# Inexact SA Method for Constrained Stochastic Convex SDP and Application in Chinese Stock Market

Authors: Shuang Chen; Li-Ping Pang; Jian Lv; Zun-Quan Xia
Journal: Journal of Function Spaces
 (2018)

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## Abstract

We propose stochastic convex semidefinite programs (SCSDPs) to handle uncertain data in applications. For these models, we design an efficient inexact stochastic approximation (SA) method and prove the convergence, complexity, and robust treatment of the algorithm. We apply the inexact method for solving SCSDPs where the subproblem in each iteration is only solved approximately and show that it enjoys the similar iteration complexity as the exact counterpart if the subproblems are progressively solved to sufficient accuracy. Numerical experiments show that the method we proposed was effective for uncertain problem.


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## Body

## 1. Introduction

In this paper, we propose a class of optimization problems called stochastic convex semidefinite programs (SCSDPs):(1)minfX=EFX,ξs.t.AX=b,X⪰0,X∈Sn,where F is a smooth convex function for every realization of ξ on S+n, ξ is a random matrix whose probability distribution P is supported on set Ω∈Rr×r, A:Sn→Rm is a linear map, b∈Rm and X⪰0 mean that X∈S+n, and Sn is the space of n×n real symmetric endowed with the standard trace inner product 〈·,·〉 and Frobenius norm ·F. S+n is the set of positive semidefinite matrices in Sn.SCSDPs may be viewed as an extension of the following stochastic models:(1) Stochastic (Linear) Semidefinite Programs (SLSDPs) ([1–5])
(2)
m
i
n
C
,
X
+
E
Q
X
,
w
s
.
t
.
A
i
,
X
=
b
i
,
i
=
1
,
…
,
m
1
,
X
⪰
0
,where Q(X,w) is the minimum of the problem (3)minDw,Ys.t.Tj,X+Wj,Y=djw,i=1,…,m2,Y⪰0,where 〈A,B〉=trace(ATB) denotes the Frobenius inner product between A and B.(2) Stochastic Convex Quadratic Semidefinite Programs (SCQSDPs)
(4)
m
i
n
1
2
X
,
E
Q
X
,
ξ
+
C
,
X
s
.
t
.
A
X
=
b
,
X
⪰
0
,where Q(·,ξ):Sn×Ω→Sn is given self-adjoint positive semidefinite linear operator for every ξ∈Ω and C∈Sn.(3) Stochastic Nearest Correlation Matrix. Now we consider an interesting example from finance industry. In stock research, sample correlation matrices (a symmetric positive semidefinite matrix with unit diagonal) constructed from vectors of stock returns are used for predictive purposes. Unfortunately, on any day when an observation is made data is rarely available for all the stocks of interest. Higham [6] proposed a method that is to compute the sample correlations of pairs of stocks using data drawn only from the days on which both stocks have data available. Compute the nearest correlation matrix and use it for the subsequent stock analysis. This is a statistical application that motivates nearest correlation matrix problem (5)min12LX-U2s.t.diagX=eX⪰0, where e∈Rn is a vector of all ones and U∈Sn is given.Furthermore, with the particularity of Chinese stock market (in Chinese stock market which is different from stock markets in other countries, stock price rise or fall during a day does not exceed ten percent), we could more accurately estimate the information of stock price in the future. We consider computing expected correlations of pairs of stocks returns for better response random factors of the stock market. In order to justify the subsequent stock analysis, it is desired to compute the nearest expectation correlation matrix and to use that matrix in the computation. We use this matrix to predict the correlation of these stocks in the future. The problem we consider is an important special case of (4)(6)min12ELX-Uξ2s.t.diagX=eX⪰0,where U(ξ) is a stochastic matrix from (F,Ω,P)→Sn.Alternatively, SCSDPs may be viewed as an extension of the following deterministic convex semidefinite programs ([7–11]): (7)minFXs.t.AX=b,X⪰0,X∈Sn,where F is a smooth convex function on S+n and A:Sn→Rm is a linear map and b∈Rm and X⪰0 mean that X∈S+n.There are several methods available for solving determinate deterministic convex semidefinite programs and their special cases, which include the accelerated proximal gradient method [8], the alternating projection method [6], the quasi-Newton method [12], the inexact semismooth Newton-CG method [13], and the inexact interior-point method [14]. However, all the methods mentioned above may not be extended to efficiently solve the SCSDPs (1) because it might not be easy to evaluate E[f(X,ξ)] in the following occasions:(a)
ξ is a random vector with a known probability distribution, but calculations of the expected value involve multidimensional integration, which is computationally expensive if not impossible.(b)
The functionf(X,ξ) is known, but the distribution of ξ is unknown and the information on ξ can only be obtained using past data or sampling.(c)
E[f(X,ξ)] is not observable and it must be approximately evaluated through simulation.Under these circumstances, the existing numerical methods for deterministic convex semidefinite programs are not applicable to SCSDPs and new methods are needed. On the other hand, Ariyawansa and Zhu [1] and Mehrotra and Gozevin [4] apply barrier decomposition algorithms and interior-point methods for solving stochastic (linear) semidefinite programs. However, these methods may not be extended for solving the convex problems.The main purpose of this paper is to design an efficient algorithm to solve the general problems (1) including all the special cases mentioned above. The algorithm we propose here is based on the classical SA algorithm whose subproblem must be solved exactly to generate the next iterate point. The SA algorithm approach originates from the pioneering work of Robbins and Monro and is discussed in numerous publications [15–18]. In this paper, we design an inexact SA method which overcomes the limitation just mentioned. Specifically, in our inexact SA method, the subproblem is only solved approximately and we allow the error in solving subproblem to be deterministic or stochastic. From the theoretical point of view we analyze the convergence, complexity, and robust treatment of the algorithm. We also give the numerical results for illustrating the effectiveness of our method.The rest of the paper is organized as follows. In Section2, we focus on theory of the inexact SA method to solve the more general model-stochastic convex matrix program over abstract set Θ, and the proof of convergence is given whether the error is determinate or stochastic. In Section 3, we provide two error estimations of the algorithm. In Section 4, we apply the theoretical results proved before for solving SCSDPs (1). Numerical experiments will be showed at last.


