Optimal Portfolio Choice in a Jump Diffusion Model with Self Exciting

ISSN Online: 2162-2442 ISSN Print: 2162-2434

DOI: 10.4236/jmf.2019.93020 Aug. 20, 2019 345 Journal of Mathematical Finance

Optimal Portfolio Choice in a Jump-Diffusion

Model with Self-Exciting

Baojun Bian

1

_{, Xinfu Chen}

2

_{, Xudong Zeng}

3

1_{Department of Mathematics, Tongji University, Shanghai, China}

2_{Department of Mathematics, University of Pittsburgh, Pittsburgh, PA, USA} 3_{School of Finance, Shanghai University of Finance and Economics, Shanghai, China}

Abstract

We solve the optimal portfolio choice problem for an investor who can trade a risk-free asset and a risky asset. The investor faces both Brownian and jump risks and the jump is modeled by a Hawkes process so that occurrence of a jump in the risky asset price triggers more sequent jumps. We obtain the op-timal portfolio by maximizing expectation of a constant relative risk aversion (CRRA) utility function of terminal wealth. The existence and uniqueness of a classical solution to the associated partial differential equation are proved, and the corresponding verification theorem is provided as well. Based on the theoretical results, we develop a numerical monotonic iteration algorithm and present an illustrative numerical example.

Keywords

Portfolio Choice, Jump Diffusion, Stochastic Volatility, Hawkes Process, Self-Exciting Jump, HJB Equation

1. Introduction

Empirical studies suggest that asset price encounters jumps and its volatility is stochastic. Further studies show that jumps occur in clusters, that is, a sequence of jumps occur in short time following a (big) jump which occurs after a rela-tively long quiet period of time. The feature of clustered jumps can be caught by a type of stochastic process known as Hawkes process. In this paper, we model occurrence of jumps by a Hawkes process hence our model is an extension of well-known jump-diffusion models, e.g. [1]. Meanwhile, we assume that such Hawkes jumps may occur in asset price itself as well as in its volatility. As a re-sult, our model merges with the vast literature of stochastic volatility.

How to cite this paper: Bian, B.J., Chen, X.F. and Zeng, X.D. (2019) Optimal Portfo-lio Choice in a Jump-Diffusion Model with Self-Exciting. Journal of Mathematical Finance, 9, 345-367.

https://doi.org/10.4236/jmf.2019.93020

Received: May 25, 2019 Accepted: August 17, 2019 Published: August 20, 2019

Copyright © 2019 by author(s) and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

DOI: 10.4236/jmf.2019.93020 346 Journal of Mathematical Finance

The contributions of the present paper are twofold. First, we solve the optimal investment problem and prove a verification result for a CRRA utility while [2]

solves for a logarithm utility function. They do discuss the CRRA case but leave proofs to some references. However, our work shows that it is highly non-trivial to prove the existence and uniqueness of a classic solution to the associated Hamilton-Jacobi-Bellman (HJB) equation, which is essential for solving the op-timal portfolio choice problem and implementing numerical methods. Second, our model may incorporate stochastic volatility or stochastic risk premium. Thus it shall be powerful to explain more financial phenomena, e.g. flight-to-quality, under-diversification, and possibly disclose more economic insights.

Technically, in the strand of the relevant literature, [3] introduces an Ornstein-Uhlenbeck (OU) type process of subordinator to model volatility. Af-ter that, there are several works about optimal portfolio selection in a model with OU type processes of subordinators. For example, [4] solves the portfolio choice problem in a model where volatility is a linear combination of Ornstein-Uhlenbeck type processes of subordinators. [5] solves the optimal in-vestment and consumption problem in a similar model where a state variable (economic factor) is an Ornstein-Uhlenbeck type process of subordinator. They deal with a more general model, compared to [4]. As a result, they arrive at a nonlinear partial integro-differential equation, instead of a linear one in the lat-ter. In both of these two papers, there are no jump components in the dynamics of asset prices. In stark contrast to the literature we incorporate jumps in the as-set price in the present paper and hence confront a new/different challenge (if not more difficult) when we solve the optimal portfolio choice problem.

In our general setting, the volatility is a function of the state variable (the jump intensity) following a Hawkes process. Hence the present paper may com-bine two strands of research: modeling volatility by an OU type of process of subordinator and modeling jumps by Hawkes processes together. There are sev-eral papers studying the case where there are jumps in volatilities, e.g. [5], etc., while the present paper considers jumps in both of asset price and volatility, and in jump intensity as well. This leads to a more complicated HJB equation than in

[5]. As a result, our model has the features of stochastic volatility and self-exciting, while the optimal investment problem is still solvable in sense of by an iteration method, given that the iteration is proved to converge correctly to the unique classical solution of the problem.

This paper is incremental to the few aforementioned papers and dedicated to solving several technical problems related to CRRA utility functions. Neverthe-less, the literature on portfolio choice is vast and is growing quickly. We would like to refer to some of them, for example, [6], which proposes a semivariance method for diversified portfolio selection; [7], that discusses a portfolio adjusting problem. The both assume that security returns are subject to experts’ estima-tions.

op-DOI: 10.4236/jmf.2019.93020 347 Journal of Mathematical Finance

timal portfolio choice problem in Section 2. The Hawkes process is introduced in this section. In Section 3, we analyze the HJB equation. The existence and un-iqueness of a classical solution to the equation are proved under some appropri-ate conditions. We also prove a verification theorem for the solution in Section 4. Two illustrative examples and one numerical example are provided in Section 5. Conclusion and further discussion are in Section 6. An extension is supplied in Appendix.

2. Problem Formulation

2.1. Hawkes Process for Self-Exciting Jumps

A Hawkes process is a counting process with self-exciting feature. Roughly speaking, it is a compounded Poisson process with stochastic intensity.

Given a complete probability space

(

Ω, , 

( )

_{t t≥0},P

)

, a counting process,

{ }

N_{t t≥0}, satisfies

(

Nt+∆t−Nt=1|t

)

= ∆ + ∆

λ

t t o t

( )

,

P

(

Nt+∆t−Nt >1|t

) ( )

=o t∆ ,

P

where the intensity process λt is given by the integrated form

(

)

( )

0 ₀

e t 1 e t t e t sd

t α α α Ns

λ

−

λ

−

λ

β

− − ∞

= + − +

_∫

or by the differentiation form

(

)

d

λ α λ

t = ∞−

λ

t dt+

β

d ;Nt (2.1)

here λ_{∞ is the long-run average of the jump intensity corresponding to the}

jump;

α >

0

is the decay rate driving the jump intensity back to the long-run average before a jump occurs; β ≥0 is a constant indicating non-negative im-pact of the jump occurrence on the jump intensity.

