The First Passage Time and the Dividend Value Function for One-Dimensional Diffusion Processes between Two Reflecting Barriers

Abstract

We consider the general one-dimensional time-homogeneous regular diffusion process between two reflecting barriers. An approach based on the Itô formula with corresponding boundary conditions allows us to derive the differential equations with boundary conditions for the Laplace transform of the first passage time and the value function. As examples, the explicit solutions of them for several popular diffusions are obtained. In addition, some applications to risk theory are considered.

1. Introduction and the Model

Diffusion processes with one or two barriers appear in many applications in economics, finance, queueing, mathematical biology, and electrical engineering. Among queueing system applications, reflected Ornstein-Uhlenbeck and reflected affine processes have been studied as approximations of queueing systems with reneging or balking [1, 2]. Motivated by Ward and Glynn’s one-sided problem, Bo et al. [3] considered a reflected Ornstein-Uhlenbeck process with two-sided barriers. In this paper, we consider the expectations of some random variables involving the first passage time and local times for the general one-dimensional diffusion processes between two reflecting barriers.

Let 𝑋={𝑋_𝑡,𝑡≥0} be a one-dimensional time-homogeneous reflected diffusion process with barriers 𝑎 and 𝑏, which is defined by the following stochastic differential equation: 𝑑𝑋_𝑡=𝜇𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡+𝑑𝐿_𝑡−𝑑𝑈_𝑡,𝑋₀=𝑥∈(𝑎,𝑏), (1.1) where 𝐵(𝑡) is a Brownian motion in ℛ, 𝐿={𝐿_𝑡,𝑡≥0} and 𝑈={𝑈_𝑡,𝑡≥0} are the regulators of point 𝑎 and 𝑏, respectively. Further, the processes 𝐿 and 𝑈 are uniquely determined by the following properties (see, e.g., [4]):(1)both 𝑡→𝐿_𝑡 and 𝑡→𝑈_𝑡 are continuous nondecreasing processes with 𝐿₀=𝑈₀=0,(2)𝐿 and 𝑈 increase only when 𝑋=𝑎 and 𝑋=𝑏, respectively, that is, ∫^𝑡₀𝐼_{{𝑋𝑠=𝑎}}𝑑𝐿_𝑠=𝐿_𝑡 and ∫^𝑡₀𝐼_{{𝑋𝑠=𝑏}}𝑑𝑈_𝑠=𝑈_𝑡, for 𝑡≥0.

It is well known that under certain mild regularity conditions on the coefficients 𝜇(𝑥) and 𝜎(𝑥), the SDE (1.1) has a unique strong solution for each starting point (see, e.g., [5]). The solution 𝑋_𝑡 is a time-homogeneous strong Markov process with infinitesimal generator (𝑥)=1𝒜𝑓2𝜎²(𝑥)𝑓^′′(𝑥)+𝜇(𝑥)𝑓^′(𝑥),𝑥∈(𝑎,𝑏), (1.2) acting on functions on [𝑎,𝑏] subject to boundary conditions: 𝑓^′(𝑎)=𝑓^′(𝑏)=0.

Define the first passage time 𝜏_𝑦=inf𝑡≥0∶𝑋_𝑡=𝑦, (1.3) where 𝜏_𝑦=∞ if 𝑋_𝑡 never reaches 𝑦.

For 𝜆>0, 𝜂>0, 𝜃>0, we consider the Laplace transform 𝜑, and the value functions 𝜓, 𝜓₁, and 𝜓₂ on 𝑥∈[𝑎,𝑏]: 𝜑(𝑥)=𝐸_𝑥𝑒^{−𝜆𝜏𝑦},𝜓(𝑥)=𝐸_𝑥𝜂^∞₀𝑒^−𝜆𝑡𝑑𝑈_𝑡−𝜃^∞₀𝑒^−𝜆𝑡𝑑𝐿_𝑡𝜓₁,(𝑥)=𝐸_𝑥^∞₀𝑒^−𝜆𝑡𝑑𝑈_𝑡𝜓₂,(𝑥)=𝐸_𝑥^∞₀𝑒^−𝜆𝑡𝑑𝐿_𝑡. (1.4)(1.5)(1.6)(1.7)

The rest of the paper is organized as follows. Section 2 studies the Laplace transform of the first passage time. Section 3 deals with the value function. Some applications in risk theory are considered in Section 4.

2. Laplace Transform

Bo et al. [3] consider the Laplace transform 𝐸_𝑥[𝑒^{−𝜆𝜏𝑦}] for a reflected Ornstein-Uhlenbeck process with two-sided barriers. In this section we consider the Laplace transform of the first passage time for the general reflected diffusion process 𝑋 defined by (1.1).