## 2. Algorithm and Convergence Analysis of General Convex Stochastic Matrix Optimization

For more generality, we first consider the following stochastic convex optimization problem:(8)minX∈ΘfX=EFX,ξ,where Θ⊂Rn×n is a nonempty closed convex set, ξ is a random matrix whose probability distribution P is supported on set Ω⊂Rn×n, and F is a smooth convex stochastic function on Θ×Ω. We assume that the expectation(9)EFX,ξ=∫ΩFX,ξdPξis well defined and finite valued for every X∈Θ. We also assume that the expected value function f(·) is continuous and convex on Θ. If for every ξ∈Ω the function F(·,ξ) is convex on X, then it follows that f(·) is convex. With these assumptions, (8) becomes a convex programming problem.Letf be any convex finite at Θ. A matrix X∗ is called ε-subgradient of f at X (where ε>0) if (10)fX≥fX-ε+X∗,Z-X,∀Z∈Θ. The set of all such ε-subgradients is denoted by ∂εf(X).Next we make the following assumptions (also in [19–21]) which will be used to analyze the convergence of algorithms in this paper:(A1)
It is possible to generate an independent identically distributed i.i.d sampleξ1,ξ2,… of realizations of random vector ξ.(A2)
There is an oracle, which, for a given input point(X,ξ)∈Θ×Ω, returns inexact stochastic subgradient-a vector G(X,ξ) such that g(X)=E[G(X,ξ)] is well defined and is a ε-subgradient of f(·) at X, that is, g(X)∈∂εf(X).(A3)
There is a positive numberM such that(11)EGX,ξF2≤M2,∀X∈Θ.Remark 1.
Assumption   (A2) is different from the assumption used in [19]. We only need to get a ε-subgradient which is more likely got in practice.Definition 2.
Theconvex hull of set S (denoted by co(S)) is the smallest convex polygon that contains all the points of S.Throughout the paper, we use the following notations.XF=∑i,jXij2 denotes the Frobenius norm, where Xij2 means the element at the ith row and jth column of X. 〈X,Y〉=Trace(XTY) denotes standard trace inner product. By ΠΘ, we denote the metric projection operator onto the set X: that is, ΠΘ(X)=argminX′∈ΘX-X′F. Note that ΠΘ is a nonexpanding operator, that is,(12)ΠΘ′-ΠΘF≤X′-XF,∀X,X′∈Rn.The notation [a] stands for the largest integer less than or equal to a∈R. By ξ[t]=(ξ1,…,ξt), we denote the history of the process ξ1,…,ξt up to time t. Unless stated otherwise, all relations between random variables are supposed to hold almost surely.In the following, we discuss theory of the inexact SA approach to the minimization problem (8).The inexact SA algorithm solves (8) by mimicking the subgradient descent method (13)Xj+1=ΠΘXj-γjf′Xj, where f′(X)∈∂f(X). These lead to an executable algorithm below.Algorithm 3.Step 1.
Give the initial pointX1∈Θ and set j=1, iterate the following steps.Step 2.
Choose suitable step sizesγj>0 and computational error ej.Step 3.
Find an approximation solution of(14)Xj+1=ΠΘXj-γjGXj,ξj-γjej.
Setj:=j+1; go back to Step  2.In order to prove the convergence of Algorithm3 we introduce the following lemma.Lemma 4 (see [22]).
LetFk be an increasing sequence of σ-algebras and let {tk}, {αk}, {βk}, and {γk} be nonnegative random variables adapted to Fk. If Σi=1∞αk<∞, Σi=1∞βk<∞, it holds almost surely that(15)tk+1≤1+αktk+βk-ρk,and then tk is convergent almost surely and Σρk<∞ almost surely.We now state the main convergence theorem.Theorem 5.
Suppose that the stochastic optimization problem (8) has an optimal solution X∗ and ej→0. Let γj=θ/j, where θ>0, and Xj⊂Θ be a sequence generated by (14); then Xj converges to X∗.Proof.
Note that the iterateXj=Xj(ξ[j-1]) is a function of the history ξ[j-1]=(ξ1,…,ξj-1) of the generated random process and hence is random.
Denote(16)Dj=12Xj-X∗F2,dj=EDj=12EXj-X∗F2.
SinceX∗∈Θ and hence ΠΘ(X∗)=X∗, we can write(17)Dj+1=12ΠΘXj-γjGXj,ξj-γjej-X∗F2=12ΠΘXj-γjGXj,ξj-γjej-ΠΘX∗F2≤12Xj-γjGXj,ξj-γjej-X∗F2=Dj+12γj2GXj,ξj+ejF2-γjXj-X∗,GXj,ξj+ej,where the second inequality is due to nonexpansionary of projection operator.
SinceXj is independent of ξj, we have(18)EXj-X∗,GXj,ξj=EEXj-X∗,GXj,ξj∣ξj-1=EXj-X∗,EGXj,ξj∣ξj-1=EXj-X∗,gXj.
Since assumption (A3) andej→0, there is a positive number M such that (19)Dj+1≤Dj+12γj2M2-γjXj-X∗,GXj,ξj+ej. Then, by taking expectation of both sides of (17) and using (18), we obtain (20)dj+1≤dj-γjEXj-X∗,gXj+ej+12γj2M2. For γj=θ/j, we known that ∑θ2/j2<∞. By Lemma 4, we know that dj=1/2E[Xj-X∗F2] is convergent almost surely and(21)∑j=1∞γjEXj-X∗,gXj+ej<∞.Suppose that Xj→X¯≠X∗ and by the convexity of f(x), we have that E[〈Xj-X∗,g(Xj)〉]≥E[f(xj)-f(X∗)]>0, when j→∞. For ∑j=1∞γj=∑j=1∞θ/j=∞, we know that (22)∑j=1∞γjEXj-X∗,gXj+ej=∞.
It is in contradiction with (21); then we have E[Xj-X∗2]→0 almost surely.Sometimes, the error at each iterationj is relevant to the random variable ξj. In this environment, we consider stochastic inexact SA algorithm.Algorithm 6.Step 1.
Give the initial pointX1∈Θ and set k=1; iterate the following steps.Step 2.
Choose suitable step sizesγj>0 and computational stochastic error ej(ξj).Step 3.
Find an approximation solution of(23)Xj+1=ΠΘXj-γjGXj,ξj-γjejξj.Setj:=j+1; go back to Step 2.Theorem 7.
Suppose that the stochastic optimization problem (8) has an optimal solution X∗. We also supposed that stochastic error ej(ξj)→0 and there is a positive number N such that ej(ξj)≤N. Let γj=θ/j, where θ>0, and Xj⊂Θ be a sequence generated by (23); then Xj converges to X∗.Proof.
Like the proof of Theorem5, we obtain(24)dj+1≤dj-γjEXj-X∗,gXj+Eejξj+12γj2M+N2.For γj=θ/j, we known that ∑θ2/j2<∞. By Lemma 4, we know that dj=1/2E[Xj-X∗F2] is convergent almost surely and(25)∑j=1∞γjEXj-X∗,gXj+Eejξj<∞.Suppose that Xj→X¯≠X∗ and by the convexity of f(x), we have that E[〈Xj-X∗,g(Xj)〉]≥E[f(xj)-f(X∗)]>0, when j→∞. For ∑j=1∞γj=∑j=1∞θ/j=∞, we know that (26)∑j=1∞γjEXj-X∗,gXj+Eejξj=∞.
It is in contradiction with (25); then we have EXj-X∗F→0 almost surely.


