Stochastic differential equations for eigenvalues and eigenvectors of a $G$-Wishart process with drift
We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the $G$-Wishart process defined according to a $G$-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the $G$-Brownian motion matr...
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| author | Boutabia, H. Meradji, S. Stihi, S. Бутабія, Г. Мераджи, С. Стихи, С. |
| author_facet | Boutabia, H. Meradji, S. Stihi, S. Бутабія, Г. Мераджи, С. Стихи, С. |
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| datestamp_date | 2019-12-05T10:12:29Z |
| description | We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the $G$-Wishart process
defined according to a $G$-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence
between the entries of the $G$-Brownian motion matrix, we assume in our model that their quadratic covariations are zero.
An intermediate result, which states that the eigenvalues never collide is also obtained. This extends Bru’s results obtained
for the classical Wishart process (1989). |
| first_indexed | 2026-03-24T02:05:41Z |
| format | Article |
| fulltext |
UDC 519.21
S. Meradji, H. Boutabia, S. Stihi (LaPS Laboratory, Badji-Mokhtar Univ., Algeria)
STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES
AND EIGENVECTORS OF A \bfitG -WISHART PROCESS WITH DRIFT
СТОХАСТИЧНI ДИФЕРЕНЦIАЛЬНI РIВНЯННЯ ДЛЯ ВЛАСНИХ ЗНАЧЕНЬ
I ВЛАСНИХ ВЕКТОРIВ \bfitG -ПРОЦЕСУ ВIШАРТА ЗI ЗНОСОМ
We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the G-Wishart process
defined according to a G-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence
between the entries of the G-Brownian motion matrix, we assume in our model that their quadratic covariations are zero.
An intermediate result, which states that the eigenvalues never collide is also obtained. This extends Bru’s results obtained
for the classical Wishart process (1989).
Запропоновано систему G-стохастичних диференцiальних рiвнянь для власних значень i власних векторiв G-
процесу Вiшарта, визначену, як i у класичному випадку, через G-броунiвську матрицю руху. З огляду на те, що
елементи G-броунiвської матрицi руху не є обов’язково незалежними, в нашiй моделi ми припускаємо, що їхнi
квадратнi коварiацiї дорiвнюють нулю. Отримано також промiжний результат про те, що власнi значення нiколи не
стикаються. Цей факт узагальнює результати Брю (1989), що отриманi для класичного процесу Вiшарта.
1. Introduction. Random matrices have been widely developed in recent years as a branch of
mathematics, but also as applications in many fields of sciences such as physics, biology, population
genetics, finance, meteorology and oceanography. The earliest studied ensemble of random matrices
is the Wishart ensemble, introduced by Wishart [7] in 1928 in the context of multivariate data
analysis, much before Wigner introduced the standard Gaussian ensembles of random matrices in the
physics literature. In physics, Wishart matrices have appeared in multiple areas: In nuclear physics,
quantum gravity and also in several problems in statistical physics. On the other hand, many studies
have been done on the asymptotic behavior of the eigenvalues of random matrices, in particular
by L. Pastur, M. Shcherbina [10]. In another context, Girko [4] used the perturbation technique to
give the stochastic differential equations (SDEs) of eigenvalues and eigenvectors for a matrix-valued
process with independent increments.
However, the notion of sublinear expectation space was introduced by Peng [3], which is a
generalization of classical probability space. The G-expectation, a type of sublinear expectation, has
played an important role in the researches of sublinear expectation space recently. Together with the
notion of G-expectations Peng also introduced the related G-normal distribution and the G-Brownian
motion. The G-Brownian motion is a stochastic process with stationary and independent increments
and its quadratic variation process is, unlike the classical case, a non deterministic process. Moreover,
an Itô calculus for the G-Brownian motion has been developed recently in [13 – 15].
The aim of this paper is to derive from the SDE of G-Wishart matrix with drift, a system of SDE
for its eigenvalues, eigenvectors and prove that the eigenvalues never collide. As in the classical case,
the G-Wishart matrix with drift is defined by Xt = (Bt + \eta t)T (Bt + \eta t) , where \eta is a deterministic
matrix and Bt is a G-Brownian motion matrix of dimension n\times n, the matrix stochastic process Xt
takes values in the space of symmetric n\times n matrices. In fact, our results are a generalization of the
works obtained by Bru [1] and by E. Mayerhofer [8] in the sense that the classical Brownian motion
is replaced by a G-Brownian motion. The main difficulties lie in the fact that the G-expectation is not
c\bigcirc S. MERADJI, H. BOUTABIA, S. STIHI, 2019
502 ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 503
linear and that the quadratic variation \langle B\rangle is not a deterministic process. The notion of independence
of random variables with respect to a non linear expectation being delicate, so we assume in our
model that
\bigl\langle
Bij , Bkl
\bigr\rangle
= 0 if (i, j) \not = (k, l) and
\bigl\langle
Bij
\bigr\rangle
depend only on j .
The remainder of this paper is organized as follows. In Section 2, we recall some notions
and properties in the G-expectation space which will be useful in this paper. Section 3 deals with
G-Brownian motion matrix and G-Wishart process with drift. In Section 4, we state the main results
of this paper, that is the study of SDEs satisfied by the eigenvalues and the eigenvectors processes
and the fact that the eigenvalues never collide. Section 5 is devoted to the proof of the main results.
2. Basic settings. For the convenience of the reader, we review some basic notions and results
of the G-expectation, the related spaces of random variables and the SDEs driven by a G-Brownian
motion (for more details see [3, 11, 12]).
Let \Omega be a given set and let \scrH be a linear space of real valued functions defined on \Omega , such
that c \in \scrH for each constant c and | X| \in \scrH if X \in \scrH . \scrH is considered as the space of random
variables.
Definition 1. A sublinear expectation on \scrH is a functional \^E : \scrH \rightarrow \BbbR satisfying the following
properties: For all X, Y \in \scrH , we have:
1) monotonicity: if X \geq Y, then \^E [X] \geq \^E[Y ];
2) preservation of constants: \^E [c] = c for all c \in \BbbR ;
3) subadditivity: \^E [X] - \^E [Y ] \leq \^E[X - Y ];
4) positive homogeneity: \^E [\lambda X] = \lambda \^E[X] for all \lambda \geq 0.