Theorem 2.1. Let 𝑥∈[𝑎,𝑏], 𝜆>0, and assume that 𝑓₁(𝑦), 𝑓₂(𝑦) satisfy the following equations, respectively: 𝒜𝑓₁𝑦=𝜆𝑓₁𝑦(),𝑓,𝑦∈𝑎,𝑏_′1(𝑎)=0,𝒜𝑓₂𝑦=𝜆𝑓₂𝑦(),𝑓,𝑦∈a,𝑏_′2(𝑏)=0. (2.1)
If 𝑓₁(𝑦)≠0 for 𝑦∈[𝑥,𝑏] and 𝑓₂(𝑦)≠0 for 𝑦∈[𝑎,𝑥], then 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=𝑓₁(𝑥)𝑓₁𝑦[],𝐸,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=𝑓₂(𝑥)𝑓₂𝑦,𝑦∈[𝑎,𝑥]. (2.2)

Proof. Applying the Itô formula for semimartingales to ℎ(𝑡,𝑋_𝑡)=𝑒^−𝜆𝑡𝑓(𝑋_𝑡) with 𝑓∈𝐶²_𝑏([𝑎,𝑏]) we obtain ℎ𝑡,𝑋_𝑡=ℎ0,𝑋₀+^𝑡₀𝜕ℎℎ𝜕𝑠𝑠,𝑋_𝑠𝑑𝑠+^𝑡₀𝜕ℎℎ𝜕𝑥𝑠,𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝜕²ℎ𝜕𝑥²ℎ𝑠,𝑋_𝑠𝑑𝑋_𝑠,𝑋_𝑠=𝑓𝑋₀+^𝑡₀−𝜆𝑒^−𝜆𝑠𝑓𝑋_𝑠𝑑𝑠+^𝑡₀𝑒^−𝜆𝑠𝑓^′𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝑒^−𝜆𝑠𝑓^′′𝑋_𝑠𝜎²𝑋_𝑠(𝑥)+𝑑𝑠=𝑓^𝑡₀𝑒^−𝜆𝑠(𝒜−𝜆)𝑓𝑋_𝑠𝑑𝑠+^𝑡₀𝑒^−𝜆𝑠𝑓^′𝑋_𝑠𝑑𝐵_𝑠+𝑓^′(𝑎)^𝑡₀𝑒^−𝜆𝑠𝑑𝐿_𝑠−𝑓^′(𝑏)^𝑡₀𝑒^−𝜆𝑠𝑑𝑈_𝑠. (2.3) Since 𝜏_𝑦<∞ is a stopping time and 𝑥∈[𝑎,𝑏], it follows from the optional sampling theorem that 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}𝑓𝑋_𝜏𝑦=𝑓(𝑥)+𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠(𝒜−𝜆)𝑓𝑋_𝑠𝑑𝑠+𝑓^′(𝑎)𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝐿_𝑠−𝑓^′(𝑏)𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝑈_𝑠. (2.4) By the definitions of 𝜏_𝑦, 𝑈, and 𝐿 we have 𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝐿_𝑠=0,𝑦∈[𝑎,𝑥];𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝑈_𝑠=0,𝑦∈[𝑥,𝑏];𝑋𝜏_𝑦=𝑦. (2.5) Substituting them into (2.4) one gets 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}𝑓𝑦=𝑓(𝑥)+𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠(𝒜−𝜆)𝑓𝑋_𝑠𝑑𝑠+𝑓^′(𝑎)𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝐿_𝑠[].𝐸,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}𝑓𝑦=𝑓(𝑥)+𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠(𝒜−𝜆)𝑓𝑋_𝑠𝑑𝑠−𝑓^′(𝑏)𝐸_𝑥^𝜏𝑦₀𝑒^−𝜆𝑠𝑑𝑈_𝑠,𝑦∈[𝑎,𝑥]. (2.6) The result follows.

Remark 2.2. Although neither 𝑓₁(⋅) nor 𝑓₂(⋅) in the Theorem 2.1 is unique, but each of their ratios is unique.

As illustrations of Theorem 2.1, we consider some examples.

