# A Stochastic Differential Equation Driven by Poisson Random Measure and Its Application in a Duopoly Market

Authors: Tong Wang; Hao Liang
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

We investigate a stochastic differential equation driven by Poisson random measure and its application in a duopoly market for a finite number of consumers with two unknown preferences. The scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. The quantity of buyers is proved to obey a stochastic differential equation (SDE) driven by Poisson random measure which exists a unique solution. We derive the Hamilton-Jacobi-Bellman (HJB) about vendors’ profits and provide a verification theorem about the problem. When all consumers believe a vendor’s guidance about their preferences, the conditions that the other vendor’s profit is zero are obtained. We give an example of this problem and acquire approximate solutions about the profits of the two vendors.


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## Body

## 1. Introduction

Consider two vendors that provide different goods for different types of consumers in a duopoly market. For instance, in the pharmaceutical market, two vendors provide different drugs for different patients with different diseases. As the commodity price is proportional to the number of consumers, pricing strategies are especially important for vendors. We study this problem in a nondurable goods duopoly market.Consumers’ preferences or types take key roles in the market and affect the pricing strategies determined by vendors. Eeckhout and Weng [1] assume that there are N≥2 consumers who have the same type either H or L. Two vendors provide two different kinds of goods for the two types, respectively. In this article, we assume that consumers’ types are diverse. This assumption is different from the assumption that all consumers’ types are identical in [1]. In general, there exist different types of consumers who need to buy the same kind of goods in the market, just like people with high and low fever who need antipyretics simultaneously. Thus, it is reasonable to assume that all consumers’ preferences are different in the market. In this situation, some consumers choose one vendor and others choose the other vendor, i.e., all consumers do not choose the same vendor at the first time.Furthermore, we assume that the price of goods is a function of the number of consumers instead of the consumers’ posterior beliefs in [1] as the price of goods is affected by the quantity of supply and demand. Since technological content of nondurable goods is lower than that of other goods in general, prices of nondurable goods fluctuate more obviously with the number of consumers. Therefore, the assumption that price of goods depends on the number of consumers is logical.Vendors need to know how many consumers choose their own goods at each timet∈0,+∞ in order to implement pricing strategies. Whether a consumer changes his decision is related to the judgment of his type and the prices of two goods. We find ranges of commodity prices and prove that the quantity of buyers who purchase one of the vendor’s goods obeys a SDE and verifies existence and uniqueness of its solutions, which is a main contribution in this article.Generally speaking, consumers do not know their own types whereas the vendors recognize. Under the setting of asymmetric information, consumers who are informed of their own types tend to buy commodity, which brings benefits for vendors. Consumers need to make choices of the types based on the guidance given by the two vendors. In certain cases, consumers are willing to follow the authoritative vendor’s guidance. We call this vendor as the type-leader vendor and the other as the following vendor, which is a new setting evolved from the Stackelberg leadership model [2]. The type-leader