# Optimal Harvesting When the Exchange Rate Is a Semimartingale

Authors: E. R. Offen; E. M. Lungu
Journal: International Journal of Stochastic Analysis
 (2011)

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## Abstract

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the processEt=E0exp{Xt}, where Xt is a semimartingale, and we ask the following question: What harvesting strategy γ* and the value function Φ maximize the expected total income of an investment? We formulate a singular stochastic control problem and give sufficient conditions for the existence of an optimal strategy. We found that, if the value function is not too sensitive to changes in the prices of the investments, the problem reduces to that of Lungu and Øksendal. However, the general solution of this problem still remains elusive.


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## Body

## 1. Introduction

This paper is concerned with an optimal harvesting strategy in the Black-Scholes Quanto Market when the exchange rate is being driven by a general semimartingale. Specifically, it is proposed that the optimal harvesting strategy can be found under certain conditions. The paper aims to make a contribution by deriving the general formula for an optimal harvesting strategy when the exchange rate is a semimartingale. This study could shed light on the application of general semimartingales in optimization of harvests from investments. Optimal harvesting is one of the crucial areas in finance because investment into stocks and bonds can be used as a source of revenue to expand business. Therefore, making investors happy through an optimal harvesting strategy could lead to more investments and consequently to further expansion of business. This study will make reference to dividend policy to illustrate an optimal harvesting strategy. Indeed a suboptimal dividend policy can result in destruction of shareholder confidence. How much to payout and still maintain growth of investments has been a challenge. For example, Miller and Modigilliani [1] claimed that a dividend policy was irrelevant in perfect markets because it had no impact on firm value. However, research on dividend policy that followed Miller and Modigilliani [1] has further examined various market imperfections and have identified the relevance of dividend policy. A number of stochastic models for optimal dividend policy can also be found in Taksar [2] and the references therein.A number of these models ([1, 3], etc.) have developed an optimal harvesting strategy as optimal stochastic control problems, and that is our approach in this paper. For example, Asmussen and Taksar [4] applied the theory of singular control in their study of a company’s optimal dividend policy that tries to maximize expected value of the total (discounted) payments to the shareholders. Asmussen and Taksar [4] made an assumption that no fixed costs are incurred during payment of dividends and that the liquid assets are modeled by Brownian Motion with drift. Others who also contributed to this problem are Højgaard and Taksar [5, 6], Radner and Shepp [7], Asmussen et al. [3], and Choulli et al. [8]. The studies [5–7] treated the problem as a classical singular stochastic control problem but allowed a control to affect both potential profits and the risks of the financial corporation. Jeanblanc-Picqué and Shiryaev [9] investigated the problem of a company that tries to maximize the expected total (discounted) amount of dividend payments by modeling dividends as a stochastic impulse control problem. They [9] looked at a situation whereby the company faced a fixed cost each time a dividend was paid out by choosing optimally the timing and the size of the payments. They [9] assumed that, when there is no intervention, the liquid asset follows a Brownian Motion process with drift. Lungu and Øksendal [10] have considered the problem of optimizing flow of dividends for a market situation with two investments and determined an optimal harvesting strategy. They concluded that the optimal strategy was to do nothing as long as the investments were in the nonintervention region but to harvest when the investment reached a certain calculated value (see Lungu and Øksendal [10]). Motivated by Lungu and Øksendal [10], this study considers a similar problem but now in the Black-Scholes Quanto market with the exchange rate modeled by a general semimartingale. The paper is structured as follows. Section 2 states the model and the necessary theory. Section 3 applies the theory, while Section 4 gives the conclusions.