Example 2.3. Bessel process: 𝑑𝑋_𝑡=((𝑑−1)/2𝑋_𝑡)𝑑𝑡+𝑑𝐵_𝑡, where 𝑑>1 is a real number.
We consider the differential equation (1/2)𝜓^′′(𝑥)+((𝑑−1)/2𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥), 𝜆>0. It is well known that the increasing and decreasing solutions are, respectively: 𝜓₊(𝑥)=𝑥^−𝜈𝐼_𝜈√2𝜆𝑥,𝜓₋(𝑥)=𝑥^−𝜈𝐾_𝜈√2𝜆𝑥, (2.7) where 𝜈=(𝑑−2)/2 and 𝐼_𝜈 and 𝐾_𝜈 are the usual modified Bessel functions.
Then, we can give 𝑓₁(𝑥),𝑓₂(𝑥) as follows: 𝑓₁(𝑥)=𝐶₁𝜓₊(𝑥)+𝐶₂𝜓₋(𝑥),𝑓₂(𝑥)=𝐶₃𝜓₊(𝑥)+𝐶₄𝜓₋(𝑥), (2.8) where the constants 𝐶₁, 𝐶₂ and 𝐶₃, 𝐶₄ can be derived from 𝑓^′₁(𝑎)=0 and 𝑓^′₂(𝑏)=0, respectively. We can obtain their ratios, respectively: 𝐶𝑀=¹𝐶₂√=−2𝜆𝑎𝐾^′_𝜈√2𝜆𝑎−𝜈𝐾_𝜈√2𝜆𝑎√2𝜆𝑎𝐼^′_𝜈√2𝜆𝑎−𝜈𝐼_𝜈√,𝐶2𝜆𝑎𝑁=³𝐶₄√=−2𝜆𝑏𝐾^′_𝜈√2𝜆𝑏−𝜈𝐾_𝜈√2𝜆𝑏√2𝜆𝑏𝐼^′_𝜈√2𝜆𝑏−𝜈𝐼_𝜈√.2𝜆𝑏 (2.9) Substituting them into (2.2), we get 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=𝑀𝑥^−𝜈𝐼_𝜈√2𝜆𝑥+𝑥^−𝜈𝐾_𝜈√2𝜆𝑥𝑀𝑦^−𝜈𝐼_𝜈√2𝜆𝑦+𝑦^−𝜈𝐾_𝜈√[],𝐸2𝜆𝑦,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=𝑁𝑥^−𝜈𝐼_𝜈√2𝜆𝑥+𝑥^−𝜈𝐾_𝜈√2𝜆𝑥𝑁𝑦^−𝜈𝐼_𝜈√2𝜆𝑦+𝑦^−𝜈𝐾_𝜈√[].2𝜆𝑦,𝑦∈𝑎,𝑥 (2.10)