The triple
\bigl(
\Omega ,\scrH , \^E
\bigr)
is called a sublinear expectation space.
We denote by Cl,Lip (\BbbR n) the space of real continuous functions defined on \BbbR n such that
| \varphi (x) - \varphi (y)| \leq C(1 + | x| k + | y| k) | x - y| for all x, y \in \BbbR n,
where k \in \BbbN and C > 0 depend only on \varphi .
Definition 2. In a sublinear expectation space
\bigl(
\Omega ,\scrH , \^E
\bigr)
, a random vector Y \in \scrH n is said to
be independent from another random vector X \in \scrH m under \^E, if
\^E [\varphi (X,Y )] = \^E
\Bigl[
\^E [\varphi (x, Y )]
\bigm| \bigm| \bigm|
x=X
\Bigr]
\forall \varphi \in Cl,Lip
\bigl(
\BbbR m+n
\bigr)
.
Let X1 and X2 be two n-dimensional random vectors defined respectively in the sublinear
expectation spaces
\bigl(
\Omega 1,\scrH 1, \^E1
\bigr)
and
\bigl(
\Omega 2,\scrH 2, \^E2
\bigr)
. They are called identically distributed, denoted
by X1
d
= X2, if
\^E1 [\varphi (X1)] = \^E2 [\varphi (X2)] for all \varphi \in Cl,Lip (\BbbR n) .
If X is independent from X and X
d
= X, then X is said to be an independent copy of X .
After the above basic definition we introduce now the central notion of G-normal distribution.
Definition 3. Let be given two reals \sigma , \sigma with 0 \leq \sigma \leq \sigma . A random variable \xi in a sublinear
expectation space
\bigl(
\Omega ,\scrH , \^E
\bigr)
is called G\sigma ,\sigma -normally distributed, denoted by \xi \sim \scrN
\bigl(
0;
\bigl[
\sigma 2;\sigma 2
\bigr] \bigr)
,
if for each \varphi \in Cl,Lip (\BbbR ) , the following function defined by
u (t, x) = \^E
\Bigl[
\varphi
\Bigl(
x+
\surd
t\xi
\Bigr) \Bigr]
, (t, x) \in [0,\infty )\times \BbbR ,
is the unique viscosity solution of the parabolic partial differential equation
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
504 S. MERADJI, H. BOUTABIA, S. STIHI
\partial tu (t, x) = G
\bigl(
\partial 2
xxu (t, x)
\bigr)
, (t, x) \in [0,\infty )\times \BbbR ,
u (0, x) = \varphi (x) .
Here G = G\sigma ,\sigma is the following sublinear function parameterized by \sigma and \sigma :
G (\alpha ) =
1
2
\bigl(
\sigma 2\alpha + - \sigma 2\alpha - \bigr) , \alpha \in \BbbR
(recall that \alpha + = \mathrm{m}\mathrm{a}\mathrm{x}\{ 0, \alpha \} and \alpha - = - \mathrm{m}\mathrm{i}\mathrm{n}\{ 0, \alpha \} ). In fact \sigma 2 = \^E
\bigl[
\xi 2
\bigr]
and \sigma 2 = - \^E
\bigl[
- \xi 2
\bigr]
.
Definition 4. A stochastic process (Bt)t\geq 0 in a sublinear expectation space
\bigl(
\Omega ,\scrH , \^E
\bigr)
is called
a G-Brownian motion if the following properties are satisfied:
(i) B0 = 0;
(ii) for each t, s \geq 0, the increment Bt+s - Bt is \scrN
\bigl(
0;
\bigl[
\sigma 2s;\sigma 2s
\bigr] \bigr)
-distributed and independent
from (Bt1 , . . . , Btn) for all n \in \BbbN and 0 \leq t1 \leq . . . \leq tn \leq t.
Definition 5. A d-dimensional random vector X = (X1, . . . , Xd) in a sublinear expectation
space
\bigl(
\Omega ,\scrH , \^E
\bigr)
is called G-normal distributed if for each a, b \geq 0:
aX + bX
d
=
\sqrt{}
a2 + b2X,
where X is an independent copy of X, and
G (A) : =
1
2
\^E [\langle AX,X\rangle ] : \BbbS d \rightarrow \BbbR ,
here \BbbS d denotes the collection of d\times d symmetric matrices.
By [12] we know that X = (X1, . . . , Xd) is G- normal distributed if and only if u (t, x) :=
:= \^E
\bigl[
\varphi
\bigl(
x+
\surd
tX
\bigr) \bigr]
, (t, x) \in [0,\infty )\times \BbbR d, \varphi \in Cl,Lip (\BbbR n) is the unique viscosity solution of the
following G-heat equation:
\partial tu (t, x) = G (Du (t, x)) , (t, x) \in [0,\infty )\times \BbbR ,
u (0, x) = \varphi (x) ,
where Du (t, x) is the Hessian of u (t, x).
The function G (.) : \BbbS d \rightarrow \BbbR is a monotonic, sublinear functional on \BbbS d, from which we can
deduce that there exists a bounded, convex and closed subset \Sigma \in \BbbS +d the collection of nonnegative
matrices in \BbbS d such that
G (A) =
1
2
\mathrm{s}\mathrm{u}\mathrm{p}
B\in \Sigma
\mathrm{t}\mathrm{r} (AB) .
We write X \sim \scrN (0; \Sigma ).
We now give the definition of d-dimensional G-Brownian motion.
Definition 6. A d-dimensional process (Bt)t\geq 0 defined on
\bigl(
\Omega ,\scrH , \widehat \BbbE \bigr) is called a d- dimensional
G-Brownian motion if the following properties are satisfied:
(i) B0 = 0;
(ii) for all t, s \geq 0, the increment Bt+s - Bt is \scrN (0; s
\sum
)-distributed and independent of
(Bt1 , . . . , Btn) , for each n \in \BbbN and each sequence 0 \leq t1 \leq . . . \leq tn \leq t.
Note that \langle a,Bt\rangle is a real G\sigma a,\sigma a -Brownian motion for each a \in \BbbR d, where \langle ., .\rangle is tihe Euclidean
inner product of \BbbR d, \sigma a
2 = \^E
\Bigl(
\langle a,B1\rangle 2
\Bigr)
and \sigma a
2 = - \^E
\Bigl(
- \langle a,B1\rangle 2
\Bigr)
(for more details see [12]).