## 2. The Model

In the absence of interventions, the dynamics ofS(t), the value of the sterling risky investment, can be modeled by the equation(2.1)St=S0exp⁡{αt+σWt},
where α∈ℝ is the riskless interest rate, the constant σ>0 is the volatility, and Wt is Brownian Motion. Let the sterling to dollar exchange be modeled by the equation(2.2)Et=E0exp⁡{Xt},
and suppose that these processes are on a filtered probability space (Ω,ℱ,𝔽,ℙ), where 𝔽=𝔽S∪𝔽E and 𝔽S={ℱtS,t≥0},ℱtS=σ(Su:0≤u≤t) is the natural filtration generated by the stock price process while 𝔽E={ℱtE,t≥0},ℱtE=σ(Eu:0≤u≤t) is the natural filtration generated by the exchange rate process. 𝔽 describes information about prices and the exchange rate revealed to investors. We assume that the probability space (Ω,ℱ,𝔽,ℙ) satisfies the usual conditions, that is, the σ-field ℱ is ℙ-complete and every ℱt contains all ℙ-null sets of ℱ. Xt∈Sem(𝔽,ℙ), that is, Xt is a càdlàg process that admits the decomposition Xt=X0+At+Mt,t≥0, where At=At∈𝒱 (a process of bounded variation), A0=0, and M=Mt∈ℳloc (a local martingale), M0=0.We consider harvesting from the investment process given (2.1) previously, and we ask the following question: what  value  functionΦ(s,y)  and  harvesting strategyγ*(t)maximise thetotal expected discounted utility harvested from a given time interval.Since our asset is in sterling but our currency of businesses is the dollar, we need first of all to find the dollar equivalence of this asset. To do this, we letYt be the dollar value of the sterling asset price given by
(2.3)Yt=Et⋅St.
Using (2.1) and (2.2), we obtain(2.4)Yt=E0S0exp⁡{αt+σWt+Xt}=Y0exp⁡{Ht},
where(2.5)Ht=αt+σWt+Xt.Ht is a semimartingale since it is the sum of two semimartingales μt+σWt and Xt. This in turn implies that Yt is a semimartingale. Our approach will be probabilistic rather than statistical; hence, it becomes reasonable to express Yt in stochastic exponential form.Theorem 2.1 (Ito’s theorem for semimartingales [11]).
LetHt be a semimartingale, and let f be a 𝒞2 real function. Then, f(Ht) is again a semimartingale and
(2.6)f(Ht)=f(H0)+∫0tf′(Hs-)dHs+12∫0tf′′(Hs-)d〈Hc〉s+∑0≤s≤t[f(Hs)-f(Hs-)-f′(Hs-)ΔHs],
where ΔHs=Hs-Hs- and 〈Hc〉t is the quadratic characteristic of the continuous martingale part Htc of Ht, that is, a predictable process such that (Hc)t2-〈Hc〉t∈ℳloc.For the proof of Theorem2.1, the reader is referred to Protter [11].In this study,  we use Theorem2.1 to rewrite Yt  in stochastic exponential form.Let(2.7)Yt=f(Ht)=Y0exp⁡(Ht),
then, using Theorem 2.1, we have(2.8)Yt=Y0[eH0+∫0t(eHs-)′dHs+12∫0t(eHs-)′′d〈Hc〉s+∑0≤s≤t(eHs-eHs--(eHs-)′ΔHs)]=Y0[eH0+∫0teHs-dHs+12∫0teHs-d〈Hc〉s+∑0≤s≤t(eHs-eHs--eHs-ΔHs)]
and, in differential form, this can be expressed as(2.9)dYt=Y0[eHt-dHt+12eHt-d〈Hc〉t+eHt-eHt--eHt-ΔHt]=Y0[eHt-dHt+12eHt-d〈Hc〉t+eHt-+ΔHt-eHt--eHt-ΔHt]=eHt-dHt+12eHt-d〈Hc〉t+eHt-(eΔHt-1-ΔHt)=eHt-d[Ht+12〈Hc〉t+∑0<s≤t(eΔHs-1-ΔHs)]=Yt-dĤt,
where(2.10)Ĥt=Ht+12〈Hc〉t+∑0<s≤t(eΔHs-1-ΔHs).
We associate with this semimartingale Yt an integer-valued random measure defined as
(2.11)μn(A;ω)=IA(ΔXn(ω)),A∈B(Rd),
where I is an indicator function, that is,(2.12)μn(A;ω)={1ifΔXn(ω)∈A,0ifΔXn(ω)∉A,
(see Shiryaev [12], for details). We define the integral-valued random measures of jumps μX=(μ(0,n]X(·))n≥1, where μ(0,n]X(A;ω)=∑k=1nμkX(A,ω) and μkX(A;ω)=I(ΔXk(ω)∈A,A∈ℬ(ℝd∖{0}) and by ν=ν(ω,ds,dx) the compensator of μ, that is, the predictable measure νn(A:ω)=ℙ(ΔXn∈A∣ℱn-1)(ω)(ℙ-a.s.) with the property that μ-ν is a local martingale measure. This means that, for each A∈ℬ(ℝ∖{0}),(μ(w:(0,t]×A)-ν(ω;(0,t]×A))t>0 is a local martingale with value 0 for t=0.Letφ be a truncation function, for example, φ(x)=xI(|x|≤1). Then ΔXs-φ(ΔXs)≠0⇔|ΔXs|>αforsomeα>0. We now denote the jump part of X corresponding to big jumps by(2.13)X̌=∑0<k≤t((ΔXk)-φ(ΔXk)).
Using random measure of jumps, (2.9) can be written in canonical form as(2.14)dYt=Yt-dHt=Y-(αdt+dC(φ)t+12d〈Hc〉+σdWt+dXtc+d(ex-1)⋆(μ-ν)+d(ex-1-φ(x))⋆μ12),