Example 2.4. The Ornstein-Uhlenbeck process [6] is as follows: 𝑑𝑋_𝑡=𝑣𝑘−𝑋_𝑡𝑑𝑡+𝜎𝑑𝐵_𝑡,𝑣,𝜎>0,𝑘∈[𝑎,𝑏]. (2.11) In mathematical finance, the Ornstein-Uhlenbeck process above is known as Vasicek model for the short-term interest rate process [7]. We consider the differential equation (1/2)𝜎²𝜓^′′(𝑥)+𝑣(𝑘−𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥).
In the case 𝑘=0, 𝜎=1, the two independent solutions to (1/2)𝜓^′′(𝑥)−𝑣𝑥𝜓^′(𝑥)=𝜆𝜓(𝑥) are 𝜓₊(𝑥)=𝐻_−𝜆/𝑣−√𝑣𝑥=2^{−𝜆/2𝑣}𝑒^{(1/2)𝑣𝑥2}𝐷_−𝜆/𝑣−√,𝜓2𝑣𝑥₋(𝑥)=𝐻_−𝜆/𝑣√𝑣𝑥=2^{−𝜆/2𝑣}𝑒^{(1/2)𝑣𝑥2}𝐷_𝜆/𝑣√2𝑣𝑥, (2.12) where 𝐻_𝜈(⋅) and 𝐷_𝜈(⋅) are, respectively, the Hermite and parabolic cylinder functions [8]. Then, as the way used in Example 2.3, we obtain the ratios of the constants 𝐶₁, 𝐶₂ and 𝐶₃, 𝐶₄, respectively: 𝐶𝑀=¹𝐶₂=−𝑣𝑎𝐷_𝜆/𝑣√2𝑣𝑎+√2𝑣𝐷^′_𝜆/𝑣√2𝑣𝑎𝑣𝑎𝐷_−𝜆/𝑣√2𝑣𝑎−√2𝑣𝐷^′_−𝜆/𝑣√,𝐶2𝑣𝑎𝑁=³𝐶₄=−𝑣𝑏𝐷_𝜆/𝑣√2𝑣𝑏+√2𝑣𝐷^′_𝜆/𝑣√2𝑣𝑏𝑣𝑏𝐷_−𝜆/𝑣√2𝑣𝑏−√2𝑣𝐷^′_−𝜆/𝑣√.2𝑣𝑏 (2.13) Substituting them into (2.2), we get 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=𝑀𝑒^{(1/2)𝑣𝑥2}𝐷_−𝜆/𝑣−√2𝑣𝑥+𝑒^{(1/2)𝑣𝑥2}𝐷_𝜆/𝑣√2𝑣𝑥𝑀𝑒^{(1/2)𝑣𝑦2}𝐷_−𝜆/𝑣−√2𝑣𝑦+𝑒^{(1/2)𝑣𝑦2}𝐷_𝜆/𝑣√[],𝐸2𝑣𝑦,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=𝑁𝑒^{(1/2)𝑣𝑥2}𝐷_−𝜆/𝑣−√2𝑣𝑥+𝑒^{(1/2)𝑣𝑥2}𝐷_𝜆/𝑣√2𝑣𝑥𝑁𝑒^{(1/2)𝑣𝑦2}𝐷_−𝜆/𝑣−√2𝑣𝑦+𝑒^{(1/2)𝑣𝑦2}𝐷_𝜆/𝑣√[].2𝑣𝑦,𝑦∈𝑎,𝑥 (2.14) For the general 𝑘 and 𝜎, the two independent solutions are, respectively 𝜓₊(𝑥)=𝐻_−𝜆/𝑣−√𝑣𝜎(𝑥−𝑘)=2^{−𝜆/2𝑣}𝑒^{(1/2)(𝑣/𝜎2)(𝑥−𝑘)2}𝐷_−𝜆/𝑣−√2𝑣𝜎(),𝜓𝑥−𝑘₋(𝑥)=𝐻_−𝜆/𝑣√𝑣𝜎(𝑥−𝑘)=2^{−𝜆/2𝑣}𝑒^{(1/2)(𝑣/𝜎2)(𝑥−𝑘)2}𝐷_−𝜆/𝑣√2𝑣𝜎(𝑥−𝑘). (2.15) Then, as the way used in Example 2.3, we obtain the ratios of the constants 𝐶₁, 𝐶₂ and 𝐶₃, 𝐶₄, respectively 𝐶𝑀=¹𝐶₂√=−𝑣(𝑎−𝑘)𝐷_−𝜆/𝑣√2𝑣/𝜎(𝑎−𝑘)+√2𝜎𝐷^′_−𝜆/𝑣√2𝑣/𝜎(𝑎−𝑘)√𝑣(𝑎−𝑘)𝐷_−𝜆/𝑣−√2𝑣/𝜎(𝑎−𝑘)−√2𝜎𝐷^′_−𝜆/𝑣−√(),𝐶2𝑣/𝜎𝑎−𝑘𝑁=³𝐶₄√=−𝑣(𝑏−𝑘)𝐷_−𝜆/𝑣√2𝑣/𝜎(𝑏−𝑘)+√2𝜎𝐷^′_−𝜆/𝑣√2𝑣/𝜎(𝑏−𝑘)√𝑣(𝑏−𝑘)𝐷_−𝜆/𝑣−√2𝑣/𝜎(𝑏−𝑘)−√2𝜎𝐷^′_−𝜆/𝑣−√(.2𝑣/𝜎𝑏−𝑘) (2.16) Substituting them into (2.2), we get 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=ℎ(𝑥)𝑀𝐷_−𝜆/𝑣−√2𝑣/𝜎(𝑥−𝑘)+𝐷_−𝜆/𝑣√2𝑣/𝜎(𝑥−𝑘)ℎ𝑦𝑀𝐷_−𝜆/𝑣−√2𝑣/𝜎𝑦−𝑘+𝐷_−𝜆/𝑣√[],𝐸2𝑣/𝜎𝑦−𝑘,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=ℎ(𝑥)𝑁𝐷_−𝜆/𝑣−√2𝑣/𝜎(𝑥−𝑘)+𝐷_−𝜆/𝑣√2𝑣/𝜎(𝑥−𝑘)ℎ𝑦𝑁𝐷_−𝜆/𝑣−√2𝑣/𝜎𝑦−𝑘+𝐷_−𝜆/𝑣√[],2𝑣/𝜎𝑦−𝑘,𝑦∈𝑎,𝑥 (2.17) where ℎ(𝑥)=𝑒^{(1/2)(𝑣/𝜎2)(𝑥−𝑘)2}.

Remark 2.5. If we take 𝑘=0, 𝜎=1, 𝑎=0 and substitute the series forms of 𝐻_𝜈(⋅) and 𝐷_𝜈(⋅) into the above result, then it is the same as Bo et al. [3].

Example 2.6. The square root process of Cox et al. [9]: 𝑑𝑋_𝑡=𝑣𝑘−𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡,𝑣,𝜎>0,𝑘∈[𝑎,𝑏]. (2.18)
Now consider the differential equation 12𝜎²𝑥𝜓^′′(𝑥)+(𝑣𝑘−𝑣𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥),𝜆>0. (2.19) If (2𝑣/𝜎²)𝑘 is not an integer, the two linear independent solutions are 𝜓₊(𝑥)𝜆=𝑀𝑣,2𝑣𝜎²𝑘,2𝑣𝜎²𝑥,𝜓₋(𝑥)𝜆=𝑈𝑣,2𝑣𝜎²𝑘,2𝑣𝜎²𝑥, (2.20) where 𝑀 and 𝑈 are the confluent hypergeometric functions of the first and second kinds, respectively. Then, as the way used in Example 2.3, we obtain the ratios of the constants 𝐶₁, 𝐶₂ and 𝐶₃, 𝐶₄, respectively: 𝑀₁=𝐶₁𝐶₂𝑈=−^′𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑎𝑀^′𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑎,𝐶𝑁=³𝐶₄𝑈=−^′𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑏𝑀^′𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑏. (2.21) Substituting them into (2.2), we get 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=𝑀₁𝑀𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑥+𝑈𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑥𝑀₁𝑀𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑦+𝑈𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑦[].𝐸,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=𝑁𝑀𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑥+𝑈𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑥𝑁𝑀𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑦+𝑈𝜆/𝑣,2𝑣/𝜎²𝑘,2𝑣/𝜎²𝑦,𝑦∈[𝑎,𝑥]. (2.22)

