Recent publications in economic Journals and other reputable Mathematics and Science Journals have brought to light, different methods of optimizing investment strategies and returns. Example, some researchers have made various contributions in this direction, particularly, in DC Pension Plan.  work on, stochastic life styling: optimal dynamic asset allocation for defined contribution pension plans. In their work, various properties and characteristics of the optimal asset allocation strategy, both with and without the presence of non-hedge able salary risk were discussed. The significance of alternative optimal strategy by pension providers was established.
In order to deal with optimal investment strategy, the need for maximization of the expected utility of the terminal wealth became necessary. Example, the Constant Relative Risk Aversion (CRRA) utility function, and (or) the Constant Absolute Risk Aversion (CARA) utility function were used to maximize the terminal wealth.  -  used CRRA to maximize terminal wealth. However,  used the CRRA and the CARA to maximize terminal wealth.
 applied the well-known H-J-B equation, Legend transform, and dual theory to obtain the explicit solutions of CRRA and CARA utility function, for the maximization of the terminal wealth.  took a different direction, where they considered an Inflationary market. In their work, the plan member made extra contribution to amortize the pension fund. The CRRA utility function was used to maximize the terminal wealth. This triggered our research. In our model; the amortization fund is a definite proportion of the plan member’s salary.
The motive of this work is to maximize the optimal investment strategy for DC Pension with stochastic salary under the affine interest rate, with extra DC contribution, which is a proportion of the plan member’s salary. We shall use the CRRA utility function to maximize our terminal wealth. Our approach is similar to that of Chubing  , though, ours is an extension to extra DC contribution.
We start with a complete and frictionless financial market that is continuously open over the fixed time interval [0, T], for T > 0 representing the retirement time of a given shareholder.
We assume that the market is made up of risk free asset (cash) a zero coupon bond and risky asset (stock). Let be a complete probability space where is a real space and P is a probability measure, is a standard two dimensional Brownian motion such that they are orthogonal to each other. F is the filtration and denotes the information structure generated by the Brownian motions .
Let denote the price of the risk free asset at time t and it is modeled as follows
is the short interest rate process and is given by the stochastic differential equation (SDE)
where a, b, , and are positive real numbers. If (resp., ) is equal to zero, we have a special case, as in   . So under these dynamics, the term structure of the short interest rates is affine, which has been studied by     .
Let denote the price of the risky asset and its dynamics is given based on a continuous time stochastic process at and the dynamics of the price process is described as follows
with , (resp., , ) being constants (resp., positive constants) see    .
A zero-coupon bond with maturity T, whose price at time t is denoted by , and
Its dynamics is given by the SDE below see  
Based on the works of     we denote the stochastic salary at time t by (t) which is described by
where are real constants, which are two volatility scale factors measuring how the risk sources of interest rate and stock affect the salary. That is to say, the salary volatility is supposed to be a hedge able volatility whose risk source belongs to the set of the financial market risk sources. This is in accordance with the assumption  , but is differs from those of   who also suggest that the salary was influence by non hedgeable risk source (i.e., non-financial market). Also  assume that the instantaneous mean of the salary is such that where is a real constant.
3.1. Hamilton-Jacobi-Bellman (HJB) Equation
Suppose, we represent as the strategy and we define the utility attained by the contributor from a given state x at time t as
where t is the time, r is the short interest rate and x is the wealth. Our interest here is to find the optimal value function
and the optimal strategy such that
3.2. Legendre Transformation
The Legendre transform and dual theory help to transform non linear partial differential equation into a linear partial differential equation.
Theorem 3.1: Let be a convex function for , define the Legendre transform
where is the Legendre dual of  .
Since is convex, from theorem 3.1 we defined the Legendre transform
where is the dual of and is the dual variable of x.
The value of x where this optimum is attained is denoted by , so that
The function and are closely related and can be refers to either one of them as the dual of G. These functions are related as follows
At terminal time, we denote
As a result
where is the inverse of the marginal utility and note that
At terminal time T, we can define
4. Model Formulation
Here the contributions are continuously paid into the pension fund at the rate of where is the mandatory rate of contribution and is the extra contribution rate which is assumed to be at constant rate. Let denote the wealth of pension fund at time . and represent the proportion of the pension fund invested in the bond and the stock respectively. This implies that the proportion of the pension fund invested in the risk-free asset . The dynamics of the pension wealth is given by
Substituting (1), (3) and (4) in (17) we have
Let the relative wealth be defined as follows
Applying product rule and Ito’s formula to (19) and making use of (6) and (18) we arrive at the following equation
The Hamilton-Jacobi-Bellman (HJB) equation associated with (21) is
where and are partial derivatives of first and second orders with respect to time, short interest rate, and relative wealth.
Differentiating (23) with respect to and , we obtain the first-order maximizing conditions for the optimal strategies and as
Solving (24) and (25) simultaneously we have
Substituting (26) and (27) into (23) we have
Applying Legendre transform to (28) we have
Differentiating Equation (30) for with respect to z we obtain a linear PDE in terms of h and its derivatives and using , we have
We will now solve (31) for h and substitute into (33) and (34) to obtain the optimal investment strategies.
5. Explicit Solution of the Optimal Investment Strategies for the CRRA Utility Function
Assume the investor takes a power utility function
The relative risk aversion of an investor with utility described in (37) is constant and (37) is a CRRA utility.
From (16) we have and from (37) we have
We assume a solution to (31) with the following form
Substituting (39) into (31) we have
Splitting (40) we have
Solving (41) with the given boundary condition
To solve (42), we conjecture a solution of the form
Substituting (44) into (42), we have
Splitting (44) we have
Hence the solution of (31) is given as
The optimal investment strategies for cash, bond and stock is given as follows
Suppose , then the salary is not stochastic and the optimal investment is given as
Furthermore, assume there is no extra contribution i.e., we have
Suppose and then
Since and then,
But and , therefore
Suppose then the strategies is without the extra contribution hence the required strategies are as given below
The result above is similar to that in  .
Suppose and then
Since and then,
But and , therefore
6. Discussion and Conclusion
From proposition 5.1, we observed that with non stochastic salary, the optimal investment strategies for bond and stock increases with extra contribution while that of cash decreases with extra contribution. Also from proposition 5.2, we observed that with stochastic salary, the optimal investment strategies for bond and stock increases with extra contribution while that of cash decreases with extra contribution. We also had case where we could not conclude on the behavior of the optimal investment strategy for bond with extra contribution. In general we observed that extra contribution to the pension fund has an effect on the optimal investment strategies in cash, bond and stock. The analysis shows that the plan member will increase the proportion of his wealth to be invested in bond and stock and will reduce the proportion to be invested in cash.
The optimal investment strategy for a prospective investor in a DC pension scheme, with stochastic salary, under the affine interest rate model has been studied. Relevant to this work, the CRRA utility function was used and we obtained the optimal investment strategies for cash, bond and stock using the Legendre transform and dual theory. More so, the effect of various parameters were analyzed, in particular, the effect of extra defined contribution was x-rayed, with significant input on the investment strategy.
 Chubing, Z. and Ximing, R. (2013) Optimal Investment Strategies for DC Pension with Stochastic Salary under Affine Interest Rate Model. Hindawi Publishing Corporation.