model contains only one type-leader vendor. In the type-leader model, the vendor directly leads to consumers’ preferences rather than the prices of goods. In some markets, such as the pharmaceutical market, consumers are more concerned about the efficacy of goods than its price. This makes it important for vendors to guide consumers’ types. Thus, the type-leader model is more reasonable than the Stackelberg leadership model. Pricing rules for goods are obtained in the type-leader model. Moreover, we derive the conditions to ensure that the following vendor’s profit is zero.The main contributions of this paper are mentioned as follows. Compared with the assumption in [1] where there exists only one type of consumers, the situation that there exist two different types of consumers in the market is explored. Considering the impact of commodity prices, we assume that the price of goods is decided by the number of consumers. As a comparison, Eeckhout and Weng [1] define that the price is a function of consumers’ posterior beliefs. Observing that the effectiveness of goods is more important than its price in our model, we use the type-leader model instead of the Stackelberg leadership model to study vendors’ optimal strategies. An example of this problem is given and approximate solutions about the profit of the two vendors are acquired.Several literatures investigate multiarmed bandit problems. Robbins [3] describes the problem as a decision-maker facing M slot machines (called arms), and the participator has to choose one of the arms at each instantaneous time. The value of pulling an arm in discrete time is calculated by Gittins and Jones [4] and Michael et al. [5]. Comparing the value to the Gittins index of all other arms, Michael et al. [5] present that the value pulling each arm itself does not depend on the cutoff. This problem is transformed into a standard optimal stopping problem in [6, 7]. Bolton and Harris [8] and Bergemann and Valimaki [9] show that choosing the products from the same vendor is the optimal strategy of consumers when there are K≥2 vendors who offer different products and M consumers whose preferences are the same (but unknown) in the market. The necessary and sufficient conditions for the existence of only two vendors in the market are obtained by Gao et al. [10]. Two-armed bandit problems in the continuous time with the property of Le´vy processes are studied by Cohen and Solan [11] (Lévy process is described in [12, 13]). Cohen and Solan [11] conclude that the optimal strategy is a cutoff strategy when the arms have two types. The problem that multiple arms can be chosen by the decision-maker is studied by Kuksov et al. [14] and Doval [15]. It is discovered that the decision-maker is indifferent to search an alternative arm which does not have the highest reservation price. When a Bayesian decision-maker makes a selection from multiple arms with uncertain payoffs and an outside arm with known payoff, maximizing his expected profit is studied in Ke et al. [16]. For other optimal strategies and control approaches, the reader is referred to [17, 18] and the references therein.The remainder of the paper is organized as follows. In Section2, we introduce a two-period example and show the scopes of prices. In Section 3, the definition of the Poisson integral is used to prove that the quantity of buyers obeys an SDE and verifies the existence and uniqueness of the solutions for the SDE in global space. In Section 4, based on the dynamic programming principle, we derive HJB equations for the vendors’ utility functions and give the verification theorem for the type-leader model. Using solutions of HJB equations, we obtain the optimal strategy for the type-leader vendor. In Section 5, we give an example of this problem and acquire approximate solutions about the profit of the two vendors.