Example 2.7. The Gompertz Brownian motion process [10] is as follows: 𝑑𝑋_𝑡=𝑣𝑋(𝑡)ln(𝑘)−ln𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡,𝑣,𝜎>0,𝑘∈[𝑎,𝑏]. (2.23)
Now consider the differential equation 12𝜎²𝑥²𝜓^′′(𝑥)+𝑣𝑥(ln𝑘−ln𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥),𝜆>0. (2.24) The increasing and decreasing solutions are, respectively: 𝜓₊(𝑥)𝜆=𝑀,12𝑣2,𝑣𝜎²𝑥ln𝑘+𝜎²2𝑣²,𝜓₋(𝑥)𝜆=𝑈,12𝑣2,𝑣𝜎²𝑥ln𝑘+𝜎²2𝑣², (2.25) where 𝑀 and 𝑈, as in Example 2.6, are the first and second Kummer’s functions, respectively. Then, as the way used in Example 2.3, we obtain the ratios of the constants 𝐶₁, 𝐶₂ and 𝐶₃, 𝐶₄, respectively: 𝑀₁=𝐶₁𝐶₂𝑈=−^′𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑎/𝑘)+𝜎²/2𝑣²𝑀^′𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑎/𝑘)+𝜎²/2𝑣²,𝐶𝑁=³𝐶₄𝑈=−^′𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑏/𝑘)+𝜎²/2𝑣²𝑀^′𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑏/𝑘)+𝜎²/2𝑣². (2.26) Substituting them into (2.2), we get 𝐸_𝑥𝑒^{−𝜆𝜏𝑦}=𝑀₁𝑀𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑥/𝑘)+𝜎²/2𝑣²+𝑈(𝔸)𝑀₁𝑀𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑥/𝑘)+𝜎²/2𝑣²(𝔸)[],𝐸+𝑈,𝑦∈𝑥,𝑏_𝑥𝑒^{−𝜆𝜏𝑦}=𝑁𝑀𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑥/𝑘)+𝜎²/2𝑣²+𝑈(𝔸)𝑁𝑀𝜆/2𝑣,1/2,𝑣/𝜎²ln(𝑥/𝑘)+𝜎²/2𝑣²(𝔸)[],+𝑈,𝑦∈𝑎,𝑥 (2.27) where (𝔸) denotes (𝜆/2𝑣,1/2,(𝑣/𝜎²)(ln(𝑥/𝑘)+𝜎²/2𝑣)²).

Remark 2.8. For a certain choice of parameters for 𝑎 and 𝑏 in Examples 2.3–2.7, we get the Laplace transform of the first passage time of one-dimensional diffusion with one-sided barrier. For example, letting 𝑎→−∞ or 𝑏→+∞ in Example 2.4, one gets the Laplace transform of the first passage time of the Ornstein-Uhlenbeck process with one-sided barrier; see Nobile et al. [11], Ricciardi and Sato [12], Alili et al. [13], and Ditlevsen [6].

3. The Value Function

In this section we study the value functions (1.5)–(1.7). Using Itô’s formula, we derive differential equation with boundary conditions for 𝜓.

Theorem 3.1. The function 𝜓 defined by (1.5) satisfies the differential equation 12𝜎²(𝑥)𝜓^′′(𝑥)+𝜇(𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥),𝑎<𝑥<𝑏, (3.1) with the boundary conditions 𝜓^′(𝑎)=𝜃, 𝜓^′(𝑏)=𝜂.