where C(φ) is a predictable process and Xc is the continuous martingale part of Xt [13].Adopting the definition of harvesting strategy from Lungu and Øksendal [10], we define a harvesting strategy as a stochastic process with the following properties: γ(t,ω)∈ℝ,t>s,ω∈Ω,(1)
γ(t) is measurable with respect to σ-algebra ℱt generated by X(s,·);s≤t (i.e., {γ(t)}γ≥0 is adapted);(2)
γ(t,ω) is nondecreasing with respect to t for almost all (a.a.) ω;(3)
γ(t,ω) is right continuous as a function of t for a.a. ω;(4)
γ(s,ω)=0 for a.a. ω.γ(t,ω) represents the total amount harvested from an initial time s up to time t. We let Γ represent a set of all harvesting strategies. If we apply the harvesting strategy γ(t,ω), then the corresponding process Yγ satisfies the equation(2.15)dYtγ=Y-γ(αdt+dC(φ)t+12d〈Hc〉t+σdWt+dXtc+d(ex-1)⋆(μ-ν)+d(ex-1-ϕ(x)⋆μ))-dγ(t),Yγ(s-)=y1.
It is important to note that the difference between Yγ(s) and Yγ(s-):Yγ(s-) is the state before harvesting starts at time t=s while Yγ(s) is the state immediately after. If γ consists of an immediate harvest of size γ at t=s, then(2.16)Y(s)=Y(s-)-Δγ.
Let the prices/utilities per unit investment when harvested at time t be given by a constant nonnegative function π. The expected total discounted payoff in this case is given by(2.17)Jγ(s,y)=Ey[∫0Tπe-ρ(s+t)⋅dγ(t)],
where 𝔼s,y denotes expectation with respect to probability law Qs,y of Ys,y(t)=(t,Yγ(t)) for t≥s, assuming that Ys,y(0)=(s,y),(2.18)T=inf⁡{t>0;πY(t)∉S},S={(s,y):πe-ρ(s+t)y≥0}
are the time of bankruptcy and S is the solvency region, respectively. The optimal harvesting problem is then to find the value function Φ(s,y) and an optimal dividend strategy γ*(t) such that(2.19)Φ(s,y)=sup⁡γ∈ΓJγ(s,y)=Jγ*(s,y).
We let 0<t1<t2⋯ denote the jumping times of the given strategy γ∈Γ, and we let Δ(tk)=γ(tk)-γ(tk-) be the jump of γ(tk). We also let(2.20)γc(t)=γ(t)-∑s≤tk≤tΔγ(tk)
be the continuous part of γ(t). We formulate the sufficient conditions for the given function ϕ(s,y) to be the value function Φ(s,y) of (2.19) and for a given strategy γ̂∈Γ to be optimal in the following theorem.Theorem 2.2 (extended Lungu and Øksendal [10]).
Suppose thatϕ≥0 is twice continuously differentiable on S with the following properties:(1)
One has(2.21)∂ϕ∂y≥πe-ρ(s+t).(2)
One has(2.22)Lϕ(t,y)=∂ϕ∂t+(αt+C(φ)t+12(σ2+〈Hc〉t)+(ey-1)⋆μ)∂ϕ∂y+12σ2∂ϕ2∂y2≤0onS.
Then
(2.23)ϕ(s,y)≥Φ(s,y)onS.(3)
Define the nonintervention region as(2.24)D={(t,y)∈S;∂ϕ∂y(t,y)>πe-ρ(s+t)∀i=1⋯n}.
Suppose
(2.25)Lϕ=0inD,
and there exists a harvesting strategy γ̂ such that the following hold:
(2.26)Yγ̂(t)∈D¯∀t>s,(∂ϕ∂y(t,Yγ̂(t))-πe-ρ(s+t))γ̂c(t)=0∀i=1⋯n,(i.e.,γ̂cincreasesonlywhen∂ϕ∂y=πe-ρ(s+t)),
where D¯ is the closure of D, that is, D¯=D∪∂D, where ∂D is the boundary of D.(4)
One has(2.27)∂ϕ∂y(t,Yγ̂(s))(αdt+dC(φ)t+12d〈Hc〉+∫-∞∞(ey-1-φ(y)dμ))-(αt+C(φ)t+12(σ2+〈Hc〉t)+∫-∞∞(ey-1-φ(y))dμ)dt≤0.(5)
One has(2.28)Δϕ(tk):=ϕ(tk,Yγ̂(tk))-ϕ(tk,Yγ̂(tk-))=-π⋅Δγ̂(tk)
at all jumping times tk≥s of γ̂(tk) and
(2.29)Es,y[ϕ(TR,Yγ̂(TR))]⟶∞,
where
(2.30)TR=T∧R∧inf⁡⁡{t>s;|Yγ̂(t)|≥R}.
Then
(2.31)ϕ(s,y)=Φ(s,y)∀(s,y)∈S
and γ=γ* is an optimal harvest strategy.Proof.
Consider the following:(2.32)ΔYt=αΔt+ΔC(φ)t+Δ〈Hc〉t+σΔWt+ΔXtc+Δ(ex-1)⋆(μ-ν)+Δ(ex-1-φ)⋆ν=Δ(ex-1)⋆(μ-ν)+Δ(ex-1-φ)⋆ν,(2.33)dYt=αdt+dC(φ)t+12d〈Hc〉+σdWt+dXtc+d(ey-1)⋆(μ-ν)+d(ex-1-φ(x)⋆μ)-dγ(t)=αdt+dC(φ)t+12d〈Hc〉+σdWt+dXtc+∫-∞∞(ex-1-φ(y))d(μ-ν)+∫-∞∞(ex-1-φ(y))dμ-dγ(t).
Choose γ∈Γ, and assume that ϕ∈C2 satisfies (2.21)-(2.22). Then, by Ito’s formula for semimartingales and then computing the expectation throughout the equation
(2.34)Esy[ϕ(TR,Yγ(TR))]=Es,y[ϕ(s,Yγ(s))]+Es,y[∫s+TR∂ϕ∂t(t,Yγ(s))dt]+Es,y[∫s+TR{∂ϕ∂y(t,Yγ(s))(αdt+dC(φ)t+12d〈Hc〉t+σdWt+dXtc+∫-∞∞(ex-1)d(μ-ν)+∫-∞∞(ex-1-φ(y))dμ)-dγ(t)}]+Es,y[∫s+TR12σ2(t,Yγ(t))∂2ϕ∂y2(t,Yγ)]+Es,y[∑s<tk≤TR{ϕ(tk,Yγ(tk))-ϕ(tk,Yγ(ttk-))-∂ϕ∂y(tk,Yϕ(tk))ΔYiγ(tk)}].
From the theory of martingales for integrals,
(2.35)Es,y[∫s+TRσ∂ϕ∂ydWt]=Es,y[∫s+TRσ∂ϕ∂ydXtc]=Es,y[∫s+TR∫-∞∞(ey-1)d(μ-ν)]=0.
Substituting (2.28), (2.32), and (2.35) in (2.34), we have