## 2. A Two Period Example

There are two vendors who offer different nondurable goods, indexed byj=1,2, and M consumers whose preferences are high or low in the market, where M is positive integer. However, consumers do not know their preferences. If the type is high, a consumer gets expected value ζ1H from buying goods 1 and ζ2H from buying goods 2. Otherwise, a consumer gets expected value ζ1L from buying goods 1 and ζ2L from buying goods 2. We assume that ζ1H>ζ2H and ζ1L<ζ2L, where ζjH and ζjL belong to 0,+∞.At any time, all market participants observe all previous outcomes. Because of the influence caused by uncertain external factors, the flow utilityujiti∈H,L has a noisy signal of the true value (for detailed discussion, refer to [1]).(1)dujit=ζjidt+σjdB˜jt,where B˜1t and B˜2t are independent Brownian motions. In the market, besides the types of consumers, there are many uncertain factors affecting the effectiveness of products to consumers. For example, there exist some subtle and unavoidable differences in the quality of the goods and these differences affect consumers’ utilities. The noisy signal of the true value is used to characterize the effects of these factors on consumers’ utilities.Assume thatxt∈0,1 is the belief that the type is high if consumers choose the first vendor and yt∈0,1 is the belief that the type is high if consumers choose the other vendor; that is to say, xt≔Pri=Hu1it and yt≔Pri=Hu2it, where ujit≔ujiττ=0t is a realized path. From [19], we have(2)dxt=xt1−xts1du1it−ζ1Hdt−ζ1Ldtσ1≔xt1−xts1dB1t,dyt=yt1−yts2du2it−ζ2Hdt−ζ2Ldtσ2≔yt1−yts2dB2t,where sj=ζjH−ζjL/σj, j=1,2, B1t and B2t are independent Brownian motions.If a consumer chooses the first vendor, his expected utility isf1xt≔xtζ1H+1−xtζ1L. Letting a1≔ζ1H−ζ1L and b1≔ζ1L, the utility is represented by(3)f1x=a1xt+b1.Similarly, if a consumer chooses the second vendor, his expected utility is represented by(4)f2x=a2yt+b2,where a2≔ζ2H−ζ2L and b2≔ζ2L. The definitions of aj and bj imply a1+b1>a2+b2, b1<b2 and a1>a2.Letnt be the number of consumers who choose the first vendor and M−nt be the number of consumers who choose the second vendor, nt∈0,1,…,M. P1nt is the price of goods from the first vendor and P2nt is the price of goods from the second one. Formally, the price of goods is a measurable function Pj:0,1,…,M⟶ℝ+∪0. We assume that the change of price has hysteresis(the product price of the second vendor is a function with respect to nt when M fixed).As consumers’ beliefs change, the lower the beliefs they have, the lower the profits they earn if they choose the first vendor. For a consumer who chooses the first vendor, if his belief is low enough at certain time, he gives up the vendor to choose the second one. Denoteαt∈0,1 as the belief, i.e., if xt<αt, a consumer transforms his choice from the first to the second vendor. Similarly, the consumer gives up buying goods from the second vendor and chooses to buy goods from the first vendor if yt>βt∈0,1.For nondurable goods, if a consumer chooses the first vendor at first time, the common belief for high type isxt at time t. The utility of the consumer is xtζ1H+1−xtζ1L−P1nt. If the consumer gives up the first vendor to choose second one, the prices for the vendors do not change due to the hysteresis of the change of price. The