Open Access
Issue
Wuhan Univ. J. Nat. Sci.
Volume 27, Number 1, March 2022
Page(s) 53 - 56
DOI https://doi.org/10.1051/wujns/2022271053
Published online 16 March 2022

© Wuhan University 2022

Licence Creative CommonsThis is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

0 Introduction

The idea of replacing Brownian motion with another Gaussian process in the usual financial models has been around for some time. In particular, fractional Brownian motion has been considered as it has better behaved tails and exhibits long-term dependence while remaining Gaussian.

Let H(0,1) be a fixed constant. The fractional Brownian motion with Hurst parameter H is the Gaussian process BH(t,ω);t0,ωΩ on the probability space (Ω,F,P) with the property thatE[BH(t)]=BH(0)=0andE[BH(t)BH(s)]=12{|t|2H+|s|2H-|t-s|2H};t,s0

Here E denotes the expectation with respect to the probability P. If H=1/2 then BH(t) coincides with the classical Brownian motion, denoted by B(t).

Rogers[1] showed that arbitrage is possible when the risky asset has a log-normal price driven by a fractional Brownian motion if stochastic integrals are defined using pointwise products. However, using the white noise approach it is clear that stochastic integrals should be defined using Wick products. When the factors are strongly independent, a Wick product reduces to a pointwise product, and in the Brownian motion case white noise integral reduces to the usual Itô integral.

In this paper we will concentrate on the BH-integral[2], defined byabf(t,ω)dBH(t)=lim|Δ|0k=0n1f(t,ω)(BH(tk+1)BH(tk))where f(t,ω):R×ΩR is Skorohod integrable, denotes the Wick product. We call these fractional Itô integral because these integrals share many of the properties of the classical Itô integrals. In particular, we haveE[Rf(t,ω)dBH(t)]=0(1)Hu and Ksendal[3] extended this fractional Itô calculus to a white-noise calculus for fractional Brownian motion and applied it to finance, still for the case 1/2<H<1 only. Then Elliott and Hoek [4] extended this theory and its application to finance to be valid for all values of H(0,1).

There are two major mathematical techniques to find optimal controls in the field of optimal control theory. Pontryagins maximum principle and dynamic programming principle are applied to obtain the Hamilton-Jacobi-Bellman (HJB) equation [5]. In this paper, we consider the method of fractional HJB equation. This equation is a partial differential equation. The solution of this equation is the value function which gives the maximum expected utility from wealth at the time horizon.

Therefore, in this article, we shall first consider the Merton’s model, in which the classical Brownian motion is replaced by a fractional Brownian motion with Hurst parameter H(0,1). Then, as an application of this derivation, two optimal problems have discussed and solved by the method of fractional HJB equation.

1 Optimization Model

In this section, we describe the portfolio optimization problem under the fractional Brownian motion and derive the HJB equation for the value function.

We consider a continuous-time financial market consisting of two assets: a bond and a stock. Assume that the stock price S(t) follows the fractional Brownian motion model and the bond price M(t) satisfies the following equation: dS(t)=μS(t)dt+σS(t)dBH(t)(2)anddM(t)=rM(t)dt(3)where r>0 is the constant interest rate, μ>r>0 and σ0 are constants. For a portfolio comprising a stock and a risk-free bond, let πt denote the percentages of wealth invested in the stock, t[0,T]. Then the wealth process {X(t)} of the portfolio evolves asdX(t)=X(t)rdt+X(t)πtσ[θdt+dBH(t)](4)where θ=μrσ, which is a real valued predictable process.

We aim to maximize the expected utility of terminal wealth:supπΠtEt[U(X(T))]  subject to (4).

To this end, we define a value functionV(t,x):=supπΠtEt[U(X(T))](5)where Et[]=Et[|X(t)=x] and the utility function U() is assumed to be strictly concave and continuously differentiable on (,+), Πt:={πs,s[t,T]} is the set of all admissible strategies over [t,T]. Since U() is strictly concave, there exists a unique optimal trading strategy. It is obvious that V satisfies boundary condition:V(T,x)=U(x),x0(6)

2 The Closed-Form Solution

In this section, we apply dynamic programming principle to derive the HJB equation for the value function and investigate the optimal investment policies for problem (5) with the boundary condition (6) in the power and logarithm utility cases.

Lemma 1  [6] Let f(x,s):R×RR belong to C1,2(R×R), and assume that the three random variablesf(t,BH(t)),0tfs(s,BH(s))ds,0t2fx2(s,BH(s))s2H1dsall belong to L2(P). Thenf(t,BH(t))=f(0,0)+0tfs(s,BH(s))ds+H0t2fx2(s,BH(s))s2H1ds+0tfx(s,BH(s))dBH(s)(7)

Theorem 1   The optimal value function V(t,x) satisfiesVt+rxVx14Ht2H1θ2Vx2Vxx=0(8)on [0,T]×[0,), with terminal condition (6), where Vt,Vx,Vxx denote partial derivative of first and second orders with respect to time and wealth.