## 3. Error Estimation

Suppose further that the expectation functionf(X) is differentiable and strongly convex on Θ; that is, there is constant c>0 such that(27)X′-X,∇fX′-∇fX≥cX′-XF2,∀X′,X∈Θ.Next we will discuss the complexity of the above algorithms; we take the second algorithm, for instance.Theorem 8.
Suppose thatf(X) is differentiable and strongly convex, ∇f(X) is Lipschitz continuous, and the conditions of Theorem 7 are satisfied; we have that(28)EfXj-fX∗≤1/2LQθj, where (29)Qθ=maxθ2M+N22cθ-1-1,X1-X∗F2.Proof.
Note that strong convexity off(X) implies that the minimizer X∗ is unique. As optimality of X∗ is unique, we have that (30)X-X∗,∇fX∗≥0∀X∈Θ,which together with (27) implies that 〈X-X∗,∇f(X)〉≥cX′-X∗F2. In turn, it follows that 〈X-X∗,g+E[ej(ξj)]〉≥cX′-X∗F2 for all X∈Θ and g(Xj)+E[ej(ξj)]∈∂f(Xj) and hence (31)EXj-X∗,gXj+Eejξj≥cEXj-X∗F2=2cdj.
Therefore, it follows from (24) that(32)dj+1≤dj1-2cγj+12γj2M+N2.
For some constantθ>1/(2c), by (32), we have (33)dj+1≤dj1-2cθj+1/2θ2j2M+N2. Next we will prove that (34)EXj-X∗F2=2dj≤Qθj, where (35)Qθ=maxθ2M+N22cθ-1-1,X1-X∗F2. Using mathematical induction, for j=1, we have (36)EX1-X∗F2=2d1≤Qθ, which holds naturally. For ∗2dj+1≤Qθj1-2cθj+θ2j2M+N2=θ2M+N2j-2cθ+θ2M+N22cθ-1j22cθ-1=θ2M+N2j-1j22cθ-1,∗∗Qθj+1=θ2M+N22cθ-1j+1,we know that ∗/∗∗ < 1; then we have E[Xj-X∗F]=2dj≤Q(θ)/j.
For∇f(X) is Lipschitz continuous, there is constant L>0 such that(37)∇fX′-∇fXF≤LX′-XF,∀X′,X∈Θ,and hence(38)EfXj-fX∗≤E-∇fXjX∗-Xj+∇fX∗X∗-Xj=E∇fX∗-∇fXjX∗-Xj≤LEXj-X∗F2≤LQθj.Whenθ>1/2c, it follows from (38) that, after j iterations, the expected error in terms of the objective value is of order O(j-1). But that the result is highly sensitive to a priori information on strongly convex factor c. In order to make the SA method applicable to general convex objectives rather than to strongly convex ones, one should replace the classical step sizes γj=θ/j, which can be too small to ensure a reasonable rate of convergence. We take appropriate averages of the search points Xj rather than these points themselves. This processing method go back to [20, 23] and we call it “robust treatment.” We will show the effectiveness of the above technique in the following theorem.Theorem 9.
Suppose thatf is convex; let νt≥0 and ∑t=ijνt=1. Consider the points X~ij=∑t=ijνtXt, and let (39)DΘ=maxX∈coXkXk-X1F. We know that (40)EfX~ij-fX∗≤DΘ25M+N+N2nγ.Proof.
Due to (24) and assumption (A3), we know that (41)γtEfXt-fX∗≤γtXt-X∗,GXt,ξj≤dt-dt+1+12γt2M+N2-γtEXt-X∗,Eetξt. It follows that whenever 1≤i≤j, we have(42)∑t=ijγtEfXt-fX∗≤∑t=ijdt-dt+1+12M+N2∑t=ijγt2-∑t=ijγtEXt-X∗,Eetξt≤di+12M+N2∑t=ijγt2-∑t=ijγtEXt-X∗,Eetξt,
and hence, settingνt=γt/∑t=ijγt,(43)E∑t=ijνtfXt-fX∗=E∑t=ijγt∑t=ijγtfXt-∑t=ijγt∑t=ijγtfX∗=E∑t=ijγt∑t=ijγtfXt-fX∗=1∑t=ijγt∑t=ijγtEfXt-fX∗≤di+1/2M+N2∑t=ijγt2-∑t=ijγtEXt-X∗,Eetξt∑t=ijγt.
By convexity ofΘ, we have X~ij∈X, and by convexity of f, we have f(X~ij)≤∑t=ijνtf(Xt). Thus, by (43) and in view of d1≤DΘ2 and di≤4DΘ2,i>1, we get(44)EfX~1j-fX∗≤DΘ2+M+N2∑t=ijγt2+∑t=ijγtDΘN2∑t=ijγt,EfX~ij-fX∗≤4DΘ2+M+N2∑t=ijγt2+∑t=ijγtDΘN2∑t=ijγt.Now we can develop step size policies along with the associated efficiency estimates based on (44). Assume that the number n of iterations of the method is fixed in advance and that γt=γ. Then (44) becomes(45)EfX~1j-fX∗≤DΘ2+M+N2nγ2+nγDΘN2nγ,(46)EfX~ij-fX∗≤4DΘ2+M+N2nγ2+nγDΘN2nγ.
Minimizing the right-hand side of (45) over γ>0, we arrive at the constant step size policy(47)γ=DΘM+Nn,t=1,…,N.Along with the associated efficiency estimate and O(E[et(ξt)])<O(1/N), we have(48)EfX~ij-fX∗≤DΘ25M+N+N2nγ.With constant step size policy (47) in the above theorem, after n iterations, the expected error is of order O(n1/2). This is worse than the rate for the inexact SA algorithm. However, the error bounds (48) are guaranteed independently of any smoothness and/or strong convexity assumptions on f.