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 505
3. \bfitG -Whishart process with drift. In the following we will identify each n \times n matrix to
a vector of n2 dimension. Let us consider \Omega = C0 (\scrM n) be the set of all \scrM n-valued continuous
functions (\omega t)t\in \BbbR + with \omega 0 = 0, where \scrM n is the space of n \times n matrices, equipped with the
distance
\rho
\bigl(
\omega 1, \omega 2
\bigr)
=
\infty \sum
i=1
2 - i
\biggl[ \biggl(
\mathrm{m}\mathrm{a}\mathrm{x}
t\in [0,i]
\bigm| \bigm| \omega 1
t - \omega 2
t
\bigm| \bigm| \biggr) \wedge 1
\biggr]
, \omega 1, \omega 2 \in \Omega .
We denote by \scrB (\Omega ) the Borel \sigma -algebra on \Omega . We also set, for each t \in [0,\infty ), \Omega t :=
:= \{ \omega .\wedge t : \omega \in \Omega \} . The spaces of Lipschitzian functions on \Omega are denoted by
\mathrm{L}\mathrm{i}\mathrm{p} (\Omega t) = \{ \varphi (Bt1\wedge t, . . . , Btn\wedge t) : t1, . . . , tn \in [0;\infty ) , \varphi \in Cl,Lip (\scrM n)\} ,
\mathrm{L}\mathrm{i}\mathrm{p} (\Omega ) =
\infty \bigcup
n=1
\mathrm{L}\mathrm{i}\mathrm{p} (\Omega n) .
Here we use Cl,Lip (\scrM n) in our framework only for convenience. In general Cl,Lip (\scrM n) can
be replaced by the following spaces of functions defined on \scrM n :
L\infty (\scrM n): the space of all bounded Borel-measurable functions;
Cunif (\scrM n): the space of all bounded and uniformly continuous functions;
Cb.Lip (\scrM n): the space of all bounded and Lipschitz continuous functions;
\mathrm{L}\mathrm{i}\mathrm{p} (\scrM n): the space of all Lipschitzian functions on \scrM n .
As in Peng [11,12], we can construct a nonlinear expectation \^E on \mathrm{L}\mathrm{i}\mathrm{p} (\Omega ) , under which the
coordinate process (Bt) ( i.e., Bt (\omega ) = \omega t) is a G-Brownian motion matrix. Thus
\Bigl(
Bij
t
\Bigr)
is
a G\sigma ij ,\sigma ij -Brownian motion where \sigma ij
2 = \^E
\biggl[ \Bigl(
Bij
1
\Bigr) 2\biggr]
and \sigma ij
2 = - \^E
\biggl[
-
\Bigl(
Bij
1
\Bigr) 2\biggr]
for each i,
j \in 1, n.
Following Peng [3] (see also [11] for a simple proof), there exists a weakly compact family \scrP of
probability measures defined on (\Omega ,\scrB (\Omega )) such that
\^E [X] = \mathrm{s}\mathrm{u}\mathrm{p}
P\in \scrP
EP [X] for each X \in \mathrm{L}\mathrm{i}\mathrm{p} (\Omega ) ,
where EP stands for the linear expectation under P . We say that a property holds quasi surely (q.s.)
if it holds P a.s. for each P \in \scrP .
Let T > 0 be a fixed time. We denote by Lp
G (\Omega T ) , p \geq 1, the completion of G-expectation
space \mathrm{L}\mathrm{i}\mathrm{p} (\Omega T ) under the norm \| X\| p,G : =
\Bigl(
\^E [| X| p]
\Bigr) 1
p
. Peng [12] defined also the conditional
expectation \^E (. | \Omega t) , which is continuous on Lp
G (\Omega T ) .
Definition 7 [12]. A process (Mt)t\geq 0 is called a G-martingale if for each t \in [0;T ] , Mt \in
\in L1
G (\Omega t) and for each s \in [0, t] we have \^E [Mt | \Omega s] = Ms q.s.
For each partition \{ t0, . . . , tN\} = \pi T of [0, T ], such that 0 = t0 < t1 < . . . < tN = T, we set
\mu (\pi T ) = \mathrm{m}\mathrm{a}\mathrm{x} \{ | ti+1 - ti| : i = 0, . . . , N - 1\} .
Definition 8. Let Mp,0
G (0, T ;\BbbR ) be the collection of processes in the following form: for a given
sequence
\bigl(
\pi N
T
\bigr)
of partitions of [0, T ] such that \mathrm{l}\mathrm{i}\mathrm{m}N\rightarrow \infty \mu
\bigl(
\pi N
T
\bigr)
= 0,
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
506 S. MERADJI, H. BOUTABIA, S. STIHI
\eta t (\omega ) =
N - 1\sum
j=0
\xi j (\omega )\bfone [tj ,tj+1) (t) ,
where \xi j \in \mathrm{L}\mathrm{i}\mathrm{p}
\bigl(
\Omega tj
\bigr)
, j = 0, . . . , N - 1.
Definition 9. The quadratic variation of
\Bigl(
Bij
t
\Bigr)
t\geq 0
is defined by
\bigl\langle
Bij
\bigr\rangle
t
: = \mathrm{l}\mathrm{i}\mathrm{m}
\mu (\pi N
t )\rightarrow 0
N - 1\sum
l=0
\Bigl(
Bij
tl+1
- Bij
tl
\Bigr) 2
=
\Bigl(
Bij
t
\Bigr) 2
- 2
t\int
0
Bij
s dB
ij
s .
It was proved in [3,12] that \sigma ij2t \leq
\bigl\langle
Bij
\bigr\rangle
t
\leq \sigma ij
2t. We denote by Mp
G (0, T ;\BbbR ) the completion
of Mp,0
G (0, T ;\BbbR ) under the norm \| \eta \| Mp
G
=
\biggl(
\^E
\biggl[ \int T
0
| \eta s| p ds
\biggr] \biggr) 1/p
for p \geq 1.
Definition 10. For each \eta \in M2,0
G (0, T ;\BbbR ) , we define
I (\eta ) =
T\int
0
\eta tdB
ij
t : =
N - 1\sum
l=0
\xi j
\Bigl(
Bij
tl+1
- Bij
tl
\Bigr)
.
