Starting from an initial wealth x 0 , an investor invests her wealth in the financial market. Suppose that the investor invests L t and K t dollar in the risk-free asset and the risky asset at time t, respectively. Then X t = L t + K t denotes the total wealth at time t. In addition, the investor is free to transfer money from the risk-free asset account to the risky asset account and conversely. Let | I t | be the total dollar amount transferred from one asset to the other asset up to time t ( t ≥ 0 ) , that is, I t ≥ 0 means to transfer I t dollar from the risk-free asset account to the risky asset account, I t < 0 means to transfer − I t dollar from the risky asset account to the risk-free asset account. In addition, define two delay variables by

Y t = ∫ − δ 0     e λ s X ( t + s ) d s , (2.3)

Z t = X ( t − δ ) ,   ∀ t ∈ [ 0 , T ] , (2.4)

where λ ≥ 0 is a constant, δ > 0 is a delay parameter. From a view of economic point, delay variable Y t and Z t reflect the average and pointwise performance information of the wealth process in the past period [ t − δ , t ] , respectively. With slight abuse of notation, we do not distinguish X t and X ( t ) . The same is true for other variables.

L t changes with the risk-free interest rate r, the dynamic of L t is described by

d L t = ( r L t − I t ) d t . (2.5)

Generally, K t changes with risky asset’s price. In addition, the historic performance affects the investor’s investment decision, further, K t is affected by the historic performance, so we formulate the dynamic of K t as following stochastic differential equation:

d K t = [ K t ( r + ν Y t + μ Z t + η V t ) + I t ] d t + V t K t d B t S , (2.6)

where ν , μ , η are real constants, { B t S , t ≥ 0 } is a one-dimension standard Brownian motion, Y t and Z t are given by (2.3) and (2.4). V t satisfies (2.2).

During the investment time horizon [ 0, T ] , the investor continuously invests her wealth in the risk-free asset and the risky asset. Let π t be the proportion of the investor’s wealth invested in the risky asset at time t. The remaining propor- tion 1 − π t is invested in the risk-free asset. Then K t = π t X t and L t = X t − K t = ( 1 − π t ) X t . The process π = { π t } t ∈ [ 0, T ] is called an investment strategy. We assume that short-selling and borrowing are prohibited, i.e., 0 ≤ π t ≤ 1 , and the investment strategies satisfy the self-financial condition, that is, d X t = d K t + d L t . Then the dynamic of the wealth { X t , t ≥ 0 } under the in- vestment strategy { π t } t ∈ [ 0, T ) is given by the following stochastic differential delay equation (SDDE):

d X t = X t [ ( π t ( ν Y t + μ Z t + η V t ) + r ) d t + π t V t d B t S ] ,   ∀ t ∈ [ 0 , T ] . (2.7)

where Y t and Z t are given by (2.3)-(2.4) and V t satisfies (2.2), respectively. We further assume that X t = x 0 > 0 , t ∈ [ − δ ,0 ] , which can be interpreted that the investor is endowed with the initial wealth x 0 at time − δ and do not start investment until time 0. Then the initial value of the delay variable Y t is

y 0 : = Y 0 = x 0 ( 1 − e − λ δ ) λ .

Definition 2.1. (ADMISSIBLE STRATEGY) For any fixed t ∈ [ 0, T ] , a stra- tegy π t is said to be admissible if it satisfies the following conditions:

i) π t is F t -measurable for any t ∈ [ 0, T ] ,

ii) For any t ∈ [ 0, T ] , | π t X t | ≤ k | X t + μ Y t | , here k ≥ 0 is a constant.

Let Π denote the set of all admissible strategies.

In this subsection, we formulate the portfolio optimization problem with delay.

Definition 2.2. A utility function U : [ 0, ∞ ) × [ 0, ∞ ) → R { ∞ } is a two varia- bles function i.e. u = U ( x , y ) , ( x , y ) ∈ ℝ 2 . u is strictly increasing, strictly con- cave, twice continuously differentiable with respect to the first variable x , and satisfies l i m x → ∞ u x ( x , y ) = 0 and l i m x → 0 u x ( x , y ) = ∞ .