utility of the consumer is xtζ2H+1−xtζ2L−P2nt. If the consumer voluntarily gives up the first vendor to choose the second vendor, it has(5)xtζ1H+1−xtζ1L−P1nt<xtζ2H+1−xtζ2L−P2nt.Inequality (5) is rewritten as(6)xt<b2−b1+P1nt−P2nta1−a2.From the definition ofαt, it has αt=b2−b1+P1nt−P2nt/a1−a2. In the same way, we obtain βt=b2−b1+P1nt−P2nt/a1−a2. After the above discussion, αt and βt satisfy(7)αt=βt=b2−b1+P1nt−P2nta1−a2.Equation (7) shows that the common belief which makes a consumer change his choice from the first vendor to the second one is equal to that which makes the changes from the second vendor to the first one. It is called cutoff in [1]. The cutoff increases as P1nt increases and decreases as P2nt increases. The cutoff is linear with both P1nt and P2nt. In a market, as P1nt increases, the utility of the consumer who chooses the first vendor is smaller. The consumers who have the same common belief tend to choose the second vendor. Thus, the cutoff cuts down. If P2nt decreases, the relative price for the second vendor increases. Similarly, the cutoff reduces. In the following, we use αt to denote the cutoff. The ranges of pricing for two vendors obtained from equation (7) are shown in Proposition 1.Proposition 1.
Suppose thatP1⋅ and P2⋅ are the prices of goods from the first and second vendors, respectively. Then P1⋅ and P2⋅ satisfy(8)0≤P1nt≤a1+b1−a2+b2+P2nt,(9)0≤P2nt≤b2−b1+P1nt,where one of the second signs of inequalities (8) or (9) is sign of strict inequality.Proof.
P
1
n
t and P2nt are greater than zero as the vendors’ costs of goods are zero. In the following, we prove P1nt≤a1+b1−a2+b2+P1nt and P2nt≤b2−b1+P2nt.
Ifαt>1, i.e., xt<αt, all consumers give up the first vendor to choose the second one. For the first vendor, the price of his goods satisfies αt≤1. It has(10)b2−b1+P1nt−P2nta1−a2≤1,i.e.,(11)P1nt≤a1+b1−a2+b2+P2nt.
In the same way, the second vendor makes the price of goods satisfyαt≥0. We have(12)P2nt≤b2−b1+P2nt.
Inequalities (8) and (9) are proved.
From inequalities (8) and (9), we obtain(13)P1nt≤a1+b1−a2+b2+P2nt≤a1+b1−a2+b2+b2−b1+P1nt=a1−a2+P1nt.
From inequality (13), we obtain a2≤a1 which contradicts the assumption a2<a1. Thus, one of the second signs of inequalities in (8) and (9) is sign of strict inequality.
Proposition1 shows the scopes of commodity prices which are decided by two monopolistic vendors. We find that the supremum of the price for a vendor increases as the price for the other vendor increases. The supremum of the price for a vendor and the price for the other vendor are linear dependence. In the duopoly market, an increase in the price of goods means a decrease of benefit to the number of consumers. Due to the substitution effect between goods, the benefit to someone who chooses the other vendor increases. The supremum of the price for the other vendor aggrandizes.Corollary 1.
P
1
n
t
<
+
∞ if and only if P2nt<+∞ while P1nt=∞ if and only if P2nt=∞.Corollary1 is easily proved from Proposition 1 and reveals the commodity prices under periods of economic prosperity and economic crisis. During periods of economic prosperity, the commodity prices are bounded while the prices are infinite on account of discontinued sale of goods during the economic crisis. In next discussion, we ignore the situation of economic crisis.


