Optimal Selling Rule in a Regime Switching Lévy Market

Abstract

This paper is concerned with a finite-horizon optimal selling rule problem when the underlying stock price movements are modeled by a Markov switching Lévy process. Assuming that the transaction fee of the selling operation is a function of the underlying stock price, the optimal selling rule can be obtained by solving an optimal stopping problem. The corresponding value function is shown to be the unique viscosity solution to the associated HJB variational inequalities. A numerical example is presented to illustrate the results.

1. Introduction

One of the major decision investors have to make on a daily basis is to identify the best time to sell or buy a particular stock. Usually if the right decision is not taken at the right time, this will generally result in large losses for the investor. Such decisions are mainly affected by various macro- and micro-economical parameters. One of the main factors that affect decision making in the marketplace is the trend of the stock market. In this paper, we study trading decision making when we assume that market trends are subject to change and that these fluctuations can be captured by a combination of a latent Markov chain and a jump process. In fact, we model the stock price dynamics with a regime switching Lévy process. Regime switching Lévy processes are obtained by combining a finite number of geometric Lévy processes modulated by a finite-state Markov chain. This type of processes clearly capture the main features of a wide variety of stock such as energy stock and commodities which usually display a lot of spikes and seasonality. Selling rule problems in general have been intensively studied in the literature, and most of the work have been done when the stock price follows a geometric Brownian motion or a simple Markov switching process. Among many others, we can cite the work of Zhang [1]; in this paper, a selling rule is determined by two threshold levels, and a target price and a stop-loss limit are considered. One makes a selling decision whenever the price reaches either the target price or the stop-loss limit. The objective is to choose these threshold levels to maximize an expected return function. In [1], such optimal threshold levels are obtained by solving a set of two-point boundary value problems. Recently Pemy and Zhang [2] studied a similar problem in the case where there is no jump process associated and the underlying dynamics is just a traditional Markov switching process built by coupling a set of geometric Brownian motions.

In this paper, we extend the result of Pemy and Zhang [2], we consider an optimal selling rule among the class of almost all stopping times under a regime switching Lévy model. We study the case when the stock has to be sold within a prespecified time limit. Given a transaction cost which is a function of the underlying stock price, the objective is to choose a stopping time so as to maximize an expected return. The optimal stopping problem was studied by McKean [3] back to the 1960s when there is no switching; see also Samuelson [4] in connection with derivative pricing and Øksendal [5] for optimal stopping in general. In models with regime switching, Guo and Zhang [6] considered the model with a two-state Markov chain. Using a smooth-fit technique, they were able to convert the optimal stopping problem to a set of algebraic equations under certain smoothness conditions. Closed-form solutions were obtained in these cases. However, it can be shown with extensive numerical tests that the associated algebraic equations may have no solutions. This suggests that the smoothness (𝐶²) assumption may not hold in these cases. Moreover, the results in [5, 6] are established on an infinite time horizon setup. However, in practice, an investor often has to sell his stock holdings by a certain date due to various nonprice-related considerations such as year-end tax deduction or the need for raising cash for major purchases. In these cases, it is necessary to consider the corresponding optimal selling with a finite horizon. It is the purpose of this paper to treat the underlying finite horizon optimization problem with possible nonsmoothness of the solutions to the associated HJB variational inequalities. We resort to the concept of viscosity solutions and show that the corresponding value function is indeed the only viscosity solution to the HJB variational inequalities. We clearly prove that the value function of this optimal stopping time problem is the unique viscosity solution to the associated HJB variational inequalities, which enables us to run some numerical schemes in order to approximate the value function and derive the both the continuation region and the stopping region. It is well known that the optimal stopping rule can be determined by the corresponding value function; see, for example, Krylov [7] and Øksendal [5] for diffusions, Pham [8] for jump diffusions, and Guo and Zhang [6] and [9] for regime switching diffusions.

The paper is organized as follows. In the next section, we formulate the problem under consideration and then present the associated HJB inequalities and their viscosity solutions. In Section 3, we obtain the continuity property of the value function and show that it is the only viscosity solution to the HJB equations. In Section 4, we give a numerical example in order to illustrate our results. To better present the results without undue technical difficulties, all proofs are moved to the appendix placed at the end of the paper.

2. Problem Formulation

Given an integer 𝑚≥2, let 𝛼(𝑡)∈ℳ={1,2,…,𝑚} denote a Markov chain with an 𝑚×𝑚 matrix generator 𝑄=(𝑞_𝑖𝑗)_𝑚,𝑚, that is, 𝑞_𝑖𝑗≥0 for 𝑖≠𝑗 and Σ^𝑚_𝑗=1𝑞_𝑖𝑗=0 for 𝑖∈ℳ and a Lévy process (𝜂_𝑡)_𝑡. Let 𝑁 be the Poisson random measure of (𝜂_𝑡)_𝑡, then it is defined as follows: for any Borel set 𝑈⊂ℝ, 𝑁(𝑡,𝑈)=_{0<𝑠≤𝑡}𝟏_𝑈𝜂_𝑠−𝜂_𝑠−. (2.1) The differential form of 𝑁 is denoted by 𝑁(𝑑𝑡,𝑑𝑧). Let 𝜈 be the Lévy measure of (𝜂_𝑡)_𝑡; we have 𝜈(𝑈)=𝐸[𝑁(1,𝑈)] for any Borel set 𝑈⊂ℝ. We define the differential form 𝑁(𝑑𝑡,𝑑𝑧) as follows: 𝑁()=𝑁()()|𝑧|𝑁()|𝑧|𝑑𝑡,𝑑𝑧𝑑𝑡,𝑑𝑧−𝜈𝑑𝑧𝑑𝑡,𝑖𝑓<1,𝑑𝑡,𝑑𝑧,𝑖𝑓≥1. (2.2) From Lévy-Khintchine formula, we have min|_ℝ𝑧|²,1𝜈(𝑑𝑧)<∞. (2.3) In our regime switching Lévy market model, the stock price denoted by 𝑋(𝑡) satisfies the following Lévy stochastic differential equation d𝑋(𝑡)=𝑋(𝑡)𝜇(𝛼(𝑡))d𝑡+𝜎(𝛼(𝑡))d𝑊(𝑡)+_ℝ𝛾(𝛼(𝑡))𝑧𝑁(),𝑋(𝑠)𝑑𝑡,𝑑𝑧=𝑥,𝑠≤𝑡≤𝑇, (2.4) where 𝑥 is the initial price and 𝑇 is a finite time. For each state 𝑖∈ℳ, 𝜇(𝑖) the rate of return, 𝜎(𝑖) the volatility and 𝛾(𝑖) the jump intensity are known and satisfied the linear growth condition. There exists a constant 𝐶>0 such that for all 𝑥∈ℝ and all 𝑡∈[0,𝑇], we have 𝑥²𝜇(𝛼(𝑡))²+𝜎(𝛼(𝑡))²+_ℝ𝛾(𝛼(𝑡))²𝑧²𝜈(𝑑𝑧)≤𝐶1+𝑥². (2.5)𝑊(𝑡) is the standard Weiner process, and 𝑁(𝑑𝑡,𝑑𝑧) represents the differential form of the jump measure of 𝜂_𝑡. The processes 𝑊(⋅), 𝛼(⋅), and 𝜂(⋅) are defined on a probability space (Ω,ℱ,𝑃) and are independent of each other.

In this paper, we consider the optimal selling rule with a finite horizon 𝑇. We assume that the transaction cost function 𝑎(⋅)>0 is the function of the stock price itself. In this case, we take into account all costs associated with the selling operation. The main objective of this selling problem is to sell the stock by time 𝑇 so as to maximize the quantity 𝐸[𝑒^{−𝑟(𝜏−𝑠)}(𝑋(𝜏)−𝑎(𝑋(𝜏)))], where 𝑟>0 is a discount rate.

Let ℱ_𝑡=𝜎{𝛼(𝑠),𝑊(s),𝜂(𝑠);𝑠≤𝑡} and let Λ_𝑠,𝑇 denote the set of ℱ_𝑡-stopping times such that 𝑠≤𝜏≤𝑇 a.s. The value function can be written as follows: 𝑣(𝑠,𝑥,𝑖)=sup_{𝜏∈Λ𝑠,𝑇}𝐸𝑒^{−𝑟(𝜏−𝑠)}(𝑋(𝜏)−𝑎(𝑋(𝜏)))∣𝑋(𝑠)=𝑥,𝛼(𝑠)=𝑖. (2.6)

Given the value function 𝑣(𝑠,𝑥,𝑖), it is typical that an optimal stopping time 𝜏^∗ can be determined by the following continuation region:𝐷={(𝑡,𝑥,𝑖)∈[0,𝑇)×ℝ×ℳ;𝑣(𝑡,𝑥,𝑖)>𝑥−𝑎(𝑥)}, (2.7) as follows:𝜏^∗=inf{𝑡>0;(𝑡,𝑋(𝑡),𝛼(𝑡))∉𝐷}. (2.8) It can be proved that if 𝜏^∗<+∞, then 𝑣(𝑠,𝑥,𝑖)=𝐸^{𝑠,𝑥,𝑖}𝑒^{−𝑟(𝜏∗−𝑠)}𝑋𝜏^∗−𝑎(𝑥). (2.9) Thus, 𝜏^∗ is the optimal stopping time; see [9].