We consider an optimization problem of the investor who starts with an initial wealth x 0 and initial historic information of y 0 and z 0 . The investor wants to select a investment strategy π ∈ Π so as to maximize the expected utility E [ U ( X T , Y T ) ] . Here, we consider an expected utility of a combination of the terminal wealth X T and the average performance information of the wealth process in the past period [ T − δ , t ] , i.e., Y T . That is, U ( X T , Y T ) is the terminal utility function which depends on both the terminal wealth X T and delay variable Y T such that

E t , x , y , v [ U ( X T , Y T ) ] < ∞ ,

for all x , y and v , where E t , x , y , v [ ⋅ ] = E [ ⋅ | X t = x , Y t = y , V t = v ] .

In mathematical terms, the portfolio optimization problem on a finite time horizon [ 0, T ] can be modeled as the following optimization problem.

Problem 2.3. (Portfolio optimization problem with delay)

s u p π ∈ Π E [ U ( X T , Y T ) ] (2.8)

s . t .         d X t = X t [ ( π t ( ν Y t + μ Z t + η V t ) + r ) d t + π t V t d Z t S ] , (2.9)

d V t = κ ( θ − V t ) d t + σ V t ( ρ d Z t S + 1 − ρ 2 d Z t V ) . (2.10)

For convenience, we first provide a notation. Let A 0 ⊂ R 3 be an open set and A = [ 0, T ] × A 0 . Denote

C 1,2,1,2 ( A ) = { ϕ ( t , x , y , z ) | ϕ ( t , ⋅ , ⋅ , ⋅ )   and   ϕ ( ⋅ , ⋅ , ⋅ , z )   areoncecontinuouslydifferentiable   on   [ 0, T ]       and   ϕ ( ⋅ , x , ⋅ , ⋅ )   and   ϕ ( t , ⋅ , y , ⋅ , )   aretwicecontinuouslydifferentiableon   A 0 } .

First, we show an important Itô’s formula. Let f ∈ C 1,2,1,2 ( ℝ 4 ) and define

G ( t ) = f ( t , x , y , v ) ,

where

x = x ( X t ) , y = y ( X t ) = ∫ − δ 0     e λ s X ( t + s ) d s ,   z = z ( X t ) = X ( t − δ ) , v = V t ,   π = π ( t ) .

Lemma 2.4. (Itô’s formula)

d G ( t ) = L π f d t + f x π x v d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V )                       + f y [ x − e − λ δ z − λ y ] d t , (2.11)

where

L π f = L π f ( t , x , y , v ) = f t + x [ π ( ν y + μ z + η v ) + r ] f x + κ ( θ − v ) f v     + 1 2 π 2 x 2 v f x x + π σ ρ x v f V x + 1 2 σ 2 v f v v . (2.12)

Proof. For each t ∈ [ 0, T ] , using the Leibnitz formula, by (2.3)-(2.4) we have

d d t ( y ( X t ) ) = d d t [ ∫ − δ 0     e λ s X ( t + s ) d s ] = d d t [ ∫ t − δ t     e λ ( u − t ) X ( u ) d u ]     ( u = t + s ) = ζ ( t ) − e − λ δ X ( t − h ) − λ ∫ t − δ t     e λ ( u − t ) X ( u ) d u = x ( ζ t ) − e − λ δ z ( X t ) − λ y ( X t ) .

Since G ( t ) = f ( t , x , y , v ) , by the classical Itô’s formula we can obtain

d G ( t ) = { f t + x [ π ( ν y + μ z + η v ) + r ] f x + κ ( θ − v ) f v + 1 2 π 2 x 2 v f x x     + π σ ρ x v f v x + 1 2 σ 2 v f v v } d t + f x π x v d B t S     + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) + f y [ x − e − λ δ z − λ y ] d t ≡ L π f d t + f x π x v d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V )       + f y [ x − e − λ δ z − λ y ] d t .

where L π f is the form of (2.12).

Remark 2.5. Lemma 2.4 yields the following useful formula for d y :

d y = ( x − e − λ δ z − λ y ) d t . (2.13)

For an admissible strategy π , define the value function

J ( t , x , y , v ) = E [ U ( X T , Y T ) | X t = x , Y t = y , V t = v ] . (3.1)

Then the optimal value function is

J π ( t , x , y , v ) = s u p π ∈ Π J ( t , x , y , v ) , (3.2)

with boundary condition J π ( T , x , y , v ) = U ( x , y ) .