Proof. Applying the Itô’s formula for semimartingales to ℎ(𝑡,𝑋_𝑡)=𝑒^−𝜆𝑡𝜓(𝑋_𝑡) with 𝜓∈𝐶²_𝑏([𝑎,𝑏]) we obtain ℎ𝑡,𝑋_𝑡=ℎ0,𝑋₀+^𝑡₀𝜕ℎℎ𝜕𝑠𝑠,𝑋_𝑠𝑑𝑠+^𝑡₀𝜕ℎℎ𝜕𝑥𝑠,𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝜕²ℎ𝜕𝑥²ℎ𝑠,𝑋_𝑠𝑑𝑋_𝑠,𝑋_𝑠=𝜓𝑋₀+^𝑡₀−𝜆𝑒^−𝜆𝑠𝜓𝑋_𝑠𝑑𝑠+^𝑡₀𝑒^−𝜆𝑠𝜓^′𝑋_𝑠𝑑𝑋_𝑠+12^𝑡₀𝑒^−𝜆𝑠𝜓^′′𝑋_𝑠𝜎²𝑋_𝑠(𝑥)+𝑑𝑠=𝜓^𝑡₀𝑒^−𝜆𝑠(𝒜−𝜆)𝜓𝑋_𝑠𝑑𝑠+^𝑡₀𝑒^−𝜆𝑠𝜓^′𝑋_𝑠𝑑𝐵_𝑠+𝜓^′(𝑎)^𝑡₀𝑒^−𝜆𝑠𝑑𝐿_𝑠−𝜓^′(𝑏)^𝑡₀𝑒^−𝜆𝑠𝑑𝑈_𝑠, (3.2) where we have used that 𝜓^′(𝑋_𝑡)𝑑𝐿_𝑡=𝜓^′(𝑎)𝑑𝐿_𝑡 and 𝜓^′(𝑋_𝑡)𝑑𝑈_𝑡=𝜓^′(𝑏)𝑑𝑈_𝑡. From (3.2) we have 𝐸_𝑥e^−𝜆𝑡𝜓𝑋_𝑡−𝜓(𝑥)=𝐸_𝑥^𝑡₀𝑒^−𝜆𝑠(𝒜−𝜆)𝜓𝑋_𝑠𝑑𝑠+𝐸_𝑥𝜓^′(𝑎)^𝑡₀𝑒^−𝜆𝑠𝑑𝐿_𝑠−𝜓^′(𝑏)^𝑡₀𝑒^−𝜆𝑠𝑑𝑈_𝑠. (3.3)
Let 𝑓 be a solution of (𝑥)(𝑥)(),𝑓𝒜𝑓=𝜆𝑓,𝑥∈𝑎,𝑏_′(𝑎)𝑓=0,_′(𝑏)=1. (3.4) In place of (3.3), we have 𝐸_𝑥e^−𝜆𝑡𝑓𝑋_𝑡−𝑓(𝑥)=−^𝑡₀𝑒^−𝜆𝑠𝑑𝑈_𝑠. (3.5) Letting 𝑡→∞, we get ^∞₀𝑒^−𝜆𝑠𝑑𝑈_𝑠=𝑓(𝑥)<∞. (3.6) Likewise ^∞₀𝑒^−𝜆𝑠𝑑𝐿_𝑠<∞. (3.7) Letting 𝑡→∞ in (3.3) and noting (3.6) and (3.7), we get the desired result.

Corollary 3.2. The function 𝜓₁(𝑥)=𝐸_𝑥[∫^∞₀𝑒^−𝜆𝑡𝑑𝑈_𝑡] is solution to the differential equation 12𝜎²(𝑥)𝜓₁^′′(𝑥)+𝜇(𝑥)𝜓^′₁(𝑥)=𝜆𝜓₁(𝑥),𝑎<𝑥<𝑏, (3.8) with the boundary conditions 𝜓^′₁(𝑎)=0,𝜓^′₁(𝑏)=1.

Corollary 3.3. The function 𝜓₂(𝑥)=𝐸_𝑥[∫^∞₀𝑒^−𝜆𝑡𝑑𝐿_𝑡] is solution to the differential equation 12𝜎²(𝑥)𝜓₂^′′(𝑥)+𝜇(𝑥)𝜓^′₂(𝑥)=𝜆𝜓₂(𝑥),𝑎<𝑥<𝑏, (3.9) with the boundary conditions 𝜓^′₂(𝑎)=−1, 𝜓^′₂(𝑏)=0.

For diffusions in Examples 2.3–2.7 we can obtain the explicit expressions for 𝜓, 𝜓₁, and 𝜓₂. Now we consider the Ornstein-Uhlenbeck process only.

Example 3.4. The Ornstein-Uhlenbeck process is as follows: 𝑑𝑋_𝑡=𝑣𝑘−𝑋_𝑡𝑑𝑡+𝜎𝑑𝐵_𝑡,𝑣,𝜎>0,𝑘∈[𝑎,𝑏]. (3.10) From Example 2.4, the two independent solutions of differential equation 12𝜎²𝜓^′′(𝑥)+𝑣(𝑘−𝑥)𝜓^′(𝑥)=𝜆𝜓(𝑥) (3.11) are, respectively, 𝜓₊(𝑥)=𝐻_−𝜆/𝑣−√𝑣𝜎(𝑥−𝑘)=2^{−𝜆/2𝑣}𝑒^{(1/2)(𝑣/𝜎2)(𝑥−𝑘)2}𝐷_−𝜆/𝑣−√2𝑣𝜎(),𝜓𝑥−𝑘₋(𝑥)=𝐻_−𝜆/𝑣√𝑣𝜎(𝑥−𝑘)=2^{−𝜆/2𝑣}𝑒^{(1/2)(𝑣/𝜎2)(𝑥−𝑘)2}𝐷_−𝜆/𝑣√2𝑣𝜎(𝑥−𝑘). (3.12) The general solution of (3.11) is of the form 𝜓(𝑥)=𝐶₁𝜓₊(𝑥)+𝐶₂𝜓₋(𝑥), (3.13) where the constants 𝐶₁ and 𝐶₂ are determined by the boundary conditions 𝜓^′(𝑎)=𝜃, 𝜓^′(𝑏)=𝜂. They are 𝐶₁=𝜃𝜓^′₊(𝑏)−𝜂𝜓^′₊(𝑎)𝜓^′₋(𝑎)𝜓^′₊(𝑏)−𝜓^′₊(𝑎)𝜓^′₋(𝑏),𝐶₂=𝜂𝜓^′₋(𝑎)−𝜃𝜓^′₋(𝑏)𝜓^′₋(𝑎)𝜓^′₊(𝑏)−𝜓^′₊(𝑎)𝜓^′₋(𝑏). (3.14)