## 4. Specialization to the Case WhereΘ={A(X)=b,X⪰0}

To illustrate the advantage of inexact SA method over classical SA method, we apply our method to SCSDPs in this section. Problem (1) can be expressed in the form (8) with Θ={A(X)=b,X⪰0,X∈Sn}. Subproblem (14) then becomes the following constrained minimization problem:(49)min12X-Xj+γjGXj,ξj+γjej2s.t.AX=b,X⪰0. The KKT corresponding to problem (49) is written as(50)Xj+1-Xj+γjGXj,ξj-A∗μj+1-λj+1+γjej=0,AXj+1-b=0,Xj+1,λj+1=εj+1≈0,Xj+1⪰0,λj+1⪰0.We can get the algorithm for problem (49).Algorithm 10.Step 1.
Given the initial pointX1∈Θ and setting j=1, iterate the following steps.Step 2.
Choose suitable step sizesγj>0 and computational error ej.Step 3.
Solve the following equation:(51)Xj+1-Xj+γjGXj,ξj-A∗μj+1-λj+1+γjej=0,AXj+1-b=0,Xj+1,λj+1=0,Xj+1⪰0,λj+1⪰0.Setj≔j+1; go back to Step 2.


## 5. Application and Numerical Results

In order to assess from a practical point of view the inexact SA method, we code the Algorithm in MATLAB and ran it on several subcategories of SDP. We implement our algorithm in MATLAB 2009a and use a computer with one 2.20 GHz processor and 2.00 GB RAM.


### 5.1. Numerical Results for Random Matrix

In our first example, we take a simple case of stochastic convex SDP(52)minfX=X-EAξFs.t.diagX=e,X⪰0,X∈S4,where f:S4+×Θ→R is a convex function for every ξ and (53)Aξ=2-100-12-100-12-100-12∘ξ,where “∘” means each element of A multiplied by random variable ξ. We let ξ mean 1 which is uniformly distributed whose interval radius is mutative from 0.5 to 4. Using inexact SA method to solve this problem, we list the results in Table 1.Table 1
Objective function is random matrix.
Radius
rP
r
D
CPU time (s)
(1)
0.5
5.4987
e
-
014
3.8803
e
-
009
0.2260
(2)
1
5.1761
e
-
014
9.8586
e
-
009
0.2868
(3)
1.5
3.4534
e
-
015
5.4413
e
-
009
0.5979
(4)
2
7.9802
e
-
016
1.8327
e
-
009
0.7178
(5)
2.5
4.5433
e
-
016
6.1692
e
-
009
0.8770
(6)
3
8.3721
e
-
015
3.4310
e
-
009
0.9087
(7)
3.5
2.6872
e
-
016
3.1458
e
-
009
1.1569
(8)
4
4.5937
e
-
016
9.1487
e
-
009
1.2725
(9)
4.5
8.6907
e
-
016
8.1782
e
-
009
1.3046
(10)
5
6.6707
e
-
016
6.1892
e
-
009
1.3068
(11)
6
5.6907
e
-
016
3.1882
e
-
009
1.5898Table1 shows that the inexact SA method for solving stochastic convex SDP is effective. The abscissa of Figure 1 represent the interval radius of ξ and ordinate of Figure 1 represents norm difference of the optimal value between ξ which is a random variable and ξ≡1. This means that the variance of random variable may impact approximation; however, this influence does not exceed the tolerance range.Figure 1
Discrete map.


### 5.2. Numerical Results for SCQSDPs

In this example, we tested the inexact SA method on the linearly constrained SCQSDPs:(54)min12Ex,ξQx+c,x:Ax=b,x⪰0, where ξ~U[0.5,1.5] is random variable, Q, and A is given as follows: (55)Qx=12Bx+xB,whereB≻0,Ax=diagx=x11,x22,…,xnnT. Then b and c can be generated at random. The MATLAB code for generating the matrix b is given by b=rand(n,1); b=b.∗b. We generated B as follows: (56)randvec=1+9∗randn,1;tmp=randnn,ceiln4B=diagrandvec+tmp∗tmp′n;B=B+B′2.We show the elementary numerical results in Table2.Table 2
Objective function is stochastic quadratic.
n
rP
rD
CPU time (s)
Iter
Pobj
(1)
100
9.2734
e
-
014
1.0529
e
-
007
17.46
18
6.8577
e
+
002
(2)
200
7.1942
e
-
014
2.4666
e
-
007
35.11
17
1.6196
e
+
003
(3)
300
5.2093
e
-
014
5.1427
e
-
007
90.36
16
2.3798
e
+
003
(4)
500
5.6795
e
-
014
1.3859
e
-
007
230.89
16
4.9247
e
+
003
(5)
800
4.5533
e
-
014
1.1092
e
-
007
756.32
20
8.5645
e
+
004
(6)
1000
4.3823
e
-
014
7.4480
e
-
007
1757.73
14
1.1458
e
+
004
(7)
1200
3.6972
e
-
014
1.1656
e
-
007
3939.16
17
4.9487
e
+
004
(8)
1500
3.6907
e
-
014
1.1882
e
-
007
7500.49
18
5.7514
e
+
005Our numerical results are reported in Table2, where n, rP, rD, iter, and pobj stand for, respectively, the matrix dimension, the primal infeasibility, the dual infeasibility, the number of outer iterations, and the primal objective value. For each instance, our algorithm can solve the convex QSDP problem efficiently with very good primal infeasibility. Experimental result shows that the exact SA algorithm can achieve a high degree of detecting reliability.