A jump process

{ }

Nt with the jump intensity described by (2.1) is called a

Hawkes process [8]. It is known that the process is stationary if β 1

α < . A

Hawkes process differs from a doubly stochastic Poisson process since its incre-ments are not independent.

{ }

Nt is not Markovian but

{

(

Nt,λt

)

}

is. The

compensated process Nt−

∫

₀t

λ

sds is a local martingale. For more information

and a formal definition of Hawkes process, we refer to [8]. Figure 1 illustrates a sample path of one self-exciting (Hawkes) process.

[image:4.595.273.483.74.240.2]

DOI: 10.4236/jmf.2019.93020 348 Journal of Mathematical Finance Figure 1. A sample path of Hawkes

{

Nt,λt

}

and stock price

{ }

St .

Meanwhile, the mean-reversion property of (2.1) prevents the jump intensity from explosion given 0≤β α< . Indeed, taking the expectation of (2.1) and us-ing 

[ ]

dNs =

[ ]

λ

s ds we obtain that

[ ]

( )

(

( )

)

0e 1 e as .

1

t t

t β α β α t

αλ λ

λ λ _β

α β

α

− ∞ − ∞

= + − → → ∞

− ₋



2.2. Asset Dynamics with Self-Exciting Jumps

We consider a market with a risk-free asset (bond) and a risky asset (stock). The price

{ }

Bt of the risk-free asset follows dynamics:

dBt =rB ttd ,

where r is the risk-free interest rate. The price

{ }

St of the risky asset follows a

stochastic differential equation

( )

( )

d t _{d d d}

t t t t t

t

S _{r t W Y N}

S−

µ λ

σ λ

 

=_ + _ + + _(2.2)

where

{ }

Nt is a Hawkes process described in the preceding section,

{ }

Wt is a

standard Brownian motion, and Yt > −1 is a random jump size independent of

the jump process and the Brownian motion. To be more precise, we define

0 d ₁

t t

t t i

i N

Y N y

≤ ≤

=

∑

∫

where

{ }

yi i₁

∞

= are i.i.d. random variables that are independent of either the

Brownian motion or the jump process.

Note that the volatility σ and the risk premium

µ

are set to be functions of the jump intensity. Given appropriate conditions about the mappings (e.g. monotonic mappings), we may write

λ λ σ

=

( )

_and

µ µ σ

=

( )

_{, so our model}

is compatible with many models studied in stochastic volatility literature.

2.3. Optimal Portfolio Selection Problem

DOI: 10.4236/jmf.2019.93020 349 Journal of Mathematical Finance

the risk-free asset in a time horizon

[ ]

0,T . In order to maximize the expected utility of the terminal wealth, the investor needs to find an optimal investment strategy.

Let Xt be the wealth of portfolio at time t and πt be the proportion of

wealth invested into the risky asset at time t. Then

(

)

(

)

d t d t ₁ d t _{d d d .}

t t t t t t t t

t t t

X S B

r t W Y N

X−

π

S−

π

B

π µ

π σ

π

= + − = + + + _(2.3)

An investment strategy is an adapted stochastic process

π

=

{ }

π

_{t t∈[ ]0,T} . It is

admissible if the associated wealth process is non-negative almost surely. The jump size Yt, in particular, is assumed to take a form of eZt −1, where Zt is a

Gaussian random variable. This setting is popularly admitted in the literature of jump-diffusion model. See, for example, the seminal paper of [1]. As a result, an admissible strategy π _{shall satisfy the constraint} 0≤πt ≤1. Thus, shorting

either stock or bond is not permitted. This constraint condition may be relaxed1

according to distribution and support set of a specific jump size Yt. We denote

all admissible strategies by .

An optimal investment strategy is a strategy that maximizes the expected util-ity of the terminal wealth. That is, the objective of an investor is to find V and

*

π _{such that}

(

)

,

( )

* ,

( )

0

, , max x , arg max x

t T T

V x t λ U X λ U X

π _π

λ

π

∈ _∈

= _ _ _ = _ _ _

 _ (2.4)

where x,

tλ

 is the expectation conditioned on Xt =x and λt =λ. In this

paper, we solve the problem for the CRRA utility _{U x}

( )

_{=x p}p _{. We prove the}

existence and uniqueness of a classical solution to the associated HJB equation when p∈

( )

0,1 _{. By a similar approach, our framework may be extended to the}

case of p<0 _{regardless of an amount of efforts. The case of logarithm utility}

(corresponding to p=0) has been studied in [2] while assuming constant vola-tility and risk premium. In Section 5, we will discuss an extension case of loga-rithm utility with stochastic volatility and stochastic risk premium, as an appli-cation of our general results.

The Hamilton-Jacobi-Bellman (HJB) equation associated with the above sto-chastic optimization problem can be derived as

(

)

(

)

1 2 2 2

(

(

)

)

max 1 , , ,

2

t

x xx

V V V

r xV x V V x Y t

λ

π

α λ λ λ

πµ π σ λ π λ β

∞

− − +

 _{ }

= _ + + + _ + + _

  

(2.5)

(

, ,

)

( )

.

V x T

λ

=U x _(2.6)

Here subscripts denote partial derivatives and Y is a random variable having the same distribution as Yt. In this paper, we shall make a full mathematical

analysis of the HJB Equation (2.5) subject to the terminal condition (2.6), and a certain growth condition such as (2.8) below.

1_{If a tight bound of Y is − ≤ ≤a Y b, where a and b are positive constants, then { }}

t

DOI: 10.4236/jmf.2019.93020 350 Journal of Mathematical Finance

There are several papers studying the case where there are jumps in volatilities, e.g. [5]. Different from only jumps in volatility, our model incorporates jumps in both of asset price and volatility, and in jump intensity as well. It is worth to mention that the HJB Equation (2.5) is more complicated than that in [5]. The framework of our model is the same as that in [2], except the setting of utility function. As well-known, CRRA utility functions generally involve much more difficult technical problems than the logarithm utility function.

Remark 2.1. The HJB Equation (2.5), together with the terminal condition

(2.6), mayhaveseveralsolutions.Fromaviewpointofpartialdifferential equa-tion, it is necessary to prescribe an asymptotic behavior of the solution as

λ → ∞

_{. Note that, as a suboptimal strategy, investing everything into the}

risk-freeassetgivesthelowerbound

(

, ,

)

( )

ert

( )

e .prt

V x λ t ≥U x =U x (2.7)

Although it is very had to estimate an upper bound, we shall prove the exis-tence of a unique solution of (2.5)-(2.6) under the following growth condition:

For some constant

C

>

0

_.