Using the Itô’s formula in Lemma 2.4, we can show the following HJB equa- tion. Assume that J ( t , x , y , v ) ∈ C 1,2,1,2 ( [ 0, T ] × ℝ × ℝ × ℝ ) , then the value func- tion J ( t , x , y , v ) solves the following HJB equation

s u p π ∈ Π { L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } = 0 (3.3)

with boundary condition

J ( T , x , y , v ) = U ( x , y ) , (3.4)

where

L π J ( t , x , y , v ) = J t + x [ π ( ν y + μ z + η v ) + r ] J x + κ ( θ − v ) J v     + 1 2 π 2 x 2 v J x x + π σ ρ x v J v x + 1 2 σ 2 v J v v . (3.5)

Though we have known that the value function J ( t , x , y , v ) is the solution of the HJB equation, we need to prove a verification theorem to ensure that a solution to the HJB equation is actually equal to the value function.

Theorem 3.1. (Verification Theorem) Let X t be a strong solution of (2.7), Y t and Z t are given by (2.3) and (2.4), and J ( t , x , y , v ) ∈ C 1,2,1,2 ( [ 0, T ] × ℝ × ℝ × ℝ ) is a solution of HJB Equation (3.3) with boundary condition (3.4) such that

E { ∫ 0 T ( v J v ) 2 d t } < ∞ ,   E { ∫ 0 T ( π t v x J x ) 2 d t } < ∞ ,   ∀ π ∈ Π . (3.6)

Then we have

J ( t , x , y , v ) ≥ s u p π ∈ Π E t , x , y , v [ U ( X T , Y T ) ] .

Further let

π * = − ( ν y + μ z + η v ) J x + σ ρ v J x v x v J x v . (3.7)

If π * ∈ Π , then π * is the optimal strategy of Problem 2.3 and

J ( t , x , y , v ) = E t , x , y , v [ U ( X T , Y T ) | π = π * ] .

Proof. Let J ( t , x , y , v ) be a solution of the HJB equation (3.3) with boundary condition (3.4). For any given admissible strategy π ∈ Π and for any ( t , x , y , v ) ∈ [ 0, T ] × ℝ × ℝ × ℝ , we must have

{ L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } ≤ 0. (3.8)

On the other hand, applying Itô’s formula (2.11) to J ( t , x , y , v ) , we have

d [ J ( t , x , y , v ) ] = [ L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) ] d t     + π t v x J x d B t S + f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) .

Integrating it from s to T, and using (3.8), we obtain

J ( T , X T , Y T , V T ) − J ( s , X s , Y s , V s ) = ∫ s T { L π J ( t , x , y , v ) + J y ( x − e − λ δ z − λ y ) } d t       + ∫ s T π t v x J x d B t S + ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) ≤ ∫ s T π t v x J x d B t S + ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) .

Therefore, by virtue of boundary condition (3.4), we have

J ( s , X s , Y s , V s ) ≥ U ( X T , Y T ) − ∫ s T π t v x J x d B t S − ∫ s T f v σ v ( ρ d B t S + 1 − ρ 2 d B t V ) . (3.9)

Using the condition (3.6), it is easy to see that

∫ s T f v v d B t S ,   ∫ s T f v v d B t V       and       ∫ s T π t v x J x d B t S

are square integrable martingale whose expectation vanish. Therefore, by taking expectations on both sides of (3.9), we derive

J ( t , x , y , v ) ≥ E t , x , y , v [ U ( X T , Y T ) ] .

Because it holds for all π ∈ Π , we must get

J ( t , x , y , v ) ≥ s u p π ∈ Π E t , x , y , v [ U ( X T , Y T ) ] .

On the other hand, if we take π = π * as defined in (3.7), and if π * ∈ Π , then all the above inequalities can be replaced by equalities. In other words, we have

J ( t , x , y , v ) = E t , x , y , v [ U ( X T , Y T ) | π = π * ] .

Now the proof is complete.

To obtain a closed-form solution of HJB equation (3.3) with boundary condition (3.4), we assume that the investor has a utility function of the following form

U ( x , y ) = 1 1 − γ ( x + μ e λ δ y ) 1 − γ , (3.10)

where γ > 0 , γ ≠ 1 , and γ is the investor’s relative risk aversion coefficient. In fact, Chang et al. [23] and Elsanousi [22] also adopt the utility function of (3.10) to study the optimal investment and consumption problem with delay.