### 5.3. Predict Correlation Coefficient in the Chinese Stock Market

In this example, we discuss how to use our inexact SA method to predict correlation coefficient in the Chinese stock market. This is the main motivation of this paper. Chinese stock market is different from other countries, and the rise or fall in a day is not more than ten percent. This feature could make us estimate some index more accurately.In this paper, our main concern is the correlation coefficient of stocks. Correlation coefficient is often useful to know if two stocks tend to move together. For a diversified portfolio, you would want stocks that are not closely related. It helps to measure the closeness of the returns. Usually, we use the determinate correlation coefficient based on historical data. This principle which we mentioned in the previous paragraph makes us discover that we can use stochastic factor to estimate the correlation coefficient in the future.According to the difference of thetotal market capitalization(=totalequity×stockprices) of company, we can divide stocks into two parts called small-cap stocks and big-cap stocks.(1) Small-Cap Stocks. They refer to stocks with a relatively small market capitalization. The definition of small cap can vary among brokerages, but generally it is a company with a market capitalization of less than ¥0.1billion.Table3 is the closing price of ten stocks of small-cap from 2017-6-3 to 2017-7-3, where CP is logogram of closing price. The second line of Table 3 is code of stocks.Table 3
Closing price of small-cap stocks.
CP
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
6.3
18.140
9.550
14.500
11.330
8.760
26.000
13.500
9.620
9.380
12.380
6.4
18.260
9.250
13.870
10.940
9.430
25.430
13.250
9.600
9.440
12.270
6.5
18.300
9.240
14.010
11.310
9.090
25.300
13.360
9.680
9.600
12.160
6.6
17.680
9.070
13.990
10.950
8.520
25.800
12.770
9.500
9.560
11.930
6.7
17.130
8.950
13.890
10.950
8.070
25.680
12.250
9.240
9.530
11.720
6.13
16.820
8.750
13.370
10.950
8.270
24.500
12.170
8.940
9.380
11.400
6.14
17.060
8.800
13.630
11.460
8.220
24.450
12.790
9.120
9.620
11.860
6.17
17.370
8.770
14.090
11.570
8.280
24.260
12.710
9.300
9.620
11.780
6.18
17.150
8.810
14.450
11.360
8.230
24.800
12.770
9.260
9.860
11.540
6.19
17.280
8.810
14.300
11.160
8.110
25.350
12.990
9.120
9.590
11.500
6.20
16.750
8.620
13.960
10.620
7.750
24.580
12.300
8.650
9.200
10.740
6.21
16.670
8.480
13.990
10.350
7.900
24.560
11.950
8.730
8.760
10.330
6.24
15.640
8.200
13.500
9.3200
7.110
22.900
10.850
8.210
8.350
9.900
6.25
15.160
8.200
13.740
9.190
7.240
23.010
11.290
8.500
8.430
10.310
6.26
15.080
8.180
14.190
9.960
7.310
23.400
11.470
8.770
8.380
10.590
6.27
14.890
8.180
13.850
9.650
7.020
22.890
11.340
8.620
8.000
10.280
6.28
14.740
8.200
14.090
9.300
6.960
22.580
11.190
9.030
8.090
10.140
7.1
14.650
8.250
14.150
9.600
7.050
22.750
11.500
9.490
8.160
10.360
7.2
15.120
8.240
13.920
10.150
7.130
23.320
11.740
9.540
8.200
10.800
7.3
14.750
8.130
14.040
10.230
7.400
23.700
11.570
9.590
8.000
10.680We know that the change of the closing price in 2017.7.4 will not exceed ten percent and seriously up or down is small probability event. According to this and model (6), we can forecast the correlation coefficient from 2017.6.4 to 2017.7.4. We list the results in Table 4.Table 4
Prediction correlation matrix.
CC
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
002001
1
0.9597
0.0690
0.8547
0.9343
0.9042
0.9003
0.4002
0.9128
0.8682
002003
0.9597
1
0.0681
0.8080
0.9129
0.9003
0.9110
0.6429
0.8139
0.9120
002004
0.0690
0.0681
1
0.2535
0.1418
0.2719
0.2700
0.4429
0.1797
0.1379
002005
0.8547
0.8080
0.2535
1
0.8080
0.8139
0.8896
0.5691
0.9100
0.8741
002006
0.9343
0.9129
0.1418
0.8080
1
0.8517
0.8964
0.5837
0.8255
0.8888
002007
0.9042
0.9003
0.2719
0.8139
0.8517
1
0.8624
0.5235
0.8381
0.8430
002008
0.9003
0.9110
0.2700
0.8896
0.8964
0.8624
1
0.6828
0.8517
0.9168
002009
0.4002
0.6429
0.4429
0.5691
0.5837
0.5235
0.6828
1
0.4691
0.7216
002010
0.9128
0.8139
0.1797
0.9100
0.8255
0.8381
0.8517
0.4691
1
0.8430
002011
0.8682
0.9120
0.1379
0.8741
0.8886
0.8430
0.9168
0.7216
0.8430
1Now we give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.From Tables4 and 5, we know that the correlation matrix we calculated is very close to the true correlation matrix. This mean that our method for predicting correlation matrix is effective.Table 5
True correlation matrix.
CC
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
002001
1
0.966
0.016
0.852
0.948
0.905
0.917
0.342
0.928
0.871
002003
0.966
1
0.043
0.833
0.957
0.908
0.931
0.493
0.908
0.929
002004
0.016
0.043
1
0.172
0.016
0.133
0.236
0.388
0.106
0.09
002005
0.852
0.833
0.172
1
0.821
0.832
0.916
0.512
0.913
0.899
002006
0.948
0.957
0.016
0.821
1
0.868
0.923
0.496
0.848
0.911
002007
0.905
0.908
0.133
0.832
0.868
1
0.87
0.437
0.864
0.849
002008
0.917
0.931
0.236
0.916
0.923
0.87
1
0.583
0.898
0.935
002009
0.342
0.493
0.388
0.512
0.496
0.437
0.583
1
0.339
0.647
002010
0.928
0.908
0.106
0.913
0.848
0.864
0.898
0.339
1
0.875
002011
0.871
0.929
0.09
0.899
0.911
0.849
0.935
0.647
0.875
1(2) Big-Cap Stocks.In China, it is a term used by the investment community to refer to companies with a market capitalization value of more than ¥0.1billion. Large cap is an abbreviation of the term “large market capitalization.” Market capitalization is calculated by multiplying the number of a company’s shares outstanding by its stock price per share.Table6 is the closing price of ten stocks of big-cap from 2017-6-3 to 2017-7-3.Table 6
Closing price of big-cap stocks.
CP
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
6.3
9.840
4.790
10.71
10.38
4.81
6.740
12.990
9.280
13.770
12.350
6.4
9.740
4.730
10.70
10.28
4.77
6.730
12.740
9.160
13.750
12.180
6.5
9.650
4.770
10.59
10.10
4.76
6.710
12.880
9.060
13.360
12.150
6.6
9.450
4.700
10.39
10.00
4.71
6.710
12.720
9.040
13.180
11.760
6.7
9.350
4.560
10.26
9.990
4.66
6.600
12.120
8.840
13.170
11.520
6.13
9.020
4.300
9.990
9.880
4.53
6.390
11.350
8.340
12.320
11.000
6.14
9.020
4.500
10.06
9.960
4.55
6.290
11.340
8.310
12.210
11.070
6.17
9.010
4.620
10.03
9.910
4.51
6.350
11.120
8.160
12.100
10.970
6.18
9.070
4.590
10.10
9.950
4.33
6.350
11.220
8.140
12.200
11.120
6.19
8.890
4.560
9.960
9.750
4.28
4.680
11.150
8.110
11.980
10.940
6.20
8.420
4.340
9.520
9.290
4.17
4.500
10.820
7.940
11.470
10.610
6.21
8.280
4.370
9.480
9.450
4.15
4.450
11.050
7.730
11.710
10.590
6.24
7.520
4.060
8.690
8.510
4.05
4.210
10.090
7.480
10.930
9.5300
6.25
7.800
3.960
8.690
8.440
4.03
4.190
10.000
7.540
10.920
9.3000
6.26
7.770
3.930
8.580
8.300
3.92
4.100
9.9900
7.490
10.660
9.3100
6.27
7.880
4.230
8.660
8.210
3.92
4.110
9.8400
7.450
10.920
9.3300
6.28
8.280
4.100
9.020
8.570
3.93
4.180
10.130
7.510
11.600
9.9100
7.1
8.170
4.030
8.920
8.590
3.91
4.270
10.180
7.420
11.380
9.9900
7.2
8.150
3.990
8.880
8.480
3.92
4.240
10.190
7.280
11.210
9.8700
7.3
8.040
3.940
8.680
8.440
3.94
4.250
9.9700
7.060
11.350
9.6600Like in the small-cap stocks, we calculate the correlation matrix as shown in Table7.Table 7
Prediction correlation matrix.
CC
600000
600010
600015
600016
600019
600028
600030
600031
600036
600048
600000
1
0.9119
0.9776
0.9497
0.9388
0.9656
0.9447
0.9427
0.9766
0.9736
600010
0.9119
1
0.9477
0.9318
0.9209
0.9298
0.9020
0.8960
0.9030
0.9318
600015
0.9776
0.9477
1
0.9825
0.9646
0.9835
0.9467
0.9408
0.9716
0.9825
600016
0.9497
0.9318
0.9825
1
0.9437
0.9666
0.9050
0.8940
0.9547
0.9597
600019
0.9388
0.9209
0.9646
0.9437
1
0.9766
0.9587
0.9716
0.9358
0.9527
600028
0.9656
0.9298
0.9835
0.9666
0.9766
1
0.9597
0.9557
0.9567
0.9756
600030
0.9447
0.9020
0.9467
0.9050
0.9587
0.9597
1
0.9776
0.9408
0.9696
600031
0.9427
0.8960
0.9408
0.8940
0.9716
0.9557
0.9776
1
0.9318
0.9487
600036
0.9766
0.9030
0.9716
0.9547
0.9358
0.9567
0.9408
0.9318
1
0.9746
600048
0.9736
0.9318
0.9825
0.9597
0.9527
0.9756
0.9696
0.9487
0.9746
1Now we also give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.From Tables7 and 8, we know that the correlation matrix we calculated is even more close to the true correlation matrix compared to the case in small-cap stocks. This means that our method for predicting correlation matrix is more effective in the company whose total market capitalization is huge.Table 8
True correlation matrix.
CC
600000
600010
600015
600016
600019
600028
600030
600031
600036
600048
600000
1
0.911
0.979
0.952
0.937
0.97
0.943
0.932
0.979
0.976
600010
0.911
1
0.951
0.932
0.92
0.925
0.901
0.9
0.887
0.93
600015
0.979
0.951
1
0.987
0.968
0.985
0.951
0.944
0.968
0.988
600016
0.952
0.932
0.987
1
0.946
0.97
0.907
0.893
0.955
0.965
600019
0.937
0.92
0.968
0.946
1
0.981
0.962
0.972
0.929
0.954
600028
0.97
0.925
0.985
0.97
0.981
1
0.965
0.952
0.96
0.982
600030
0.943
0.901
0.951
0.907
0.962
0.965
1
0.975
0.934
0.973
600031
0.932
0.9
0.944
0.893
0.972
0.952
0.975
1
0.909
0.942
600036
0.979
0.887
0.968
0.955
0.929
0.96
0.934
0.909
1
0.973
600048
0.976
0.93
0.988
0.965
0.954
0.982
0.973
0.942
0.973
1


