Optimal Harvesting When the Exchange Rate Is a Semimartingale

Abstract

We consider harvesting in the Black-Scholes Quanto Market when the exchange rate is being modeled by the process 𝐸_𝑡=𝐸₀exp{𝑋_𝑡}, where 𝑋_𝑡 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.

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 of 𝑆(𝑡), the value of the sterling risky investment, can be modeled by the equation𝑆_𝑡=𝑆₀exp𝛼𝑡+𝜎𝑊_𝑡, (2.1) where 𝛼∈ℝ is the riskless interest rate, the constant 𝜎>0 is the volatility, and 𝑊_𝑡 is Brownian Motion. Let the sterling to dollar exchange be modeled by the equation𝐸_𝑡=𝐸₀exp𝑋_𝑡, (2.2) and suppose that these processes are on a filtered probability space (Ω,ℱ,𝔽,ℙ), where 𝔽=𝔽^𝑆∪𝔽^𝐸 and 𝔽^𝑆={ℱ^𝑆_𝑡,𝑡≥0},ℱ^𝑆_𝑡=𝜎(𝑆_𝑢∶0≤𝑢≤𝑡) is the natural filtration generated by the stock price process while 𝔽^𝐸={ℱ^𝐸_𝑡,𝑡≥0},ℱ^𝐸_𝑡=𝜎(𝐸_𝑢∶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 ℱ_𝑡 contains all ℙ-null sets of ℱ. 𝑋_𝑡∈Sem(𝔽,ℙ), that is, 𝑋_𝑡 is a càdlàg process that admits the decomposition 𝑋_𝑡=𝑋₀+𝐴_𝑡+𝑀_𝑡,𝑡≥0, where 𝐴_𝑡=𝐴_𝑡∈𝒱 (a process of bounded variation), 𝐴₀=0, and 𝑀=𝑀_𝑡∈ℳ_loc (a local martingale), 𝑀₀=0.

We consider harvesting from the investment process given (2.1) previously, and we ask the following question: what value function Φ(𝑠,𝑦)  and harvesting strategy 𝛾^∗(𝑡)maximise the total 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 let 𝑌_𝑡 be the dollar value of the sterling asset price given by 𝑌_𝑡=𝐸_𝑡⋅𝑆_𝑡. (2.3) Using (2.1) and (2.2), we obtain𝑌_𝑡=𝐸₀𝑆₀exp𝛼𝑡+𝜎𝑊_𝑡+𝑋_𝑡=𝑌₀exp𝐻_𝑡, (2.4) where𝐻_𝑡=𝛼𝑡+𝜎𝑊_𝑡+𝑋_𝑡. (2.5)𝐻_𝑡 is a semimartingale since it is the sum of two semimartingales 𝜇𝑡+𝜎𝑊_𝑡 and 𝑋_𝑡. This in turn implies that 𝑌_𝑡 is a semimartingale. Our approach will be probabilistic rather than statistical; hence, it becomes reasonable to express 𝑌_𝑡 in stochastic exponential form.

Theorem 2.1 (Ito’s theorem for semimartingales [11]). Let 𝐻_𝑡 be a semimartingale, and let 𝑓 be a 𝒞² real function. Then, 𝑓(𝐻_𝑡) is again a semimartingale and 𝑓𝐻_𝑡=𝑓𝐻₀+^𝑡₀𝑓^′𝐻_𝑠−𝑑𝐻_𝑠+12^𝑡₀𝑓^′′𝐻_𝑠−𝑑𝐻^𝑐_𝑠+_{0≤𝑠≤𝑡}𝑓𝐻_𝑠−𝑓𝐻_𝑠−−𝑓^′𝐻_𝑠−Δ𝐻_𝑠, (2.6) where Δ𝐻_𝑠=𝐻_𝑠−𝐻_𝑠− and ⟨𝐻^𝑐⟩_𝑡 is the quadratic characteristic of the continuous martingale part 𝐻^𝑐_𝑡 of 𝐻_𝑡, that is, a predictable process such that (𝐻^𝑐)²_𝑡−⟨𝐻^𝑐⟩_𝑡∈ℳ_loc.

For the proof of Theorem 2.1, the reader is referred to Protter [11].

In this study, we use Theorem 2.1 to rewrite 𝑌_𝑡 in stochastic exponential form.