## 3. Duopoly Market

If there existstΘ0≤Θ≤d such that 0=t0<t1<⋯<td=t, a consumer chooses a vendor, denoted by j, t∈t2k,t2k+1 and chooses the other vendor, denoted by −j, t∈t2k+1,t2k+2, k=0,1,…,d/2−1. Let(14)pt≔Pri=Hujit0,t1,u−j,it1,t2,…,ujit2k,t2k+1,u−j,it2k+1,t2k+2,…,ujitd−2,td−1,u−j,itd−1,t,where ujiπ1,π2≔ujiττ=π1π2. Using the arguments in [8, 9], we know that pt satisfies(15)dpt=ntpt1−pts1dW1t+M−ntpt1−pts2dW2t,where W1t and W2t are independent Winner processes.For any timel and small ε>0, a consumer gives up the second vendor and chooses the first vendor when(16)l∈s>0ps=αs,αs−ε−ps−ε∈A2,αs+ε−ps+ε∈A1⊆s>0ps=αs,Δαs−ps∈A1⊆s>0Δαs−ps∈A1,where A1≔x≤0/0, A2≔x≥0/0 and ΔXs≔Xs−Xs− with Xs−≔limt⟶s−Xt. Similarly, a consumer replaces the goods of the first vendor with the goods of the second vendor if(17)l∈s>0ps=αs,αs−ε−ps−ε∈A1,αs+ε−ps+ε∈A2⊆s>0ps=αs,Δαs−ps∈A2⊆s>0Δαs−ps∈A2.We assume that the number of consumers who choose one vendor to replace the other one is related toΔαl−pl, pl and nl at time l. All of them reflect the relationship between αt and pt at t∈l−ε,l+ε. Let GjΔαl−pl,pl,nl∈0,1,…,M be the number of consumers who give up the other vendor to choose the vendor j, j=1,2. If nt is known, the number of consumers who choose the first vendor at t+dt is expressed by(18)nt+dt=nt+∑0≤u≤dtG1Δαu−pu,pu,nu♯0≤s≤ups=αs,αs−ε−ps−ε∈A2,αs+ε−ps+ε∈A1−∑0≤u≤dtG2Δαu−pu,pu,nu♯0≤s≤ups=αs,αs−ε−ps−ε∈A1,αs+ε−ps+ε∈A2,where ♯ represents the number of elements in a set. Let(19)ϖ1Δαu−pu,pu=♯0≤s≤ups=αs,αs−ε−ps−ε∈A2,αs+ε−ps+ε∈A1♯0≤s≤uΔαu−pu∈A1,ϖ2Δαu−pu,pu=♯0≤s≤ups=αs,αs−ε−ps−ε∈A1,αs+ε−ps+ε∈A2♯0≤s≤uΔαu−pu∈A2.If0≤s≤ups=αs,αs−ε−ps−ε∈A2,αs+ε−ps+ε∈A1≠ϕ, we require that γ1Δαu−pu,pu equals to ϖ1Δαu−pu,pu while γ1Δαu−pu,pu≔0 for another case. Similarly, if 0≤s≤ups=αs,αs−ε−ps−ε∈A1,αs+ε−ps+ε∈A2 is not empty, let γ2Δαu−pu,pu be equal to ϖ2Δαu−pu,pu and zero otherwise. Defining(20)g1Δαu−pu,pu,nu=G1Δαu−pu,pu,nuγ1Δαu−pu,pu,(21)g2Δαu−pu,pu,nu=G2Δαu−pu,pu,nuγ2Δαu−pu,pu,we have(22)nt+dt=nt+∑0≤u≤dtg1Δαu−pu,pu,nu♯0≤s≤uΔαu−pu∈A1−∑0≤u≤dtg2Δαu−pu,pu,nu♯0≤s≤uΔαu−pu∈A2,where gjΔαu−pu,pu,nu≤GjΔαu−pu,pu,nu due to γj⋅,⋅∈0,1, i.e., gjΔαu−pu,pu,nu is bounded. Proposition 2 can be obtained by using the definition of Poisson stochastic integral as dt⟶0 and s⟶u.Proposition 2.
Ifαt−pt is a stochastic process which has independent and stationary increments with ca`dla`g paths6 (paths are continuous on the right and have limits on the left [20, 21]), then(23)dnt=∫A1g1h,pt−,nt−Ndt,dh−∫A2g2h,pt−,nt−Ndt,dh,where g1h,pt−,nt− and g2h,pt−,nt− are defined in Equations (20) and (21). Moreover, SDE (23) has a unique solution.Proof.
Asg1⋅,⋅,⋅ and g2⋅,⋅,⋅ are bounded, A1 and A2 are bounded below, from [22], we obtain that nt is convergent in L2 for any measurable random variable n0. Besides, g1⋅,⋅,⋅ and g2⋅,⋅,⋅ can be verified to satisfy(i)
Lipschitz condition:tThere existsK1>0, for any y1, y2∈0,1,…,M and x∈0,1 such that(24)∫A1g1h,x,y1−g1h,x,y22νdh≤K1y1−y22,∫A2g2h,x,y1−g2h,x,y22νdh≤K1y1−y22.(ii)
Growth condition: there existsK2>0, for any y∈0,1,…,M and x∈0,1, such that(25)2y∫A1g1h,x,yνdh+∫A1g1h,x,y|2νdh≤K21+y2,2y∫A1g1h,x,yνdh+∫A2g2h,x,y|2νdh≤K21+y2.
Equation (23) is proved to exist a unique solution by using the results in [23, 24].
Proposition2 shows the existence and uniqueness of global solutions for the SDE (23) which describes the quantity of buyers who choose the first vendor in the duopoly market. It is presented as a pure jump process. gj⋅,⋅,⋅ is defined as the rate of change of the number of consumers who give up the other vendor and choose the jth one.