The process (𝑋(𝑡),𝛼(𝑡)) is a Markov process with generator 𝒜 defined as follows:()=1𝒜𝑓𝑠,𝑥,𝑖2𝑥²𝜎²(𝑖)𝜕²𝑓(𝑠,𝑥,𝑖)𝜕𝑥²(𝑖)()+𝑥𝜇𝜕𝑓𝑠,𝑥,𝑖(𝑖)+𝜕𝑥+𝑄𝑓𝑠,𝑥,⋅)(_ℝ𝑓𝑠,𝑥+𝛾(𝑖)𝑥𝑧−𝑓(𝑠,𝑥)−𝛾(𝑖)𝑥𝑧𝟏_{{|𝑧|<𝟏}}(𝑧)𝜕𝑓𝜈(),𝜕𝑥𝑑𝑧 (2.10) where 𝑄𝑓(𝑠,𝑥,⋅)(𝑖)=_𝑗≠𝑖𝑞_𝑖𝑗𝑓𝑠,𝑥,𝑗−𝑓(𝑠,𝑥,𝑖). (2.11) The corresponding Hamiltonian has the following form: ℋ𝑖,𝑠,𝑥,𝑢,D_𝑠𝑢,D_𝑥𝑢,D²_𝑥𝑢()−()=min𝑟𝑢𝑠,𝑥,𝑖𝜕𝑢𝑠,𝑥,𝑖−()()−((𝑥𝜕𝑠𝒜𝑢)(𝑠,𝑥,𝑖,𝑢𝑠,𝑥,𝑖𝑥−𝑎))=0. (2.12) Note that 𝑋(𝑡)>0 for all 𝑡. Let ℝ⁺=(0,∞). Formally, the value function 𝑣(𝑠,𝑥,𝑖) satisfies the HJB equation ℋ𝑖,𝑠,𝑥,𝑣,D_𝑠𝑣,D_𝑥𝑣,D²_𝑥𝑣=0,for(𝑠,𝑥,𝑖)∈[0,𝑇)×ℝ⁺𝑣((𝑇=((𝑥.×ℳ,𝑇,𝑥,𝛼))𝑥−𝑎)) (2.13) In order to study the possibility of existence and uniqueness of a solution of (2.12), we use a notion of viscosity solution introduced by Crandall et al. [10].

Definition 2.1. We say that 𝑓(𝑠,𝑥,𝑖) is a viscosity solution of ℋ𝑖,𝑠,𝑥,𝑓,𝜕𝑓,𝜕𝑠𝜕𝑓,𝜕𝜕𝑥²𝑓𝜕𝑥²=0,for𝑖∈ℳ,𝑠∈[0,𝑇),𝑥∈ℝ⁺,𝑓(𝑇,𝑥,𝛼(𝑇))=(𝑥−𝑎(𝑥)). (2.14) If(1)for all 𝑥∈ℝ⁺𝑓(𝑇,𝑥,𝛼(𝑇))=(𝑥−𝑎(𝑥)), and for each 𝑖∈ℳ, 𝑓(𝑠,𝑥,𝑖) is continuous in (𝑠,𝑥), moreover, there exist constants 𝐾 and 𝜅 such that 𝑓(𝑠,𝑥,𝑖)≤𝐾1+|𝑥|^𝜅, (2.15)(2)for each 𝑖∈ℳ, ℋ𝑖,𝑠₀,𝑥₀,𝑓,𝜕𝜙,𝜕𝑠𝜕𝜙,𝜕𝜕𝑥²𝜙𝜕𝑥²≤0 (2.16) whenever 𝜙(𝑠,𝑥)∈𝐶² such that 𝑓(𝑠,𝑥,𝑖)−𝜙(𝑠,𝑥) has local maximum at (𝑠,𝑥)=(𝑠₀,𝑥₀), (3)and for each 𝑖∈ℳ, ℋ𝑖,𝑠₀,𝑥₀,𝑓,𝜕𝜓,𝜕𝑠𝜕𝜓,𝜕𝜕𝑥²𝜓𝜕𝑥²≥0 (2.17) whenever 𝜓(𝑠,𝑥)∈𝐶² such that 𝑓(𝑠,𝑥,𝑖)−𝜓(𝑠,𝑥) has local minimum at (𝑠,𝑥)=(𝑠₀,𝑥₀).
Let 𝑓 be a function that satisfies (2.3). It is a viscosity subsolution (resp. supersolution) if it satisfies (2.4) (resp. (2.5)).

3. Properties of Value Functions

In this section, we study the continuity of the value function, show that it satisfies the associated HJB equation as a viscosity solution, and establish the uniqueness. We first show the continuity property.

Lemma 3.1. For each 𝑖∈ℳ, the value function 𝑣(𝑠,𝑥,𝑖) is continuous in (𝑠,𝑥). Moreover, it has at most linear growth rate, that is, there exists a constant 𝐶 such that |𝑣(𝑠,𝑥,𝑖)|≤𝐶(1+|𝑥|).

The continuity of the value function and its at most linear growth will be very helpful in deriving the maximum principle which itself guarantees the uniqueness of the value function. The following lemma is a simple version of the dynamic programming principle in optimal stopping. A similar result has been proven in Pemy [9]. For general dynamic programming principle, see Krylov [7] for diffusions, Pham [8] for jump diffusions, and Yin and Zhang [11] for dynamic models with regime switching.

Definition 3.2. For each 𝜀>0, a stopping time 𝜏_𝜀∈Λ_𝑠,𝑇 is said to be 𝜀-optimal if 0≤𝑣(𝑠,𝑥,𝑖)−𝐸𝑒^{−𝑟(𝜏𝜀−𝑠)}𝑣𝜏_𝜀,𝑋𝜏_𝜀,𝛼𝜏_𝜀≤𝜀. (3.1)

Lemma 3.3. (1) Let 𝛽,𝛾∈Λ_𝑠,𝑇 two stopping time such that 𝑠≤𝛽≤𝛾 a.s., then one has 𝐸𝑒^{−𝑟(𝛽−𝑠)}𝑣𝛽,𝑋𝛽,𝛼𝛽≥𝐸𝑒^{−𝑟(𝛾−𝑠)}𝑣𝛾,𝑋𝛾,𝛼𝛾. (3.2)
In particular for any stopping time 𝜃∈Λ_𝑠,𝑇, one has 𝑣(𝑠,𝑥,𝑖)≥𝐸^{𝑠,𝑥,𝑖}𝑒^{−𝑟(𝜃−𝑠)}𝑣(𝜃,𝑋(𝜃),𝛼(𝜃)). (3.3)
(2)Let 𝜃∈Λ_𝑠,𝑇 such that 𝑠≤𝜃≤𝜏_𝜖 for any 𝜀>0, where 𝜏_𝜖 an 𝜀-optimal stopping time. Then, one has 𝑣(𝑠,𝑥,𝑖)=𝐸^{𝑠,𝑥,𝑖}𝑒^{−𝑟(𝜃−𝑠)}𝑣(𝜃,𝑋(𝜃),𝛼(𝜃)). (3.4)
With Lemma 3.3 at our hand, we proceed and show that the value function 𝑣(𝑠,𝑥,𝑖) is a viscosity solution of the variational inequality (2.13).

Theorem 3.4. The value function 𝑣(𝑠,𝑥,𝑖) is a viscosity solution of (2.13).

3.1. Uniqueness of the Viscosity Solution

In this subsection, we will prove that the value function defined in (2.6) is the unique viscosity solution of the HJB equation (2.13). We begin by recalling the definition of parabolic superjet and subjet.

Definition 3.5. Let 𝑓(𝑠,𝑥,𝑖)∶[0,𝑇]×ℝ×ℳ→ℝ. Define the parabolic superjet by 𝒫^2,+𝑓()=()()+1𝑠,𝑥,𝑖𝑝,𝑞,𝑀∈ℝ×ℝ∶𝑓𝑡,𝑦,𝑖≤𝑓𝑠,𝑥,𝑖+𝑝𝑡−𝑠+𝑞𝑦−𝑥2𝑦−𝑥²𝑀+𝑜𝑦−𝑥²as𝑡,𝑦⟶(𝑠,𝑥), (3.5) and its closure is 𝒫^2,+𝑓(𝑠,𝑥,𝑖)=𝑝,𝑞,𝑀=lim_𝑛→∞𝑝_𝑛,𝑞_𝑛,𝑀_𝑛with𝑝_𝑛,𝑞_𝑛,𝑀_𝑛∈𝒫^2,+𝑓𝑠_𝑛,𝑥_𝑛,𝑖andlim_𝑛→∞𝑠_𝑛,𝑥_𝑛,𝑓𝑠_𝑛,𝑥_𝑛,𝑖=𝑥,𝑓(𝑠,𝑥,𝑖). (3.6) Similarly, we define the parabolic subjet 𝒫^2,−𝑓(𝑠,𝑥,𝑖)=−𝒫^2,+(−𝑓)(𝑠,𝑥,𝑖) and its closure 𝒫^2,−𝑓(𝑠,𝑥,𝑖)=−𝒫^2,+(−𝑓)(𝑠,𝑥,𝑖)

We have the following result.