(

)

( )

, ,

[ ]

1 V x t _{, 0, 0, .}

C t T

C U x

λ

λ

≤ ≤ ∀ ≥ ∈ (2.8)

3. Mathematical Analysis of the HJB Equation

3.1. Scaling Invariance

Note that applying the same strategy for two initial portfolios with initial condi-tion

(

X0,

λ

0

) ( )

= x,

λ

and

(

X0,

λ

0

) ( )

= 1,

λ

respectively, we find that the cor-responding wealth of the two portfolios differ by a factor of x at any time

[ ]

0,

t∈ T _{. Hence, optimal strategies do not depend on x, and we have the scaling}

invariance

(

, ,

)

( ) (

,

)

,

V x t

λ

=U x H

λ

T t− _(3.1)

where

τ

= −

T t

is the time to expiry and H

( )

λ τ

, :=V

(

1, ,x T−

τ

) ( )

U 1 _.

Plug-ging (3.1) into (2.5) and using Remark 2.2 we obtain the equation for H:

(

)

max_{0 1}

{

(

,

) (

,

)

(

,

) (

,

)

}

,

Hτ +α λ λ− ∞ Hλ = _{≤ ≤π} A λ π H λ τ +B λ π H λ β τ+ (3.2)

( )

,0 1,

H

λ

= _(3.3)

where

(

_,

)

( )

(

1

)

2 2 _,

(

_,

)

(

₁

)

_.

2

p

p p

A

λ π

₌ pr p₊

µ λ π

₋ −

σ

π

₋

λ

B

λ π

₌

λ

 ₊

π

Y 

 

 (3.4)

It is easy to check that both A and B are concave function of π_{. Hence, if}

0

H

>

, then there exists a unique _π* _{such that}2

( )

{

(

) ( ) (

) (

)

}

*

0 1

, arg max A , H , B , H , .

π

π λ τ

λ π

λ τ

λ π

λ β τ

≤ ≤

= + + _(3.5)

In fact, with integrability of the jump size Y the first order condition brings us

DOI: 10.4236/jmf.2019.93020 351 Journal of Mathematical Finance

that

( )

(

) ( ) (

) ( )

(

)

(

(

)

)

1

* *

2 2

,

1 .

,

1 1

p H

Y Y

H

p p

µ λ λ λ β τ

π π

λ τ

σ λ σ λ

− +

 

= + _ + _

 

− −  (3.6)

The second term in the right hand side of the above formula is the hedging demand for the self-exciting jump risks.

In the rest of this section, we shall impose certain conditions on

σ

( ) ( )

⋅ ,

µ

⋅

and Y and show that the HJB Equation (3.2) subject to the initial condition (3.3) and a certain growth condition admits a unique solution. In the next section we prove a verification result showing that the solution obtained solves the optimal investment problem.

For the case of CRRA utility, [2] suggest to prove the existence of a solution by verifying a contracting mapping as [5]. Our attempts show that it is not a trivial task to do that, instead, we use a different analytic method to accomplish the mission.

3.2. Basic Assumptions

First of all, we state some necessary assumptions as follow.

( ) ( )

_{0,1 , ,} 2

( )

2

(

[

_0,

)

)

_.

p∈ µ ⋅ σ ⋅ ∈C ∞ (3.7)

Recall that we assume ~ eZt 1

t

Y Y = − _{. Thus} Y > −1 a.s., _Y2_{ < ∞}

 

 . To explain our idea in a clear manner, we assume that

( )

[ ]

limsup Y 0.

λ→∞ 

µ λ

+

λ

 < (3.8)

Intuitively, the assumption (3.8) claims that the excess return of the asset will be negative if the jump frequency is high enough. In the literature of jump-diffusion models, the excess return is usually assumed with a compensa-tion of jump risk, see, e.g. [10]. Hence the assumption (3.8) is equivalent to say that

µ λ

( )

_{may not be enough compensated when (negative) jumps occur at a}

high frequency. That does not sound unreasonable. The assumption may be relaxed to

( )

[ ]

( )

limsup Y .

λ

µ λ

λ

σ λ

→∞

+

< ∞ 

(3.9)

We shall discuss this later in the Appendix. Under (3.8) we define

[

) ( )

[ ]

[

)

{

}

*: min s , | Y 0, s, .

λ = ∈ λ∞ ∞ µ λ +λ ≤ ∀ ∈λ ∞ (3.10)

3.3. An Integral Formulation

Let

τ

= −

T t

_{. We study the problem}

(

)

( ) ( )

[ ]

( )

, 0, , 0,

,0 1 0,

H H H T

H

τ

α λ λ

λ

τ

λ

τ

λ

λ

λ

∞

 + − =  ⋅  ∀ ∈ ≥

  



= ∀ ≥





(3.11)

DOI: 10.4236/jmf.2019.93020 352 Journal of Mathematical Finance

[ ]( )

u λ : max_{0π 1}

{

A

(

λ π,

) ( )

u λ B

(

λ π,

) (

u λ β

)

}

.

≤ ≤

= + +



Note that



satisfies, for any positive constant c and continuous functions u

and v,

[ ]

cu =c u

[ ]

,

[ ]

u −

[ ]

v ≤

[

u v−

]

.

     (3.12)

We shall use characteristic curves to convert (3.11) into an equivalent integral formulation. For this we introduce

[

_0,

)

[

_{0, ,}

]

(

_{, , :}

)

(

)

_e (t )_.

D T _{λ τ} t _{λ λ λ} α −τ

∞ ∞

= ∞ × Λ = + − (3.13)

For notational simplicity, in the sequel, we write Λ

(

λ τ

, ,t

)

_{simply as} Λ_. Let M be a positive constant to be determined. For fixed

( )

λ τ

, ∈D _and

along the characteristic curve

{

(

(

, , ,

)

)

}

₀

t

t t _τ

λ τ _{≤ ≤}

Λ _{, we obtain from (3.11) that}

(

)

(

)

( )

( )

(

) ( )

( )

( ) ( )

{

}

[ ]

d e , , ,

d

e , , ,

e , , , 0, .

Mt Mt Mt

H t t

t

MH t H t H t

MH t H t t

τ λ

λ τ

α

λ

τ

∞

 Λ 

 

= _ Λ + Λ + Λ − Λ _

= Λ +_ ⋅ _ Λ ∀ ∈

Integration over t∈

[ ]

0,

τ

gives

( )

{

(

(

)

)

( )

(

(

)

)

}

( )

0

, e M , , , , , , eM t d .