(2.36)Esy[ϕ(TR,Yγ(TR))]=Esy[ϕ(s,y)]+Es,y[∫s+TR∂ϕ∂t(t,Yγ(s))dt]+Es,y[∫s+TR{∂ϕ∂y(t,Yγ(s))(αdt+dC(ϕ)t+12d〈Hc〉t+∫-∞∞(ex-1-φ(x))dμ)-dγ(t)}]+Es,y[∫s+TR12σ2(t,Xγ(t))∂2ϕ∂y2(t,Yγ)]+Es,y[∑s<tk≤TR{(Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yϕ(tk))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-φ)⋆ν))}].
Now from (2.22),
(2.37)∂ϕ∂t=Lϕ(t,x)-(αt+C(φ)t+12(σ2+〈Hc〉t)+(ex-1-φ(y))⋆μ)∂ϕ∂y-12σ2(t,x)∂2ϕ∂y2.
Using (2.37) in (2.36) gives
(2.38)Esy[ϕ(TR,Yγ)]=Esy[ϕ(TR,Yγ)]+Es,y[∫s+TRLϕ(t,x)dt]+Es,y[∫s+TR{∫s+TR∂ϕ∂y(t,Yγ(s)(αdt+dC(φ)t+12d〈Hc〉+∫-∞∞(ey-1-φ(y)dμ))-(αt+C(φ)t+12(σ2+〈Hc〉t)+∫-∞∞(ey-1-φ(y))dμdt}]-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγ(t)]+Esx[∑s<tk≤TR{(Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yϕ(tk))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-φ)⋆ν))}]
Using the fact that 𝔼sy[ϕ(TR,Yγ)]≤ϕ(s,x) and that Lϕ(s,x)≤0 (refer to (2.27)) yields the inequality
(2.39)Esy[ϕ(TR,Yγ)]≤ϕ(s,y)-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγ(t)]+Esx[∑s<tk≤TR{(Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yϕ(tk))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-ϕ)⋆ν))}].
From the relation
(2.40)γc(t)=γ(t)-∑s≤tk≤tΔγ(tk),
we deduce that
(2.41)dγ(t)=dγc(t)+Δ(t),ΔY(t)=-Δγ(t).
Using (2.41), the right-hand side of (2.32) becomes
(2.42)Δ(ex-1)⋆(μ-ν)+Δ(ex-1-ϕ)⋆ν=-Δγ(t).
Furthermore, using (2.42) we can achieve the simplification of (2.34) as follows: taking the last three terms in (2.39) and using the fact that
(2.43)∑s≤tk≤TRf(tk)=∑s<tk≤TRf(tk)+Δf(s),
we have
(2.44)Esy[∑s<tk≤TR{(Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yϕ(tk))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-ϕ)⋆ν))}]=Esy[∑s≤tk≤TR{(Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yϕ(tk))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-ϕ)⋆ν))}]-Es,y[Δϕ(s,Yγ(s))]-Es,y[∂ϕ∂y(s,Yϕ(s))(Δ(ex-1)⋆(μ-ν)+Δ(ex-1-ϕ)⋆ν)]=Esy[∑s≤tk≤TR{Δϕ(tk,Yγ(tk))-∂ϕ∂y(tk,Yγ(tk))Δγ(t)}]-Es,y[Δϕ(s,Yγ(s))]+Es,y[∂ϕ∂y(s,Yγ(s))Δγ(s)].
Substituting (2.44) into (2.39), inequality (2.39) becomes
(2.45)Esy[ϕ(TR,Yγ)]≤ϕ(s,y)-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγ(t)]-Es,y[∑s≤tk≤TR∂ϕ∂y(t,Yγ(s))Δγ(t)]+Esy[∑s≤tk≤TRΔϕ(tk,Yγ(tk))]+Esy[∑s≤<tk≤TR∂ϕ∂y(tk,Yϕ(tk))Δγ(t)]-Es,y[Δϕ(s,Yγ(s))]+Es,y[∂ϕ∂y(s,Yγ(s))Δγ(s)].
This will then lead to the inequality
(2.46)Esy[ϕ(TR,Yγ(TR))]≤ϕ(s,y)-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγc(t)]+Esy[∑s≤tk≤TRΔϕ(tk,Yγ(tk))].
By the Mean-Value property, we have
(2.47)Δϕ(tk,Ŷγ(tk))=∂ϕ∂y(tk,Ŷkγ)ΔŶiγ(tk)
for some point Ŷkγ on the line connecting the points Yγ(tk-) and Yγ(tk), and, using (2.41), we have
(2.48)Δϕ(tk,ŷγ(tk))=∂ϕ∂y(tk,Ŷkγ)ΔŶiγ(tk)=-∂ϕ∂y(tk,X̂(k)γ)Δγ(t(k))
which leads to
(2.49)Esy[ϕ(TR,Yγ(TR))]≤ϕ(s,y)-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγc(t)]-Esy[∑s≤tk≤TR∂ϕ∂y(tk,X̂(k)γ)Δγ(t(k))].
From (2.21), we obtain
(2.50)ϕ(s,y)-∫s+TR∂ϕ∂y(t,Yγ(s))dγc(t)≤ϕ(s,x)-∫s+TRπ⋅e-(ρ+s)tdγc(t).
Equation (2.21) can also be expressed in discrete form as
(2.51)∑s≤tk≤TR∂ϕ∂y(tk,X̂(k)γ)Δγ(t(k))≥∑s≤tk≤TRπe-(ρ+s)tΔγ(t(k))
Combining (2.50) and (2.51) gives the inequality
(2.52)ϕ(s,y)-∫s+TR∂ϕ∂y(t,Yγ(s))dγc(t)-∑s≤tk≤TR∂ϕ∂y(tk,Ŷ(k)γ)Δγ(t(k))≤ϕ(s,y)-∫s+TRπ⋅e-(ρ+s)tdγc(t)-∑s≤tk≤TRπe-(ρ+s)tΔγ(t(k)).
Taking the expectation of (2.52) yields the inequality
(2.53)Es,y[ϕ(TR,Yγ(TR))]≤ϕ(s,y)-Es,y[∫s+TR∂ϕ∂y(t,Yγ(s))dγc(t)]-Esy[∑s≤tk≤TR∂ϕ∂y(tk,X̂(k)γ)Δγ(t(k))]≤ϕ(s,y)-Esy[∫s+TRπ⋅e-(ρ+s)tdγc(t)]-Es,y[∑s≤tk≤TRπe-(ρ+s)tΔγ(t(k))],
from which we obtain
(2.54)ϕ(s,y)≥Es,y[∫s+TRπ⋅e-(ρ+s)tdγc(t)]+Esy[∑s≤tk≤TRπe-(ρ+s)tΔγ(t(k))]+Esy[ϕ(TR,Yγ(TR))]≥Esy[∫sTRπ⋅e-(ρ+s)tdγ(t)]+Esy[ϕ(TR,Yγ(TR))].
Since R<∞,γ∈Γ were arbitrary and ϕ≥0, this proves that
(2.55)ϕ(s,y)≥Φ(s,y)andγ̂isoptimal.
Let us now assume that D is given by (2.24) and that (2.25)–(2.28) hold. If we replace γ in the above calculation by γ̂, then equality holds everywhere and we end up with the relation
(2.56)ϕ(s,y)=Es,y[∫sTRπ⋅e-(ρ+s)tdγ̂+Eϕ(TR,Yγ̂(TR))].
Letting R→∞ and using (2.29), we get
(2.57)ϕ(s,y)=Es,y[∫s∞π⋅e-(ρ+s)tdγ̂].Combining (eqntawina) with (2.23), we have(2.58)ϕ(s,y)=Φ(s,y),γ̂isoptimal.
The strategy γ̂ can be found by solving the Skorohod stochastic differential equations (see Lungu and Øksendal [10]).