Lemma 3.6. 𝒫^2,+𝑓(𝑠,𝑥,𝑖) (resp. 𝒫^2,−𝑓(𝑠,𝑥,𝑖)) consist of the set of (𝜕𝜙(𝑠,𝑥)/𝜕s,𝜕𝜙(𝑠,𝑥)/𝜕𝑥,𝜕²𝜙(𝑠,𝑥)/𝜕𝑥²) where 𝜙∈𝒞²([0,𝑇]×ℝ) and 𝑓−𝜙 has a global maximum (resp. minimum) at (𝑠,𝑥).

A proof can be found in Fleming and Soner [12].

The following result from Crandall et al. [10] is crucial for the proof of the uniqueness.

Theorem 3.7 (Crandall et al. [10]). For 𝑖=1,2, let Ω_𝑖 be locally compact subsets of ℝ, and Ω=Ω₁×Ω₂, and let 𝑢_𝑖 be upper semicontinuous in [0,𝑇]×Ω_𝑖, and 𝒫_Ω^2,+_𝑖𝑢_𝑖(𝑡,𝑥) the parabolic superjet of 𝑢_𝑖(𝑡,𝑥), and 𝜙 twice continuously differentiable in a neighborhood of [0,𝑇]×Ω.
Set 𝑤𝑡,𝑥₁,𝑥₂=𝑢₁𝑡,𝑥₁+𝑢₂𝑡,𝑥₂ (3.7) for (𝑡,𝑥₁,𝑥₂)∈[0,𝑇]×Ω, and suppose (𝑡,̂𝑥₁,𝑥₂)∈[0,𝑇]×Ω is a local maximum of 𝑤−𝜙 relative to [0,𝑇]×Ω. Moreover, let us assume that there is an 𝑟>0 such that for every 𝑀>0 there exists a 𝐶 such that for 𝑖=1,2𝑏_𝑖≤𝐶whenever𝑏_𝑖,𝑞_𝑖,𝑋_𝑖∈𝒫_Ω^2,+_𝑖𝑢_𝑖𝑡,𝑥_𝑖,𝑥_𝑖−𝑥_𝑖+𝑡−𝑡̂≤𝑟,𝑢_𝑖𝑡,𝑥_𝑖+𝑞_𝑖+𝑋_𝑖≤𝑀. (3.8) Then, for each 𝜖>0, there exists 𝑋_𝑖∈𝒮(1)=ℝ such that
(1)𝑏_𝑖,𝐷𝑥_𝑖𝜙𝑡,̂𝑥,𝑋_𝑖∈𝒫_Ω^2,+_𝑖𝑢_𝑖𝑡,̂𝑥_𝑖for𝑖=1,2, (3.9)(2)−1𝜖+𝐷²𝜙(𝑥𝑋)𝐼≤¹00𝑋₂≤𝐷²𝜙(𝑥)+𝜖𝐷²𝜙(𝑥)², (3.10)(3)𝑏₁+𝑏₂=𝜕𝜙𝑡,̂𝑥,𝑦.𝜕𝑡 (3.11)

We have the following maximum principle.

Theorem 3.8 (Comparison Principle). If 𝑣₁(𝑡,𝑥,𝑖) and 𝑣₂(𝑡,𝑥,𝑖) are continuous in (𝑡,𝑥) and are, respectively, viscosity subsolution and supersolution of (2.13) with at most a linear growth. Then, 𝑣₁(𝑡,𝑥,𝑖)≤𝑣₂(𝑡,𝑥,𝑖)∀(𝑡,𝑥,𝑖)∈[0,𝑇]×ℝ⁺×ℳ. (3.12)

The following lemma is be very useful in derivation of the maximum principle.

Lemma 3.9. Let 𝐶_𝐿𝑖𝑝([0,𝑇]×ℝ×ℳ) be the set of functions 𝑣(𝑠,𝑥,𝑖) on [0,𝑇]×ℝ×ℳ which are continuous with respect to 𝑠 and Lipschitz continuous with respect to the variable 𝑥. For fixed 𝑡_𝛿∈[0,𝑇] and 𝛼₀∈ℳ, let the operator ℱ on ℝ×𝐶_𝐿𝑖𝑝([0,𝑇]×ℝ×ℳ)×ℝ×ℝ be defined as follows: ℱ1𝑥,𝑣,𝛽,𝑋=−2𝑥²𝜎²𝛼₀𝑋−𝑥𝜇𝛼₀𝛽−𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝛾𝛼₀𝑥𝑧𝟏_{|𝑧|<1}(𝑧)𝛽𝜈(𝑑𝑧). (3.13) Then, there exits a constant 𝐶>0 such that ℱ𝑥𝑦,𝑤,𝑎𝑥−𝑦−𝑏𝑦,𝑌−ℱ𝑥,𝑣,𝑎𝑥−𝑦+𝑏𝑥,𝑋≤𝐶²𝑋−𝑦²𝑌+𝐶𝑎𝑥−𝑦²+𝐶𝑏1+𝑥²+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀+_ℝ𝑛𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑧𝑥,𝛼₀−𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑧𝑦,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀+𝑤𝑡_𝛿,𝑦,𝛼₀𝜈(𝑑𝑧), (3.14) for any v,𝑤∈𝐶_𝐿𝑖𝑝([0,𝑇]×ℝ×ℳ) and 𝑥,𝑦,𝑎,𝑏,𝑋,𝑌∈ℝ.

Remark 3.10. Theorem 3.8 obviously implies the uniqueness of the viscosity solution of the variational inequality (2.13). If we assume that (2.13) has two solutions 𝑣₁ and 𝑣₂ with linear growth, then they are both viscosity subsolution and supersolution of (2.13). Therefore, using the fact that 𝑣₁ is subsolution and 𝑣₂ is supersolution, Theorem 3.8 implies that 𝑣₁(𝑡,𝑥,𝑖)≤𝑣₂(𝑡,𝑥,𝑖) for all (𝑡,𝑥,𝑖)∈[0,𝑇]×ℝ⁺×ℳ. And conversely, we also have 𝑣₂(𝑡,𝑥,𝑖)≤𝑣₁(𝑡,𝑥,𝑖) for all (𝑡,𝑥,𝑖)∈[0,𝑇]×ℝ⁺×ℳ, since 𝑣₂ is subsolution and 𝑣₁ is supersolution. Consequently, we have 𝑣₁(𝑡,𝑥,𝑖)=𝑣₂(𝑡,𝑥,𝑖) for all (𝑡,𝑥,𝑖)∈[0,𝑇]×ℝ⁺×ℳ, which confirms the fact the value function defined on (2.6) is the unique solution of the variational inequality (2.13).

4. Numerical Example

This example is for a stock which share price roughly around $55 in average; in this example we assume that the market has two main movements: an uptrend and a downtrend. Thus the Markov chain 𝛼 takes two states ℳ={1,2}, where 𝛼(𝑡)=1 denotes the uptrend and 𝛼(𝑡)=2 denotes the downtrend. The transaction fee 𝑎=0.5, the discount rate 𝑟=0.05, the return vector is 𝜇=(0.01,−0.01), the volatility vector is 𝜎=(0.4,0.2), the intensity vector is 𝜆=(0.25,0.5), the time 𝑇=0.35 (in year), and the generator of the Markov chain is𝑄=−0.0230.0231.023−1.023. (4.1)

Figures 1 and 2 represent the value function 𝑣(𝑡,𝑥,𝑖) computed by solving the nonlinear system of equations (2.13) and the free boundary curves in both trends. These curves divide the plane in two regions. The region below the free boundary curve is the continuation region, and the region above the free boundary curve is the stopping region. In other terms, whenever the share price at any given time within the interval [0,𝑇] is below the free boundary curve, it is optimal for the investor to hold the stock. And whenever the share price is above the free boundary, it is optimal for the investor to sell. This selling rule is obviously easy to implement. This example clearly shows that the selling rule derived by our method can be very attractive to practitioners in an automated trading setting.

(a)

(b)

Figure 1 

Value functions 𝑣(𝑡,𝑥,1) for the uptrend and 𝑣(𝑡,𝑥,2) for the downtrend.

(a)

(b)

Figure 2 

Free boundary curves.

Appendix

A. Proofs of Results

A.1. Proof of Lemma 3.3

We have 𝑣𝛽,𝑋𝛽,𝛼𝛽=sup_{𝜏∈Λ𝛽,𝑇}𝐸𝑒^{−𝑟(𝜏−𝛽)}𝑔(𝛼(𝜏)(𝜏,𝐸𝑒,𝑋))_{−𝑟(𝛽−𝑠)}𝑣𝛽,𝑋𝛽,𝛼𝛽≥sup_{𝜏∈Λ𝛽,𝑇}𝐸𝑒^{−𝑟(𝛽−𝑠)}𝑒^{−𝑟(𝜏−𝛽)}𝑔(𝛼(𝜏),𝑋(𝜏))≥sup_{𝜏∈Λ𝛽,𝑇}𝐸𝑒^{−𝑟(𝜏−𝑠)}𝑔(𝛼(𝜏),𝑋(𝜏))≥sup_{𝜏∈Λ𝛾,𝑇}𝐸𝑒^{−𝑟(𝛾−𝑠)}𝑒^{−𝑟(𝜏−𝛾)}𝑔(𝛼(𝜏)(𝜏𝑒,𝑋))=𝐸^{−𝑟(𝛾−𝑠)}𝑣𝛾,𝑋𝛾,𝛼𝛾. (A.1)

This proves (3.2).