Let𝑌_𝑡=𝑓𝐻_𝑡=𝑌₀exp𝐻_𝑡, (2.7) then, using Theorem 2.1, we have𝑌_𝑡=𝑌₀𝑒^𝐻0+^𝑡₀𝑒^{𝐻𝑠−′}𝑑𝐻_𝑠+12^𝑡₀𝑒^{𝐻𝑠−′′}𝑑𝐻^𝑐_𝑠+_{0≤𝑠≤𝑡}𝑒^𝐻𝑠−𝑒^𝐻𝑠−−𝑒^{𝐻𝑠−′}Δ𝐻_𝑠=𝑌₀𝑒^𝐻0+^𝑡₀𝑒^𝐻𝑠−𝑑𝐻_𝑠+12^𝑡₀𝑒^𝐻𝑠−𝑑𝐻^𝑐_𝑠+_{0≤𝑠≤𝑡}𝑒^𝐻𝑠−𝑒^𝐻𝑠−−𝑒^𝐻𝑠−Δ𝐻_𝑠 (2.8) and, in differential form, this can be expressed as𝑑𝑌_𝑡=𝑌₀𝑒^𝐻𝑡−𝑑𝐻_𝑡+12𝑒^𝐻𝑡−𝑑𝐻^𝑐_𝑡+𝑒^𝐻𝑡−𝑒^𝐻𝑡−−𝑒^𝐻𝑡−Δ𝐻_𝑡=𝑌₀𝑒^𝐻𝑡−𝑑𝐻_𝑡+12𝑒^𝐻𝑡−𝑑𝐻^𝑐_𝑡+𝑒^{𝐻𝑡−+Δ𝐻𝑡}−𝑒^𝐻𝑡−−𝑒^𝐻𝑡−Δ𝐻_𝑡=𝑒^𝐻𝑡−𝑑𝐻_𝑡+12𝑒^𝐻𝑡−𝑑𝐻^𝑐_𝑡+𝑒^𝐻𝑡−𝑒^Δ𝐻𝑡−1−Δ𝐻_𝑡=𝑒^𝐻𝑡−𝑑𝐻_𝑡+12𝐻^𝑐_𝑡+_{0<𝑠≤𝑡}𝑒^Δ𝐻𝑠−1−Δ𝐻_𝑠=𝑌_𝑡−𝑑𝐻_𝑡, (2.9) where𝐻_𝑡=𝐻_𝑡+12𝐻^𝑐_𝑡+_{0<𝑠≤𝑡}𝑒^Δ𝐻𝑠−1−Δ𝐻_𝑠. (2.10) We associate with this semimartingale 𝑌_𝑡 an integer-valued random measure defined as 𝜇_𝑛(𝐴;𝜔)=𝐼_𝐴Δ𝑋_𝑛(𝜔),𝐴∈ℬℝ^𝑑, (2.11) where 𝐼 is an indicator function, that is,𝜇_𝑛(𝐴;𝜔)=1ifΔ𝑋_𝑛(𝜔)∈𝐴,0ifΔ𝑋_𝑛(𝜔)∉𝐴, (2.12) (see Shiryaev [12], for details). We define the integral-valued random measures of jumps 𝜇^𝑋=(𝜇^𝑋_(0,𝑛](⋅))_𝑛≥1, where 𝜇^𝑋_(0,𝑛](𝐴;𝜔)=∑^𝑛_𝑘=1𝜇^𝑋_𝑘(𝐴,𝜔) and 𝜇^𝑋_𝑘(𝐴;𝜔)=𝐼(Δ𝑋_𝑘(𝜔)∈𝐴,𝐴∈ℬ(ℝ^𝑑⧵{0}) and by 𝜈=𝜈(𝜔,𝑑𝑠,𝑑𝑥) the compensator of 𝜇, that is, the predictable measure 𝜈_𝑛(𝐴∶𝜔)=ℙ(Δ𝑋_𝑛∈𝐴∣ℱ_𝑛−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.