The mapping I : M2,0
G (0, T ;\BbbR ) \rightarrow L2
G (\Omega T ) is continuous and thus can be continuously extended
to M2
G (0, T ;\BbbR ).
Definition 11. The integral of a process \eta \in M1,0
G (0, T ;\BbbR ) with respect to
\bigl\langle
Bij
\bigr\rangle
t
is defined by
Q (\eta ) =
T\int
0
\eta td
\bigl\langle
Bij
\bigr\rangle
t
: =
N - 1\sum
l=0
\xi l
\Bigl( \bigl\langle
Bij
\bigr\rangle
tl+1
-
\bigl\langle
Bij
\bigr\rangle
tl
\Bigr)
.
The mapping Q : M1,0
G (0, T ;\BbbR ) \rightarrow L1
G (\Omega T ) is continuous and thus can be continuously ex-
tended to M1
G (0, T ;\BbbR ).
Unlike the classical theory, the quadratic covariation process of B is not always a deterministic
process and it can be formulated in L2
G (\Omega t) by
\Bigl\langle
Bij , Bkl
\Bigr\rangle
t
: = Bij
t B
kl
t -
t\int
0
Bij
s dB
kl
s -
t\int
0
Bkl
s dBij
s .
For the following generalized Itô formula (see [11] for the vectorial case), we use Einstein’s
notation.
Theorem 1. Let \varphi \in C2 (\scrM n) and its first and second derivatives are in Cb,Lip (\scrM n). Let
X =
\bigl(
Xij
\bigr)
be a matrix process on [0, T ] with the form
Xpq
t = Xpq
0 +
t\int
0
\alpha pq (s) ds+
t\int
0
\theta pqijkl (s) d
\Bigl\langle
Bij , Bkl
\Bigr\rangle
s
+
t\int
0
\beta pq
kl (s) dB
kl
s ,
where \alpha pq, \theta pqijkl \in M1
G (0, T ) and \beta pq
kl \in M2
G (0, T ). Then for each t \in [0, T ], we have
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 507
\varphi (Xt) - \varphi (X0) =
=
t\int
0
\partial xpq\varphi (Xu)\beta
pq
kl (u) dB
kl
u +
t\int
0
\partial xpq\varphi (Xu)\alpha
pq (u) du+
+
t\int
0
\biggl[
\partial xpq\varphi (Xu) \theta
pq
ijkl (u) +
1
2
\partial 2
xp\prime q\prime xpq\varphi (Xu)\beta
pq
ij (u)\beta p\prime q\prime
kl (u)
\biggr]
d
\Bigl\langle
Bij , Bkl
\Bigr\rangle
u
q.s.
Note that this formula remains valid if X is not a square matrix.
Remark 1. By a direct application of the G-Itô’s formula, we find the integration formula by
parts, that is
d (Xpq
t Xmn
t ) = dXpq
t Xmn
t +Xpq
t dXmn
t + dXpq
t dXmn
t ,
where dXpq
t dXmn
t =
\sum
i,j,k,l \beta
pqij
t \beta mnkl
t d
\bigl\langle
Bij , Bkl
\bigr\rangle
t
.
Then we have d
\bigl\langle
Bij , Bkl
\bigr\rangle
= dBijdBkl .
In the rest of this paper, we write
\bigl\langle
Bij
\bigr\rangle
t
instead of
\bigl\langle
Bij , Bij
\bigr\rangle
t
and we assume that B satisfies
the following assumption:
(A) There exist an increasing real process bj such that
\bigl\langle
Bij , Bkl
\bigr\rangle
t
= \delta ik\delta jlb
j
t q.s. for each i,
j, k, l \in 1, n, where \delta uv is the Kronecker symbol.
Then we get \sigma 2t \leq bjt \leq \sigma 2t, where \sigma := \mathrm{m}\mathrm{a}\mathrm{x}i,j \sigma ij and \sigma := \mathrm{m}\mathrm{i}\mathrm{n}i,j \sigma ij . Note that in the
classical case the assumption (A) is satisfied with bjt = t.
Definition 12. A G-Wishart process with drift is defined by Xt = (Bt + \eta t)T (Bt + \eta t) , where
\eta is a deterministic matrix, and Y T denotes the transpose of a matrix Y .
Note that if B is the classical Brownian motion, then X is the classical Wishart process with drift,
which appeared in many different applications such as communication technology, nuclear physics,
quantum chromodynamics, statistical physics of directed polymers in random media.
As in the classical case, we define the Stratonovich differential \circ for two matrices X and Y :
X \circ dY = XdY +
1
2
dXdY and dX \circ Y = dXY +
1
2
dXdY,
where dXdY is the matricial product. The following proposition holds.
Proposition 1. For each matrices X,Y defined as in the above theorem, we have:
(i) the integration formula by parts:
d (XY ) = XdY + dXY + dXdY,
(ii) the formulae
d (XY ) = dX \circ Y +X \circ dY,
dX \circ (Y Z) = (dX \circ Y ) \circ Z
and
(X \circ dY )T = dY T \circ XT .
Proof. (i) follows from the Remark 1.
(ii) follows from (i) and the definition of the Stratonovich differential.
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
508 S. MERADJI, H. BOUTABIA, S. STIHI
4. Main results. We wish to find SDEs of eigenvalues and eigenvectors for a G-Wishart
process with drift. Our approach is the same as that used in [1]. The idea is simply to set
Mt = Bt + \eta t.
Then we have
Xt = MT
t Mt,
and
dXt = dBT
t Mt +MT
t dBt +
\bigl(
\eta TMt +MT
t \eta
\bigr)
dt+ dBT
t dBt.
Note that dBT
t dBt = nd \langle B\rangle t . Since dXiidXii = 4Xiidbi, then
dXii = 2
\surd
Xiid\kappa i + 2
\bigl(
\eta TM
\bigr) ii
dt+ ndbi, i = 1, . . . , n,
where \kappa i are G-Brownian motions, such that
\bigl\langle
\kappa i, \kappa j
\bigr\rangle
= \delta ijb
j for each i, j .