## 5.1. Numerical Results for Random Matrix

In our first example, we take a simple case of stochastic convex SDP(52)minfX=X-EAξFs.t.diagX=e,X⪰0,X∈S4,where f:S4+×Θ→R is a convex function for every ξ and (53)Aξ=2-100-12-100-12-100-12∘ξ,where “∘” means each element of A multiplied by random variable ξ. We let ξ mean 1 which is uniformly distributed whose interval radius is mutative from 0.5 to 4. Using inexact SA method to solve this problem, we list the results in Table 1.Table 1
Objective function is random matrix.
Radius
rP
r
D
CPU time (s)
(1)
0.5
5.4987
e
-
014
3.8803
e
-
009
0.2260
(2)
1
5.1761
e
-
014
9.8586
e
-
009
0.2868
(3)
1.5
3.4534
e
-
015
5.4413
e
-
009
0.5979
(4)
2
7.9802
e
-
016
1.8327
e
-
009
0.7178
(5)
2.5
4.5433
e
-
016
6.1692
e
-
009
0.8770
(6)
3
8.3721
e
-
015
3.4310
e
-
009
0.9087
(7)
3.5
2.6872
e
-
016
3.1458
e
-
009
1.1569
(8)
4
4.5937
e
-
016
9.1487
e
-
009
1.2725
(9)
4.5
8.6907
e
-
016
8.1782
e
-
009
1.3046
(10)
5
6.6707
e
-
016
6.1892
e
-
009
1.3068
(11)
6
5.6907
e
-
016
3.1882
e
-
009
1.5898Table1 shows that the inexact SA method for solving stochastic convex SDP is effective. The abscissa of Figure 1 represent the interval radius of ξ and ordinate of Figure 1 represents norm difference of the optimal value between ξ which is a random variable and ξ≡1. This means that the variance of random variable may impact approximation; however, this influence does not exceed the tolerance range.Figure 1
Discrete map.


## 5.2. Numerical Results for SCQSDPs

In this example, we tested the inexact SA method on the linearly constrained SCQSDPs:(54)min12Ex,ξQx+c,x:Ax=b,x⪰0, where ξ~U[0.5,1.5] is random variable, Q, and A is given as follows: (55)Qx=12Bx+xB,whereB≻0,Ax=diagx=x11,x22,…,xnnT. Then b and c can be generated at random. The MATLAB code for generating the matrix b is given by b=rand(n,1); b=b.∗b. We generated B as follows: (56)randvec=1+9∗randn,1;tmp=randnn,ceiln4B=diagrandvec+tmp∗tmp′n;B=B+B′2.We show the elementary numerical results in Table2.Table 2
Objective function is stochastic quadratic.
n
rP
rD
CPU time (s)
Iter
Pobj
(1)
100
9.2734
e
-
014
1.0529
e
-
007
17.46
18
6.8577
e
+
002
(2)
200
7.1942
e
-
014
2.4666
e
-
007
35.11
17
1.6196
e
+
003
(3)
300
5.2093
e
-
014
5.1427
e
-
007
90.36
16
2.3798
e
+
003
(4)
500
5.6795
e
-
014
1.3859
e
-
007
230.89
16
4.9247
e
+
003
(5)
800
4.5533
e
-
014
1.1092
e
-
007
756.32
20
8.5645
e
+
004
(6)
1000
4.3823
e
-
014
7.4480
e
-
007
1757.73
14
1.1458
e
+
004
(7)
1200
3.6972
e
-
014
1.1656
e
-
007
3939.16
17
4.9487
e
+
004
(8)
1500
3.6907
e
-
014
1.1882
e
-
007
7500.49
18
5.7514
e
+
005Our numerical results are reported in Table2, where n, rP, rD, iter, and pobj stand for, respectively, the matrix dimension, the primal infeasibility, the dual infeasibility, the number of outer iterations, and the primal objective value. For each instance, our algorithm can solve the convex QSDP problem efficiently with very good primal infeasibility. Experimental result shows that the exact SA algorithm can achieve a high degree of detecting reliability.