H

_{λ τ}

₌ − τ₊ τ MH _Λ

_{λ τ}

t t _{+ }H t_{⋅  Λ}

_{λ τ}

t −τ t

 

∫



Substituting the definition of



into the expression we obtain the integral formulation:

(

) ( ) (

) (

)

{

}

( )

0 0 1

e M max , , , , eM t d ,

H τ τ M A H t B H t τ t

π

π

π

β

− −

≤ ≤

= +

_∫

_ + Λ _ Λ + Λ Λ + _(3.14)

for every

( )

λ τ

, ∈D _{where D and} Λ = Λ:

(

λ τ

, ,t

)

_{are as in (3.13).}

In the sequel we shall choose an appropriate positive constant M and solve the above integral equation by a monotonic iteration technique.

3.4. Determination of the Constant

M

First we consider the function

(

, :

)

(

,

)

(

, ,

)

0,

[ ]

0,1 .

f

λ π

=A

λ π

+B

λ π

∀ ≥

λ

π

∈

Direct calculation yields

( )

,0 ,

f

λ

= pr

(

_,

)

( ) (

₁

)

2

( )

(

₁

)

p 1 _,

fπ λ π p µ λ p πσ λ λ y πy

−

  

= _ − − + _ + _

  

(

_,

)

(

₁

)

{

2

( )

2

(

₁

)

p 2

}

_0.

fππ λ π p p σ λ λ y πy

−

 

= − − + _ + _ <

Hence, we have the following:

Lemma 1. Foreach

λ ≥

0

, thereexistsaunique _{π λ ∈}*

( )

[ ]

_{0,1 suchthat}

( )

(

)

(

*

( )

)

(

)

0 1

: max f , f , f ,0 pr.

π

ζ λ λ π λ π λ λ

≤ ≤

= = ≥ = _(3.15)

DOI: 10.4236/jmf.2019.93020 353 Journal of Mathematical Finance

1) If λ∈I0:=

{

λ≥0 |µ λ

( )

+λ

[ ]

y ≤0

}

, then π λ =*

( )

0 and

ζ λ

( )

= pr;

2) If

{

( ) (

) ( )

2

(

)

1

}

1: 0 | 1 1 0

p

I p y y

λ_{∈ =} λ_≥ µ λ _{− −} σ λ ₊λ _{ +} − _≥

 

 , then

( )

* ₁

π λ = ;

3) If

λ

∈ =I: 0,

[

∞

) (

\ I1I2

)

, then π λ ∈*

( ) ( )

0,1 . In addition, π ⋅*

( )

is

smooth on I.

Consequently, if (3.8) holds, then for λ∗ as in (3.10),

( )

( )

* _{0, pr, .}

π λ = ζ λ = ∀ ≥λ λ_{∗ (3.16)}

We now define

( )

{

(

)

}

0 1,0 max eT , .

M _α A

π λ λ∞ λ λ∗ ∞ λ π

≤ ≤ ≤ ≤ + −

=

This implies that

(

,

)

0,

[ ]

0,1 , 0,

(

)

e T .

M A _{λ π π λ λ λ λ} α

∞ ∗ ∞

 

+ ≥ ∀ ∈ ∈_ + − _ (3.17)

3.5. Monotonic Iteration

We now solve (3.14) by the following iteration: for each non-negative integer n

and

( )

λ τ

, ∈D, we define iteratively Hn by

( )

0 , : e ,pr

H _{λ τ =} τ _(3.18)

(

)

{

(

)

( )

(

) (

)

}

( )

1 , : e _{0 0}max₁ , ,

, , e d .

M

n n

M t n

H M A H t

B H t t

τ τ

π

τ

λ τ π

π β

−

+ _{≤ ≤}

−

= + _ + Λ _ Λ

+ Λ Λ +

∫

_(3.19)

Since

(

_{λ τ, ,t}

)

_{λ 1 e}α(t−τ) _λeα(t−τ) ₀

∞ 

Λ = Λ = _ − _+ ≥ for every

( )

λ τ

, ∈D and

[ ]

0,

t∈

τ

_{, we see that}

{ }

H_{n n}∞₌₀ is a well-defined family of continuous functions on D. We introduce

(

)

[

]

(

)

{

}

: , | 0, , e .

Q _{λ τ τ} T _{λ λ λ λ} ατ

∞ ∗ ∞

= ∈ ≥ + −

Lemma 2. Foreachinteger

n

≥

0

_andevery

( )

λ τ

, ∈Q_,

( )

, epr n

H _{λ τ =} τ_.

Proof. We use a mathematical induction. Assume that

( )

, eprt n

H λ t = _{for all}

( )

λ

,t ∈Q_{. Then when}

( )

λ τ

, ∈Q _and t∈

[ ]

0,

τ

_{, we have}

(

)

_eα(t τ)

₍

₎

_eαt _.

λ λ λ − λ λ λ λ

∞ ∞ ∞ ∗ ∞ ∗

Λ = + − ≥ + − ≥

Thus,

( )

Λ ∈,t Q and

(

Λ +

β

,t

)

∈Q for all t∈

[ ]

0,

τ

. Consequently,

( )

,

(

,

)

eprt

n n

H Λ t =H Λ +β t = _{. Also, by (3.16),}

ζ

( )

Λ =pr _{. It then follows}

from (3.19) that

( )

{

(

)

(

)

}

( )

( )

{

}

( )

1 _{0 0 1}

0

, e max , , e d

e e d e .

prt M t M

n

M pr t M

M pr

H M A B t

M t

τ _τ

τ π

τ _τ

τ τ

λ τ π π

ζ

+ − −

+ _{≤ ≤}

+ − −

= + + Λ + Λ

= + + Λ =

∫

∫

Hence, by mathematical induction,

( )

, epr

n

H _{λ τ =} τ _{for every}

( )

_{λ τ}

_{, ∈Q and}

every non-negative integer n. This completes the proof.

In the sequel, we focus on the case

( )

λ τ

, ∈D Q\ . Note that

(

_{λ τ}, ,_t

)

_λ

(

_{λ λ}

)

e ,αt

( )

_{λ τ}, _{D Q t}\ ,

[ ]

0, ._τ

∞ ∗ ∞

DOI: 10.4236/jmf.2019.93020 354 Journal of Mathematical Finance

Hence,

(

,

)

0,

[ ]

0,1 , ,

( )

\ ,

[ ]

0, .