In what follows, let HT
t XtHt = \Lambda t := \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{g} (\lambda i (t)) be the diagonalization form of Xt, where
Ht is an orthonormal matrix. The following result is the equivalent of Theorem 18.5.1 [4], proved
in the linear context for a matrix-valued process with independent increments.
Theorem 2. Suppose that at time t = 0 all the eigenvalues are distinct. Then the following
G-stochastic differential system holds:
d\lambda i = 2
\sqrt{}
\lambda i
\sum
l
H lid\nu l + 2 (\eta H)ii
\sqrt{}
\lambda idt+
\sum
l
\Bigl(
H li
\Bigr) 2
dbl
\left[ \sum
p \not =i
\lambda p
\lambda i - \lambda p
+ n
\right] +
+\lambda i
\sum
p \not =i
1
\lambda i - \lambda p
\sum
l
\Bigl(
H lp
\Bigr) 2
dbl, (1)
where \nu 1, \nu 2, . . . , \nu n are G-Brownian motions, satisfying
\bigl\langle
\nu i, \nu j
\bigr\rangle
= \delta ijb
j for each i, j \in 1, n.
Corollary 1. Assume that bi = b for each i = 1, . . . , n. Then we have the following SDEs: for
t < T0: = \mathrm{i}\mathrm{n}\mathrm{f} \{ t:\mathrm{d}\mathrm{e}\mathrm{t} (Xt) = 0\}
d (\mathrm{t}\mathrm{r} (Xt)) = 2
\sqrt{}
\mathrm{t}\mathrm{r} (Xt)d\gamma t + 2\mathrm{t}\mathrm{r}
\Bigl(
\eta T
\sqrt{}
Xt
\Bigr)
dt+ n2dbt, (2)
d (\mathrm{d}\mathrm{e}\mathrm{t} (Xt)) = 2 \mathrm{d}\mathrm{e}\mathrm{t}Xt
\sqrt{}
\mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
d\beta t + 2\mathrm{d}\mathrm{e}\mathrm{t}Xt\mathrm{t}\mathrm{r}
\Bigl(
\eta HX - 1
2H
\Bigr)
dt+
+\mathrm{d}\mathrm{e}\mathrm{t}Xt\mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
dbt, (3)
d (\mathrm{l}\mathrm{o}\mathrm{g} (\mathrm{d}\mathrm{e}\mathrm{t} (Xt))) = 2
\sqrt{}
\mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
d\beta t + 2\mathrm{t}\mathrm{r}
\Bigl(
\eta HX - 1
2H
\Bigr)
dt - \mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
dbt, (4)
d (\mathrm{d}\mathrm{e}\mathrm{t} (Xt)
r) = 2r (\mathrm{d}\mathrm{e}\mathrm{t}Xt)
r
\sqrt{}
\mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
d\beta t + 2r (\mathrm{d}\mathrm{e}\mathrm{t}Xt)
r \mathrm{t}\mathrm{r}
\Bigl(
\eta HX - 1
2H
\Bigr)
dt+
+r (2r - 1) (\mathrm{d}\mathrm{e}\mathrm{t}Xt)
r \mathrm{t}\mathrm{r}
\bigl(
X - 1
t
\bigr)
dbt for r \in \BbbR , (5)
where \gamma (resp. \beta ) is a G-real Brownian motion, such that \langle \gamma \rangle = b (resp. \langle \beta \rangle = b).
Remark 2. It is easy to derive from this corollary, the corresponding of these SDEs in the
classical case which are obtained by Demni [2].
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STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 509
As in [1], the matrix Ht satisfy
dH = H \circ dA = HdA+
1
2
HdAdA,
where A is the skew-symmetric n\times n matrix, such that
dA = HT \circ dH.
Theorem 3. The eigenvectors satisfy the following equations:
dH ij =
\sum
k \not =j
H ik 1
\lambda j - \lambda k
\Biggl( \sqrt{}
\lambda k
\sum
l
H ljd\beta jl +
\sqrt{}
\lambda j
\sum
l
H lkd\beta kl
\Biggr)
+
1
2
\sum
k
H ikdV kj , (6)
where
dV kj =
\sum
p \not =j,p\not =k
1
(\lambda p - \lambda k) (\lambda j - \lambda p)
\Biggl[
\lambda p
\sum
l
H lkH ljdbl + \delta kj\lambda k
\sum
l
H lpH lpdbl
\Biggr]
,
and
\bigl(
\beta ij
\bigr)
is a G-Brownian motion matrix satisfying the assumption (A).
Collision time. Let us consider the first collision time \tau = \mathrm{i}\mathrm{n}\mathrm{f}
\bigl\{
t \geq 0 : \lambda j (t) - \lambda i (t) = 0 for
some i \not = j
\bigr\}
.
Corollary 2. If, at time t = 0, the eigenvalues of X are distinct \lambda 1 < \lambda 2 < . . . < \lambda n, then they
will never collide, that is, \tau = +\infty q.s.
Proof. Let (\lambda j (t) - \lambda i (t))t<\tau be the \BbbR +\setminus \{ 0\} valued stochastic process. As in [1], by applying
the G-Itô formula to U = -
\sum
i<j
\mathrm{l}\mathrm{o}\mathrm{g} (\lambda j - \lambda i) and by using the fact that the quadratic covariation
of \lambda i, \lambda j is equal to 4\delta ij
\surd
\lambda i
\sqrt{}
\lambda jdb
i, we get
dU =
\sum
i<j
d\lambda i - d\lambda j
\lambda j - \lambda i
+ 2
\sum
i<j
\lambda idb
i + \lambda jdb
j
(\lambda j - \lambda i)
2 .
It follows from the SDE satisfied by \lambda i that
\langle U\rangle t = 4
\sum
i<j
t\int
0
\lambda i (s)
\sum
l
\Bigl(
H li
s
\Bigr) 2
dbls + \lambda j (s)
\sum
l
\Bigl(
H lj
s
\Bigr) 2
dbls
(\lambda j (s) - \lambda i (s))
2 .