## 3. Application of the Theory

From the form of our discounted utility function (2.17), it becomes reasonable to look for the function Φ of the form(3.1)Φ(s,y)=e-ρsΨ(y).
Let(3.2)ϕ(s,y)=e-ρsψ(y).
Equations (2.21), (2.22) become(3.3)∂ψ∂y≥π,(3.4)Lψ(t,y)=-ρψ+(αt+C(φ)t+12(σ2+〈Hc〉t)+(ex-1)⋆μ)∂ψ∂y+12σ2∂2ψ∂y2≤0on⁡S,
respectively. We try a solution ψ of the form(3.5)ψ(y)=F(z),wherez=πy.
Equations (3.2) and (3.3) become
(3.6)πF′(z)≥πorF′(z)≥1,(3.7)AF(z)=-ρF(z)+(αt+C(φ)t+12(σ2+〈Hc〉t)+(ex-1)⋆μ)πF′(z)+12σ2π2F′′(z)=-ρF(z)+κF′(z)+βF′′(z)<0,
where(3.8)κ=(αt+C(φ)t+12(σ2+〈Hc〉t)+(ex-1)⋆μ)π,β=12σ2π2.
Inequality (3.7) is similar to inequality (3.11) in Lungu and Øksendal [10]. The major difference is that in Lungu and Øksendal [10] the coefficients are constants whereas in our case the coefficient κ is not a constant. Since κ(t,z) is a process with jumps, the general solution of(3.9)βF′′(z)+κF′(z)-ρF(z)=0
is elusive. We still try to explore the behavior of the solution after fixing the value of t; that is, we want to see the general behavior of the solution with respect to z.


### 3.1. Examples

#### 3.1.1. Example  1 (κ Constant with respect to z)

The caseκ is a constant that reduces to the problem in Lungu and Øksendal [10]. The auxiliary equation for (3.9) is(3.10)βm2+κm-ρ=0.
The solutions are(3.11)m1,2=-κ±κ2-4ρβ2β.
Let us suppose the nonintervention region B to be of the form(3.12)B={z;0<z<z*}
for some z*>0. From (2.21) and (2.22), we try a solution of the form(3.13)F(z)={z+G,forz≥z*,α1em1z+α2em2z,0<z<z*,
for G∈ℝ. We want to determine parameters α1,α2,G, and z* such that F becomes a C2 at z=z*. Using the continuity and differentiability of F(z) at z=z* and taking α=α1=-α2, we obtain(3.14)z*=2ln⁡[m2/m1](m1-m2)>0,α=(m1em1z*+m2em2z*)-1,G=α(em1z*-em2z*)-z*.
With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we have(3.15)Φ(s,y)={α1e-ρs(em1z+em2z),for0≤z<z*,e-ρs(z+G),z*≤z.
Similarly, as discussed in Lungu and Øksendal [10], the optimal strategy is obtained by doing nothing as long as Y(t)∈(0,z*) (i.e., Y(t)∈B) and to harvest a total amount γ*=γ̂ of the reflected process Yγ̂ in the direction of -π.