M A+ Λ

π

≥ ∀ ∈

π

λ τ

∈D Q t∈

τ

_(3.20)

Lemma 3. Foreachinteger

n

≥

0

, Hn+1≥Hn onD.

Proof. We need only consider the case

( )

λ τ

, ∈D Q\ _{. When}

n

=

0

_{, we}

ob-tain from (3.18), (3.19), and the definition of ζ in (3.15) that

(

)

( )

( )

[

]

( )

₍

₎

1 ₀

0 0

, e e d

e e d e , .

pr M t M M

pr M t M

M pr

H M t

M pr t H

τ _τ

τ

τ _τ

τ τ

λ τ ζ

λ τ + −

−

+ − −

= + _ + Λ _

≥ + + = =

∫

∫

Thus, H1≥H0. Next using (3.20) we can show by a mathematical induction that Hn+1≥Hn on D Q\ . This completes the proof.

Now we establish an upper bound. We define

(

)

( )

_{[ ]}

( )

0 0 1 [0, ) 0,

: max max , max max .

k f pr

λ≥ ≤ ≤π

λ π

λ∈ ∞

ζ λ

λ∈ λ∗

ζ λ

= = = ≥ (3.21)

Lemma 4. Foreachinteger

n

≥

0

_and

( )

λ τ

, ∈D_,

(

,

)

ek n

H _{λ τ ≤} τ_.

Proof. We use an induction argument. Assume that the assertion holds for some

n

≥

0

_{. Then for}

( )

λ τ

, ∈D Q\ _{, by (3.19)-(3.20) we obtain}

( )

{

(

)

(

)

}

( )

{

}

( )

1 _{0 0 1}

0

, e max , , e d

e e d e .

k M t M M

n

k M t M

M k

H M A B t

M k t

τ _τ

τ π

τ _τ

τ τ

λ τ − π π + −

+ _{≤ ≤}

+ − −

≤ + + Λ + Λ

≤ + + =

∫

∫

Thus, the assertion of the Lemma holds for Hn+1. This completes the proof.

3.6. Solution of the Integral Equation

The family

{ }

H_{n n}∞₀

= is a bounded monotonic family, so H: lim= n→∞Hn exists.

We wish to prove a uniform convergence. For this, we introduce a norm ⋅ _by

[0, )

( )

: sup .

u u

λ∈ ∞ λ

=

Note that when 0,_λ

(

_{λ λ}

)

eαT

∞ ∗ ∞

 

Λ ∈_ + − _,

(

)

(

)

{

}

(

)

(

)

{

}

( )

0 1 0 1

max , ,

max , ,

.

M A B

M A B

M M k

π π π π π π ζ ≤ ≤ ≤ ≤

+ Λ + Λ

= + Λ + Λ

= + Λ ≤ +

Hence, using max

{ }

f −max

{ }

g ≤max

{

f g−

}

we obtain, when

n

≥

1

and

( )

λ τ

, ∈D Q\ _,

( )

( )

(

)

( )

( )

{

(

)

(

)

(

)

}

( )

(

)

(

)

{

}

( )

( )

( )

[

]

( )

( )

1 1

0 0 1

1

1

0 0 1

1 0

0 , ,

max , , ,

, , , e d

max , , , , e d

, , d .

n n n n M t n n M t n n n n H H

M A H t H t

B H t H t t

M A B H t H t t

M k H t H t t

τ π τ τ _τ π τ

λ τ λ τ

π

π β β

π π + − ≤ ≤ − − − − ≤ ≤ − ≤ −

≤ _ + Λ  _{ } Λ − Λ _

+ Λ _ Λ + − Λ + _

≤ + Λ + Λ ⋅ − ⋅

= + ⋅ − ⋅

∫

∫

∫

DOI: 10.4236/jmf.2019.93020 355 Journal of Mathematical Finance

( )

( ) (

)

[ ]

1

e

, , , 0, , 0,1,

!

n k n

n n

M k

H H T n

n τ

τ τ τ τ

+

+

⋅ − ⋅ ≤ ∀ ∈ = _

Hence, we have the following:

Lemma 5. Thereexistsafunction H C D∈

( )

_suchthat

( ),

( )

( )

lim sup , n , 0.

n→∞_{λt D∈} H λ τ −H λ τ =

In addition, H is a solution of (3.14) and has the following properties:

( )

( )

( )

( )

eprτ _{≤H λ τ}, _≤e ,kτ _{∀ λ τ}, _{∈D H}, _{λ τ}, ₌e ,prτ _{∀ λ τ}, _∈Q.

3.7. Lipschitz Continuity

For each

τ

∈

[ ]

0,T _{, we define}

( )

{

1

(

[

)

)

( )

( )

(

)

}

1 τ = u C∈ 0,∞ | eprτ ≤u ⋅ ≤e ,kτ u λ =eprτ ∀ ≥λ λ∞+ λ λ∗− ∞ eατ . X

For u∈X1

( )

τ

, we consider the function

(

,

)

(

,

) ( )

(

,

) (

)

,

[

0,

)

,

[ ]

0,1 .

u

F λ π =_M A+ λ π _u λ +B λ π u λ β+ ∀ ∈λ ∞ π∈

Lemma 6. Let

τ

∈

[ ]

0,T _and u∈X₁

( )

τ

.Thenforeach

λ ≥

0

_thereexista

unique _{π λ ∈}u

( )

[ ]

0,1 _suchthat

( )

(

)

(

( )

)

0 1

: max , , .

u u u u

F∗ λ = _{≤ ≤π} F λ π =F λ π λ

In addition, _{π ⋅}u

( )

_{is Lipschitz continuous on}

[ )

_{0,∞ , F}u

_{( )}

_C1

₍

_[

_0,

₎

₎

∗ ⋅ ∈ ∞ ,

and

( )

(

( )

)

( )

(

( )

)

( )

(

,

)

d

d _{, , .}

d d ,

u u

u

u u u

u u

F

F F

F

πλ λ

ππ

λ π λ

π λ

λ

λ π λ

λ

∗ =

λ

≤

λ π λ

(3.22)

Proof. When _{λ λ}

(

_{λ λ}

)

eατ

∞ ∗ ∞

≥ + − _{, we have u}

( )

_{λ =u}

(

_{λ β+}

)

₌ekπτ _so

( )

0,

( )

e .

u _Fu prτ

π λ = _∗ λ =

It remains to consider the case _λ 0,_λ

(

_{λ λ}

)

eατ ∞ ∗ ∞

 

∈_ + − _. Note that for

[

0,1

)