## 4. The Optimal Strategy for Vendors

As consumers do not know their types, they choose one of two vendors referred by the prices of goods at initial time. We assume that there aren consumers who choose the first vendor, i.e., n0=n. After that, vendors inform consumers’ types, denoted by p≔p0, to guide consumers to make subsequent choices. Ignoring the interaction between the two vendors, denoting Vj⋅ as the jth vendor’s optimal utility and r as risk-free interest rate, for the first vendor, we have(26)V1n=maxp∈0,1E∫0+∞e−rtntP1ntdts.tdnt=∫A1g1h,pt−,nt−Ndt,dh−∫A2g2h,pt−,nt−Ndt,dh.The HJB equation is obtained as follows:(27)rV1n=nP1n+maxp∈0,1∫A1V1n+g1h,p,n−V1nνdh−∫A2V1n+g2h,p,n−V1nνdh,where ν, the intensity measure of N, is the finite intensity measure as A1 and A2 are bounded below. Suppose that wn is a solution of equation (27), the verification result is obtained as follows.Proposition 3.
If there exists an integrable functionϕ⋅ such that w⋅≤ϕ⋅.(i)
Suppose that(28)rwn−nP1n−maxp∈0,1∫A1wn+g1h,p,n−wnνdh−∫A2wn+g2h,p,n−wnνdh≥0,(29)limsupT⟶+∞e−rTEwnT≥0.
Thenwn≥vn if n∈1,2,…,M.(ii).
Suppose that for alln∈1,2,…,M, there exists a p∗ such that(30)rwn−nP1n−∫A1wn+g1h,p∗,n−wnνdh−∫A2wn+g2h,p∗,n−wnνdh=0,the stochastic differential equation(31)dnt=∫A1g1h,pt−∗,nt−Ndt,dh−∫A2g2h,pt−∗,nt−Ndt,dh,admits a unique solution and(32)liminft⟶∞e−rTEwnT≤0,then(33)wn=vn,n∈1,2,…,M.Proof.
Three steps are divided to prove Proposition3.Step 1.
We proveE∫0+∞e−rtntP1ntdt<+∞. From the assumptions, it has(34)E∫0+∞e−rtntP1ntdt≤ME∫0+∞e−rtP1ntdt.
Hence, from∫0+∞e−rtdt<+∞ and P1nt which is a monotone bounded function, we derive(35)E∫0+∞e−rtP1ntdt<+∞.
Thus,E∫0+∞e−rtntP1ntdt<+∞ as M<+∞.Step 2.
The proof of (i) in Proposition3. Let(36)τk≔inft≥0∫A1e−rtwnt−+g1h,pt,nt−wnt−νdh≥kor∫A2e−rtwnt−+g1h,pt,nt−wnt−νdh≥k.
Using Ito^’s lemma for e−rT∧τkwnT∧τk, where T∧τk≔minT,τk, we obtain(37)e−rT∧τkwnT∧τk=wn−∫0T∧τkre−rswnsds+∫0T∧τk∫A1e−rswns−+g1h,ps,ns−wns−Nds,dh−∫0T∧τk∫A2e−rswns−+g2h,ps,ns−wns−Nds,dh.
Introducing compensation Poisson random measure, equation (37) is rewritten as(38)e−rT∧τkwnT∧τk=wn−∫0T∧τkre−rswnsds+∫0T∧τk∫A1e−rswns−+g1h,ps,ns−wns−N˜ds,dh−∫0T∧τk∫A2e−rswns−+g2h,ps,ns−wns−N˜ds,dh+∫0T∧τkds∫A1e−rswns−+g1h,ps,ns−wns−vdh−∫0T∧τk∫A2e−rswns−+g2h,ps,ns−wns−vdh,where the compensation Poisson random measure N˜s,dh≔Ns,dh−s⋅νdh is a martingale. Taking expectation for both sides of equation (38) yields(39)Ee−rT∧τkwnT∧τk=wn−E∫0T∧τkre−rswnsds+E∫0T∧τkds∫A1e−rswns−+g1h,ps,ns−wns−νdh−E∫0T∧τk∫A2e−rswns−+g2h,ps,ns−wns−νdh.
From inequality (28), we obtain(40)Ee−rT∧τkwnT∧τk≤wn−E∫0T∧τke−rsnsP1ns.
Lettingk⟶+∞, using the dominated convergence theorem, inequality (40) becomes(41)Ee−rTwnT≤wn−E∫0Te−rsnsP1nsds.
AsT⟶+∞, in accordance with inequality (29), for any p∈0,1, the inequality(42)wn≥E∫0Te−rsnsP1nsds,is obtained. Hence,(43)wn≥maxp∈0,1E∫0Te−rsnsP1nsds=Vn.
The (i) in Proposition3 is proved by Steps 1 and 2.Step 3.
The proof of (ii) in Proposition3. As p∗ is the optimal choice for the vendor, we obtain(44)Ee−rTwnT=wn−E∫0Te−rsnsP1nsds,where(45)dnt=∫A1g1h,pt−∗,nt−Ndt,dh−∫A2g2h,pt−∗,nt−Ndt,dh,has a unique solution. Letting T⟶+∞, from inequality (32), we have(46)wn≤E∫0Te−rsnsP1nsds=Vn.
From inequalities (43) and (46), we have wn=Vn. The proof is completed.
Proposition3 shows that the solution of HJB equation (27) is V1n.
Similarly, for the second vendor, it has(47)rV2n=M−nP2n+maxp∈0,1∫A1V2n+g1h,p,n−V2nνdh−∫A2V2n+g2h,p,n−V2nνdh.
