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

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. -e scopes of pricing for two monopolistic vendors are illustrated when the prices of items are determined by the number of buyers in the market. -e 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.


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.
is 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. us, 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. erefore, 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 time t ∈ [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 typeleader vendor and the other as the following vendor, which is a new setting evolved from the Stackelberg leadership model [2]. e 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.
is makes it important for vendors to guide consumers' types.
us, 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. e 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.
e 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. is 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. e 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. e 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. e remainder of the paper is organized as follows. In Section 2, 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.

A Two Period Example
ere are two vendors who offer different nondurable goods, indexed by j � 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 ζ 1H from buying goods 1 and ζ 2H from buying goods 2. Otherwise, a consumer gets expected value ζ 1L from buying goods 1 and ζ 2L from buying goods 2. We assume that ζ 1H > ζ 2H and ζ 1L < ζ 2L , where ζ jH and ζ jL belong to (0, +∞).
At any time, all market participants observe all previous outcomes. Because of the influence caused by uncertain external factors, the flow utility u ji (t)(i ∈ H, L { }) has a noisy signal of the true value (for detailed discussion, refer to [1]).
where B 1 (t) and B 2 (t) 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. e noisy signal of the true value is used to characterize the effects of these factors on consumers' utilities.
Assume that x t ∈ [0, 1] is the belief that the type is high if consumers choose the first vendor and y t ∈ [0, 1] is the belief that the type is high if consumers choose the other vendor; that is to say, where s j � (ζ jH − ζ jL )/σ j , j � 1, 2, B 1 (t) and B 2 (t) are independent Brownian motions.
If a consumer chooses the first vendor, his expected utility is Similarly, if a consumer chooses the second vendor, his expected utility is represented by where a 2 ≔ ζ 2H − ζ 2L and b 2 ≔ ζ 2L . e definitions of a j and b j imply Let n t be the number of consumers who choose the first vendor and M − n t be the number of consumers who choose the second vendor, n t ∈ 0, 1, . . . , M { }. P 1 (n t ) is the price of goods from the first vendor and P 2 (n t ) is the price of goods from the second one. Formally, the price of goods is a measurable function P j : We assume that the change of price has hysteresis(the product price of the second vendor is a function with respect to n t 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. Denote α t ∈ [0, 1] as the belief, i.e., if x t < α t , 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 y t > β t ∈ [0, 1].
For nondurable goods, if a consumer chooses the first vendor at first time, the common belief for high type is x t at time t. e utility of the consumer is x t ζ 1H + (1 − x t )ζ 1L − P 1 (n t ). 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. e utility of the consumer is x t ζ 2H + (1 − x t )ζ 2L − P 2 (n t ). If the consumer voluntarily gives up the first vendor to choose the second vendor, it has Inequality (5) is rewritten as From the definition of α t , it has α t � ((b 2 − b 1 + P 1 (n t ) − P 2 (n t ))/(a 1 − a 2 )). In the same way, we obtain β t � ((b 2 − b 1 + P 1 (n t ) − P 2 (n t ))/(a 1 − a 2 )). After the above discussion, α t and β t satisfy 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]. e cutoff increases as P 1 (n t ) increases and decreases as P 2 (n t ) increases. e cutoff is linear with both P 1 (n t ) and P 2 (n t ). In a market, as P 1 (n t ) increases, the utility of the consumer who chooses the first vendor is smaller. e consumers who have the same common belief tend to choose the second vendor. us, the cutoff cuts down. If P 2 (n t ) decreases, the relative price for the second vendor increases. Similarly, the cutoff reduces. In the following, we use α t to denote the cutoff. e ranges of pricing for two vendors obtained from equation (7) are shown in Proposition 1.

Proposition 1.
Suppose that P 1 (·) and P 2 (·) are the prices of goods from the first and second vendors, respectively. en P 1 (·) and P 2 (·) satisfy where one of the second signs of inequalities (8) or (9) is sign of strict inequality.
Proof. P 1 (n t ) and P 2 (n t ) are greater than zero as the vendors' costs of goods are zero. In the following, we prove all consumers give up the first vendor to choose the second one. For the first vendor, the price of his goods satisfies α t ≤ 1. It has i.e., In the same way, the second vendor makes the price of goods satisfy α t ≥ 0. We have Inequalities (8) and (9) are proved. From inequalities (8) and (9), we obtain From inequality (13), we obtain a 2 ≤ a 1 which contradicts the assumption a 2 < a 1 . us, one of the second signs of inequalities in (8) and (9) is sign of strict inequality.
Proposition 1 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. e 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.
Mathematical Problems in Engineering e supremum of the price for the other vendor aggrandizes. □ Corollary 1. P 1 (n t ) < + ∞ if and only if P 2 (n t ) < +∞ while P 1 (n t ) � ∞ if and only if P 2 (n t ) � ∞.
Corollary 1 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.