π ∈ _,

(

)

{

( ) ( )

(

)

(

)

}

(

)

{

( )

_{{ } { }}

}

2

2 2

2 2

0 0

1 1

e 1 0.

p u

pt p

y y

F p p u y y u

p p y y

ππ

τ

σ λ λ π λ β

σ λ

−

> <

 

− = − + _ + _ +

 

≥ − +_ + _ >

E

1 1

Thus, there exists a unique

_{π λ ∈}

u

( )

[ ]

0,1

_{such that}

_F

u

( )

_λ

_,

_⋅

_on

_{[ ]}

_0,1

at-tains its maximum at

π λ

u

( )

. Note that

( )

{

(

)

}

*

0

0 _I : 0 |_Fu ,0 0 ,

π

π λ = ⇔ ∈λ = λ≥ λ ≤

( )

{

( )

}

*

1

1 _I : 0 |_Fu ,1 0 ,

π

π λ = ⇔ ∈λ = λ≥ λ ≥

( ) ( )

{

(

)

( )

}

* _{0,1 I: 0 |F}u _{,0 0 F}u _{,1 .}

π π

π λ ∈ ⇔ ∈ =λ λ≥ λ > > λ

Here we define

_F

u

( )

,1

π

λ

= −∞

if _

(

1−y

)

p−1_= ∞; in this case we have

( )

[ )

0,1

u

π λ ∈

for any

λ >

0

. Hence, as _Fu 0

ππ < , we see that π* is

theo-DOI: 10.4236/jmf.2019.93020 356 Journal of Mathematical Finance

rem,

( )

(

,

( )

)

,

(

,

( )

)

0,

u u u u

F∗ λ =F λ π λ Fπ λ π λ =

( )

(

( )

)

( )

(

,,

)

,

u u

u

u u

F F

λπ λ

ππ

λ π λ

π λ

λ π λ

= −

( )

(

,

( )

)

(

,

( )

)

( )

(

,

( )

)

.

u u u u u u u u

F∗λ λ =Fλ λ π λ +Fπ λ π λ π λλ =Fλ λ π λ

Since _Fu

( )

_Fu

(

, u

( )

)

λ λ λ λ π λ

∗ = for almost every λ ∈I0I1, by the continuity

of

π ⋅

u

( )

, we hence know that _Fu _C1

(

[

_0,

)

)

∗ ∈ ∞ and F∗uλ

( )

λ =Fλu

(

λ π λ, u

( )

)

.

This completes the proof.

We now calculate the norm of _Fu

λ

∗ . Notice that

(

)

(

) ( )

(

) (

)

(

)

( )

(

) (

)

, , ,

, , .

u

F A u B u s

M A u B u

λ λ λ

λ λ

λ π

λ π

λ

λ π

λ

λ π

λ

λ π

λ β

= + +

+_ + _ + +

When _{λ λ}

(

_{λ λ}

)

e ,ατ

)

∞ ∗ ∞



∈_ + − ∞ , we have u_λ

( )

λ =u_λ

(

λ β+

)

=0 so

(

,

)

0

u

F

λ

λ π =

. When λ∈I

( )

τ :=0,λ∞+

(

λ∗−λ∞

)

eατ and π ∈

[ ]

0,1 ,

(

)

_{( ) ( )}

[

]

,0 1 ,0 1

1

, max e max

e .

k

I I

k

F A B M A B u

k M k u

τ

λ _{λ τ π} λ λ _{λ τ π} λ

τ

λ

λ π λ

∈ ≤ ≤   ∈ ≤ ≤

≤ _ + _ + _ + + _

= + +

where

( )

{

(

)

(

)

}

1

0 1,0 e

: max _T , , .

k _α Aλ Bλ

π λ λ∞ λ λ∗ ∞ λ π λ π

≤ ≤ ≤ ≤ + −

= +

Hence, we have

Lemma 7. UndertheconditionofLemma 6, wehave

[

]

1e .

u k

F k τ M k u

λ λ

∗ ≤ + +

Now, applying this estimate for (3.19) and using

d d

_Λ

_λ

₌

e

α( )t−τ

and

(

)

d dΛ τ α= Λ −λ∞ we obtain

( )

( ),

(

( ),

_{( )}

)

( )

1 , e M ₀ Hn t , Hn t eM t d ,

n

H

_{λ τ}

− τ τF ⋅

_π

⋅ −τ t

+ = +

∫

Λ Λ

( ),

(

( ),

_{( )}

)

( )( )

1, ₀ Hn t , Hn t e M t d ,

n

H τF α τ t

λ λ ⋅

π

⋅ + −

+ =

∫

Λ Λ (3.23)

(

)

( ) ( )

1, 1, , .

n n n

H + τ =α λ∞−λ H + λ+H ⋅τ  λ (3.24)

Thus,

( )

{

[

]

( )

}

( )( )

1, , ₀ 1ekt , , e M t d .

n n

H τ k M k H t α τ t

λ

τ

λ

+ −

+ ⋅ ≤

∫

+ + ⋅

Lemma 8. For every non-negative integer n,

H

n

∈

C D

1

( )

and for each

[ ]

0,T τ ∈ ,

( )

1

, , e .

k

n k

H λ ⋅

τ

≤

_α

τ (3.25)

DOI: 10.4236/jmf.2019.93020 357 Journal of Mathematical Finance

holds for some

n

≥

0

. Then we have

( )

[

]

1 ( )( ) 1

1, , ₀ 1e e e d e .

k M t

kt kt

n k k

H τ k M k α τ t τ

λ τ _α + − _α

+ ⋅ ≤  + +  ≤

 

∫

Thus, by mathematical induction, the assertion of the Lemma holds. Similarly, using (3.24), we can obtain an

L D

∞

( )

_{estimate for}

,

n

H τ. We omit

the details.

3.8.

W2,∞

_Estimate

Now assume that

( )

{

( )

(

(

)

)

1

}

2 1 | 0, , 1 e .k

u u u L u k τ

λλ λ

τ τ ∞ α−

∈X = ∈X ∈ ∞ ≤

Differentiating _Fu

( )

_Fu

(

, u

( )

)

λ λ λ λ π λ

∗ = with respect to

λ

we obtain

( )

(

,

( )

)

(

,

( )

)

( )

u u u u u u

F∗λλ λ =Fλλ λ π λ +Fλπ λ π λ π λλ , a.e.

Hence, using estimate (3.22) for

π λ

λu

( )

, we find that

( )

(

( )

)

(

( )

)

(

)

2

,

, .