#### 3.1.2. Example  2 (Whenκ Is Not a Constant)

We now look at a more general case corresponding toκ not a constant. Suppose that we can write F(z) as(3.16)F(z)=V(z)exp⁡(-12∫κβdz),
then (3.9) can be transformed into its canonical form given by(3.17)V′′(z)+η(z)V(z)=0,
where(3.18)η=-ρβ-14(κβ)2-14(κβ)z′
and the jumps are embedded in η(z).To solve (3.17), we use the jump transfer matrix method [14]. The auxiliary equation of (3.17) is(3.19)m2+η(z)=0
with solutions(3.20)m1=im(z),m2=-im(z),wherem(z)=η(z).
The general solution for V(z) can be written as(3.21)V(z)=h1(z)exp⁡(imz)+h2(z)exp⁡(-imz),
where h1(z) and h2(z) are functions to be determined. The solution (3.21) can be expressed as(3.22)V(z)=exp⁡[Φ(z)]tH(z),
where(3.23)H(z)=[h1(z)h2(z)],Φ(z)=[im(z)-im(z)]
and the superscript t denotes transpose of a matrix. F is the solution of the equation(3.24)dH(z)=U(z)H(z)dz,
where(3.25)U=-zK′(z)-exp⁡[-zK(z)][D(z)-1C(z)K′(z)exp⁡[zK(z)]],(3.26)C(z)=[(p-1)mqp-2]n×n,D(x)=[m(z)qp-1(z)]n×n,K(z)=[mp(z)δpq]n×n,K′(x)=[mp′(z)δpq]n×n
(cf. [14]).Clearly, for the case corresponding ton=2, the matrices (3.26) are given by(3.27)C=(0011),D=(11m1m2),K=(m100m2),henceK′=(m1′00m2′).Using power series expansion for exponential square matrices and truncating the expansion at second-order terms, we have(3.28)exp⁡[zK]=exp⁡[(zm100zm2)]=(1001)+(zm100zk2)+O(2)terms=(1+zm1001+zm2)+O(2)terms.
Similarly,(3.29)exp⁡[-zK]=(1-zm1001-zm2).
From (3.25) and (3.28), we obtain an approximation for U(z) as(3.30)U(z)=[-(z+1m1-m2)m1′m2′m2-m1exp⁡[-z(m1-m2)]m1′m1-m2exp⁡[+z(m1-m2)]-(z+1m2-m1)m2′].
From (3.19) and (3.20), we obtain(3.31)V(z)=m′(z)2m(z)[-1+i2m(z)zexp⁡⁡[i2zm(z)]exp⁡⁡[-2izm(z)]-1-i2m(z)z].
The jump transfer matrix from region 1 to region 2 across the interface z=ς in our case takes the form(3.32)Q1→2=[m1+m22m2e+iς(m2-m1)m2-m12m2e+iς(m1+m2)m2-m12m2e-iς(m1+m2)m2+m12m2e-iς(m2-m1)].
This jump transfer matrix over singularities given by Qς-δz→ς+δz is simplified as in [14] to(3.33)Qς-δz→ς+δz=[1+i21-i21-i21+i2]typeA,Qς-δz→ς+δz=[1-i21+i21+i21-i2]typeB,Qς-δz→ς+δz=[1001]typeC.
The reader is referred to [14] and the references therein for details regarding classification of singularities.


#### 3.1.3. Analysis of the Results

Letz0=0 and our solutions (3.21) have jumps at points z1,z2,z3,…,zn, and we define(3.34)Wj={z:zj-1<z<zj}fori=1,2,3,…,n,Bj={z:zj-1<z<z*j}
for some z*i, and we define(3.35)B=⋃j=1nBj.
For z∈Wi, we define(3.36)Fj(z)={z+Gj,forz≥zj*,hj1(z)emj1z+hj2emj2z,0<z<zj*,F={F0,F1,F2,…,Fn},Φ(s,z)j={hj(z)e-ρs(emj1z+emj2z),for0≤z<zj*,e-ρs(z+Gj),zj*≤z,Φ(s,z)={Φ1(s,z),Φ2(s,z),…,Φn(s,z)}.


#### 3.1.4. Conjecture

The optimal strategy is achieved by doing nothing during jumps and as long asY(t)∈Bj but to harvest according to local time γj*=γ̂ at the boundary ∂Bj.Remarks. In each nonintervention region Bj, Yγ̂ is a reflected process at ∂Bj, since, as Yγ̂ hits the boundary, a certain amount is harvested thereby forcing the process to go below zj*.


## 3.1. Examples

### 3.1.1. Example  1 (κ Constant with respect to z)

The caseκ is a constant that reduces to the problem in Lungu and Øksendal [10]. The auxiliary equation for (3.9) is(3.10)βm2+κm-ρ=0.
The solutions are(3.11)m1,2=-κ±κ2-4ρβ2β.
Let us suppose the nonintervention region B to be of the form(3.12)B={z;0<z<z*}
for some z*>0. From (2.21) and (2.22), we try a solution of the form(3.13)F(z)={z+G,forz≥z*,α1em1z+α2em2z,0<z<z*,
for G∈ℝ. We want to determine parameters α1,α2,G, and z* such that F becomes a C2 at z=z*. Using the continuity and differentiability of F(z) at z=z* and taking α=α1=-α2, we obtain(3.14)z*=2ln⁡[m2/m1](m1-m2)>0,α=(m1em1z*+m2em2z*)-1,G=α(em1z*-em2z*)-z*.
With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we have(3.15)Φ(s,y)={α1e-ρs(em1z+em2z),for0≤z<z*,e-ρs(z+G),z*≤z.
Similarly, as discussed in Lungu and Øksendal [10], the optimal strategy is obtained by doing nothing as long as Y(t)∈(0,z*) (i.e., Y(t)∈B) and to harvest a total amount γ*=γ̂ of the reflected process Yγ̂ in the direction of -π.


