## 2 The Model of Second Order Stochastic Dominance
### Portfolio Optimization Model  
In the portfolio optimization problem, our aim is to invest our capital in some assets in order to obtain some desirable characteristics of the total return on investment.  
Let \(n\) denote the number of assets available for investment at the beginning of a fixes time period and assume that we have a fixed capital to be invested in \(n\) assets. To simplify the discussion, we use \(x=(x_{1},\ldots,x_{n})^{T}\) to denote the fractions of initial capital invested in different assets. \({}_{i}=\frac{w_{i}}{w}(i=1,\ldots,n)\) where \(w_{i}\) is the capital invested in asset \(i\) and \(w\) is the total amount of capital to be invested. Let \(X\) denote a set of feasible portfolios and it is clear that \(X\in\mathbb{R}^{n}\) is a bound convex polyhedron. Let \(R_{i}(\xi),i=1,\ldots,n\) denote the return of asset \(i\) in the case of discrete distribution, we assume that \(E[\mid R_{j}\mid]<\infty\) for all \(j=1,\ldots,n\). Therefore, if we have a fixed capital, the return of portfolio can then be formulated as:  
\[g(x,\xi):=R_{1}(\xi)x_{1}+R_{2}(\xi)x_{2}+\cdots+R_{n}(\xi)x_{n} \tag{14}\]  
In economics and related disciplines, the transaction cost is a cost in making any economic trades when participating in the market. If we take the transaction cost into account, let \(c(x)\) denote the V-type function of transaction cost and the return of portfolio can be formulated as:  
\[g(x,\xi)=x^{T}\xi-c(x) \tag{15}\]  
where \(x\) is the investment proportion vector and \(\xi\) is a vector represents the return of assets. Clearly, the set of possible asset allocations can be defined as follows:  
\[X=\{(x_{1},\ldots,x_{n})\mid\sum_{j=1}^{n}x_{j}=1,x_{j}\geq 0,\forall j\in 1, \ldots,n\} \tag{16}\]  
In order to ensure the diversity of portfolio, the bounds are always set to 0.6 and 0:  
\[X=\{(x_{1},\ldots,x_{n})\mid\sum_{j=1}^{n}x_{j}=1,0\leq x_{j}\leq 0.6,\forall j \in 1,\ldots,n\} \tag{17}\]The two constraints proposed above based on the short-selling is prohibited. Short-selling is the sale of a security that is not owned by the seller, or the seller has borrowed. If the short-selling is allowed and investors do not want to invest very small amounts in an asset, the upper and lower bounds on the fraction of capital invested in each assets are set to \(2\) and \(-1\)[9]:  
\[X=\{(x_{1},\ldots,x_{n})\ |\ \sum_{j=1}^{n}x_{j}=1,-1\leq x_{j}\leq 2,\forall j \in 1,\ldots,n\} \tag{18}\]  
Assumed that a reference random return \(Y\) having a finite expected value which may be an index such as FTSE 100 index, or a current portfolio. Let \(y\) denote a benchmark investment, normally \(y_{i}=\frac{1}{m}\), for \(i=1,\ldots,n\). Based on the model (13) and the discussion above, we propose the following portfolio optimization model with SSD constraints:  
\[\begin{array}{lll}min&&f(x,\xi)\\ s.t.&&\sum_{i=1}^{m}\frac{1}{m}(g(y,\xi^{i})-g(x,\xi^{i}))_{+}\leq \sum_{i=1}^{m}\frac{1}{m}(g(y,\xi^{i})-g(y,\xi^{i}))_{+}\\ &&x\in X\end{array} \tag{19}\]  
Firstly we consider the simplest case:  
\[f(x,\xi)=-g(x,\xi) \tag{20}\]  
Then we consider a sightly more complicated condition. In view of the impact of transaction cost and the small amounts investment in portfolio, we consider the following performance function [9]:  
\[f(x,\xi)=-g(x,\xi)-\sum_{i=1}^{n}x_{i}^{2} \tag{21}\]