Let 𝜑 be a truncation function, for example, 𝜑(𝑥)=𝑥𝐼(|𝑥|≤1). Then Δ𝑋_𝑠−𝜑(Δ𝑋_𝑠)≠0⇔|Δ𝑋_𝑠|>𝛼forsome𝛼>0. We now denote the jump part of 𝑋 corresponding to big jumps by𝑋̌=_{0<𝑘≤𝑡}Δ𝑋_𝑘−𝜑Δ𝑋_𝑘. (2.13) Using random measure of jumps, (2.9) can be written in canonical form as𝑑𝑌_𝑡=𝑌_𝑡−𝑑𝐻_𝑡=𝑌₋𝛼𝑑𝑡+𝑑𝐶𝜑_𝑡+12𝑑𝐻^𝑐+𝜎𝑑𝑊_𝑡+𝑑𝑋^𝑐_𝑡+𝑑𝑒^𝑥⋆𝑒−1𝜇−𝜈+𝑑^𝑥−1−𝜑(𝑥)⋆𝜇, (2.14) where 𝐶(𝜑) is a predictable process and 𝑋^𝑐 is the continuous martingale part of 𝑋_𝑡 [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 ℱ_𝑡 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 𝑌^𝛾 satisfies the equation𝑑𝑌^𝛾_𝑡=𝑌^𝛾₋𝛼𝑑𝑡+𝑑𝐶𝜑_𝑡+12𝑑𝐻^𝑐_𝑡+𝜎𝑑𝑊_𝑡+𝑑𝑋^𝑐_𝑡+𝑑𝑒^𝑥⋆𝑒−1𝜇−𝜈+𝑑^𝑥(𝑥)(𝑡),𝑌−1−𝜙⋆𝜇−𝑑𝛾_𝛾(𝑠⁻)=𝑦₁. (2.15) It is important to note that the difference between 𝑌^𝛾(𝑠) and 𝑌^𝛾(𝑠−)∶𝑌^𝛾(𝑠−) is the state before harvesting starts at time 𝑡=𝑠 while 𝑌^𝛾(𝑠) is the state immediately after. If 𝛾 consists of an immediate harvest of size 𝛾 at 𝑡=𝑠, then𝑌(𝑠)=𝑌(𝑠⁻)−Δ𝛾. (2.16) 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 by𝐽^𝛾𝑠,𝑦=𝔼^𝑦𝑇₀𝜋𝑒^{−𝜌(𝑠+𝑡)}⋅𝑑𝛾(𝑡), (2.17) where 𝔼^𝑠,𝑦 denotes expectation with respect to probability law 𝑄^𝑠,𝑦 of 𝑌^𝑠,𝑦(𝑡)=(𝑡,𝑌^𝛾(𝑡)) for 𝑡≥𝑠, assuming that 𝑌^𝑠,𝑦(0)=(𝑠,𝑦),𝑇=inf{𝑡>0;𝜋𝑌(𝑡)∉𝑆},𝑆=𝑠,𝑦∶𝜋𝑒^{−𝜌(𝑠+𝑡)}𝑦≥0 (2.18) 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 thatΦ𝑠,𝑦=sup_𝛾∈Γ𝐽^𝛾𝑠,𝑦=𝐽^𝛾∗𝑠,𝑦. (2.19) We let 0<𝑡₁<𝑡₂⋯ denote the jumping times of the given strategy 𝛾∈Γ, and we let Δ(𝑡_𝑘)=𝛾(𝑡_𝑘)−𝛾(𝑡⁻_𝑘) be the jump of 𝛾(𝑡_𝑘). We also let𝛾^𝑐(𝑡)=𝛾(𝑡)−_{𝑠≤𝑡𝑘≤𝑡}Δ𝛾𝑡_𝑘 (2.20) 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 𝛾∈Γ to be optimal in the following theorem.

Theorem 2.2 (extended Lungu and Øksendal [10]). Suppose that 𝜙≥0 is twice continuously differentiable on 𝑆 with the following properties:(1)One has 𝜕𝜙𝜕𝑦≥𝜋𝑒^{−𝜌(𝑠+𝑡)}. (2.21)(2)One has 𝐿𝜙𝑡,𝑦=𝜕𝜙+𝜑𝜕𝑡𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐_𝑡+𝑒^𝑦−1⋆𝜇𝜕𝜙+1𝜕𝑦2𝜎²𝜕𝜙²𝜕𝑦²≤0𝑜𝑛𝑆. (2.22) Then 𝜙𝑠,𝑦≥Φ𝑠,𝑦𝑜𝑛𝑆. (2.23)(3)Define the nonintervention region as 𝐷=𝑡,𝑦∈𝑆;𝜕𝜙𝜕𝑦𝑡,𝑦>𝜋𝑒^{−𝜌(𝑠+𝑡)}∀𝑖=1⋯𝑛. (2.24) Suppose 𝐿𝜙=0in𝐷, (2.25) and there exists a harvesting strategy 𝛾 such that the following hold: 𝑌𝛾(𝑡)∈𝐷∀𝑡>𝑠,𝜕𝜙𝜕_𝑦𝑡,𝑌𝛾(𝑡)−𝜋𝑒^{−𝜌(𝑠+𝑡)}𝛾^𝑐(𝑡)𝛾=0∀𝑖=1⋯𝑛,i.e.,^𝑐increasesonlywhen𝜕𝜙𝜕𝑦=𝜋𝑒^{−𝜌(𝑠+𝑡)}, (2.26) where 𝐷 is the closure of 𝐷, that is, 𝐷=𝐷∪𝜕𝐷, where 𝜕𝐷 is the boundary of 𝐷.(4)One has𝜕𝜙𝛾(𝑠)𝜑𝜕𝑦𝑡,𝑌𝛼𝑑𝑡+𝑑𝐶_𝑡+12𝑑𝐻^𝑐+^∞_−∞𝑒^𝑦𝑦−𝜑−1−𝜑𝑑𝜇𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐_𝑡+^∞_−∞𝑒^𝑦−1−𝜑𝑦𝑑𝜇𝑑𝑡≤0. (2.27)(5)One hasΔ𝜙𝑡_𝑘∶=𝜙𝑡_𝑘,𝑌𝛾𝑡_𝑘−𝜙𝑡_𝑘,𝑌𝛾𝑡⁻_𝑘=−𝜋⋅Δ𝛾𝑡_𝑘 (2.28) at all jumping times 𝑡_𝑘≥𝑠 of 𝛾(𝑡_𝑘) and 𝔼^𝑠,𝑦𝜙𝑇_𝑅,𝑌𝛾𝑇_𝑅⟶∞, (2.29) where 𝑇_𝑅=𝑇∧𝑅∧inf𝑡>𝑠;𝑌𝛾(𝑡)≥𝑅. (2.30) Then 𝜙𝑠,𝑦=Φ𝑠,𝑦∀𝑠,𝑦∈𝑆 (2.31) and 𝛾=𝛾^∗ is an optimal harvest strategy.