4. Applications to Risk Theory

Let 𝑋_𝑡 denote the surplus of the company. If no dividends were paid, the surplus process follows the stochastic differential equation 𝑑𝑋_𝑡=𝜇𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡,𝑡≥0,𝑋₀=𝑥, (4.1) where 𝐵 is a Brownian motion and 𝜇 and 𝜎 are Lipschitz-continuous functions.

The company will pay dividends to its shareholders according to barrier strategy with parameter 𝑏>0. Whenever the surplus is about to go above the level 𝑏, the excess will be paid as dividends, and when the surplus is below 𝑏 nothing is paid out. Let 𝐷(𝑡) denote the aggregate dividends by time 𝑡. Thus the resulting surplus process 𝑌 is given by 𝑑𝑌_𝑡=𝜇𝑌_𝑡𝑑𝑡+𝜎𝑌_𝑡𝑑𝐵_𝑡−𝑑𝐷(𝑡),𝑡≥0. (4.2) Let 𝑇=inf{𝑡≥0∶𝑌_𝑡=0} be the time of ruin. Note that when 𝑏<∞ ruin is certain, that is, 𝑃(𝑇<∞)=1. We are interested in the Laplace transform of 𝑇. This model can be found in Paulsen [14], and some important special cases can be found in Gerber and Shiu [15], Cai et al. [16]. It follows from Theorem 2.1 that, for 0<𝑥<𝑏 and 𝜆>0, 𝐸_𝑥(e^−𝜆𝑇)=ℎ(𝑥), where ℎ is the solution of (𝑥)(𝑥)(),ℎ𝒜ℎ=𝜆ℎ,𝑥∈0,𝑏_′(𝑏)=0,ℎ(0)=1. (4.3)

Assume that an insurance company is not allowed to go bankrupt and the beneficiary of the dividends is required to inject capital into the insurance company to keep its risk process stays nonnegative. Under such a dividend policy the controlled risk process with initial reserve 𝑥>0 satisfies 𝑑𝑋_𝑡=𝜇𝑋_𝑡𝑑𝑡+𝜎𝑋_𝑡𝑑𝐵_𝑡+𝑑𝐿_𝑡−𝑑𝐷_𝑡,𝑡≥0, (4.4) where 𝐿_𝑡 and 𝐷_𝑡 are local times at 0 and 𝑏, respectively. 𝐿_𝑡 ensures the insurance company will not ruin and 𝐷_𝑡 is the aggregate amount of paid dividends by time 𝑡. We consider the total expected discounted dividends minus the total expected discounted costs of injected capital: 𝑉(𝑥;𝑏)=𝐸_𝑥^∞₀𝑒^−𝜆𝑡𝑑𝐷_𝑡−𝑘^∞₀𝑒^−𝜆𝑡𝑑𝐿_𝑡, (4.5) where 𝜆>0 is discounted factor and 𝑘 is the cost per unit injected capital. Avram et al. [17] consider the problem in a Levy processes setting.

According to Theorem 3.1, we obtain the formula 𝑉(𝑥,𝑏), as long as we solve the equation 12𝜎²(𝑥)𝑉^′′(𝑥)+𝜇(𝑥)𝑉^′(𝑥)=𝜆𝑉(𝑥),0<𝑥<𝑏, (4.6) with the boundary conditions 𝑉^′(0)=𝑘, 𝑉^′(𝑏)=1.

For 0<𝑥<𝑏, let 𝑇_𝑥=inf{𝑡∣𝑋_𝑡=𝑥} be the first time when the surplus reaches the level 𝑥. It follows from Theorem 2.1 that, for 0<𝑥<𝑏 and 𝜆>0, 𝐸_𝑥(e^{−𝜆𝑇𝑥})=𝑔(𝑥), where 𝑔 is the solution of (𝑥)(𝑥)(),𝑔𝒜𝑔=𝜆𝑔,𝑥∈0,𝑏_′(0)=0,𝑔(𝑏)=1. (4.7)

We now give two examples.