## 5.3. Predict Correlation Coefficient in the Chinese Stock Market

In this example, we discuss how to use our inexact SA method to predict correlation coefficient in the Chinese stock market. This is the main motivation of this paper. Chinese stock market is different from other countries, and the rise or fall in a day is not more than ten percent. This feature could make us estimate some index more accurately.In this paper, our main concern is the correlation coefficient of stocks. Correlation coefficient is often useful to know if two stocks tend to move together. For a diversified portfolio, you would want stocks that are not closely related. It helps to measure the closeness of the returns. Usually, we use the determinate correlation coefficient based on historical data. This principle which we mentioned in the previous paragraph makes us discover that we can use stochastic factor to estimate the correlation coefficient in the future.According to the difference of thetotal market capitalization(=totalequity×stockprices) of company, we can divide stocks into two parts called small-cap stocks and big-cap stocks.(1) Small-Cap Stocks. They refer to stocks with a relatively small market capitalization. The definition of small cap can vary among brokerages, but generally it is a company with a market capitalization of less than ¥0.1billion.Table3 is the closing price of ten stocks of small-cap from 2017-6-3 to 2017-7-3, where CP is logogram of closing price. The second line of Table 3 is code of stocks.Table 3
Closing price of small-cap stocks.
CP
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
6.3
18.140
9.550
14.500
11.330
8.760
26.000
13.500
9.620
9.380
12.380
6.4
18.260
9.250
13.870
10.940
9.430
25.430
13.250
9.600
9.440
12.270
6.5
18.300
9.240
14.010
11.310
9.090
25.300
13.360
9.680
9.600
12.160
6.6
17.680
9.070
13.990
10.950
8.520
25.800
12.770
9.500
9.560
11.930
6.7
17.130
8.950
13.890
10.950
8.070
25.680
12.250
9.240
9.530
11.720
6.13
16.820
8.750
13.370
10.950
8.270
24.500
12.170
8.940
9.380
11.400
6.14
17.060
8.800
13.630
11.460
8.220
24.450
12.790
9.120
9.620
11.860
6.17
17.370
8.770
14.090
11.570
8.280
24.260
12.710
9.300
9.620
11.780
6.18
17.150
8.810
14.450
11.360
8.230
24.800
12.770
9.260
9.860
11.540
6.19
17.280
8.810
14.300
11.160
8.110
25.350
12.990
9.120
9.590
11.500
6.20
16.750
8.620
13.960
10.620
7.750
24.580
12.300
8.650
9.200
10.740
6.21
16.670
8.480
13.990
10.350
7.900
24.560
11.950
8.730
8.760
10.330
6.24
15.640
8.200
13.500
9.3200
7.110
22.900
10.850
8.210
8.350
9.900
6.25
15.160
8.200
13.740
9.190
7.240
23.010
11.290
8.500
8.430
10.310
6.26
15.080
8.180
14.190
9.960
7.310
23.400
11.470
8.770
8.380
10.590
6.27
14.890
8.180
13.850
9.650
7.020
22.890
11.340
8.620
8.000
10.280
6.28
14.740
8.200
14.090
9.300
6.960
22.580
11.190
9.030
8.090
10.140
7.1
14.650
8.250
14.150
9.600
7.050
22.750
11.500
9.490
8.160
10.360
7.2
15.120
8.240
13.920
10.150
7.130
23.320
11.740
9.540
8.200
10.800
7.3
14.750
8.130
14.040
10.230
7.400
23.700
11.570
9.590
8.000
10.680We know that the change of the closing price in 2017.7.4 will not exceed ten percent and seriously up or down is small probability event. According to this and model (6), we can forecast the correlation coefficient from 2017.6.4 to 2017.7.4. We list the results in Table 4.Table 4
Prediction correlation matrix.
CC
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
002001
1
0.9597
0.0690
0.8547
0.9343
0.9042
0.9003
0.4002
0.9128
0.8682
002003
0.9597
1
0.0681
0.8080
0.9129
0.9003
0.9110
0.6429
0.8139
0.9120
002004
0.0690
0.0681
1
0.2535
0.1418
0.2719
0.2700
0.4429
0.1797
0.1379
002005
0.8547
0.8080
0.2535
1
0.8080
0.8139
0.8896
0.5691
0.9100
0.8741
002006
0.9343
0.9129
0.1418
0.8080
1
0.8517
0.8964
0.5837
0.8255
0.8888
002007
0.9042
0.9003
0.2719
0.8139
0.8517
1
0.8624
0.5235
0.8381
0.8430
002008
0.9003
0.9110
0.2700
0.8896
0.8964
0.8624
1
0.6828
0.8517
0.9168
002009
0.4002
0.6429
0.4429
0.5691
0.5837
0.5235
0.6828
1
0.4691
0.7216
002010
0.9128
0.8139
0.1797
0.9100
0.8255
0.8381
0.8517
0.4691
1
0.8430
002011
0.8682
0.9120
0.1379
0.8741
0.8886
0.8430
0.9168
0.7216
0.8430
1Now we give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.From Tables4 and 5, we know that the correlation matrix we calculated is very close to the true correlation matrix. This mean that our method for predicting correlation matrix is effective.Table 5
True correlation matrix.
CC
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
002001
1
0.966
0.016
0.852
0.948
0.905
0.917
0.342
0.928
0.871
002003
0.966
1
0.043
0.833
0.957
0.908
0.931
0.493
0.908
0.929
002004
0.016
0.043
1
0.172
0.016
0.133
0.236
0.388
0.106
0.09
002005
0.852
0.833
0.172
1
0.821
0.832
0.916
0.512
0.913
0.899
002006
0.948
0.957
0.016
0.821
1
0.868
0.923
0.496
0.848
0.911
002007
0.905