Obviously Ut is a classical local martingale with respect to its natural filtration under each probability
measure P \in \scrP . If two eigenvalues collide, then there exists P 0 \in \scrP such that P 0 (\tau = +\infty ) < 1,
\mathrm{l}\mathrm{i}\mathrm{m}t\uparrow \tau Ut = - \infty and U is continuous on [0, \tau ]. Let \xi t be the inverse of \langle U\rangle t . By the argument
of McKean [9, p. 47], and Bru [1], the process \widetilde Bt = U\xi t is a Brownian motion on [0, \langle U\rangle \tau [ P 0 a.s.
on \{ \tau < +\infty \} [5, p. 92]. Thus,
\mathrm{l}\mathrm{i}\mathrm{m}
t\uparrow \langle U\rangle (\tau )
\widetilde Bt = \mathrm{l}\mathrm{i}\mathrm{m}
\xi t\uparrow \tau
U\xi t = \mathrm{l}\mathrm{i}\mathrm{m}
t\uparrow \tau
Ut = - \infty ,
which is impossible for a Brownian motion. Hence \tau = +\infty q.s.
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510 S. MERADJI, H. BOUTABIA, S. STIHI
5. Proofs of theorems. Proof of Theorem 2. Since\bigl(
dMTdM
\bigr) ij
=
\sum
k
dBkidBkj = n\delta ijdb
i,
then
dXij =
\sum
k
\Bigl(
MkjdBki +MkidBkj
\Bigr)
+
\sum
k
\Bigl(
\eta kiMkj +Mki\eta kj
\Bigr)
dt+ n\delta ijdb
i,
which implies that
dXijdXkm =
\sum
p
\bigl(
MpjdBpi +MpidBpj
\bigr) \sum
q
\Bigl(
M qmdBqk +M qkdBqm
\Bigr)
=
=
\Bigl(
Xjm\delta ik +Xjk\delta im
\Bigr)
dbi +
\Bigl(
Xim\delta jk +Xik\delta jm
\Bigr)
dbj .
As in [1], we have d\Lambda = dN - dA \circ \Lambda +\Lambda \circ dA, where dN : = HT \circ dX \circ H and introduce the
skew-symmetric matrix A such that, A0 = 0, dA = HT \circ dH . Then we obtain
dH = HdA+
1
2
HdAdA.
The process \Lambda \circ dA - dA \circ \Lambda is zero on the diagonal, consequently d\lambda i = dN ii and
0 = dN ij + (\lambda i - \lambda j) dA
ij , when i \not = j. Thus, for \{ t < \tau \} ,
dAij =
1
\lambda j - \lambda i
dN ij .
In fact dN = HTdXH +
1
2
HTdXdH +
1
2
dHTdXH, which implies that the G-martingale part
of dN equals the G-martingale part of HTdXH given by
dN ijdNkm =
\sum
p,q
HpidXpqHqj
\sum
p\prime ,q\prime
Hp\prime kdXp\prime q\prime Hq\prime m =
=
\sum
p,q,p\prime ,q\prime
HpiHqjHp\prime kHq\prime m
\Bigl[ \Bigl(
Xqq\prime \delta pp\prime +Xqp\prime \delta pq\prime
\Bigr)
dbp +
+
\Bigl(
Xpq\prime \delta qp\prime +Xpp\prime \delta qq\prime
\Bigr)
dbq
\Bigr]
=
=
\sum
p
\left[ HpiHpkdbp
\sum
q,q\prime
HqjXqq\prime Hq\prime m +
+HpiHpmdbp
\sum
q,p\prime
HqjXqp\prime Hp\prime k
\right] +
+
\sum
q
\left[ HqjHqkdbq
\sum
p,q\prime
HpiXpq\prime Hq\prime m +
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STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 511
+HqjHqmdbq
\sum
p,p\prime
HpiXpp\prime Hp\prime k
\right] =
=
\sum
p
HpiHpkdbp\Lambda jm +
\sum
p
HpiHpmdbp\Lambda jk+
+
\sum
q
HqjHqkdbq\Lambda im +
\sum
q
HqjHqmdbq\Lambda ik (7)
and
dN iidN jj = 4\Lambda ij
\sum
p
HpiHpjdbp =
= 4\lambda i\delta ij
\sum
p
\bigl(
Hpi
\bigr) 2
dbp.
The finite-variation part of dN
dF = HT
\bigl(
\eta TM +MT \eta
\bigr)
Hdt,
since M = \Lambda
1
2HT , then
dF =
\Bigl(
HT \eta T\Lambda
1
2HTH +HTH\Lambda
1
2 \eta H
\Bigr)
dt =
=
\Bigl(
HT \eta T\Lambda
1
2 + \Lambda
1
2 \eta H
\Bigr)
dt.
It follows that
dF ii = 2 (\eta H)ii
\sqrt{}
\lambda idt.
Now we compute the integral part dQ of dN, with respect to dbi :
dQ = HTdBTdBH +
1
2
HTdXdH +
1
2
dHTdXH =
= HTdBTdBH +
1
2
\bigl( \bigl(
dHTH
\bigr) \bigl(
HTdXH
\bigr)
+
\bigl(
HTdXH
\bigr) \bigl(
HTdH
\bigr) \bigr)
=
= HTdBTdBH +
1
2
\Bigl(
dNdA+ (dNdA)T
\Bigr)
.
Since \bigl(
HTdBTdBH
\bigr) ij
=
\sum
p,q
Hpi
\bigl(
dBTdB
\bigr) pq
Hqj
and \bigl(
dBTdB
\bigr) pq
=
\sum
l
dBlpdBlq =
\sum
l
\delta pqdb
p = n\delta pqdb
p,
then \bigl(
HTdBTdBH
\bigr) ij
=
\sum
p,q
Hpin\delta pqdb
pHqj = n
\sum
p
HpiHpjdbp. (8)
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512 S. MERADJI, H. BOUTABIA, S. STIHI
On the other hand,
(dNdA)ij =
\sum
p
dN ipdApj =
\sum
p \not =j
dN ip 1
\lambda j - \lambda p
dNpj =
=
\sum
p \not =j
1
\lambda j - \lambda p
\Biggl[ \sum
l
H liH lpdbl\Lambda pj +
\sum
l
H liH ljdbl\Lambda pp +
+
\sum
l
H lpH lpdbl\Lambda ij +
\sum
l
H lpH ljdbl\Lambda ip
\Biggr]
=
=
\sum
p \not =j
1
\lambda j - \lambda p
\Biggl[
\lambda p
\sum
l
H liH ljdbl + \lambda i\delta ij
\sum
l
\Bigl(
H lp
\Bigr) 2
dbl +
+\lambda i\delta ip
\sum
l
H lpH ljdbl
\Biggr]
. (9)
It now follows from (8) and (9), that
dQii =
\sum
l
\Bigl(
H li
\Bigr) 2
dbl
\left[ \sum
p \not =i
\lambda p
\lambda i - \lambda p
+ n
\right] + \lambda i
\sum
p \not =i
1
\lambda i - \lambda p
\sum
l
\Bigl(
H lp
\Bigr) 2
dbl.