Proof. Consider the following: Δ𝑌_𝑡=𝛼Δ𝑡+Δ𝐶𝜑_𝑡+Δ𝐻^𝑐_𝑡+𝜎Δ𝑊_𝑡+Δ𝑋^𝑐_𝑡+Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥𝑒−1−𝜑⋆𝜈=Δ^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜑⋆𝜈, (2.32)𝑑𝑌_𝑡=𝛼𝑑𝑡+𝑑𝐶𝜑_𝑡+12𝑑𝐻^𝑐+𝜎𝑑𝑊_𝑡+𝑑𝑋^𝑐_𝑡+𝑑𝑒^𝑦⋆𝑒−1𝜇−𝜈+𝑑^𝑥(𝑥)(𝑡)𝜑−1−𝜑⋆𝜇−𝑑𝛾=𝛼𝑑𝑡+𝑑𝐶_𝑡+12𝑑𝐻^𝑐+𝜎𝑑𝑊_𝑡+𝑑𝑋^𝑐_𝑡+^∞_−∞𝑒^𝑥𝑦𝑑+−1−𝜑𝜇−𝜈_∞−∞𝑒^𝑥−1−𝜑𝑦𝑑𝜇−𝑑𝛾(𝑡). (2.33) 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 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅=𝔼^𝑠,𝑦𝜙𝑠,𝑌^𝛾(𝑠)+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑡𝑡,𝑌^𝛾(𝑠)𝑑𝑡+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝛼𝑑𝑡+𝑑𝐶𝜑_𝑡+12𝑑𝐻^𝑐_𝑡+𝜎𝑑𝑊_𝑡+𝑑𝑋^𝑐_𝑡+^∞_−∞𝑒^𝑥−1𝑑𝜇−𝜈+^∞_−∞𝑒^𝑥−1−𝜑𝑦𝑑𝜇−𝑑𝛾(𝑡)+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+12𝜎²𝑡,𝑌^𝛾(𝑡)𝜕²𝜙𝜕𝑦²𝑡,𝑌^𝛾+𝔼^𝑠,𝑦_{𝑠<𝑡𝑘≤𝑇𝑅}𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜙𝑡_𝑘,𝑌^𝛾𝑡⁻_𝑡𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝑌^𝛾_𝑖𝑡_𝑘. (2.34) From the theory of martingales for integrals, 𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜎𝜕𝜙𝜕𝑦𝑑𝑊_𝑡=𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜎𝜕𝜙𝜕𝑦𝑑𝑋^𝑐_𝑡=𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+^∞_−∞𝑒^𝑦−1𝑑𝜇−𝜈=0. (2.35) Substituting (2.28), (2.32), and (2.35) in (2.34), we have 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅=𝔼^𝑠𝑦𝜙𝑠,𝑦+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑡𝑡,𝑌^𝛾(𝑠)𝑑𝑡+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝛼𝑑𝑡+𝑑𝐶𝜙_𝑡+12𝑑𝐻^𝑐_𝑡+^∞_−∞𝑒^𝑥−1−𝜑(𝑥)𝑑𝜇−𝑑𝛾(𝑡)+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+12𝜎²𝑡,𝑋^𝛾(𝑡)𝜕²𝜙𝜕𝑦²𝑡,𝑌^𝛾+𝔼^𝑠,𝑦_{𝑠<𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜑⋆𝜈. (2.36) Now from (2.22), 𝜕𝜙()−𝜑𝜕𝑡=𝐿𝜙𝑡,𝑥𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐_𝑡+𝑒^𝑥−1−𝜑𝑦⋆𝜇𝜕𝜙−1𝜕𝑦2𝜎²()𝜕𝑡,𝑥²𝜙𝜕𝑦². (2.37) Using (2.37) in (2.36) gives 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾=𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝐿𝜙(𝑡,𝑥)𝑑𝑡+𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝛼𝑑𝑡+𝑑𝐶𝜑_𝑡+12𝑑𝐻^𝑐+^∞_−∞𝑒^𝑦𝑦−𝜑−1−𝜑𝑑𝜇𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐_𝑡+^∞_−∞𝑒^𝑦−1−𝜑𝑦𝑑𝜇𝑑𝑡−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾(𝑡)+𝔼^𝑠𝑥_{𝑠<𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜑⋆𝜈 (2.38) Using the fact that 𝔼^𝑠𝑦[𝜙(𝑇_𝑅,𝑌^𝛾)]≤𝜙(𝑠,𝑥) and that 𝐿𝜙(𝑠,𝑥)≤0 (refer to (2.27)) yields the inequality 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾≤𝜙𝑠,𝑦−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾(𝑡)+𝔼^𝑠𝑥_{𝑠<𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘(Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜙⋆𝜈. (2.39) From the relation 𝛾^𝑐(𝑡)=𝛾(𝑡)−_{𝑠≤𝑡𝑘≤𝑡}Δ𝛾𝑡_𝑘, (2.40) we deduce that 𝑑𝛾(𝑡)=𝑑𝛾^𝑐(𝑡)(𝑡),(𝑡)(𝑡).+ΔΔ𝑌=−Δ𝛾 (2.41) Using (2.41), the right-hand side of (2.32) becomes Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜙⋆𝜈=−Δ𝛾(𝑡). (2.42) 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 _{𝑠≤𝑡𝑘≤𝑇𝑅}𝑓𝑡_𝑘=_{𝑠<𝑡𝑘≤𝑇𝑅}𝑓𝑡_𝑘+Δ𝑓(𝑠), (2.43) we have 𝔼^𝑠𝑦_{𝑠<𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜙⋆𝜈=𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜙⋆𝜈−𝔼^𝑠,𝑦Δ𝜙𝑠,𝑌^𝛾(𝑠)−𝔼^𝑠,𝑦𝜕𝜙𝜕𝑦𝑠,𝑌^𝜙(𝑠)Δ𝑒^𝑥−1⋆𝜇−𝜈+Δ𝑒^𝑥−1−𝜙⋆𝜈=𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝛾𝑡_𝑘Δ𝛾(𝑡)−𝔼^𝑠,𝑦Δ𝜙𝑠,𝑌^𝛾(𝑠)+𝔼^𝑠,𝑦𝜕𝜙𝜕𝑦𝑠,𝑌^𝛾(𝑠)Δ𝛾(𝑠). (2.44) Substituting (2.44) into (2.39), inequality (2.39) becomes 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾≤𝜙𝑠,𝑦−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾(𝑡)−𝔼^𝑠,𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)Δ𝛾(𝑡)+𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘+𝔼^𝑠𝑦_{𝑠≤<𝑡𝑘≤𝑇𝑅}𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝜙𝑡_𝑘Δ𝛾(𝑡)−𝔼^𝑠,𝑦Δ𝜙𝑠,𝑌^𝛾(𝑠)+𝔼^𝑠,𝑦𝜕𝜙𝜕𝑦𝑠,𝑌^𝛾(𝑠)Δ𝛾(𝑠). (2.45) This will then lead to the inequality 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅≤𝜙𝑠,𝑦−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾^𝑐(𝑡)+𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘. (2.46) By the Mean-Value property, we have Δ𝜙𝑡_𝑘,𝑌^𝛾𝑡_𝑘=𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝛾_𝑘Δ𝑌^𝛾_𝑖𝑡_𝑘 (2.47) for some point 𝑌^𝛾_𝑘 on the line connecting the points 𝑌^𝛾(𝑡⁻_𝑘) and 𝑌^𝛾(𝑡_𝑘), and, using (2.41), we have Δ𝜙𝑡_𝑘,𝑦^𝛾𝑡_𝑘=𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝛾_𝑘Δ𝑌^𝛾_𝑖𝑡_𝑘=−𝜕𝜙𝑡𝜕𝑦^𝑘,𝑋^𝛾_(𝑘)Δ𝛾𝑡_(𝑘) (2.48) which leads to 𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅≤𝜙𝑠,𝑦−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾^𝑐(𝑡)−𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜕𝜙𝑡𝜕𝑦^𝑘,𝑋^𝛾_(𝑘)Δ𝛾𝑡_(𝑘). (2.49) From (2.21), we obtain 𝜙𝑠,𝑦−^𝑇𝑅_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾^𝑐(𝑡)≤𝜙(𝑠,𝑥)−^𝑇𝑅_𝑠+𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾^𝑐(𝑡). (2.50) Equation (2.21) can also be expressed in discrete form as _{𝑠≤𝑡𝑘≤𝑇𝑅}𝜕𝜙𝑡𝜕𝑦^𝑘,𝑋^𝛾_(𝑘)Δ𝛾𝑡_(𝑘)≥_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜋𝑒^{−(𝜌+𝑠)𝑡}Δ𝛾𝑡_(𝑘) (2.51) Combining (2.50) and (2.51) gives the inequality 𝜙𝑠,𝑦−^𝑇𝑅_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾^𝑐(𝑡)−_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜕𝜙𝑡𝜕𝑦^𝑘,𝑌^𝛾_(𝑘)Δ𝛾𝑡_(𝑘)≤𝜙𝑠,𝑦−^𝑇𝑅_𝑠+𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾^𝑐(𝑡)−_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜋𝑒^{−(𝜌+𝑠)𝑡}Δ𝛾𝑡_(𝑘). (2.52) Taking the expectation of (2.52) yields the inequality 𝔼^𝑠,𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅≤𝜙𝑠,𝑦−𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝑑𝛾^𝑐(𝑡)−𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜕𝜙𝑡𝜕𝑦^𝑘,𝑋^𝛾_(𝑘)Δ𝛾𝑡_(𝑘)≤𝜙𝑠,𝑦−𝔼^{𝑠𝑦𝑇𝑅}_𝑠+𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾^𝑐(𝑡)−𝔼^𝑠,𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜋𝑒^{−(𝜌+𝑠)𝑡}Δ𝛾𝑡_(𝑘), (2.53) from which we obtain 𝜙𝑠,𝑦≥𝔼^{𝑠,𝑦𝑇𝑅}_𝑠+𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾^𝑐(𝑡)+𝔼^𝑠𝑦_{𝑠≤𝑡𝑘≤𝑇𝑅}𝜋𝑒^{−(𝜌+𝑠)𝑡}Δ𝛾𝑡_(𝑘)+𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅≥𝔼^{𝑠𝑦𝑇𝑅}_𝑠𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾(𝑡)+𝔼^𝑠𝑦𝜙𝑇_𝑅,𝑌^𝛾𝑇_𝑅. (2.54) Since 𝑅<∞,𝛾∈Γ were arbitrary and 𝜙≥0, this proves that 𝜙𝑠,𝑦≥Φ𝑠,𝑦and𝛾isoptimal. (2.55) 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 𝛾, then equality holds everywhere and we end up with the relation 𝜙𝑠,𝑦=𝔼^{𝑠,𝑦𝑇𝑅}_𝑠𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾+𝔼𝜙𝑇_𝑅,𝑌𝛾𝑇_𝑅. (2.56) Letting 𝑅→∞ and using (2.29), we get 𝜙𝑠,𝑦=𝔼^𝑠,𝑦∞_𝑠𝜋⋅𝑒^{−(𝜌+𝑠)𝑡}𝑑𝛾. (2.57)