,

u u

u u u

u

F

F F

F λπ

λλ λλ

ππ

λ π λ

λ λ π λ

λ π

∗ ≤ +

We can calculate, when λ∈I

( )

τ _and π ∈

[ ]

0,1 _,

(

)

(

) ( )

(

) ( )

(

) (

)

(

)

( )

(

) (

)

, , 2 , ,

, , .

u

F A u A u B u

M A u B u

λλ λλ λ λ λ λ

λλ λλ

λ π

λ π

λ

λ π

λ

λ π

λ β

λ π

λ

λ π

λ β

= + _ + + _

+_ + _ + +

Hence,

(

,

)

2 2 1

[

]

,

u

Fλλ λ π ≤k u + k uλ + M k u+ λλ

where

( )

(

)

2: ₀ max_{1, I T} , .

k = _{≤ ≤π λ∈} Aλλ λ π

Next, we calculate for λ∈I

( )

τ _and π ∈

[

0,1

)

,

( )

(

,

) (

)

( )

,

(

)

,

u

F

λπ

=

A u

λπ

λ

+

B

λπ

λ π

u

λ β

+

+

Au

λ

λ

t Bu

+

λ

λ β

+

(

) ( ) ( )

(

) (

)

(

) ( )

(

)

2

2

1 ,

1 , .

u

F p p u B u

p p B

ππ ππ

ππ

σ λ

λ

λ π

λ β

σ λ

λ π

− = − − +

≥ − +

Note that, for π ∈

[

0,1

)

_,

(

_,

)

(

₁

)

(

₁

)

p 2 2 _,

Bππ λ π p p λ πy y

−

 

− = − _ + _

(

)

(

)

(

)

(

)

{ } { }

(

)

2

2 2 2 1 2 2 2 2 2

2

2 2 2 2 2 2

0 0

2 2 2

, 1 1

1

, .

1

p p

p

y y

B p y y p y y

p y y p y

p _{B p y}

p π

ππ

λ π λ π λ π

λ π λ

λ _{λ π λ}

− −

−

< >

   

= _ + _ ≤ _ + _

   

≤ _ + _+ _{ }

 

≤ + _{ }

−

 

 



DOI: 10.4236/jmf.2019.93020 358 Journal of Mathematical Finance

Hence,

( )

(

)

[

]

(

) ( )

(

)

2 1

3 _{0 1,} 2

, 1

: max .

1 ,

I T

A B k

k

p p B

λπ π π λ

ππ

λ π λ α

σ λ λ π

≤ < ∈

 + + 

 

= < ∞

− +

It then follows that

[

]

2

2 1

2 3

2

ek k ek e .k

F k τ τ M k u k τ

λλ

_α

λλ

∗ ≤ + + + +

Thus, we obtain

( )

[

]

( )

( )( )

1,

2

2

2 1

3 2 ,

0 ,

2

e e e , e d .

n

t _{k k k M t}

n

H

k

k k M k H t s

λλ

α τ

τ τ τ

λλ

λ τ

α

+

+ −

 

≤  + + + + ⋅ 

 

∫

Hence, we have the following lemma.

Lemma 9. There exists a constant k4 >0 such that for each non-negative

integern,

( )

( )

( )

2

[

]

, , , , , , 4e ,k 0, .

n n n

H H H k τ T

λλ ⋅τ + τλ ⋅τ + ττ ⋅τ ≤ ∀ ∈τ (3.26)

Proof. The estimate for Hn,λλ follows from a mathematical induction. The

estimate for Hn,τλ and Hn,ττ follows by differentiating (3.24).

Now sending n→ ∞_{, obtain the following:}

Theorem 1. Assume (3.7) and (3.8). Then theproblem (3.11) admitsa clas-sicalsolutionHthathasthefollowingproperties:

1) For each τ ∈

[ ]

0,T _and

_{λ λ}

(

_{λ λ}

)

e

ατ

∞ ∗ ∞

≥

+

−

_,

_H

( )

_{λ τ =}

,

e

prτ

; 2) For each τ ∈

[ ]

0,T _,

( )

( )

1

1

eprτ _H , e ,kτ _H , _k e ,kτ λ

τ τ α−

≤ ⋅ ≤ ⋅ ≤

( )

( )

( )

2 4

, , , e .k

H H H k τ

ττ ⋅τ + λτ ⋅τ + λλ ⋅τ ≤

3.9. Uniqueness

Theorem 2. Thereexistsauniquesolutionof (3.11) inthefollowingclass

( )

{

1

}

: | sup ,inf_D 0 .

D

H C D H H

= ∈ < ∞ >

X (3.27)

Proof. Let H and Hˆ be two solutions of in (3.11) in X. Suppose H Hˆ ≤ is not true. Then there exists

(

λ τ ∈0, 0

)

D such that

H

ˆ

(

λ τ

0

,

0

)

>

H

(

λ τ

0

,

0

)

. Set

(

0 0

)

(

0 0

)

_{( )}

(1 )

_{( )}

0 ˆ

ln , ln ,

, , e , .

1

H H

Hε ε τ H

λ τ λ τ

ε λ τ λ τ

τ

+

−

= =

+

Then

H

ε

(

λ τ

0

,

0

)

=

H

ˆ

0

(

λ τ

0

,

0

)

. Now for each z≥λ0 we define

( )

, e ,t

( ) (

:

{

, |

)

[ ]

0, , 0,

( )

,

}

,

s z t _λ z α D z _{λ τ τ} T _λ s z_τ

∞

= + = ∈ ∈ _ _

( )

_{( )}

(

)

( )

ˆ ˆ

max , z, z arg max .

D z _{D z}

H H

M z t

Hε

λ

Hε

= ∈

Then M z

( )

≥1_{. Since}

H

( )

_⋅

,0 1

_{= <}

H

ε

( )

_⋅

, we see that

_t

z

_∈

(

0,

_T

]

_{. Denote}

by _sz

( )

_τ : _λ

(

_λz _λ

)

eα τ

( )

−tz

∞ ∞

DOI: 10.4236/jmf.2019.93020 359 Journal of Mathematical Finance

(

_λz,_tz

)

_{. Then}

( )

(

)

( )

(

( )

)

( ) ( )

( )

( )

( ) ( )

( )

(

)

( )

( )

( ) ( )

( )

(

)

(

) (

)

( )

(

)

{

}

( )

(

)

1 0 1 d ˆ

0 , ,

d

ˆ , e , ,

, , ,

ˆ

max , , ,

, . z z z z t t t

z z z z z z

z z z z z

z z z z z

z z

H s t t M z H s t t

t

H t M z H t M z H t

H t M z H t M z H t

B H t M z H t

M z H t

ε ε _ε ε ε ε π ε

λ λ ε λ

λ ε λ

λ π λ β λ β

ε λ = + ≤ ≤   ≤ _ − _     = _ ⋅ _ − _ ⋅ _ −   ≤ _ ⋅ − ⋅ _ −   = _ + − + _ −   

Hence, denoting *

(

)

0 1

max 1 p

B = ≤ ≤π _ +πY _ we obtain

(

)

( )

(

)

( )

(

)

* ˆ _{, , , .}

z_{B H} z _tz _{M z H}ε z _tz _{M z H}ε z _tz

λ _ λ +β − λ +β _≥ε λ

This implies that

(

)

(

)

( )

*

(

(

)

)

ˆ _{, ,}

1 .