Now let us prove (3.4). Since 𝜃≤𝜏_𝜖 for any 𝜀>0, using (3.2), we have 𝐸𝑒^{−𝑟(𝜃−𝑠)}𝑣(𝜃,𝑋(𝜃),𝛼(𝜃))≥𝐸𝑒^{−𝑟(𝜏𝜖−𝑠)}𝑣𝜏_𝜖,𝑋𝜏_𝜖,𝛼𝜏_𝜖. (A.2) So using (3.1) and (3.3), we obtain 0≤𝑣(𝑠,𝑥,𝑖)−𝐸𝑒^{−𝑟(𝜃−𝑠)}𝑣(𝜃,𝑋(𝜃),𝛼(𝜃))≤𝑣(𝑠,𝑥,𝑖)−𝐸𝑒^{−𝑟(𝜏𝜖−𝑠)}𝑣𝜏_𝜖,𝑋𝜏_𝜖,𝛼𝜏_𝜖≤𝜀, (A.3) for any 𝜀>0. Thus, letting 𝜀→0, we obtain (3.4).

A.2. Proof of Lemma 3.1

Given 𝑥₁ and 𝑥₂, let 𝑋₁ and 𝑋₂ be two solutions of (2.4) with 𝑋₁(𝑠)=𝑥₁ and 𝑋₂(𝑠)=𝑥₂, respectively. For each 𝑡∈[𝑠,𝑇], we have 𝑋₁(𝑡)−𝑋₂(𝑡)=𝑥₁−𝑥₂+^𝑡_𝑠𝑋₁(𝜉)−𝑋₂(𝜉)𝜇(𝛼(𝜉))𝑑𝜉+^𝑡_𝑠𝑋₁(𝜉)−𝑋₂(𝜉)𝜎(𝛼(𝜉(𝜉)+))𝑑𝑊_𝑡𝑠ℝ𝑋₁(𝜉)−𝑋₂(𝜉)𝛾(𝛼(𝜉))𝑧𝑁(𝑑𝜉,𝑑𝑧). (A.4) Using the Itô-Lévy isometry, we have 𝐸𝑋₁(𝑡)−𝑋₂(𝑡)²≤𝐶₀𝑥₁−𝑥₂²+𝐶₁^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉+𝐶₂^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉+𝐶₃^𝑡_𝑠ℝ𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑧²𝜈(𝑑𝑧)𝑑𝜉≤𝐶₀𝑥₁−𝑥₂²+max𝐶₁,𝐶₂^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉+𝐶₃^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉_ℝ𝑧²𝜈(𝑑𝑧). (A.5) Taking into account (2.5), we can find 𝐾<∞ such that 𝐶₃∫_ℝ𝑧²𝜈(𝑑𝑧)<𝐾. The inequality (A.5) becomes 𝐸𝑋₁(𝑡)−𝑋₂(𝑡)²≤𝐶₀𝑥₁−𝑥₂²+max𝐶₁,𝐶₂^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉+𝐾^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉. (A.6) Let 𝐶=max(𝐶₀,𝐶₁,𝐶₂,𝐾), then (A.7) becomes 𝐸𝑋₁(𝑡)−𝑋₂(𝑡)²≤𝐶𝑥₁−𝑥₂²+𝐶^𝑡_𝑠𝐸𝑋₁(𝜉)−𝑋₂(𝜉)²𝑑𝜉. (A.7) Applying Gronwall's inequality, we have𝐸𝑋₁(𝑡)−𝑋₂(𝑡)²≤𝐶𝑥₁−𝑥₂²𝑒^𝐶𝑡. (A.8) This implies, in view of Cauchy-Schwarz inequality, that 𝐸𝑋₁(𝑡)−𝑋₂(𝑡)≤𝐶𝑥₁−𝑥₂𝑒^𝐶𝑡. (A.9) Using this inequality, we have 𝑣𝑠,𝑥₁,𝑖−𝑣𝑠,𝑥₂,𝑖≤sup_{𝜏∈Λ𝑠,𝑇}𝐸𝑒^{−𝑟(𝜏−𝑠)}𝑋₁(𝜏)−𝑎−𝑋₂(𝜏)−𝑎≤sup_{𝜏∈Λ𝑠,𝑇}𝐸𝑋₁(𝜏)−𝑋₂(𝜏𝑥)≤𝐶₁−𝑥₂𝑒^𝐶𝑇. (A.10) This implies the (uniform) continuity of 𝑣(𝑠,𝑥,𝑖) with respect to 𝑥.

We next show the continuity of 𝑣(𝑠,𝑥,𝑖) with respect to 𝑠. Let 𝑋_𝑡 be the solution of (2.4) that starts at 𝑡=𝑠 with 𝑋(𝑠)=𝑥 and 𝛼(𝑠)=𝑖. Let 0≤𝑠≤𝑠′≤𝑇, we define 𝑋^′(𝑡),𝛼=𝑋𝑡−𝑠′−𝑠_′(𝑡)=𝛼𝑡−𝑠′−𝑠. (A.11) It is easy to show that𝐸𝑋(𝑡)−𝑋^′(𝑡)²≤𝐶𝑠′−𝑠forsomeconstant𝐶>0. (A.12)

Given 𝜏∈Λ_𝑠,𝑇, let 𝜏′=𝜏+(𝑠′−𝑠). Then, 𝜏′≥𝑠′ and 𝑃(𝜏′>𝑇)→0 as 𝑠′−𝑠→0.

Let 𝑔(𝑡,𝑥)=𝑒^−𝑟𝑡(𝑥−𝑎(𝑥)). Then, 𝑣(𝑠,𝑥,𝑖)=𝑒^𝑟𝑠sup_{𝜏∈Γ𝑠,𝑇}𝐸𝑔(𝜏,𝑋(𝜏)). It is easy to show that 𝑔(𝑠,𝑥)−𝑔𝑠′,𝑥′≤𝑥−𝑥′+𝐶𝑥′−𝑎𝑥′𝑠−𝑠′, (A.13) for some constant 𝐶.

We define𝐽(𝑠,𝑥,𝑖,𝜏)=𝑒^𝑟𝑠𝐸𝑔(𝜏,𝑋(𝜏)). (A.14) We have𝐽(𝑠,𝑥,𝑖,𝜏)=𝑒^𝑟𝑠𝐸𝑔𝜏′−𝑠′−𝑠,𝑋^′𝜏′=𝑒^𝑟𝑠′𝐸𝑔𝜏′,𝑋^′𝜏′+𝑜(1)=𝑒^𝑟𝑠′𝐸𝑔𝜏′,𝑋^′𝜏′𝐼_{{𝜏′≤𝑇}}+𝑒^𝑟𝑠′𝐸𝑔𝜏′,𝑋^′𝜏′𝐼_{{𝜏′>𝑇}}(1)(1),+𝑜=𝐽𝑠′,𝑥,𝑖,𝜏′∧𝑇+𝑜 (A.15) where 𝑜(1)→0 as 𝑠′−𝑠→0. It follows that𝑣𝑠′,𝑥,𝑖−𝑣(𝑠,𝑥,𝑖)≤sup_{𝜏∈Λ𝑠,𝑇}𝐽𝑠′,𝑥,𝑖,𝜏′−𝐽(𝑠,𝑥,𝑖,𝜏)⟶0. (A.16)

Therefore, we have lim_{𝑠′−𝑠→0}𝑣𝑠′,𝑥,𝑖−𝑣(𝑠,𝑥,𝑖)=0. (A.17) This gives the continuity of 𝑣 with respect to 𝑠.

The joint continuity of 𝑣 follows from (A.10) and (A.17). Finally, the linear growth inequality follows from (A.10) and |𝑣(𝑠,𝑥,𝑖)|≤|𝑥|+|𝑣(𝑠,0,𝑖)|≤𝐶(1+|𝑥|). (A.18) This completes the proof.