We assume that one of the two vendors is a type-leader vendor. Without loss of generality, denote the first vendor as the type-leader vendor. In this situation, we acquire(48)κV1n=nP1n+∫A1V1n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,nνdh.
For the second vendor, his payoff function satisfies(49)κV2n=M−nP2n+∫A1V2n+g1h,p∗,nνdh−∫A2V2n+g2h,p∗,nνdh,where κ≔r−∫A1vdh−∫A1vdh<∞.
As the second vendor considers that the first vendor informs all consumers whose types areH and consumers tend to buy goods from the first vendor, the second vendor’s goods are not sold. From Corollary 2.1, if the vendor wants more consumers to buy his goods, he prices his goods P2nt, based on the price P1nt of the first vendor, such that P1nt=a1+b1−a2+b2+P2nt., i.e.,(50)P2nt=a2+b2−a1+b1+P1nt.
From equation (50), we find that P2nt<0 if P1nt<a1+b1−a2+b2. Thus, the price of the second vendor’s goods is zero. For the type-leader vendor, the price of their goods is(51)P1n=a1+b1−a2+b2−ε,where ε>0 is an arbitrary small number. Denoted by P1ε as a1+b1−a2+b2−ε, then P1ε<a1+b1−a2+b2+0 and 0=P2nt≤a1−a2−ε satisfy pricing ranges for both vendors in Proposition 1. Equations (48) and (49) are equivalent to(52)κV1n=nP1ε+∫A1V1n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,nνdh,(53)κV2n=∫A1V2n+g1h,p∗,nνdh−∫A2V2n+g2h,p∗,nνdh,respectively. In this situation, the payoffs of the leader and the following vendors are shown in Proposition 4.Proposition 4.
If there existP1ε, the rate of change gj⋅,⋅,⋅, a finite intensity measure ν and p∗∈0,1 such that the equation(54)κV1n=nP1ε+∫A1V1n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,nνdh,has solutions; then its solution is unique if and only if V2n≡0, i.e., the type-leader vendor makes the other vendor’s profit be zero with pricing strategy.Proof.
Firstly, we prove that if there existsP1ε, the rate of change gj⋅,⋅,⋅, a finite intensity measure ν and p∗∈0,1 such that the equation(55)κV1n=nP1ε+∫A1V1n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,nνdh,has a unique solution, then V2n≡0.
AsV1n and V2n are the maximum utilities of the first and second vendor, we have V1n≥0 and V2n≥0. Denote V⋅=V1⋅+V2⋅, where V⋅ represents the sum of two vendors’ profits. It is straightforward to verify Vn≥V1n and Vn≥V2n. From equations (52) and (53), we have(56)κVn=nP1ε+∫A1V1n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,nνdh+∫A1V2n+g1h,p∗,nνdh−∫A2V2n+g2h,p∗,nνdh=nP1ε+∫A1V1n+g1h,p∗,n+V2n+g1h,p∗,nνdh−∫A2V1n+g2h,p∗,n+V2n+g2h,p∗,nνdh.
From the definition ofV⋅, we derive that(57)κVn=nP1ε+∫A1Vn+g1h,p∗,nνdh−∫A2Vn+g2h,p∗,nνdh.
Comparing Eqs. (54) and (57) finds that V1⋅ and V⋅ have the same structure. If there exists a unique solution to equation (57), we have V⋅≡V1⋅. Therefore, V2⋅≡0. The necessity of Proposition 3 is proved.
Now, we prove the sufficiency. IfV2⋅≡0 and the solution of equation (54) exists, then it is unique. Assume that there exist two solutions v1 and v2, v1≡v2, i.e., there exists m∈0,1,…,M such that v1m≠v2m. Without loss of generality, we assume that v1m<v2m. From V⋅>V1⋅, v2m=Vm and v1m=V1m are obtained. Moreover, V2m=Vm−V1m≠0, contradictory with V2≡0. Therefore, if the solution of equation (54) exists, its solution is unique when V2⋅≡0. The sufficiency of Proposition 3 is proved.
Proposition4 shows the condition that the type-leader vendor obtains surplus of all producers. In this case, the following vendor is unprofitable and gradually withdraws from the market. Eventually, the market will be monopolized by the type-leader vendor.