Combining (eqntawina) with (2.23), we have𝜙𝑠,𝑦=Φ𝑠,𝑦,𝛾isoptimal. (2.58) The strategy 𝛾 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 formΦ𝑠,𝑦=𝑒^−𝜌𝑠Ψ𝑦. (3.1) Let𝜙𝑠,𝑦=𝑒^−𝜌𝑠𝜓𝑦. (3.2) Equations (2.21), (2.22) become𝜕𝜓𝜑𝜕𝑦≥𝜋,𝐿𝜓𝑡,𝑦=−𝜌𝜓+𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐_𝑡+𝑒^𝑥−1⋆𝜇𝜕𝜓+1𝜕𝑦2𝜎²𝜕²𝜓𝜕𝑦²≤0on𝑆, (3.3)(3.4) where𝜅=𝛼𝑡+𝐶𝜑_𝑡+12𝜎²+𝐻^𝑐_𝑡+𝑒^𝑥1−1⋆𝜇𝜋,𝛽=2𝜎²𝜋². (3.8) 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 of𝛽𝐹^′′(𝑧)+𝜅𝐹^′(𝑧)−𝜌𝐹(𝑧)=0 (3.9) 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 case 𝜅 is a constant that reduces to the problem in Lungu and Øksendal [10]. The auxiliary equation for (3.9) is𝛽𝑚²+𝜅𝑚−𝜌=0. (3.10) The solutions are𝑚_1,2=−𝜅±𝜅²−4𝜌𝛽.2𝛽 (3.11) Let us suppose the nonintervention region 𝐵 to be of the form𝐵=𝑧;0<𝑧<𝑧^∗ (3.12) for some 𝑧^∗>0. From (2.21) and (2.22), we try a solution of the form𝐹(𝑧)=𝑧+𝐺,for𝑧≥𝑧_∗,𝛼₁𝑒^𝑚1𝑧+𝛼₂𝑒^𝑚2𝑧,0<𝑧<𝑧_∗, (3.13) 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 obtain𝑧_∗=2ln𝑚₂/𝑚₁𝑚₁−𝑚₂𝑚>0,𝛼=₁𝑒^𝑚1𝑧∗+𝑚₂𝑒^{𝑚2𝑧∗−1},𝐺=𝛼𝑒^𝑚1𝑧∗−𝑒^𝑚2𝑧∗−𝑧_∗. (3.14) With this choice of parameters, all the conditions of Theorem 2.2 are satisfied, and we haveΦ=𝛼𝑠,𝑦¹𝑒^−𝜌𝑠𝑒^𝑚1𝑧+𝑒^𝑚2𝑧,for0≤𝑧<𝑧_∗,𝑒^−𝜌𝑠(𝑧+𝐺),𝑧_∗≤𝑧. (3.15) 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^∗=𝛾 of the reflected process 𝑌𝛾 in the direction of −𝜋.