The maximum in HJB Equation (3.3) is obtained when

∂ ∂ π ( L π J ( t , x , y , v ) + J Y ( x − e − λ δ z − λ y ) ) = 0 ,

furthermore,

π * = − ( ν y + μ z + η v ) J x + σ ρ v J v x v x J x x . (3.11)

We try to seek a function J satisfying the HJB equation (3.3) in the form

J ( t , x , y , v ) = ( x + μ e λ δ y ) 1 − γ 1 − γ e x p { γ h ( T − t ) + γ H ( T − t ) v } . (3.12)

Define

u ≡ x + μ e λ δ y ,   H ˜ ( t ) = exp { γ h ( T − t ) + γ H ( T − t ) v } .

It is easy to verify that

J t = u 1 − γ 1 − γ ( − γ h ′ − γ H ′ v ) H ˜ ( t ) ,   J y = μ e λ δ H ˜ ( t ) u − γ ,   J x = H ˜ ( t ) u − γ , J v = γ 1 − γ H H ˜ ( t ) u 1 − γ ,   J x x = ( − γ ) H ˜ ( t ) u − γ − 1 ,   J x v = γ H H ˜ ( t ) u − γ , J v v = u 1 − γ 1 − γ ( γ H ) 2 H ˜ ( t ) . (3.13)

Plugging them into (3.11), we can derive

π * = ( ν y + μ z + η v + γ σ ρ v H ) u γ v x . (3.14)

Substituting (3.13)-(3.14) into (3.3) yields

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = x ( ( ν y + μ z + η v ) ( ν y + μ z + η v + γ σ ρ v H ) u γ v x + r ) u − γ     + u − γ μ e λ δ ( x − e − λ δ z − λ y ) + u 1 − γ 1 − γ γ κ ( θ − v ) H     + 1 2 u − γ − 1 ( ν y + μ z + η v + γ σ ρ v H ) 2 u 2 γ v     + u − γ γ σ ρ v x H ( ν y + μ z + η v + γ σ ρ v H ) u γ v x     + 1 2 u 1 − γ 1 − γ σ 2 γ 2 v H 2 . (3.15)

Let Q = ν y + η v . From (3.15) we have

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = ( Q + μ z ) ( Q + γ σ ρ v H + μ z ) γ v u 1 − γ + r x u − γ + μ e λ δ ( x − λ y ) u − γ     − μ z u − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ − 1 2 ( Q + γ σ ρ v H + μ z ) 2 γ v u 1 − γ     + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + μ z + γ σ ρ v H ) ρ σ H u 1 − γ , (3.16)

or equivalently,

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = Q ( Q + γ σ ρ v H ) γ V u 1 − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ     + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + γ σ ρ v H ) ρ σ H u 1 − γ     + [ ( r + μ e λ δ ) x − λ μ e λ δ y ] u − γ − 1 2 ( Q + γ σ ρ v H ) 2 γ v u 1 − γ     + ( μ z Q γ v + 1 2 μ 2 z 2 γ v + μ ρ σ z H ) u 1 − γ − μ z u − γ . (3.17)

Since (3.12) has a solution that does not depend on z, we have the following condition

( μ z Q γ v + 1 2 μ 2 z 2 γ v + μ ρ σ z H ) u 1 − γ − μ z u − γ = 0. (3.18)

By (3.18), (3.17) becomes

γ 1 − γ ( h ′ + v H ′ ) u 1 − γ = Q ( Q + γ σ ρ v H ) γ v u 1 − γ + γ 1 − γ κ ( θ − v ) H u 1 − γ     + 1 2 γ 2 σ 2 v H 2 1 − γ u 1 − γ + ( Q + γ σ ρ v H ) ρ σ H u 1 − γ     − 1 2 ( Q + γ σ ρ v H ) 2 γ v u 1 − γ + [ ( r + μ e λ δ ) x − λ μ e λ δ y ] u − γ .