0.908
0.133
0.832
0.868
1
0.87
0.437
0.864
0.849
002008
0.917
0.931
0.236
0.916
0.923
0.87
1
0.583
0.898
0.935
002009
0.342
0.493
0.388
0.512
0.496
0.437
0.583
1
0.339
0.647
002010
0.928
0.908
0.106
0.913
0.848
0.864
0.898
0.339
1
0.875
002011
0.871
0.929
0.09
0.899
0.911
0.849
0.935
0.647
0.875
1(2) Big-Cap Stocks.In China, it is a term used by the investment community to refer to companies with a market capitalization value of more than ¥0.1billion. Large cap is an abbreviation of the term “large market capitalization.” Market capitalization is calculated by multiplying the number of a company’s shares outstanding by its stock price per share.Table6 is the closing price of ten stocks of big-cap from 2017-6-3 to 2017-7-3.Table 6
Closing price of big-cap stocks.
CP
002001
002003
002004
002005
002006
002007
002008
002009
002010
002011
6.3
9.840
4.790
10.71
10.38
4.81
6.740
12.990
9.280
13.770
12.350
6.4
9.740
4.730
10.70
10.28
4.77
6.730
12.740
9.160
13.750
12.180
6.5
9.650
4.770
10.59
10.10
4.76
6.710
12.880
9.060
13.360
12.150
6.6
9.450
4.700
10.39
10.00
4.71
6.710
12.720
9.040
13.180
11.760
6.7
9.350
4.560
10.26
9.990
4.66
6.600
12.120
8.840
13.170
11.520
6.13
9.020
4.300
9.990
9.880
4.53
6.390
11.350
8.340
12.320
11.000
6.14
9.020
4.500
10.06
9.960
4.55
6.290
11.340
8.310
12.210
11.070
6.17
9.010
4.620
10.03
9.910
4.51
6.350
11.120
8.160
12.100
10.970
6.18
9.070
4.590
10.10
9.950
4.33
6.350
11.220
8.140
12.200
11.120
6.19
8.890
4.560
9.960
9.750
4.28
4.680
11.150
8.110
11.980
10.940
6.20
8.420
4.340
9.520
9.290
4.17
4.500
10.820
7.940
11.470
10.610
6.21
8.280
4.370
9.480
9.450
4.15
4.450
11.050
7.730
11.710
10.590
6.24
7.520
4.060
8.690
8.510
4.05
4.210
10.090
7.480
10.930
9.5300
6.25
7.800
3.960
8.690
8.440
4.03
4.190
10.000
7.540
10.920
9.3000
6.26
7.770
3.930
8.580
8.300
3.92
4.100
9.9900
7.490
10.660
9.3100
6.27
7.880
4.230
8.660
8.210
3.92
4.110
9.8400
7.450
10.920
9.3300
6.28
8.280
4.100
9.020
8.570
3.93
4.180
10.130
7.510
11.600
9.9100
7.1
8.170
4.030
8.920
8.590
3.91
4.270
10.180
7.420
11.380
9.9900
7.2
8.150
3.990
8.880
8.480
3.92
4.240
10.190
7.280
11.210
9.8700
7.3
8.040
3.940
8.680
8.440
3.94
4.250
9.9700
7.060
11.350
9.6600Like in the small-cap stocks, we calculate the correlation matrix as shown in Table7.Table 7
Prediction correlation matrix.
CC
600000
600010
600015
600016
600019
600028
600030
600031
600036
600048
600000
1
0.9119
0.9776
0.9497
0.9388
0.9656
0.9447
0.9427
0.9766
0.9736
600010
0.9119
1
0.9477
0.9318
0.9209
0.9298
0.9020
0.8960
0.9030
0.9318
600015
0.9776
0.9477
1
0.9825
0.9646
0.9835
0.9467
0.9408
0.9716
0.9825
600016
0.9497
0.9318
0.9825
1
0.9437
0.9666
0.9050
0.8940
0.9547
0.9597
600019
0.9388
0.9209
0.9646
0.9437
1
0.9766
0.9587
0.9716
0.9358
0.9527
600028
0.9656
0.9298
0.9835
0.9666
0.9766
1
0.9597
0.9557
0.9567
0.9756
600030
0.9447
0.9020
0.9467
0.9050
0.9587
0.9597
1
0.9776
0.9408
0.9696
600031
0.9427
0.8960
0.9408
0.8940
0.9716
0.9557
0.9776
1
0.9318
0.9487
600036
0.9766
0.9030
0.9716
0.9547
0.9358
0.9567
0.9408
0.9318
1
0.9746
600048
0.9736
0.9318
0.9825
0.9597
0.9527
0.9756
0.9696
0.9487
0.9746
1Now we also give the true correlation coefficient of the stocks from 2017.6.4 to 2017.7.4 for comparison.From Tables7 and 8, we know that the correlation matrix we calculated is even more close to the true correlation matrix compared to the case in small-cap stocks. This means that our method for predicting correlation matrix is more effective in the company whose total market capitalization is huge.Table 8
True correlation matrix.
CC
600000
600010
600015
600016
600019
600028
600030
600031
600036
600048
600000
1
0.911
0.979
0.952
0.937
0.97
0.943
0.932
0.979
0.976
600010
0.911
1
0.951
0.932
0.92
0.925
0.901
0.9
0.887
0.93
600015
0.979
0.951
1
0.987
0.968
0.985
0.951
0.944
0.968
0.988
600016
0.952
0.932
0.987
1
0.946
0.97
0.907
0.893
0.955
0.965
600019
0.937
0.92
0.968
0.946
1
0.981
0.962
0.972
0.929
0.954
600028
0.97
0.925
0.985
0.97
0.981
1
0.965
0.952
0.96
0.982
600030
0.943
0.901
0.951
0.907
0.962
0.965
1
0.975
0.934
0.973
600031
0.932
0.9
0.944
0.893
0.972
0.952
0.975
1
0.909
0.942
600036
0.979
0.887
0.968
0.955
0.929
0.96
0.934
0.909
1
0.973
600048
0.976
0.93
0.988
0.965
0.954
0.982
0.973
0.942
0.973
1


## 6. Conclusion

In this paper, we propose stochastic convex semidefinite programs (SCSDPs) to handle data uncertainty in financial problem. An efficient inexact stochastic approximation method is designed. We proved the convergence, complexity, and robust treatment of the algorithm. Numerical experiments show that the method we proposed was effective for SCSDP and also for its special cases. We also numerically demonstrated that the method is more effective in big-cap stocks.



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*Source: 3742575-2018-01-23.xml*