Duopoly Market
If there exists t Θ 0≤Θ≤d such that 0 � t 0 < t 1 < · · · < t d � t, a consumer chooses a vendor, denoted by j, t ∈ (t 2k , t 2k+1 ) and chooses the other vendor, denoted by − j, t ∈ (t 2k+1 , t 2k+2 ), where u . Using the arguments in [8,9], we know that p t satisfies dp t � �� n t √ p t 1 − p t s 1 dW 1 (t) where W 1 (t) and W 2 (t) are independent Winner processes. For any time l and small ε > 0, a consumer gives up the second vendor and chooses the first vendor when where We assume that the number of consumers who choose one vendor to replace the other one is related to Δ(α l − p l ), p l and n l at time l. All of them reflect the relationship between α t and p t at t ∈ (l − ε, l + ε). Let G j (Δ(α l − p l ), p l , n l ) (∈ 0, 1, . . . , M { }) be the number of consumers who give up the other vendor to choose the vendor j, j � 1, 2. If n t is known, the number of consumers who choose the first vendor at (t + dt) is expressed by where ♯ represents the number of elements in a set. Let Similarly, if 0 ≤ s ≤ u | p s � α s , α s− ε − p s− ε ∈ A 1 , α s+ε − p s+ε ∈ A 2 } is not empty, let c 2 (Δ(α u − p u ), p u ) be equal to ϖ 2 (Δ(α u − p u ), p u ) and zero otherwise. Defining we have where g j (Δ(α u − p u ), p u , n u ) ≤ G j (Δ(α u − p u ), p u , n u ) due to c j (·, ·) ∈ [0, 1], i.e., g j (Δ(α u − p u ), p u , n u ) is bounded. Proposition 2 can be obtained by using the definition of Poisson stochastic integral as dt ⟶ 0 and s ⟶ u.

Proposition 2. If (α t − p t ) is a stochastic process which has independent and stationary increments with ca
(paths are continuous on the right and have limits on the left [20,21]), then where g 1 (h, p t− , n t− ) and g 2 (h, p t− , n t− ) are defined in Equations (20) and (21). Moreover, SDE (23) has a unique solution.
Proposition 2 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. g j (·, ·, ·) is defined as the rate of change of the number of consumers who give up the other vendor and choose the jth one. □

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 are n consumers who choose the first vendor, i.e., n 0 � n. After that, vendors inform consumers' types, denoted by p ≔ p 0 , to guide consumers to make subsequent choices. Ignoring the interaction between the two vendors, denoting V j (·) as the jth vendor's optimal utility and r as risk-free interest rate, for the first vendor, we have

(26)
e HJB equation is obtained as follows:

Mathematical Problems in Engineering
where ], the intensity measure of N, is the finite intensity measure as A 1 and A 2 are bounded below. Suppose that w(n) is a solution of equation (27), the verification result is obtained as follows.
ree steps are divided to prove Proposition 3. □ Step 1. We prove E[ +∞ 0 e − rt n t P 1 (n t )dt] < +∞. From the assumptions, it has E +∞ 0 e − rt n t P 1 n t dt ≤ ME +∞ 0 e − rt P 1 n t dt .
Step 2. e proof of (i) in Proposition 3. Let Using Ito's lemma for e − r(T∧τ k ) w(n T∧τ k ), where T ∧ τ k ≔ min T, τ k , we obtain Letting k ⟶ +∞, using the dominated convergence theorem, inequality (40) becomes E e − rT w n T ≤ w(n) − E T 0 e − rs n s P 1 n s ds . (41) As T ⟶ +∞, in accordance with inequality (29), for any p ∈ [0, 1], the inequality w(n) ≥ E T 0 e − rs n s P 1 n s ds , Mathematical Problems in Engineering 7 is obtained. Hence, e (i) in Proposition 3 is proved by Steps 1 and 2.
Step 3. e proof of (ii) in Proposition 3. As p * is the optimal choice for the vendor, we obtain has a unique solution. Letting T ⟶ +∞, from inequality (32), we have From inequalities (43) and (46), we have w(n) � V(n). e proof is completed. Proposition 3 shows that the solution of HJB equation Similarly, for the second vendor, it has 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 For the second vendor, his payoff function satisfies As the second vendor considers that the first vendor informs all consumers whose types are H 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 P 2 (n t ), based on the price P 1 (n t ) of the first vendor, such that P 1 (n t ) � (a 1 + b 1 ) − (a 2 + b 2 ) + P 2 (n t )., i.e., From equation (50), we find that P 2 (n t ) < 0 if P 1 (n t ) < (a 1 + b 1 ) − (a 2 + b 2 ). us, the price of the second vendor's goods is zero. For the type-leader vendor, the price of their goods is where ε > 0 is an arbitrary small number. Denoted by P ε 1 as (a 1 + b 1 ) − (a 2 + b 2 ) − ε, then P ε 1 < (a 1 + b 1 ) − (a 2 + b 2 ) + 0 and 0 � P 2 (n t ) ≤ a 1 − a 2 − ε satisfy pricing ranges for both vendors in Proposition 1. Equations (48) and (49) are equivalent to respectively. In this situation, the payoffs of the leader and the following vendors are shown in Proposition 4.

Proposition 4.
If there exist P ε 1 , the rate of change g j (·, ·, ·), a finite intensity measure ] and p * ∈ [0, 1] such that the equation has solutions; then its solution is unique if and only if V 2 (n) ≡ 0, i.e., the type-leader vendor makes the other vendor's profit be zero with pricing strategy.
Proposition 4 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.
If q 3 � 0, the solutions of V 1 (n) and V 2 (n) are respectively.

(68)
If q 2 � 0, V 1 (n) and V 2 (n) satisfy It is easy to verify that the solutions in the four cases satisfy the verification theorem in Proposition 3. In particular, from equations (69) and (70), V 1 (·) has a unique solution and V 2 (n) ≡ 0 in this situation, which satisfies the conclusion in Proposition 4.

Conclusion
is 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 typeleader model. In the type-leader model, we find vendors' price strategies and verify that the commodity prices are in the price ranges in Proposition 1. e 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.

Data Availability
No data were used in this manuscript.

Conflicts of Interest
e authors declare that they have no conflicts of interests.

Authors' Contributions
e article is a joint work of two authors who contributed equally to the final version of the paper. Both authors read and approved the final manuscript.