Equation (3.17) has a solution depending only on t and u if

− λ = μ e λ δ + r ,   λ < 0. (3.19)

Plugging Q and (3.19) into (3.17), we have

h ′ + v H ′ = 1 − γ γ ( ν 2 y 2 2 γ v + ν η y γ + ν ρ σ y H ) + ( κ θ H + 1 − γ γ ( r + μ e λ δ ) )     + ( 1 2 σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) H 2 + [ ( 1 − γ ) η ρ σ γ − κ ] H + 1 − γ 2 γ 2 η 2 ) v . (3.20)

Equation (3.20) is equivalent to the following ordinary differential equations

{ ν 2 y 2 2 γ v + ν η y γ + ν ρ σ y H = 0 h ′ = κ θ H + 1 − γ γ ( r + μ e λ δ ) H ′ = 1 2 σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) H 2 + [ ( 1 − γ ) η ρ σ γ − κ ] H + 1 − γ 2 γ 2 η 2 (3.21)

From boundary condition (3.4), we have

H ( T ) = 0 ,   h ( T ) = 0. (3.22)

Solving equations (3.21) with boundary condition (3.22), we derive the fol- lowing Theorem.

Theorem 3.2. (Optimal strategy of Problem 2.3) Given wealth X t and CRRA utility (3.10), the solution to HJB Equation (3.3) is given by

#Math_178# (3.23)

where h ( ⋅ ) and H ( ⋅ ) are time-dependent coefficients that are independent of the state variables. That is, for any 0 ≤ τ ≤ T ,

H ( τ ) = exp ( k 3 τ ) − 1 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ϑ , (3.24)

h ( τ ) = 2 κ θ σ 2 ln ( 2 k 3 exp ( ( k 1 + k 3 ) τ / 2 ) 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ) + 1 − γ γ ( r + μ exp ( λ δ ) ) τ , (3.25)

where

ϑ = 1 − γ γ 2 η 2 ,   k 1 = κ − 1 − γ γ η ρ σ ,   k 2 = σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) ,   k 3 = k 1 2 − ϑ k 2 . (3.26)

The optimal investment proportion in the risky asset of Problem 2.3 is given by

π t * = ( ν Y t + μ Z t + η V t + γ σ ρ V t H ( T − t ) ) ( X t + μ e λ δ Y t ) γ V t X t . (3.27)

Remark 3.3. It is interesting that our results are similar to the results of Liu and Pan [28] . Liu and Pan study optimal investment strategies given an investor access not only to bond and stock markets but also to the derivatives market and the price process of the risk asset is associated with stochastic volatility and jump. Here are some comparisons between them.

i) In our results, the proportion in the risky asset depends on wealth X t , delay variables Y t and Z t , and stochastic volatility V t at time t. However, in Liu and Pan [28] the optimal strategy is a deterministic function and does not depend on wealth X t and stochastic volatility V t . Moreover, our results are consistent with the results of Liu and Pan [28] , when the delay variables are not considered in our model (i.e. ν = μ = 0 ). The details are given in Lemma 3.4.

ii) Let the delay δ approach 0 then Y t → 0 , in this case the delay variable Y t vanishes. Assuming that (3.19) holds and λ = − r , then μ = 0 . At this time, the dynamics of wealth (2.7) degenerates as

d X t = X t [ ( π t η V t + r ) d t + π t V t d Z t S ] ,

which is the case without delay. The corresponding problem without delay and its optimal strategy are given in Lemma 3.4.

iii) The value function J ( t , X t , Y t , V t ) of (3.23) depends only on delay vari- able Y t and does not depend on delay variable Z t .

Now we consider a special case of our model. Suppose that the dynamics of risky asset does not depend on historic performance, then our model degenerates into a CIR SV model without delay. The results of our model will be reduced to the following special case.

Proposition 3.4. (Optimal strategy without delay) Consider the following problem without delay:

m a x π t ∈ Π E [ U ( X T ) ] = E [ X T 1 − γ 1 − γ ]

s . t .       d X t = X t [ ( π t η V t + r ) d t + π t V t d B t S ] , (3.28)

d V t = κ ( θ − V t ) d t + σ V t ( ρ d B t S + 1 − ρ 2 d B t V ) . (3.29)

The value function is given by

J ( t , x , v ) = m a x π t ∈ Π E [ U ( X T ) | X t = x , V t = v ] (3.30)

and HJB equation is given by

max { J t + x ( π η v + r ) J x + κ ( θ − v ) J v + 1 2 π 2 x 2 v J x x + π σ ρ v x J v x + 1 2 σ 2 v J v v } = 0 (3.31)

with boundary condition J ( T , x , v ) = x 1 − γ 1 − γ .