3.1.2. Example  2 (When 𝜅 Is Not a Constant)

We now look at a more general case corresponding to 𝜅 not a constant. Suppose that we can write 𝐹(𝑧) as𝐹(𝑧)(𝑧)−1=𝑉exp2𝜅𝛽𝑑𝑧, (3.16) then (3.9) can be transformed into its canonical form given by𝑉^′′(𝑧)+𝜂(𝑧)𝑉(𝑧)=0, (3.17) where𝜌𝜂=−𝛽−14𝜅𝛽²−14𝜅𝛽^′_𝑧 (3.18) 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) is𝑚²+𝜂(𝑧)=0 (3.19) with solutions𝑚₁=𝑖𝑚(𝑧),𝑚₂=−𝑖𝑚(𝑧),where𝑚(𝑧)=𝜂(𝑧). (3.20) The general solution for 𝑉(𝑧) can be written as𝑉(𝑧)=ℎ₁(𝑧)exp(𝑖𝑚𝑧)+ℎ₂(𝑧)exp(−𝑖𝑚𝑧), (3.21) where ℎ₁(𝑧) and ℎ₂(𝑧) are functions to be determined. The solution (3.21) can be expressed as𝑉(𝑧)=exp[Φ(𝑧)]^𝑡𝐇(𝑧), (3.22) where𝐇(z)=ℎ₁(𝑧)ℎ₂(𝑧)(𝑧)=(𝑧)(𝑧),Φ𝑖𝑚−𝑖𝑚 (3.23) and the superscript 𝑡 denotes transpose of a matrix. 𝐅 is the solution of the equation𝑑𝐇(𝑧)=𝐔(𝑧)𝐇(𝑧)𝑑𝑧, (3.24) where𝐔=−𝑧𝐊^′(𝑧)−exp[−𝑧𝐊(𝑧)]𝐃(𝑧)⁻¹𝐂(𝑧)𝐊^′(𝑧)[(𝑧𝐂(𝑧)=𝑚exp𝑧𝐊)],𝑝−1_𝑞^𝑝−2_𝑛×𝑛,𝐃(𝑥)=𝑚(𝑧)_𝑞^𝑝−1(𝑧)_𝑛×𝑛,𝐊(𝑧)=𝑚_𝑝(𝑧)𝛿_{𝑝𝑞𝑛×𝑛},𝐊^′(𝑥)=𝑚^′_𝑝(𝑧)𝛿_{𝑝𝑞𝑛×𝑛} (3.25)(3.26) (cf. [14]).

Clearly, for the case corresponding to 𝑛=2, the matrices (3.26) are given by𝑚𝐂=0011,𝐃=11₁𝑚₂,𝑚𝐊=¹00𝑚₂,hence𝐊^′=𝑚₁′00𝑚₂′. (3.27)