A.3. Proof of Theorem 3.4

First we prove that 𝑣(𝑠,𝑥,𝑖) is a viscosity supersolution of (2.13). Given (𝑠,𝑥_𝑠)∈[0,𝑇]×ℝ⁺, let 𝜓∈𝐶²([0,𝑇]×ℝ⁺) such that 𝑣(𝑡,𝑥,𝛼)−𝜓(𝑡,𝑥) has local minimum at (𝑠,𝑥_𝑠) in a neighborhood 𝑁(𝑠,𝑥_𝑠). We define a function 𝜑()=𝜓()𝑡,𝑥,𝑖𝑡,𝑥+𝑣𝑠,𝑥_𝑠,𝛼_𝑠−𝜓𝑠,𝑥_𝑠if𝑖=𝛼_𝑠,𝑣(𝑡,𝑥,𝑖)if𝑖≠𝛼_𝑠. (A.19) Let 𝛾≥𝑠 be the first jump time of 𝛼(⋅) from the initial state 𝛼_𝑠, and let 𝜃∈[𝑠,𝛾] be such that (𝑡,𝑋(𝑡)) starts at (𝑠,𝑥_𝑠) and stays in 𝑁(𝑠,𝑥_𝑠) for 𝑠≤𝑡≤𝜃. Moreover, 𝛼(𝑡)=𝛼_𝑠, for 𝑠≤𝑡≤𝜃. Using Dynkin's formula, we have 𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝜑𝜃,𝑋(𝜃),𝛼_𝑠−𝜑𝑠,𝑥_𝑠,𝛼_𝑠=𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}−𝑟𝜑𝑡,𝑋(𝑡),𝛼_𝑠+𝜕𝜑𝑡,𝑋(𝑡),𝛼_𝑠+1𝜕𝑡2𝑋²_𝑡𝜎²𝛼_𝑠𝜕²𝜑𝑡,𝑋(𝑡),𝛼_𝑠𝜕𝑥²+𝑋_𝑡𝜇𝛼_𝑠𝜕𝜑𝑡,𝑋(𝑡),𝛼_𝑠((𝑡))𝛼𝜕𝑥+𝑄𝜑𝑡,𝑋,⋅_𝑠+_ℝ𝜑𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧−𝜑(𝑡,𝑋(𝑡))−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜑(𝜕𝑥×𝜈𝑑𝑧)𝑑𝑡. (A.20) Recall that (𝑠,𝑥_𝑠) is the minimum of 𝑣(𝑡,𝑥,𝛼_𝑠)−𝜓(𝑡,𝑥) in 𝑁(𝑠,𝑥_𝑠). For 𝑠≤𝑡≤𝜃, we have𝑣𝑡,𝑋_𝑡,𝛼_𝑠≥𝜓𝑡,𝑋_𝑡+𝑣𝑠,𝑥_𝑠,𝛼_𝑠−𝜓𝑠,𝑥_𝑠=𝜑𝑡,𝑋_𝑡,𝛼_𝑠. (A.21) Using (A.19) and (A.21), we have𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣𝜃,𝑋_𝜃,𝛼_𝑠−𝑣𝑠,𝑥_𝑠,𝛼_𝑠≥𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}−𝑟𝑣𝑡,𝑋(𝑡),𝛼_𝑠+𝜕𝜓(𝑡,𝑋(𝑡))+1𝜕𝑡2𝑋²_𝑡𝜎²𝛼_𝑠𝜕²𝜓𝑡,𝑋_𝑡𝜕𝑥²+𝑋_𝑡𝜇𝛼_𝑠𝜕𝜓𝑡,𝑋_𝑡𝜕𝑥+𝑄𝜑𝑡,𝑋_𝑡,⋅𝛼_𝑠×_ℝ𝜑𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−𝜑𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜓(𝑡,𝑋(𝑡))𝜈()𝜕𝑥𝑑𝑧𝑑𝑡. (A.22) Moreover, we have𝑄𝜑𝑡,𝑋_𝑡,⋅𝛼_𝑠=_{𝛽≠𝛼𝑠}𝑞_𝛼𝑠𝛽𝜑𝑡,𝑋_𝑡,𝛽−𝜑𝑡,𝑋_𝑡,𝛼_𝑠≥_{𝛽≠𝛼𝑠}𝑞_𝛼𝑠𝛽𝑣𝑡,𝑋_𝑡,𝛽−𝑣𝑡,𝑋_𝑡,𝛼_𝑠≥𝑄𝑣𝑡,𝑋_𝑡,⋅𝛼_𝑠. (A.23) Combining (A.22) and (A.23), we have 𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣𝜃,𝑋_𝜃,𝛼_𝑠−𝑣𝑠,𝑥_𝑠,𝛼_𝑠≥𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}−𝑟𝑣𝑡,𝑋(𝑡),𝛼_𝑠+𝜕𝜓(𝑡,𝑋(𝑡))+1𝜕𝑡2𝑋²_𝑡𝜎²𝛼_𝑠𝜕²𝜓𝑡,𝑋_𝑡𝜕𝑥²+𝑋_𝑡𝜇𝛼_𝑠𝜕𝜓𝑡,𝑋_𝑡𝜕𝑥+𝑄𝑣𝑡,𝑋_𝑡,⋅𝛼_𝑠+_ℝ𝜑𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−𝜑𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜓(𝑡,𝑋(𝑡))𝜈(𝜕𝑥𝑑𝑧)𝑑𝑡. (A.24) It follows from Lemma 3.3 that𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}−𝑟𝑣𝑡,𝑋(𝑡),𝛼_𝑠+𝜕𝜓(𝑡,𝑋(𝑡))+1𝜕𝑡2𝑋²(𝑡)𝜎²𝛼_𝑠𝜕²𝜓(𝑡,𝑋(𝑡))𝜕𝑥²+𝑋(𝑡)𝜇𝛼_𝑠𝜕𝜓(𝑡,𝑋(𝑡))𝜕𝑥+𝑄𝑣𝑡,𝑋_𝑡,⋅𝛼_𝑠+_ℝ𝜑𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−𝜑𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜓(𝑡,𝑋(𝑡))𝜈()𝜕𝑥𝑑𝑧𝑑𝑡≤0. (A.25) Dividing both sides by 𝐸𝜃>0 and sending 𝜃→𝑠 lead to −𝑟𝑣𝑠,𝑥_𝑠,𝛼_𝑠+𝜕𝜓𝑠,𝑥_𝑠+1𝜕𝑡2𝑥²_𝑠𝜎²𝛼_𝑠𝜕²𝜓𝑠,𝑥_𝑠𝜕𝑥²+𝑥_𝑠𝜇𝛼_𝑠𝜕𝜓𝑠,𝑥_𝑠𝜕𝑥+𝑄𝑣𝑠,𝑥_𝑠,⋅𝛼_𝑠+_ℝ𝑣𝑠,𝑥_𝑠+𝛾(𝑖)𝑥_𝑠𝑧,𝛼_𝑠−𝑣𝑠,𝑥_𝑠,𝛼_𝑠−𝛾(𝑖)𝑥_𝑠𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜓𝑠,𝑥_𝑠𝜈()𝜕𝑥𝑑𝑧≤0. (A.26)

By definition, 𝑣(𝑠,𝑥,𝑖)≥𝑥−𝑎(𝑥). The supersolution inequality follows from this inequality and (A.27).

Now, let us prove the subsolution inequality, namely, that let 𝜙∈𝒞^1,2([𝑠,𝑇]×ℝ⁺) and 𝑣(𝑡,𝑥,𝛼_𝑠)−𝜙(𝑡,𝑥) has a local maximum at (𝑠,𝑥_𝑠)∈[𝑠,𝑇]×ℝ⁺, then we can assume without loss of generality that 𝑣(𝑠,𝑥_𝑠,𝛼_𝑠)−𝜙(𝑠,𝑥_𝑠)=0.

