010): 45G10, 65R20.
Образец цитирования: Zeghdane, R. New Numerical Method for Solving Nonlinear Stochastic Integral Equations // Владикавк. мат. журн.—2020.—Т. 22, № 4.—C. 68-86 (in English). DOI: 10.46698/n8076-2608-1378-r.