### 3.1.2. Example  2 (Whenκ Is Not a Constant)

We now look at a more general case corresponding toκ not a constant. Suppose that we can write F(z) as(3.16)F(z)=V(z)exp⁡(-12∫κβdz),
then (3.9) can be transformed into its canonical form given by(3.17)V′′(z)+η(z)V(z)=0,
where(3.18)η=-ρβ-14(κβ)2-14(κβ)z′
and the jumps are embedded in η(z).To solve (3.17), we use the jump transfer matrix method [14]. The auxiliary equation of (3.17) is(3.19)m2+η(z)=0
with solutions(3.20)m1=im(z),m2=-im(z),wherem(z)=η(z).
The general solution for V(z) can be written as(3.21)V(z)=h1(z)exp⁡(imz)+h2(z)exp⁡(-imz),
where h1(z) and h2(z) are functions to be determined. The solution (3.21) can be expressed as(3.22)V(z)=exp⁡[Φ(z)]tH(z),
where(3.23)H(z)=[h1(z)h2(z)],Φ(z)=[im(z)-im(z)]
and the superscript t denotes transpose of a matrix. F is the solution of the equation(3.24)dH(z)=U(z)H(z)dz,
where(3.25)U=-zK′(z)-exp⁡[-zK(z)][D(z)-1C(z)K′(z)exp⁡[zK(z)]],(3.26)C(z)=[(p-1)mqp-2]n×n,D(x)=[m(z)qp-1(z)]n×n,K(z)=[mp(z)δpq]n×n,K′(x)=[mp′(z)δpq]n×n
(cf. [14]).Clearly, for the case corresponding ton=2, the matrices (3.26) are given by(3.27)C=(0011),D=(11m1m2),K=(m100m2),henceK′=(m1′00m2′).Using power series expansion for exponential square matrices and truncating the expansion at second-order terms, we have(3.28)exp⁡[zK]=exp⁡[(zm100zm2)]=(1001)+(zm100zk2)+O(2)terms=(1+zm1001+zm2)+O(2)terms.
Similarly,(3.29)exp⁡[-zK]=(1-zm1001-zm2).
From (3.25) and (3.28), we obtain an approximation for U(z) as(3.30)U(z)=[-(z+1m1-m2)m1′m2′m2-m1exp⁡[-z(m1-m2)]m1′m1-m2exp⁡[+z(m1-m2)]-(z+1m2-m1)m2′].
From (3.19) and (3.20), we obtain(3.31)V(z)=m′(z)2m(z)[-1+i2m(z)zexp⁡⁡[i2zm(z)]exp⁡⁡[-2izm(z)]-1-i2m(z)z].
The jump transfer matrix from region 1 to region 2 across the interface z=ς in our case takes the form(3.32)Q1→2=[m1+m22m2e+iς(m2-m1)m2-m12m2e+iς(m1+m2)m2-m12m2e-iς(m1+m2)m2+m12m2e-iς(m2-m1)].
This jump transfer matrix over singularities given by Qς-δz→ς+δz is simplified as in [14] to(3.33)Qς-δz→ς+δz=[1+i21-i21-i21+i2]typeA,Qς-δz→ς+δz=[1-i21+i21+i21-i2]typeB,Qς-δz→ς+δz=[1001]typeC.
The reader is referred to [14] and the references therein for details regarding classification of singularities.


### 3.1.3. Analysis of the Results

Letz0=0 and our solutions (3.21) have jumps at points z1,z2,z3,…,zn, and we define(3.34)Wj={z:zj-1<z<zj}fori=1,2,3,…,n,Bj={z:zj-1<z<z*j}
for some z*i, and we define(3.35)B=⋃j=1nBj.
For z∈Wi, we define(3.36)Fj(z)={z+Gj,forz≥zj*,hj1(z)emj1z+hj2emj2z,0<z<zj*,F={F0,F1,F2,…,Fn},Φ(s,z)j={hj(z)e-ρs(emj1z+emj2z),for0≤z<zj*,e-ρs(z+Gj),zj*≤z,Φ(s,z)={Φ1(s,z),Φ2(s,z),…,Φn(s,z)}.


### 3.1.4. Conjecture

The optimal strategy is achieved by doing nothing during jumps and as long asY(t)∈Bj but to harvest according to local time γj*=γ̂ at the boundary ∂Bj.Remarks. In each nonintervention region Bj, Yγ̂ is a reflected process at ∂Bj, since, as Yγ̂ hits the boundary, a certain amount is harvested thereby forcing the process to go below zj*.


## 3.1.1. Example  1 (κ Constant with respect to z)

The caseκ is a constant that reduces to the problem in Lungu and Øksendal [10]. The auxiliary equation for (3.9) is(3.10)βm2+κm-ρ=0.
The solutions are(3.11)m1,2=-κ±κ2-4ρβ2β.
Let us suppose the nonintervention region B to be of the form(3.12)B={z;0<z<z*}
for some z*>0. From (2.21) and (2.22), we try a solution of the form(3.13)F(z)={z+G,forz≥z*,α1em1z+α2em2z,0<z<z*,
for G∈ℝ. We want to determine parameters α1,α2,G, and z* such that F becomes a C2 at z=z*. Using the continuity and differentiability of F(z) at z=z* and taking α=α1=-α2, we obtain(3.14)z*=2ln⁡[m2/m1](m1-m2)>0,α=(m1em1z*+m2em2z*)-1,G=α(em1z*-em2z*)-z*.
With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we have(3.15)Φ(s,y)={α1e-ρs(em1z+em2z),for0≤z<z*,e-ρs(z+G),z*≤z.
Similarly, as discussed in Lungu and Øksendal [10], the optimal strategy is obtained by doing nothing as long as Y(t)∈(0,z*) (i.e., Y(t)∈B) and to harvest a total amount γ*=γ̂ of the reflected process Yγ̂ in the direction of -π.