Define Φ()=𝜙()𝑡,𝑥,𝑖𝑡,𝑥if𝑖=𝛼_𝑠,𝑣(𝑡,𝑥,𝑖)if𝑖≠𝛼_𝑠. (A.27) Let 𝛾 be the first jump time of 𝛼(⋅) from the state 𝛼_𝑠, and let 𝜃₀∈[𝑠,𝛾] be such that (𝑡,𝑋(𝑡)) starts at (𝑠,𝑥_𝑠) and stays in 𝑁(𝑠,𝑥_𝑠) for 𝑠≤𝑡≤𝜃₀. Since 𝜃₀≤𝛾 we have 𝛼(𝑡)=𝛼_𝑠, for 𝑠≤𝑡≤𝜃₀, and let 𝜏_𝐷 be the optimal stopping time, and for 𝑠≤𝜃≤min(𝜏_𝐷,𝜃₀), we have from Lemma 3.3 (The appendix) 𝑣𝑠,𝑥_𝑠,𝛼_𝑠≤𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣(𝜃,𝑋(𝜃),𝛼(𝜃)). (A.28) Moreover, since 𝑣(𝑠,𝑥_𝑠,𝛼_𝑠)−𝜙(𝑠,𝑥_𝑠)=0 and attains its maximum at (𝑠,𝑥_𝑠) in 𝑁(𝑠,𝑥_𝑠), then𝑣(𝜃,𝑋(𝜃),𝛼(𝜃))≤𝜙(𝜃,𝑋(𝜃)). (A.29) Thus, we also have 𝑣(𝜃,𝑋(𝜃),𝛼(𝜃))≤Φ(𝜃,𝑋(𝜃),𝛼(𝜃)). (A.30) This implies, using Dynkin's formula, that 𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣𝜃,𝑋(𝜃),𝛼_𝑠≤𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}Φ𝜃,𝑋(𝜃),𝛼_𝑠=Φ𝑠,𝑥_𝑠,𝛼_𝑠+𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}𝜕𝜙(𝑡,𝑋(𝑡))((𝑡)(𝑡(𝑡)𝜇𝛼𝜕𝑡−𝑟Φ𝑡,𝑋,𝛼))+𝑋_𝑠𝜕𝜙(𝑡,𝑋(𝑡))((𝑡))𝛼𝜕𝑥+𝑄Φ𝑡,𝑋,⋅_𝑠+12𝑋(𝑡)²𝜎²𝛼_𝑠𝜕²𝜙(𝑡,𝑋(𝑡))𝜕𝑥²+_ℝΦ𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−Φ𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜙(𝑡,𝑋(𝑡))𝜈(𝜕𝑥𝑑𝑧)𝑑𝑡. (A.31) Note that 𝑄Φ(𝑡,𝑋(𝑡),⋅)𝛼_𝑠=_{𝛽≠𝛼𝑠}𝑞_𝛼𝑠𝛽𝑣𝑡,𝑋(𝑡),𝛽−𝜙(𝑡,𝑋(𝑡))≤_{𝛽≠𝛼𝑠}𝑞_𝛼𝑠𝛽𝑣𝑡,𝑋(𝑡),𝛽−𝑣𝑡,𝑋(𝑡),𝛼_𝑠≤𝑄𝑣(𝑡,𝑋(𝑡),⋅)𝛼_𝑠. (A.32) Using (A.27) and (A.32), we obtain 𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣𝜃,𝑋(𝜃),𝛼_𝑠≤𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟𝜃−𝑠}Φ𝜃,𝑋(𝜃),𝛼_𝑠=𝜙𝑠,𝑥_𝑠+𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^{−𝑟(𝑡−𝑠)}𝜕𝜙(𝑡,𝑋(𝑡))(𝑡)𝜇𝛼𝜕𝑡+𝑋_𝑠𝜕𝜙(𝑡,𝑋(𝑡))(𝑡)𝜕𝑥−𝑟𝑣𝑡,𝑋,𝛼_𝑠+12𝑋(𝑡)²𝜎²𝛼_𝑠𝜕²𝜙(𝑡,𝑋(𝑡))𝜕𝑥²+𝑄𝑣(𝑡,𝑋(𝑡),⋅)𝛼_𝑠+_ℝΦ𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−Φ𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜙(𝑡,𝑋(𝑡))𝜈(𝜕𝑥𝑑𝑧)𝑑𝑡. (A.33) Recall that, 𝑣(𝑠,𝑥_𝑠,𝛼_𝑠)=𝜙(𝑠,𝑥_𝑠) by assumption. From (A.28), we deduce 0≤𝐸^{𝑠,𝑥𝑠,𝛼𝑠}𝑒^{−𝑟(𝜃−𝑠)}𝑣𝜃,𝑋(𝜃),𝛼_𝑠−𝜙𝑠,𝑥_𝑠≤𝐸^{𝑠,𝑥𝑠,𝛼𝑠𝜃}_𝑠𝑒^−𝑟𝑡−𝑟𝑣𝑡,𝑋(𝑡),𝛼_𝑠+𝜕𝜙(𝑡,𝑋(𝑡))+1𝜕𝑡2𝑋(𝑡)²𝜎²𝛼_𝑠𝜕²𝜙𝑋(𝑡),𝛼_𝑠𝜕𝑥²+𝑋(𝑡)𝜇𝛼_𝑠𝜕𝜙𝑋(𝑡),𝛼_𝑠((𝑡))𝛼𝜕𝑥+𝑄𝑣𝑡,𝑋,⋅_𝑠+_ℝΦ𝑡,𝑋(𝑡)+𝛾(𝑖)𝑋(𝑡)𝑧,𝛼_𝑠−Φ𝑡,𝑋(𝑡),𝛼_𝑠−𝛾(𝑖)𝑋(𝑡)𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜙(𝑡,𝑋(𝑡))𝜈(𝜕𝑥𝑑𝑧)𝑑𝑡. (A.34) Dividing the last inequality by 𝐸𝜃>0 and sending 𝜃↓𝑠 give 𝑟𝑣𝑠,𝑥_𝑠,𝛼_𝑠−𝜕𝜙𝑠,𝑥_𝑠−1𝜕𝑡2𝑥²_𝑠𝜎²𝛼_𝑠𝜕²𝜙𝑥_𝑠,𝛼_𝑠𝜕𝑥²−𝑥_𝑠𝜇𝛼_𝑠𝜕𝜙𝑥_𝑠,𝛼_𝑠𝜕𝑥−𝑄𝑣𝑠,𝑥_𝑠,⋅𝛼_𝑠−_ℝ𝑣𝑠,𝑥_𝑠+𝛾(𝑖)𝑥_𝑠𝑧,𝛼_𝑠−𝑣𝑠,𝑥_𝑠,𝛼_𝑠−𝛾(𝑖)𝑥_𝑠𝑧𝟏_{|𝑧|<1}(𝑧)𝜕𝜙𝑠,𝑥_𝑠𝜈()𝜕𝑥𝑑𝑧≤0. (A.35) This gives the subsolution inequality. Therefore, 𝑣(𝑡,𝑥,𝛼) is a viscosity solution of (2.13).

A.4. Proof of Lemma 3.9

Let 𝑣,𝑤∈𝐶_Lip([0,𝑇]×ℝ×ℳ) and 𝑥,𝑦,𝜆,𝑎,𝑏,𝑋,𝑌∈ℝ, then we have ℱ1𝑦,𝑤,𝑎𝑥−𝑦−𝑏𝑦,𝑌−ℱ𝑥,𝑣,𝑎𝑥−𝑦+𝑏𝑥,𝑋=−2𝑦²𝜎²𝛼₀𝑌−𝑦𝜇𝛼₀𝑎𝑥−𝑦−𝑏𝑦−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀−_ℝ𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑦𝑧,𝛼₀−𝑤𝑡_𝛿,𝑦,𝛼₀−𝛾𝛼₀𝑦𝑧𝟏_{|𝑧|<1}(𝑧𝑎𝜈()+1)𝑥−𝑦−𝑏𝑦𝑑𝑧2𝑥²𝜎²𝛼₀𝑋+𝑥𝜇𝛼₀𝑎𝑥−𝑦+𝑏𝑥+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀+_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝛾𝛼₀𝑥𝑧𝟏_{|𝑧|<1}(𝑧𝑎𝜈()=1)𝑥−𝑦+𝑏𝑥𝑑𝑧2𝜎²𝛼₀𝑥²𝑋−𝑦²𝑌+𝜇𝛼₀𝑎𝑥−𝑦²+𝑏𝑥²+𝑦²+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀+_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑦𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝑤𝑡_𝛿,𝑦,𝛼₀−𝑧𝛾𝛼₀𝟏_{|𝑧|<1}(𝑧)𝑎𝑥−𝑦²+𝑏𝑥²+𝑦²𝜈()𝛼𝑑𝑧≤𝜇₀𝑎𝑥−𝑦²+𝜇𝛼₀𝑏𝑥²+𝑦²+12𝜎²𝛼₀𝑥²𝑋−𝑦²𝑌+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀+_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑦𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝑤𝑡_𝛿,𝑦,𝛼₀+𝑧𝛾𝛼₀𝟏_{|𝑧|<1}(𝑧)𝑎𝑥−𝑦²|+|𝑏𝑥²+𝑦²𝜈(𝑑𝑧). (A.36) Note that from the Lévy-Khintchine inequality (2.3), one can prove ∫_{|𝑧|<1}|𝑧|𝜈(𝑑𝑧)<∞; therefore, there exists a constant 𝐶>0 such that ℱ𝑎𝑦,𝑤,𝑎𝑥−𝑦−𝑏𝑦,𝑌−ℱ𝑥,𝑣,𝑎𝑥−𝑦+𝑏𝑥,𝑋≤𝐶𝑥−𝑦²+𝑏𝑥²+𝑦²+𝑥²𝑋−𝑦²𝑌+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀+_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑦𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝑤𝑡_𝛿,𝑦,𝛼₀𝜈(𝑑𝑧). (A.37) This proves (3.14).

Now, let us proof the theorem.