, ,

z z z z

z z z z z

H t H t

M z

Hε t B H t

λ β ε λ

λ β λ λ β

  +   ≥ +   + _ + _

Now define N =supDH infDH. Then we derive that

(

)

(

)

(

)

( )

_* 0

ˆ _,

1 , .

e , z z T z z H t

M z M z z

z B N

Hε t α

λ β _ε

β λ

λ

λ β _∞

  +   + ≥ ≥ + ∀ ≥    +  + _{   }

This implies that, for z=λ0,

(

)

( )

(

)

1

* 0

lim ln lim ln 1

e

n

T

n n _i

M z n

M z z i α B N

β ε λ β − →∞ _{→∞ =}   +   ≥ + = ∞    _ + + _   

∑

which contradicts the boundedness of H Hˆ . Thus we must have H Hˆ ≤ . Si-milarly, we can show that H H≤ ˆ _{, so} H H= ˆ _{. This completes the proof.}

4. Verification Theorem and Optimality of the Solution

That the optimal investment problem can be solved through the preceding re-sults is based on a verification result which guarantees that the solution to the HJB equation is the value function corresponding to the optimal investment problem.

Theorem 3. Suppose σ λ

( )

=O

( )

λ _.Assumethat

[

)

[

]

(

)

1,1 _{0, 0,}

H C∈ ∞ × T X solves (2)--(3). Set

V x t U x H



(

, ,

λ

)

=

( ) (

λ

,

T t

−

)

andset

_{π λ τ}

*

( )

_,

_{by (5). Then V V= , thevaluefunctiondefinedin(2.4);also}

theoptimalinvestmentstrategyisgivenby *

(

_,

)

t t

T t

π

=

π λ

−

_.

4.1. Preliminary Results

Lemma 10. Let α β, be positive constants and

{

(

λt,Nt

)

}

be stochastic

processsatisfying

(

)

dλ α λt= ∞−λt dt+βd ,Nt (4.1)

(

dNt= =1

)

λtd ,t

(

dNt =0

)

= −1 λtd .t

DOI: 10.4236/jmf.2019.93020 360 Journal of Mathematical Finance

Then: 1) With i= −1, for each constant ξ∈_{ and}

t

>

0

_, ( ), 0( ), d

0

e |ξλt

λ

λ

eb tξ λ αλ+ ∞∫tbξs s,

 = =

 

 i

where b

( )

ξ,t _{is the solutions of the o.d.e}.

( )

e b 1 , ,0 .

b _{b b}

t

β _{α ξ ξ}

∂ _{= − − =}

∂ i (4.3)

2) Assume that β α< _{. Denote by ξ}* _{the unique positive root of}

* _*

e

βξ

_{= +}

1

_αξ

. Then ( ) ( )d 0

e |ξλt

λ

λ

eb t+αλ∞b t t

 = =

 

 where a) if _{ξ ξ≤} *_{, then}

( )

*

b t

≤

ξ

_{for every}

t

>

0

, b) if _{ξ ξ>} *_{, then b t}

( )

_{= ∞ at time}

d _.

e s 1

s t s β ξ _α ∞ = − −

∫

Proof. The assertion of the Lemma follows from a general result for the joint characteristic function of NT and λT (see e.g. [9]):

(

)

(

)

0

euNT ivT | exp ; , ; ,

E_ + λ _{λ =λ= }K T u v ₊L T u v _λ

 

 i  i (4.4)

where K L, satisfy the Riccati equation:

( )

; 0 0;

K _{L K}

T αλ∞

∂

= =

∂ (4.5)

(

eu i L 1 ;

)

( )

0 .

L _{L L v}

T

β

α +

∂ _{= − − − =}

∂ i i (4.6)

4.2. Proof of the Verification Theorem

Let

{ }

πt be an admissible strategy and

{ }

Xt be the resulting wealth of the

portfolio. Let

V x t U x H



(

, ,

λ

)

=

( ) (

λ

,

T t

−

)

_{be the solution of HJB equation}

obtained in the previous section. Subject to

(

Xτ,λτ

) (

= x,λ

)

, by Itô Lemma, we

have

( )

(

)

(

)

(

)

(

)

(

)

(

)

(

)

{

}

2 2 2

, , d , ,

1

, , d

2

d , , , , d .

T

T t t

T

t t t t t x t t xx

T T

t x t t t t t t t t t

U X V x V X t

V x V V r X V X V t

X V W V X X Y t V X t N

τ

λ τ

τ τ

λ τ λ

λ τ α λ λ π µ π σ

σ π λ β λ

∞ − − − − − = +   = +  + − + + +    + + + + −

∫

∫

∫

∫

          (4.7)

Under the condition σ λ

( )

=O

( )

λ , since [ ] *

0,

sup es t

t∈ T _ λ  < ∞_ and H is

bounded, we can verify that

[ ]

{

( )

(

)

}

[ ]

{

( )

(

)

}

2 2

0, 0,

sup , , sup p , .

t t x t t t t t

t∈ T σ λ X V X λ t t∈ T σ λ X H λ T t

 

 _{ = − < ∞}

   

    

 

Therefore, taking expectation of (4.7) we obtain

( )

_{( )}

₍

₎

(

)

{

(

)

(

)

(

)

{

}

(

)

{

{

(

)

(

)

}

,

2 2 2

0 1

2 2 2

, ,

1 2

, , , , d

max

1 _{, , d 0.}

2

x

T T

t t s t t x t t xx

t t t t t t t

T

t s x

xx

U X V x

V V r X V X V

V X X Y V X t t

V V V r XV

X V V X Y t t

λ τ λ τ λ τ π λ τ

α λ λ π µ π σ

λ π λ β τ λ

α λ λ λ πµ

π σ λ π λ β