## 3.1.2. Example  2 (Whenκ Is Not a Constant)

We now look at a more general case corresponding toκ not a constant. Suppose that we can write F(z) as(3.16)F(z)=V(z)exp⁡(-12∫κβdz),
then (3.9) can be transformed into its canonical form given by(3.17)V′′(z)+η(z)V(z)=0,
where(3.18)η=-ρβ-14(κβ)2-14(κβ)z′
and the jumps are embedded in η(z).To solve (3.17), we use the jump transfer matrix method [14]. The auxiliary equation of (3.17) is(3.19)m2+η(z)=0
with solutions(3.20)m1=im(z),m2=-im(z),wherem(z)=η(z).
The general solution for V(z) can be written as(3.21)V(z)=h1(z)exp⁡(imz)+h2(z)exp⁡(-imz),
where h1(z) and h2(z) are functions to be determined. The solution (3.21) can be expressed as(3.22)V(z)=exp⁡[Φ(z)]tH(z),
where(3.23)H(z)=[h1(z)h2(z)],Φ(z)=[im(z)-im(z)]
and the superscript t denotes transpose of a matrix. F is the solution of the equation(3.24)dH(z)=U(z)H(z)dz,
where(3.25)U=-zK′(z)-exp⁡[-zK(z)][D(z)-1C(z)K′(z)exp⁡[zK(z)]],(3.26)C(z)=[(p-1)mqp-2]n×n,D(x)=[m(z)qp-1(z)]n×n,K(z)=[mp(z)δpq]n×n,K′(x)=[mp′(z)δpq]n×n
(cf. [14]).Clearly, for the case corresponding ton=2, the matrices (3.26) are given by(3.27)C=(0011),D=(11m1m2),K=(m100m2),henceK′=(m1′00m2′).Using power series expansion for exponential square matrices and truncating the expansion at second-order terms, we have(3.28)exp⁡[zK]=exp⁡[(zm100zm2)]=(1001)+(zm100zk2)+O(2)terms=(1+zm1001+zm2)+O(2)terms.
Similarly,(3.29)exp⁡[-zK]=(1-zm1001-zm2).
From (3.25) and (3.28), we obtain an approximation for U(z) as(3.30)U(z)=[-(z+1m1-m2)m1′m2′m2-m1exp⁡[-z(m1-m2)]m1′m1-m2exp⁡[+z(m1-m2)]-(z+1m2-m1)m2′].
From (3.19) and (3.20), we obtain(3.31)V(z)=m′(z)2m(z)[-1+i2m(z)zexp⁡⁡[i2zm(z)]exp⁡⁡[-2izm(z)]-1-i2m(z)z].
The jump transfer matrix from region 1 to region 2 across the interface z=ς in our case takes the form(3.32)Q1→2=[m1+m22m2e+iς(m2-m1)m2-m12m2e+iς(m1+m2)m2-m12m2e-iς(m1+m2)m2+m12m2e-iς(m2-m1)].
This jump transfer matrix over singularities given by Qς-δz→ς+δz is simplified as in [14] to(3.33)Qς-δz→ς+δz=[1+i21-i21-i21+i2]typeA,Qς-δz→ς+δz=[1-i21+i21+i21-i2]typeB,Qς-δz→ς+δz=[1001]typeC.
The reader is referred to [14] and the references therein for details regarding classification of singularities.


## 3.1.3. Analysis of the Results

Letz0=0 and our solutions (3.21) have jumps at points z1,z2,z3,…,zn, and we define(3.34)Wj={z:zj-1<z<zj}fori=1,2,3,…,n,Bj={z:zj-1<z<z*j}
for some z*i, and we define(3.35)B=⋃j=1nBj.
For z∈Wi, we define(3.36)Fj(z)={z+Gj,forz≥zj*,hj1(z)emj1z+hj2emj2z,0<z<zj*,F={F0,F1,F2,…,Fn},Φ(s,z)j={hj(z)e-ρs(emj1z+emj2z),for0≤z<zj*,e-ρs(z+Gj),zj*≤z,Φ(s,z)={Φ1(s,z),Φ2(s,z),…,Φn(s,z)}.


## 3.1.4. Conjecture

The optimal strategy is achieved by doing nothing during jumps and as long asY(t)∈Bj but to harvest according to local time γj*=γ̂ at the boundary ∂Bj.Remarks. In each nonintervention region Bj, Yγ̂ is a reflected process at ∂Bj, since, as Yγ̂ hits the boundary, a certain amount is harvested thereby forcing the process to go below zj*.


## 4. Conclusion

Most of the conditions in Theorem2.2 are extensions from Lungu and Øksendal [10] with exception of condition (2.5), that is,(4.1)∂ϕ∂y(t,Yγ(s))(αdt+dC(ϕ)t+12d〈Hc〉+∫-∞∞(ey-1-ϕ(y)dμ))-(αt+C(ϕ)t+12(σ2+〈Hc〉t′)+∫-∞∞(ey-1-ϕ(y))dμ)dt≤0.
We note that this condition is achieved if ∂ϕ/∂y is small. It is found under additional condition (2.27) that our problem can be reduced to a second-order differential equation similar to that in Lungu and Øksendal [10] though with some jumps. What is observed is that if the investment process is being modeled by a semimartingale in general, optimal value function Φ(s,y) and the optimal dividend strategy can be found if the rate of change of the value function ϕ(s,y) with respect to the investment process itself, that is, ∂ϕ/∂y, is small enough. In other words, ϕ(s,y) should not be too sensitive to variations in investments. Our results further show that the general solution to this problem is still elusive.



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*Source: 942478-2012-01-26.xml*