A.5. Proof of Theorem 3.8

For any 0<𝛿<1 and 0<𝜂<1, we define Φ𝑡,𝑥,𝑦,𝑖=𝑣₁(𝑡,𝑥,𝑖)−𝑣₂−1𝑡,𝑦,𝑖𝛿𝑥−𝑦²−𝜂𝑒^{(𝑇−𝑡)}𝑥²+𝑦²,𝜙=1𝑡,𝑥,𝑦𝛿𝑥−𝑦²+𝜂𝑒^{(𝑇−𝑡)}𝑥²+𝑦². (A.38) Note that 𝑣₁(𝑡,𝑥,𝑖) and 𝑣₂(𝑡,𝑥,𝑖) satisfy the linear growth. Then, we have for each 𝑖∈ℳlim_{|𝑥|+𝑦→∞}Φ𝑡,𝑥,𝑦,𝑖=−∞ (A.39) and since Φ is a continuous in (𝑡,𝑥,𝑦), therefore it has a global maximum at a point (𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀). Observe thatΦ𝑡_𝛿,𝑥_𝛿,𝑥_𝛿,𝛼₀+Φ𝑡_𝛿,𝑦_𝛿,𝑦_𝛿,𝛼₀≤2Φ𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀. (A.40) It implies 𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑥_𝛿,𝛼₀−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑣₁𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑦²_𝛿≤2𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−2𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−2𝛿𝑥_𝛿−𝑦_𝛿²−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿. (A.41) Then, −𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−2𝑒^{(𝑇−𝑡𝛿)}𝜂𝑥²_𝛿+𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑦²_𝛿≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−2𝛿𝑥_𝛿−𝑦_𝛿²−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿. (A.42) Consequently, we have 2𝛿𝑥_𝛿−𝑦_𝛿²≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₁𝑡_𝛿,𝑦_𝛿,𝛼₀+𝑣₂𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.43) By the linear growth condition, we know that there exist 𝐾₁,𝐾₂ such that 𝑣₁(𝑡,𝑥,𝑖)≤𝐾₁(1+|𝑥|) and 𝑣₂(𝑡,𝑥,𝑖)≤𝐾₂(1+|𝑥|). Therefore, there exists 𝐶 such that we have 2𝛿𝑥_𝛿−𝑦_𝛿²≤𝐶1+𝑥_𝛿+𝑦_𝛿. (A.44) So, 𝑥⁰_𝛿1−𝑥⁰_𝛿2²≤𝛿𝐶1+𝑥⁰_𝛿1^𝜅1+𝑥⁰_𝛿1^𝜅2. (A.45) We also have Φ(𝑠,0,0,𝛼₀)≤Φ(𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀) and |Φ(𝑠,0,0,𝛼₀)|≤𝐾(1+|𝑥_𝛿|+|𝑦_𝛿|). This leads to 𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−1𝛿𝑥_𝛿−𝑦_𝛿²−Φ𝑠,0,0,𝛼₀≤3𝐶1+𝑥_𝛿+𝑦_𝛿. (A.46) It comes that 𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿1+𝑥_𝛿+𝑦_𝛿≤3𝐶, (A.47) therefore there exists 𝐶_𝜂 such that 𝑥_𝛿+𝑦_𝛿≤𝐶_𝜂,𝑡_𝛿∈[𝑠,𝑇]. (A.48) The inequality (A.48) implies the sets {𝑥_𝛿,𝛿>0} and {𝑦_𝛿,𝛿>0} are bounded by 𝐶_𝜂 independent of 𝛿, so we can extract convergent subsequences that we also denote (𝑥_𝛿)_𝛿, (𝑦_𝛿)_𝛿, and (𝑡_𝛿)_𝛿. Moreover, from the inequality (A.45), it comes that the exists 𝑥₀ such that lim_𝛿→0𝑥_𝛿=𝑥₀=lim_𝛿→0𝑦_𝛿,lim_𝛿→0𝑡_𝛿=𝑡₀. (A.49) Using (A.43) and the previous limit, we obtain lim_𝛿→02𝛿𝑥_𝛿−𝑦_𝛿²=0. (A.50)Φ achieves its maximum at (𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀), so by the Theorem 3.7 for each 𝜖>0, there exist 𝑏_1𝛿, 𝑏_2𝛿, 𝑋_𝛿, and 𝑌_𝛿 such that 𝑏_1𝛿,2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿,𝑋_𝛿∈𝒫^2,+𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀,−𝑏_2𝛿2,−𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑦_𝛿,−𝑌_𝛿∈𝒫^2,+−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.51)(A.52) But, we know that 𝒫^2,+−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀=−𝒫^2,−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.53) Therefore, we obtain 𝑏_2𝛿,2𝛿𝑥_𝛿−𝑦_𝛿−2𝜂𝑒^{(𝑇−𝑡)}𝑦_𝛿,𝑌_𝛿∈𝒫^2,−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.54) Equation (A.51) implies by the definition of the viscosity solution that min𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑏_1𝛿−12𝑥²_𝛿𝜎²𝛼₀𝑋_𝛿−𝑥_𝛿𝜇𝛼₀2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿−𝑄𝑣₁𝑡_𝛿,𝑥_𝛿,⋅𝛼₀−_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧,𝛼₀−𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝛾𝛼₀𝑥_𝛿𝑧𝟏_{|𝑧|<1}(𝑧)×2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿𝜈(𝑑𝑧),𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑥_𝛿−𝑎≤0. (A.55) Consequently, we have two cases; either𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑥_𝛿−𝑎≤0 (A.56) or 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑏_1𝛿−12𝑥²_𝛿𝜎²𝛼₀𝑋_𝛿−𝑥_𝛿𝜇𝛼₀2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿−𝑄𝑓𝑠,𝑥_𝛿,⋅𝛼₀−_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧−𝑣₁𝑡_𝛿,𝑥_𝛿−𝛾𝛼₀𝑥_𝛿𝑧𝟏_{|𝑧|<1}(𝑧)×2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿𝜈(𝑑𝑧)≤0. (A.57) First of all, we assume that 𝑣₁(𝑡_𝛿,𝑥_𝛿,𝛼₀)−(𝑥_𝛿−𝑎)≤0. And similarly, (A.54) implies by the definition of the viscosity solution that min𝑟𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑏_2𝛿−12𝑦²_𝛿𝜎²𝛼₀𝑌_𝛿−𝑦_𝛿𝜇𝛼₀2𝛿𝑥_𝛿−𝑦_𝛿−2𝜂𝑒^{(𝑇−𝑡)}𝑦_𝛿−𝑄𝑣₂𝑡_𝛿,𝑦_𝛿,⋅𝛼₀−_ℝ𝑣₂𝑡_𝛿,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝛾𝛼₀𝑦_𝛿𝑧𝟏_{|𝑧|<1}(𝑧)×2𝛿𝑥_𝛿−𝑦_𝛿−2𝜂𝑒^{(𝑇−𝑡)}𝑦_𝛿𝜈(𝑑𝑧),𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑦_𝛿−𝑎≥0. (A.58) Therefore, we have𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑦_𝛿−𝑎≥0. (A.59) It comes that𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑥_𝛿−𝑦_𝛿≤0. (A.60) Letting 𝛿→0, we obtain 𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀≤0. (A.61) Note that the function Φ reaches its maximum at (𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀). It follows that for all 𝑥∈ℝ, 𝑡∈[𝑠,𝑇], and 𝑖∈ℳ, we have 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)−2𝜂𝑒^{(𝑇−𝑡)}𝑥²=Φ(𝑥,𝑥,𝑖)≤Φ𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿. (A.62) Again letting 𝛿→0 and using (A.61), we obtain 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)−2𝜂𝑒^{(𝑇−𝑡)}𝑥²≤𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀−2𝜂𝑒^{(𝑇−𝑡0)}𝑥₀²≤0. (A.63) so, we have 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)≤2𝜂𝑒^{(𝑇−𝑡)}𝑥². (A.64) Second of all, let assume that 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑏_1𝛿−12𝑥²_𝛿𝜎²𝛼₀𝑋_𝛿−𝑥_𝛿𝜇𝛼₀2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿−𝑄𝑓𝑠,𝑥_𝛿,⋅𝛼₀−_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧−𝑣₁𝑡_𝛿,𝑥_𝛿−𝛾𝛼₀𝑥_𝛿𝑧𝟏_{|𝑧|<1}(𝑧)2𝛿𝑥_𝛿−𝑦_𝛿+2𝜂𝑒^{(𝑇−𝑡)}𝑥_𝛿×𝜈(𝑑𝑧)≤0, (A.65) and, moreover, we have 𝑟𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝑏_2𝛿−12𝑦²_𝛿𝜎²𝛼₀𝑌_𝛿−𝑦_𝛿𝜇𝛼₀2𝛿𝑥_𝛿−𝑦_𝛿−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑦_𝛿−𝑄𝑣₂𝑡_𝛿,𝑦_𝛿,⋅𝛼₀−_ℝ𝑣₂𝑡_𝛿,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−𝛾𝛼₀𝑦_𝛿𝑧𝟏_{|𝑧|<1}(𝑧)×2𝛿𝑥_𝛿−𝑦_𝛿−2𝜂𝑒^{(𝑇−𝑡)}𝑦_𝛿𝜈(𝑑𝑧)≥0. (A.66) Let