Then
d\lambda i = 2
\sqrt{}
\lambda i
\sum
l
H lid\nu l + dF ii + dQii,
where \nu l is a G-Brownian motion such that d\nu id\nu j = \delta ijdb
i .
Theorem 2 is proved.
Proof of Corollary 1. We have
d (\mathrm{t}\mathrm{r} (Xt)) =
\sum
i
dXii
t =
=
\sum
i
\biggl(
2
\sqrt{}
Xii
t d\kappa
i
t + 2
\Bigl(
\eta T
\sqrt{}
Xt
\Bigr) ii
dt+ ndbt
\biggr)
.
On the other hand, since the quadratic variation of \mathrm{t}\mathrm{r} (X) is 4
\sum
i
Xiidb = 4\mathrm{t}\mathrm{r} (X) db, then
d (\mathrm{t}\mathrm{r} (Xt)) = 2
\sqrt{}
\mathrm{t}\mathrm{r} (Xt)d\gamma t + 2\mathrm{t}\mathrm{r}
\Bigl(
\eta T
\sqrt{}
Xt
\Bigr)
dt+ n2dbt.
Firstly, observe that, by using the formula (1), we obtain
d\lambda i = 2
\sqrt{}
\lambda idv
i + 2 (\eta H)ii
\sqrt{}
\lambda idt+
\left[ \sum
p \not =i
\lambda p + \lambda i
\lambda i - \lambda p
+ n
\right] db.
By using the G-Itô formula, we have
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STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 513
d (\mathrm{d}\mathrm{e}\mathrm{t}X) =
\sum
i
\mathrm{d}\mathrm{e}\mathrm{t}X
\lambda i
d\lambda i +
1
2
\sum
i \not =j
\mathrm{d}\mathrm{e}\mathrm{t}X
\lambda i\lambda j
d\lambda id\lambda j .
It follows, from the fact that d\lambda id\lambda j = 4
\surd
\lambda i
\sqrt{}
\lambda j\delta ijdb, that
d (\mathrm{d}\mathrm{e}\mathrm{t}X) = \mathrm{d}\mathrm{e}\mathrm{t}X
\sum
i
d\lambda i
\lambda i
=
= 2\mathrm{d}\mathrm{e}\mathrm{t}X
\sum
i
dvi\surd
\lambda i
+ 2\mathrm{d}\mathrm{e}\mathrm{t}X
\sum
i
(\eta H)ii\surd
\lambda i
dt+
+\mathrm{d}\mathrm{e}\mathrm{t}X
\sum
i
1
\lambda i
\left[ \sum
p\not =i
\lambda p + \lambda i
\lambda i - \lambda p
+ n
\right] db.
The formula (3) follows from the following facts:
\sum
i
(\eta H)ii\surd
\lambda i
= \mathrm{t}\mathrm{r}
\Bigl(
\eta H\Lambda - 1
2
\Bigr)
= \mathrm{t}\mathrm{r}
\Bigl(
\eta HX - 1
2H
\Bigr)
,
\sum
i
1
\lambda i
\left[ \sum
p \not =i
\lambda p + \lambda i
\lambda i - \lambda p
+ n
\right] =
\sum
i
n
\lambda i
+
\sum
i
\sum
p \not =i
\biggl(
- 1
\lambda i
+
2
\lambda i - \lambda p
\biggr)
=
= n\mathrm{t}\mathrm{r}
\bigl(
X - 1
\bigr)
- (n - 1) \mathrm{t}\mathrm{r}
\bigl(
X - 1
\bigr)
+ 2
\sum
i
\sum
p \not =i
1
\lambda i - \lambda p
=
= \mathrm{t}\mathrm{r}
\bigl(
X - 1
\bigr)
,
and the quadratic variation of \mathrm{d}\mathrm{e}\mathrm{t}X is 4 (\mathrm{d}\mathrm{e}\mathrm{t}X)2
\sum
i,j
1\surd
\lambda i
1\sqrt{}
\lambda j
\delta ijdb = 4\mathrm{d}\mathrm{e}\mathrm{t}X2\mathrm{t}\mathrm{r}
\bigl(
X - 1
\bigr)
db.
Equations (4) and (5) follows from (3), the G-Itô formula and the quadratic variation of \mathrm{d}\mathrm{e}\mathrm{t}X .
Proof of Theorem 3. In order to find SDEs for Ht on \{ t < \tau \} , we deduce from the definition
of dA that
dH = H \circ dA = HdA+
1
2
HdAdA.
By using the formula (7), we have, for i \not = j,
dN ijdN ij = \lambda i
\sum
l
\Bigl(
H lj
\Bigr) 2
dbl + \lambda j
\sum
l
\Bigl(
H li
\Bigr) 2
dbl,
which implies that
dN ij =
\sqrt{}
\lambda i
\sum
l
H ljd\beta jl +
\sqrt{}
\lambda j
\sum
l
H lid\beta il,
where
\bigl(
\beta il
\bigr)
is a G-Brownian motion matrix satisfying the assumption (A). It follows that
dAij =
1
\lambda j - \lambda i
\Biggl( \sqrt{}
\lambda i
\sum
l
H ljd\beta jl +
\sqrt{}
\lambda j
\sum
l
H lid\beta il
\Biggr)
.