Then, for given wealth X t , the solution to HJB equation is given by

J ( t , X t , V t ) = 1 1 − γ X t 1 − γ exp { γ h ( T − t ) + γ H ( T − t ) V t } (3.32)

where h ( ⋅ ) and H ( ⋅ ) are time-dependent coefficients that are independent of the state variables. That is, for any 0 ≤ τ ≤ T ,

H ( τ ) = exp ( k 3 τ ) − 1 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ϑ , (3.33)

h ( τ ) = 2 κ θ σ 2 ln ( 2 k 3 exp ( ( k 1 + k 3 ) τ / 2 ) 2 k 3 + ( k 1 + k 3 ) ( exp ( k 3 τ ) − 1 ) ) + 1 − γ γ r τ , (3.34)

where

ϑ = 1 − γ γ 2 η 2 ,   k 1 = κ − 1 − γ γ η ρ σ ,   k 2 = σ 2 ( γ ( 1 − ρ 2 ) + ρ 2 ) ,   k 3 = k 1 2 − ϑ k 2 . (3.35)

The optimal investment proportion in the risky asset is given by

π ¯ t * = η + γ σ ρ H ( T − t ) γ . (3.36)

Proof. Let μ = ν = 0 in Theorem 4.5, we can easily obtain the results of this lemma.

Remark 3.5. Our results without considering delay (i.e. ν = μ = 0 ) is similar to the results of Liu and Pan [28] . Their results are consistent with our results without considering delay, when there is only stock and not derivatives consi- dering in Liu and Pan’s model. In some sense, we extend the results of Liu and Pan [28] .

In general, the dynamic changes of wealth X t must depend on both delay variables Y t and Z t at the same time in a similar manner, i.e., ν μ ≥ 0 .

(1) From (3.27), for given X > 0 and V > 0 , we have

∂ π * ∂ Y = ν X + μ e λ δ γ V X { = 0 ,   ν = μ = 0 ,   > 0 ,   ν > 0 , μ > 0 < 0 ,   ν < 0 , μ < 0 ,

∂ π * ∂ Z = μ γ V X { = 0 ,   μ = 0 ,   > 0 ,   μ > 0 , < 0 ,   μ < 0 ,

According to the above results, we know that i) if ν = μ = 0 , then ∂ π * ∂ Y = ∂ π * ∂ Z = 0 , which is the case without delay, the optimal investment strategy π * dose not depend on Y and Z; ii) if ν < 0 and μ < 0 , then ∂ π * ∂ Y > 0 and ∂ π * ∂ Z > 0 , which mean that the delay factors take a positive effect on the optimal investment strategy; iii) if ν < 0 and μ < 0 , then ∂ π * ∂ Y < 0 and ∂ π * ∂ Z < 0 , which mean that the delay factors take a passive effect on the optimal investment strategy.

In Figure 1, we draw the optimal investment strategy π * with t under the different delay. Let ν = μ = 0 (or Y = 0 , Z = 0 ) denote the case without delay, i.e., the curves denoted by “ Y = 0 ” in two Figures. In Figure 1(a), we set ν = 0.01 , μ = 0.01 , Z = 2 , and Y = 2 or Y = 20 respectively. Here, ν = 0.01 > 0 and μ = 0.01 > 0 mean that delay variables Y and Z take a positive effect on optimal investment strategy, that is, the curves are arranged by Y = 20 , Y = 2 and Y = 0 from top to bottom. In other words, the higher positive historic performance of the portfolio promote the higher proportion of an investor’s wealth invested on stocks. From another point of view, the result implies that a higher historic stock price encourages investor to invest more money to the stock. Conversely, in Figure 1(b), let ν = − 0.01 , μ = − 0.01 , Z = 2 , and Y = 2 or Y = 20 respectively. And ν = 0.01 < 0 and μ = 0.01 < 0 mean that delay variables Y and Z take a negative effect on optimal investment strategy, that is, the curves are arranged by Y = 0 , Y = 2 and Y = 20 from top to bottom.

Figure 1(b) shows that the higher historic performance of the portfolio gives the investors a chance to harvest their gain by cut down the proportion invested on stocks, which implies that a higher historic stock price cause investor to invest less money on the stock so as to avoid to chase the high price. This seems to be consistent with the facts.