us use the operator ℱ(𝑥,𝑢,𝛽,𝑀) defined on the Lemma 3.9; thus, ℱ1𝑥,𝑣,𝛽,𝑀=−2𝑥²𝜎²𝛼₀𝑀−𝑥𝜇𝛼₀𝛽−𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝛾𝛼₀𝑥𝑧𝛽𝜈(𝑑𝑧). (A.67) Using the operator ℱ, thus (A.65) becomes 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀+ℱ𝑥_𝛿,𝑣₁,2𝛿𝑥_𝛿−𝑦_𝛿+2𝑥_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑋_𝛿−𝑏_1𝛿≤0, (A.68) and (A.66) becomes 𝑟𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀+ℱ𝑦_𝛿,𝑣₂,2𝛿𝑥_𝛿−𝑦_𝛿−2𝑦_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑌_𝛿−𝑏_2𝛿≥0. (A.69) Combining the last two inequalities, we obtain 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀≤ℱ𝑦_𝛿,𝑣₂,2𝛿𝑥_𝛿−𝑦_𝛿−2𝑦_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑌_𝛿−ℱ𝑥_𝛿,𝑣₁,2𝛿𝑥_𝛿−𝑦_𝛿+2𝑥_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑋_𝛿+𝑏_2𝛿−𝑏_1𝛿. (A.70) From Lemma 3.9, there exists a constant 𝐶>0 such that ℱ𝑦_𝛿,𝑣₂,2𝛿𝑥_𝛿−𝑦_𝛿−2𝑦_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑌_𝛿−ℱ𝑥_𝛿,𝑣₁,2𝛿𝑥_𝛿−𝑦_𝛿+2𝑥_𝛿𝜂𝑒^{(𝑇−𝑡𝛿)},𝑋_𝛿2≤𝐶𝛿𝑥_𝛿−𝑦_𝛿²+2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿+𝑥²_𝛿𝑋_𝛿−𝑦²_𝛿𝑌_𝛿+𝑄𝑣𝑡_𝛿,𝑥,⋅𝛼₀−𝑄𝑤𝑡_𝛿,𝑦,⋅𝛼₀+_ℝ𝑣𝑡_𝛿,𝑥+𝛾𝛼₀𝑥𝑧,𝛼₀−𝑤𝑡_𝛿,𝑦+𝛾𝛼₀𝑦𝑧,𝛼₀−𝑣𝑡_𝛿,𝑥,𝛼₀−𝑤𝑡_𝛿,𝑦,𝛼₀𝜈(𝑑𝑧). (A.71) Recall that 𝑄𝑣(𝑡,𝑥,⋅)(𝑖)=∑_𝑗≠𝑖𝑣(𝑡,𝑥,𝑗)−𝑣(𝑡,𝑥,𝑖), so we have 𝑄𝑣₁𝑡_𝛿,𝑥_𝛿,⋅𝛼₀−𝑄𝑣₂𝑡_𝛿,𝑦_𝛿,⋅𝛼₀=_{𝑖≠𝛼𝛿}𝑣₁𝑡_𝛿,𝑥_𝛿,𝑖−𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝑖−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.72) Since Φ attains its maximum at (𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀), we have Φ(𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝑖)≤Φ(𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀); therefore 𝑣₁𝑡_𝛿,𝑥_𝛿,𝑖−𝑣₂𝑡_𝛿,𝑦_𝛿−1,𝑖𝛿𝑥_𝛿−𝑦_𝛿²+𝜂𝑒^{𝑇−𝑡𝛿}𝑥_𝛿²+𝑦_𝛿²≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−1𝛿𝑥_𝛿−𝑦_𝛿²−𝜂𝑒^{𝑇−𝑡𝛿}𝑥_𝛿²+𝑦_𝛿². (A.73) Thus, we have 𝑣₁𝑡_𝛿,𝑥_𝛿,𝑖−𝑣₂𝑡_𝛿,𝑦_𝛿,𝑖≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀. (A.74) Consequently, from (A.74) it comes that (A.72) implies 𝑄𝑣₁𝑡_𝛿,𝑥_𝛿,⋅𝛼₀−𝑄𝑣₂𝑡_𝛿,𝑦_𝛿,⋅𝛼₀≤0. (A.75) Note that from (3.11), we have𝑏_1𝛿−𝑏_2𝛿=𝜕𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿𝜕𝑡=𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥_𝛿²+𝑦_𝛿². (A.76) Therefore, we have 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀2≤𝐶𝛿𝑥_𝛿−𝑦_𝛿²+2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿+𝑥²_𝛿𝑋_𝛿−𝑦²_𝛿𝑌_𝛿+_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀−𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀𝜈(𝑑𝑧). (A.77) Similarly, for any 𝑧∈ℝ, we haveΦ𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀≤Φ𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀; (A.78) consequently, we derive that _ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀𝜈()≤𝑑𝑧_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼_𝛿−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼_𝛿+1𝛿𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧−𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧²+𝜂𝑒^{𝑇−𝑡𝛿}𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧²−𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧²𝜈()≤𝑑𝑧_ℝ𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀+1𝛿𝑥_𝛿−𝑦_𝛿²+𝛾𝛼₀𝑥_𝛿𝑧−𝛾𝛼₀𝑦_𝛿𝑧²+𝜂𝑒^{𝑇−𝑡𝛿}𝑥_𝛿²+𝛾(𝛼₀)𝑥_𝛿𝑧²+𝑦_𝛿²+𝛾(𝛼₀)𝑦_𝛿𝑧²𝜈(𝑑𝑧). (A.79) Therefore, using (2.3), we obtain _ℝ𝑣₁𝑡_𝛿,𝑥_𝛿+𝛾𝛼₀𝑥_𝛿𝑧,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿+𝛾𝛼₀𝑦_𝛿𝑧,𝛼₀−𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀𝜈()≤𝑑𝑧_ℝ1𝛿𝑥_𝛿−𝑦_𝛿²+𝛾𝛼₀𝑥_𝛿𝑧−𝛾𝛼₀𝑦_𝛿𝑧²+𝜂𝑒^{𝑇−𝑡𝛿}𝑥_𝛿²+𝛾𝛼₀𝑥_𝛿𝑧²+𝑦_𝛿²+𝛾𝛼₀𝑦_𝛿𝑧²𝜈()1𝑑𝑧≤𝐶𝛿𝑥_𝛿−𝑦_𝛿²+𝜂𝑒^{𝑇−𝑡𝛿}1+𝑥_𝛿²+𝑦_𝛿², (A.80) for some constant 𝐶>0. Taking into account (A.75) and (A.80), thus (A.77) becomes 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀2≤𝐶𝛿𝑥_𝛿−𝑦_𝛿²+2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿+𝑥²_𝛿𝑋_𝛿−𝑦²_𝛿𝑌_𝛿. (A.81) We know from the Maximum principle that −1𝜖+𝐷²_𝑥,𝑦𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿𝑋𝐼≤^𝛿00−𝑌_𝛿≤𝐷²_𝑥,𝑦𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿+𝜖𝐷²_𝑥,𝑦𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿². (A.82) Moreover,𝐷²_𝑥,𝑦𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿=2𝛿1−1−11+2𝜂𝑒^{(𝑇−𝑡𝛿)},𝐷1001_2𝑥,𝑦𝜙𝑡_𝛿,𝑥_𝛿,𝑦_𝛿²=8𝛿²+1−1−118𝜂𝑒^{(𝑇−𝑡𝛿)}𝛿1−1−11+4𝜂²𝑒^{2(𝑇−𝑡𝛿)}=10018+8𝜂𝛿𝑒^{(𝑇−𝑡𝛿)}𝛿²1−1−11+4𝜂²𝑒^{2(𝑇−𝑡𝛿)}.1001 (A.83) Note that 𝑥_𝛿²𝑋_𝛿−𝑦_𝛿²𝑌_𝛿=𝑥_𝛿,𝑦_𝛿𝑋_𝛿00−𝑌_𝛿𝑥_𝛿𝑦_𝛿≤𝑥_𝛿,𝑦_𝛿2𝛿+1−1−112𝜂𝑒^{(𝑇−𝑡𝛿)}+4𝜖𝜂²𝑒^{2(𝑇−𝑡𝛿)}1001+𝜖8+8𝜂𝛿𝑒^{(𝑇−𝑡𝛿)}𝛿²1−1𝑥−11𝑦^𝛿_𝛿. (A.84) Letting 𝜂→0, we obtain 𝑥_𝛿²𝑋_𝛿−𝑦_𝛿²𝑌_𝛿≤𝑥_𝛿,𝑦_𝛿2𝛿8+𝜖𝛿²1−1𝑥−11𝑦^𝛿_𝛿. (A.85) Take 𝜖=𝛿/4, this leads to 𝑥_𝛿²𝑋_𝛿−𝑦_𝛿²𝑌_𝛿≤𝑥_𝛿,𝑦_𝛿4𝛿1−1𝑥−11𝑦^𝛿_𝛿=4𝛿𝑥_𝛿−𝑦_𝛿². (A.86) Using (A.50), we obtain limsup_𝛿↓0𝑥_𝛿²𝑋_𝛿−𝑦_𝛿²𝑌_𝛿≤limsup_𝛿↓0𝑥_𝛿,𝑦_𝛿4𝛿1−1𝑥−11𝑦^𝛿_𝛿=limsup_𝛿↓04𝛿𝑥_𝛿−𝑦_𝛿²=0. (A.87) Letting 𝜂→0 in (A.81), we have 𝑟𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀2≤𝐶𝛿𝑥_𝛿−𝑦_𝛿²+𝑥²_𝛿𝑋_𝛿−𝑦²_𝛿𝑌_𝛿 (A.88) and taking the limsup as 𝛿 goes to zero and using (A.87), we obtain 𝑟𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀≤0. (A.89) Recall that (𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀) is maximum of Φ. Then, for all 𝑥∈ℝ, 𝑡∈[𝑠,𝑇], and for all 𝑖∈ℳ, we have Φ(𝑡,𝑥,𝑥,𝑖)≤Φ𝑡_𝛿,𝑥_𝛿,𝑦_𝛿,𝛼₀. (A.90) It comes that 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)−2𝜂𝑒^{(𝑇−𝑡)}𝑥²≤𝑣₁𝑡_𝛿,𝑥_𝛿,𝛼₀−𝑣₂𝑡_𝛿,𝑦_𝛿,𝛼₀−2𝜂𝑒^{(𝑇−𝑡𝛿)}𝑥²_𝛿+𝑦²_𝛿. (A.91) Letting 𝛿→0, we obtain 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)−2𝜂𝑒^{(𝑇−𝑡)}𝑥²≤𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀−2𝜂𝑒^{(𝑇−𝑡)}𝑥²₀. (A.92)

Using (A.89), we have𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀≤0. (A.93) Therefore, using (A.92), we conclude that 𝑣₁(𝑡,𝑥,𝑖)−𝑣₂(𝑡,𝑥,𝑖)−2𝜂𝑒^{(𝑇−𝑡)}𝑥²≤𝑣₁𝑡₀,𝑥₀,𝛼₀−𝑣₂𝑡₀,𝑥₀,𝛼₀−2𝜂𝑒^{(𝑇−𝑡0)}𝑥²₀≤0. (A.94) Letting 𝜂→0 in (A.64) and the previous inequality, we have𝑣₁(𝑡,𝑥,𝑖)≤𝑣₂(𝑡,𝑥,𝑖). (A.95) This completes the proof of the theorem.

Acknowledgment

The author is grateful to the editor and the referee for their helpful suggestions and comments.