Proof   Using the dynamic programming principle[7], Eq. (5) can be read as ,V(t,X(t))=supπEt[supπEt+Δt[U(X(T))]]=supπEt[V(t+Δt,X(t+Δt))](9)

According to Lemma 1 and the dynamic of {X(t)} given by (4), we haveV(t,X(t))=supπtEt[V(t,X(t))+tt+ΔtVs(s,X(s))ds+tt+ΔtVx(s,X(s))(r+θσπs)X(s)ds    +Htt+Δt2Vx2(s,X(s))σ2πs2X(s)2s2H1ds+tt+ΔtVx(s,X(s))σπsX(s)dBH(s)](10)The stochastic integral in above equation is a Quasi-Martingale, and we getEt[tt+Δtfx(s,X(s))σπsX(s)dBH(s)]=0

Now, suppose Δt0. Then st,X(s)X(t)=x and by intermediate value theorem for integral, the integrals in Eq.(10) are evaluated asV(t,X(t))=supπt[V(t,X(t))+VtΔt+Vx(r+θσπt)xΔt    +HVxxσ2πt2x2t2H1Δt](11)

Dividing both sides of Eq.(11) by Δt, we get the following partial differential equation (PDE)0=supπt[Vt+Vx(r+θσπt)x+HVxxσ2πt2x2t2H1](12)on [0,T]×[0,), with its first-order condition leading to the optimal strategyπ=θVx2Ht2H1Vxxσx(13)By substitution, we can then obtain PDE (8). The boundary condition follows immediately from (5). Furthermore, it follows from the standard verification theorem that the solution to the PDE (8) is indeed the function V(t,x).

Remark 1   It follows (13) that the optimal investment strategy π has an analogical form of the optimal policy under a generalized Bass model (GBM) (H=1/2).

Here, we notice that the stochastic control problem has been transformed into a nonlinear second-order partial differential equation; yet it is difficult to solve it. In the following subsection, we choose power utility and logarithm utility for our analysis, respectively, and try to obtain the closed-form solutions to (8).

2.1 Power Utility

Consider power utilityU(x)=xpp,0<p<1The boundary condition V(T,x)=U(x) suggests that our value function has the following formV(t,x)=f(t)xpp(14)and the function f to be determined with terminal condition f(T)=1. Therefore, we getVt=dfdtxpp,Vx=f(t)xp1,2Vx2=f(t)(p1)xp2(15)Replacing (15) into Eq. (8), yieldsdfdtxpp+rf(t)xpθ24Ht2H1f(t)xpp1=0Eliminating the dependence on x, we obtaindfdt=f(t)[θ2p4Ht2H1(p1)rp]with f(T)=1, we obtainf(t)=exp{rp(Tt)θ2p(T22Ht22H)4H(22H)(p1)}(16)By plugging (16) into Eq. (14), we obtain the value functionV(t,x)=xppexp{rp(Tt)θ2p(T22Ht22H)4H(22H)(p1)}(17)and the optimal investment strategyπ=θ2Hσ(1p)t2H1

Theorem 2   If the utility function is given byU(x)=xpp;0<p<1the value function V(t,x) for problem (5) is given byV(t,x)=xppexp{rp(Tt)θ2p(T22Ht22H)4H(22H)(p1)}And the corresponding optimal strategy can be obtained as follows:π=θ2Hσ(1p)t2H1

Remark 2   It is natural to ask how the value function V(t,x):=VH(t,x) in (17) is related to the value function V1/2(t,x) for the corresponding problem for standard Brownian motion (H=1/2). In this case it is well-known that (see Ref. [8])V1/2(t,x)=xppexp{(rpθ2p2(p1))(Tt)}Therefore we see that, as was to be expectedlimH1/2VH(t,x)=V1/2(t,x)

2.2 Logarithm Utility

Now let us consider the following utility functionU(x)=lnx

We can assume that our value function has the following structure:V(t,x)=g(t)+ln xand the function g to be determined with terminal condition g(T)=0. Hence, we haveVt=dgdt,Vx=1x,2Vx2=1x2(18)Substituting (18) back into (8) yieldsdgdt=rθ24Ht2H1Adding to g(T)=0, we have,g(t)=r(Tt)+θ2(T22Ht22H)4H(22H)Hence, we obtain the value functionV(t,x)=lnx+r(Tt)+θ2(T22Ht22H)4H(22H)with the optimal investment strategyπ=θ2Hσt2H1 When H=1/2, we obtainV1/2(t,x)=lnx+(r+θ22)(Tt)At the end, we can conclude the optimal portfolio problem for logarithm utility in the following theorem.

Theorem 3   If the utility function is given byU(x)=lnxthe value function V(t,x) for problem (5) is given byV(t,x)=lnx+r(Tt)+θ2(T22Ht22H)4H(22H)and the corresponding optimal controls can be obtained as followsπ=θ2Hσt2H1.

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