## 5. Example

For the convenience of explanation, we add two assumptions.Assumption 1.
Letgjh,p,n≔ψjhϕjp,n where ψj⋅ and ϕj⋅,⋅ are bounded and ϕj⋅,⋅ is twice continuously differentiable for the first variable.
Letξj≔∫Ajνdh, ωj≔∫Ajψjhνdh and ηj≔∫Ajψj2hνdh. ξj, ωj and ηj are finite as A1 and A2 are bounded below. The first order condition for p from HJB equation (27) is(58)∫A1V1′n+ψ1hϕ1p,n∂ϕ1p,n∂pψ1hνdh−∫A2V2′n+ψ2hϕ2p,n∂ϕ2p,n∂pψ2hνdh=0.
As the exact analytic solution of HJB equation (27) is difficult to be obtained, we consider the approximate solution of equation (27). Let Vjn+ψ1hϕ1p,n be approximately equal to Vjn+Vj′nψjhϕjp,n+1/2Vj′′nψj2hϕj2p,n and Vj′n+ψ1hϕ1p,n be approximately equal to Vj′n+Vj′′nψjhϕjp,n. Substituting them into (58) yields(59)V˜1′nω1∂ϕ1p,n∂p−ω2∂ϕ2p,n∂p+V˜1″nη1ϕ1p,n∂ϕ1p,n∂p−η2ϕ2p,n∂ϕ2p,n∂p=0,where V˜j⋅ is an approximate solution of Vj⋅. For acquiring the type-leader vendor’s optimal strategy, Assumption 2 is given.Assumption 2.
There existωj, ηj, p∗, ϕjp,n to satisfy Assumption 1, for any p∈0,1, such that(60)V˜1′nω1∂2ϕ1p,n∂p2−ω2∂2ϕ2p,n∂p2+V˜1″nη1ϕ1p,n∂2ϕ1p,n∂p2+η1∂ϕ1p,n∂p2−η2ϕ2p,n∂ϕ2p,n∂p−η2∂ϕ2p,n∂p2<0∂ϕ1p,n∂pp=p∗=∂ϕ2p,n∂pp=p∗=0.
Assumption2 implies that p∗ is a unique optimal strategy for the type-leader vendor. Combining with pricing strategy, HJB equation (27) can be approximately written by(61)q1V˜1′′n+q2V˜1′n+q3V˜1n+nP1ε=0,where q1≔1/2η1ϕ12p∗,n−η2ϕ22p∗,n, q2≔ω1ϕ1p∗,n−ω2ϕ2p∗,n, and q3≔ξ1−ξ2−κ. Similarly,(62)q1V˜2′′n+q2V˜2′n+q3V˜2n=0.
In order to simplify the analytical solution of equations (61) and (62), we assume that q1 and q2 are independent to n. Denote CΥ as constants which do not depend on n Y=1,2,…,11. Four cases are divided to solve equations (61) and (62).Case 1.
Considerq1≠0 and q22−4q1q3>0. If q3≠0, Solving equations (61) and (62) yields(63)V˜1n=−nP1εq3+q2P1εq32+C1e−q2+q22−4q1q3/2q1n+C2e−q2+q22−4q1q3/2q1n,V˜2n=C1e−q2+q22−4q1q3/2q1n+C2e−q2+q22−4q1q3/2q1n.
Ifq3=0, the solutions of V˜1n and V˜2n are(64)V˜1n=C3e−q2/q1n+C4−nP1ϵq2V˜2n=C3e−q2/q1n+C4,respectively.Case 2.
Considerq1≠0 and q22−4q1q3=0. If q2≠0, solving equations (61) and (62), we have(65)V˜1n=−nP1εq3+q2P1εq32+C5+nC6e−q2/q1n,V˜2n=C5+nC6e−q2/q1n.
Ifq2=0, V˜1n and V˜2n are verified to satisfy(66)V˜1n=−n3P1ε6q1+C7n+C8,V˜2n=C7n+C8.Case 3.
Considerq1≠0 and q22−4q1q3<0. In this case, we solve equations (61) and (62) to obtain(67)V˜1n=−nP1εq3+q2P1εq32+e−q2/2q1nC9cos4q1q3−q222q1n+C10sin4q1q3−q222q1n,V˜2n=e−q2/2q1nC9cos4q1q3−q222q1n+C10sin4q1q3−q222q1n.Case 4.
Considerq1=0. If q2≠0, Solving equations (61) and (62) yields(68)V˜1n=C11e−q3/q2n−P1εq3n−q2q3,V˜2n=C11e−q3/q2n.
Ifq2=0, V˜1n and V˜2n satisfy(69)V˜1n=−nq3P1ε,(70)V˜2n≡0.
It is easy to verify that the solutions in the four cases satisfy the verification theorem in Proposition3. In particular, from equations (69) and (70), V˜1⋅ has a unique solution and V˜2n≡0 in this situation, which satisfies the conclusion in Proposition 4.


## 6. Conclusion

This paper explores a stochastic differential equation driven by Poisson random measure and its application in a duopoly market which exists two different types of consumers. We assume that prices of goods are decided by the number of consumers. To study vendors’ optimal pricing strategies, scopes of goods prices are obtained from the cutoff. In addition, we prove that the quantity of buyers obeys a SDE resorting to the definition of Poisson stochastic measure and maximizing the vendor payoff. We also verify that the SDE exists a unique solution. Given the SDE, the corresponding HJB equation reflecting the profits of vendor is derived by using the dynamic programming principle.In certain markets where the effectiveness of goods is more important than its price, we introduce the type-leader model. In the type-leader model, we find vendors’ price strategies and verify that the commodity prices are in the price ranges in Proposition1. The conditions that the type-leader vendor obtains surplus of all producers are acquired by existence and uniqueness of solutions of HJB equations. An example of this problem is given and approximate solutions for the profit of the two vendors are obtained.



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