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514 S. MERADJI, H. BOUTABIA, S. STIHI
Now we compute (dAdA)ij :
(dAdA)ij =
\sum
p
dAipdApj =
=
\sum
p \not =j,p\not =i
1
(\lambda p - \lambda i) (\lambda j - \lambda p)
dN ipdNpj =
=
\sum
p \not =j,p\not =i
1
(\lambda p - \lambda i) (\lambda j - \lambda p)
\Biggl[ \sum
l
H liH lpdbl\Lambda pj +
\sum
l
H liH ljdbl\Lambda pp +
+
\sum
l
H lpH lpdbl\Lambda ij +
\sum
l
H lpH ljdbl\Lambda ip
\Biggr]
=
=
\sum
p \not =j,p\not =i
1
(\lambda p - \lambda i) (\lambda j - \lambda p)
\Biggl[
\lambda p
\sum
l
H liH ljdbl + \delta ij\lambda i
\sum
l
H lpH lpdbl
\Biggr]
.
Similarly as in [1], we obtain the formula (6).
Theorem 3 is proved.
Example 1. We consider the case of the classical Wishart process, which corresponds to \eta = 0
and Xt = BT
t Bt, where (Bt) is the classical Brownian motion matrix. It was shown in [1] that
d\lambda i = 2
\sqrt{}
\lambda id\nu
i +
\left[ \sum
p \not =i
\lambda p + \lambda i
\lambda i - \lambda p
+ n
\right] dt,
where \nu i are classical Brownian motions. We can obtain this formula by the formula (1) with bit = t
and the fact that
\sum
l
\Bigl(
H li
\Bigr) 2
= 1. The same is true for the SDE of the eigenvectors.
Remark 3. If we consider the G-Wishart process Xt = Y T
t Yt where Y is the G-Ornstein –
Uhlenbeck matrix, that is the solution of the G-SDE
dYt = - 1
2
Ytdt+ adBt, a > 0,
where B is a G-Brownian motion satisfying the assumption (A), we can obtain with the same manner
(see [6]) that
d\lambda i = 2a
\sqrt{}
\lambda i
\sum
l
H lid\nu l - \lambda idt+ a2
\left[ \sum
p \not =i
\lambda p
\lambda i - \lambda p
+ n
\right] \sum
l
\Bigl(
H li
\Bigr) 2
dbl+
+a2\lambda i
\sum
p \not =i
1
\lambda i - \lambda p
\sum
l
\Bigl(
H lp
\Bigr) 2
dbl,
and
dH ij = a
\sum
k \not =j
H ik 1
\lambda j - \lambda k
\Biggl( \sqrt{}
\lambda k
\sum
l
H ljd\beta jl +
\sqrt{}
\lambda j
\sum
l
H lkd\beta kl
\Biggr)
+
a2
2
\sum
k
H ikdV kj .
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STOCHASTIC DIFFERENTIAL EQUATIONS FOR EIGENVALUES AND EIGENVECTORS . . . 515
6. Conclusion. In this paper, the system of the SDEs of eigenvalues and eigenvectors for a
G-Wishart process with drift defined by using a G-Brownian motion matrix was given. This system
has been difficult to obtain because of the fact that the quadratic variation of the G-Brownian motion
is not deterministic. Added to that, our main difficulty lies in the fact that the entries of the G-
Brownian motion matrix are not independent in general. To avoid these difficulties, it was assumed
in our model, that the quadratic covariations of the entries of the G-Brownian motion matrix are zero
and that the quadratic variations depend only on the index of column. The G-formula of integration
by parts was the key of this work. An intermediate result of the non collision of the eigenvalues was
also proven.
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Received 25.04.16
ISSN 1027-3190. Укр. мат. журн., 2019, т. 71, № 4
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| id | umjimathkievua-article-1454 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:05:41Z |
| publishDate | 2019 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/81/ecf2ce64d67b9370af74a2ee865c5e81.pdf |
| spelling | umjimathkievua-article-14542019-12-05T10:12:29Z Stochastic differential equations for eigenvalues and eigenvectors of a $G$-Wishart process with drift Стохастичнi диференцiальнi рiвняння для власних значень i власних векторiв $G$ -процесу вiшарта зi зносом Boutabia, H. Meradji, S. Stihi, S. Бутабія, Г. Мераджи, С. Стихи, С. We propose a system of G-stochastic differential equations for the eigenvalues and eigenvectors of the $G$-Wishart process defined according to a $G$-Brownian motion matrix as in the classical case. Since we do not necessarily have the independence between the entries of the $G$-Brownian motion matrix, we assume in our model that their quadratic covariations are zero. An intermediate result, which states that the eigenvalues never collide is also obtained. This extends Bru’s results obtained for the classical Wishart process (1989). УДК 519.21 Запропоновано систему $G$-стохастичних диференцiальних рiвнянь для власних значень i власних векторiв $G$- процесу Вiшарта, визначену, як i у класичному випадку, через $G$-броунiвську матрицю руху. З огляду на те, що елементи $G$-броунiвської матрицi руху не є обов’язково незалежними, в нашiй моделi ми припускаємо, що їхнi квадратнi коварiацiї дорiвнюють нулю. Отримано також промiжний результат про те, що власнi значення нiколи не стикаються. Цей факт узагальнює результати Брю (1989), що отриманi для класичного процесу Вiшарта. Institute of Mathematics, NAS of Ukraine 2019-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1454 Ukrains’kyi Matematychnyi Zhurnal; Vol. 71 No. 4 (2019); 502-515 Український математичний журнал; Том 71 № 4 (2019); 502-515 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1454/438 Copyright (c) 2019 Boutabia H.; Meradji S.; Stihi S. |
| spellingShingle | Boutabia, H. Meradji, S. Stihi, S. Бутабія, Г. Мераджи, С. Стихи, С. Stochastic differential equations for eigenvalues and eigenvectors of a $G$-Wishart process with drift |
| title | Stochastic differential equations for eigenvalues
and eigenvectors of a $G$-Wishart process with drift |
| title_alt | Стохастичнi диференцiальнi рiвняння для власних значень
i власних векторiв $G$ -процесу вiшарта зi зносом |
| title_full | Stochastic differential equations for eigenvalues
and eigenvectors of a $G$-Wishart process with drift |
| title_fullStr | Stochastic differential equations for eigenvalues
and eigenvectors of a $G$-Wishart process with drift |
| title_full_unstemmed | Stochastic differential equations for eigenvalues
and eigenvectors of a $G$-Wishart process with drift |
| title_short | Stochastic differential equations for eigenvalues
and eigenvectors of a $G$-Wishart process with drift |
| title_sort | stochastic differential equations for eigenvalues
and eigenvectors of a $g$-wishart process with drift |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1454 |
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