(2) From (3.27), for given Y > 0 and Z > 0 , we have

∂ π * ∂ V = − ( ν Y + μ Z ) ( X + μ e λ δ ) γ X V 2 < 0 , ∂ π * ∂ X = − ν Y + μ Z + η V + γ σ ρ H ( T − τ ) μ e λ δ γ V < 0.

So, the optimal investment strategy π * decrease w.r.t. stochastic volatility V and wealth X, respectively.

Figure 2 plot the optimal investment strategy π * with stochastic volatility V under the different delay. Let X = 100 , γ = 10 . ν = μ = 0 or Y = 0 , Z = 0 means the case without delay, i.e., the curve denoted by “ Y = 0 ” in two Figures. Figure 2 shows that the optimal investment strategy π * decreases with increas- ing stochastic volatility V. That is to say, the higher stochastic volatility take a passive effect on the investment enthusiasm. Moreover, in Figure 2(a), we set ν = 0.01 , μ = 0.01 , Z = 2 , and Y = 2 or Y = 20 respectively. Here, ν = 0.01 > 0 and μ = 0.01 > 0 mean that delay variables Y and Z take a posi- tive effect on optimal strategy, that is, the curves are arranged by Y = 20 , Y = 2 and Y = 0 from top to bottom. Conversely, in Figure 2(b), let ν = − 0.01 , μ = − 0.01 , Z = 2 , and Y = 2 or Y = 20 respectively. ν = 0.01 < 0 and μ = 0.01 < 0 mean that delay variables Y and Z take a negative effect on optimal investment strategy, that is, the curves are arranged by Y = 0 , Y = 2 and Y = 20 from top to bottom. That is to say, Figure 1 and Figure 2 have the same result about the effect of delay variables on the optimal investment strategy.

In Figure 3, we demonstrates the optimal investment strategy π * w.r.t.

different wealth X under the different volatility V. Let V = 0.1 , V = 0.15 , and V = 0.2 respectively, and with delay variables Y = 2 , Z = 2 , μ = 0.01 , ν = 0.1 . The curves are arranged by V = 0.1 , V = 0.15 , and V = 0.2 from top to bottom. The two figure show the larger is stochastic volatility V and the smaller is the proportion of the investor’s wealth invested on stocks. This fits with the fact that market instability has negative impact on investors. Furthermore, in Figure 3(a) the curves show that the optimal investment strategy π * decreases with the accumulation of wealth X. This suggests that an investor has an increas- ing preference to risk aversion with the increase of wealth, thereby, an investor cut down the stockholding.

In this subsection, we analyze the effect of delay variable Y and stochastic volatility V on the value function. According to (3.23), the value function de- pends on t , X , Y , V and we have

J X = ( X + μ e λ δ Y ) − γ exp { γ h ( T − t ) + γ H ( T − t ) V } > 0 ,

J V = ( X + μ e λ δ Y ) 1 − γ 1 − γ γ H ( T − t ) exp { γ h ( T − t ) + γ H ( T − t ) V } > 0 ,

J Y = μ e λ δ ( X + μ e λ δ Y ) − γ > 0   ( μ > 0 ) ,

Let X = 100 , ν = 0.01 , μ = 0.01 , Z = 2 , Y = 2 or Y = 200 , and ν = μ = 0 (or Y = 0 , Z = 0 ) denotes without delay in Figure 5 and Figure 6 (i.e., the curve denoted by “ Y = 0 ”).

Figure 4 plot the optimal value function J ( t , X , Y , V ) with t when the risk aversion coefficient γ = 0.3 and γ = 3 . They shows that the value functions are almost the same with different delay variables.

Figure 5 plot the optimal value function J ( t , X , Y , V ) with wealth X when the risk aversion coefficient γ = 0.3 and γ = 3 . They shows that the optimal value functions are almost the same with different delay variables. Figure 6 plot the optimal value function J ( t , X , Y , V ) w.r.t. stochastic volatility V when the risk aversion coefficient γ = 0.3 and γ = 3 . However, it is interesting enough that the higher stochastic volatility seemed to induce the higher value of the value functions. That is, the curves are arranged by V = 0.1 , V = 0.15 , and V = 0.2 from top to bottom. The two figures show that the optimal value func- tions increase with the increasing of stochastic volatility V.