Using power series expansion for exponential square matrices and truncating the expansion at second-order terms, we haveexp[𝑧𝐊]=exp𝑧𝑚₁00𝑧𝑚₂=+1001𝑧𝑚₁00𝑧𝑘₂(2)=+𝑂terms1+𝑧𝑚₁001+𝑧𝑚₂+𝑂(2)terms. (3.28) Similarly,exp[−𝑧𝐊]=1−𝑧𝑚₁001−𝑧𝑚₂. (3.29) From (3.25) and (3.28), we obtain an approximation for 𝐔(𝑧) as𝐔(𝑧)=−1𝑧+𝑚₁−𝑚₂𝑚₁′𝑚₂′𝑚₂−𝑚₁exp−𝑧𝑚₁−𝑚₂𝑚₁′𝑚₁−𝑚₂exp+𝑧𝑚₁−𝑚₂−1𝑧+𝑚₂−𝑚₁𝑚₂′. (3.30) From (3.19) and (3.20), we obtain𝐕(𝑧)=𝑚^′(𝑧)(𝑧)(𝑧)[(𝑧2𝑚[(𝑧(𝑧)𝑧−1+𝑖2𝑚𝑧exp𝑖2𝑧𝑚)]exp−2𝑖𝑧𝑚)]−1−𝑖2𝑚. (3.31) The jump transfer matrix from region 1 to region 2 across the interface 𝑧=𝜍 in our case takes the form𝐐_1→2=𝑚₁+𝑚₂2𝑚₂𝑒^{+𝑖𝜍(𝑚2−𝑚1)}𝑚₂−𝑚₁2𝑚₂𝑒^{+𝑖𝜍(𝑚1+𝑚2)}𝑚₂−𝑚₁2𝑚₂𝑒^{−𝑖𝜍(𝑚1+𝑚2)}𝑚₂+𝑚₁2𝑚₂𝑒^{−𝑖𝜍(𝑚2−𝑚1)}. (3.32) This jump transfer matrix over singularities given by 𝐐_{𝜍−𝛿𝑧→𝜍+𝛿𝑧} is simplified as in [14] to𝐐_{𝜍−𝛿𝑧→𝜍+𝛿𝑧}=1+𝑖21−𝑖21−𝑖21+𝑖2𝐐typeA,_{𝜍−𝛿𝑧→𝜍+𝛿𝑧}=1−𝑖21+𝑖21+𝑖21−𝑖2𝐐typeB,_{𝜍−𝛿𝑧→𝜍+𝛿𝑧}=1001typeC. (3.33) The reader is referred to [14] and the references therein for details regarding classification of singularities.

3.1.3. Analysis of the Results

Let 𝑧₀=0 and our solutions (3.21) have jumps at points 𝑧₁,𝑧₂,𝑧₃,…,𝑧_𝑛, and we define𝑊_𝑗=𝑧∶𝑧_𝑗−1<𝑧<𝑧_𝑗𝐵for𝑖=1,2,3,…,𝑛,_𝑗=𝑧∶𝑧_𝑗−1<𝑧<𝑧_∗𝑗 (3.34) for some 𝑧_∗𝑖, and we define𝐵=^𝑛_𝑗=1𝐵_𝑗. (3.35) For 𝑧∈𝑊_𝑖, we define𝐹_𝑗(𝑧)=𝑧+𝐺_𝑗,for𝑧≥𝑧_𝑗∗,ℎ_𝑗1(𝑧)𝑒^{𝑚𝑗1𝑧}+ℎ_𝑗2𝑒^{𝑚𝑗2𝑧},0<𝑧<𝑧_𝑗∗,𝐹=𝐹₀,𝐹₁,𝐹₂,…,𝐹_𝑛,Φ(𝑠,𝑧)_𝑗=ℎ_𝑗(𝑧)𝑒^−𝜌𝑠𝑒^{𝑚𝑗1𝑧}+𝑒^{𝑚𝑗2𝑧},for0≤𝑧<𝑧_𝑗∗,𝑒^−𝜌𝑠𝑧+𝐺_𝑗,𝑧_𝑗∗Φ()=Φ≤𝑧,𝑠,𝑧₁(𝑠,𝑧),Φ₂(𝑠,𝑧),…,Φ_𝑛(𝑠,𝑧). (3.36)

3.1.4. Conjecture

The optimal strategy is achieved by doing nothing during jumps and as long as 𝑌(𝑡)∈𝐵_𝑗 but to harvest according to local time 𝛾^∗_𝑗=𝛾 at the boundary 𝜕𝐵_𝑗.

Remarks. In each nonintervention region 𝐵_𝑗, 𝑌𝛾 is a reflected process at 𝜕𝐵_𝑗, since, as 𝑌𝛾 hits the boundary, a certain amount is harvested thereby forcing the process to go below 𝑧_𝑗∗.

4. Conclusion

Most of the conditions in Theorem 2.2 are extensions from Lungu and Øksendal [10] with exception of condition (2.5), that is,𝜕𝜙𝜕𝑦𝑡,𝑌^𝛾(𝑠)𝛼𝑑𝑡+𝑑𝐶𝜙_𝑡+12𝑑𝐻^𝑐+^∞_−∞𝑒^𝑦𝑦−𝜙−1−𝜙𝑑𝜇𝛼𝑡+𝐶_𝑡+12𝜎²+𝐻^𝑐′_𝑡+^∞_−∞𝑒^𝑦−1−𝜙𝑦𝑑𝜇𝑑𝑡≤0. (4.1) 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.