Example 4.1. In this example we consider the uncontrolled surplus of insurance company satisfying 𝑋_𝑡=𝑥+𝜇𝑡+𝜎𝑊_𝑡, where 𝑊_𝑡 is Brownian motion. The controlled surplus process 𝑋 at time 𝑡 follows the equation 𝑋_𝑡=𝑥+𝜇𝑡+𝜎𝑊_𝑡+𝐿_𝑡−𝐷_𝑡. (4.8)
Now we consider the differential equation 12𝜎²𝑉^′′(𝑥)+𝜇𝑉^′(𝑥)=𝜆𝑉(𝑥),0<𝑥<𝑏, (4.9) with the boundary conditions 𝑉^′(0)=𝑘, 𝑉^′(𝑏)=1. Then 𝑉(𝑥,𝑏)=𝑘𝑒^𝑟𝑏−1𝑠𝑒^𝑟𝑏−𝑒^𝑠𝑏𝑒^𝑠𝑥−𝑘𝑒^𝑠𝑏−1𝑟𝑒^𝑟𝑏−𝑒^𝑠𝑏𝑒^𝑟𝑥,0<𝑥<𝑏, (4.10) where 𝑟 and 𝑠 are the positive root and negative root of the equation (𝜎²/2)𝜉²+𝜇𝜉−𝜆=0, respectively, that is, 𝑟=−𝜇+𝜇²+2𝜎²𝜆𝜎²,𝑠=−𝜇−𝜇²+2𝜎²𝜆𝜎². (4.11)
For 0<𝑥<𝑏 and 𝜆>0, 𝐸_𝑥(e^{−𝜆𝑇𝑥})∶=𝑔(𝑥) is the solution of 12𝜎²𝑔^′′(𝑥)+𝜇𝑔^′(𝑥)=𝜆𝑔(𝑥),0<𝑥<𝑏, (4.12) with the boundary conditions 𝑔^′(0)=0, 𝑔(𝑏)=1. Solving it gives 𝐸_𝑥e^{−𝜆𝑇𝑥}=𝑟e^𝑠𝑥−𝑠e^𝑟𝑥𝑟e^𝑠𝑏−𝑠e^𝑟𝑏,0<𝑥<𝑏. (4.13)

Example 4.2. In this example we consider the Ornstein-Uhlenbeck-type model. The company’s surplus evolves according to 𝑋_𝑡=𝑥+^𝑡₀𝜇+𝜌𝑋_𝑠𝑑𝑠+𝜎𝐵_𝑡+𝐿_𝑡−𝐷_𝑡. (4.14) The model is considered in Cai et al. [16] for the special case where 𝐿_𝑡=0.
The diffusion and drift coefficients are 𝜎(𝑥)≡𝜎, 𝜇(𝑥)=𝜇+𝜌𝑥. We consider the differential equation 12𝜎²𝑉^′′(𝑥,𝑏)+𝜇+𝜌𝑥𝑉^′(𝑥,𝑏)=𝜆𝑉(𝑥,𝑏),0<𝑥<𝑏, (4.15) with the boundary conditions 𝑉^′(0)=𝑘,𝑉^′(𝑏)=1. In Cai et al. [16], they pointed out that the solution is given by 𝑉(𝑥,𝑏)=𝐻(𝑡)=𝐶₁(−𝑡)^1−𝑐𝑒^𝑡𝑀(1−𝑎,2−𝑐;−𝑡)+𝐶₂𝑒^𝑡𝑈(𝑐−𝑎,𝑐;−𝑡) (4.16) for certain coefficients 𝐶₁ and 𝐶₂, with 𝑐=1/2, 𝑎=−𝜆/2𝜌, 𝑡=−(1/𝜌𝜎²)(𝜇+𝜌𝑥)². Here 𝑀 and 𝑈 are called the confluent hypergeometric functions of the first and second kinds, respectively. For more details on confluent hypergeometric functions, see Abramowitz and Stegun [8]. It follows from (3.7) in Cai et al. [16] that 𝐻^′(𝑡)=𝐻(𝑡)−𝐶₁(1−𝑐)(−𝑡)^−𝑐𝑒^𝑡𝑀(1−𝑎,2−𝑐;−𝑡)−𝐶₁1−𝑎()2−𝑐−𝑡^1−𝑐𝑒^𝑡𝑀(2−𝑎,3−𝑐;−𝑡)+𝐶₂(𝑐−𝑎)𝑒^𝑡𝑈(𝑐−𝑎+1,𝑐+1;−𝑡). (4.17) The conditions 𝐶₁ and 𝐶₂ can be determined by 𝑉^′(0)=𝑘, 𝑉^′(𝑏)=1, where 𝑉^′(𝑥)=−(2(𝜇+𝜌𝑥)/𝜎²)𝐻^′(𝑡). Solving it gives 𝐶₁=𝜎²2𝐶/𝜇+𝜌𝑥𝑒^{(𝜇+𝜌𝑏)2/𝜌𝜎2}−𝑘𝐷/𝜇𝑒^{𝜇2/𝜌𝜎2},𝐶𝐴𝐷−𝐵𝐶₂=𝜎²2𝑘𝐵/𝜇𝑒^{𝜇2/𝜌𝜎2}−𝐴/𝜇+𝜌𝑏𝑒^{(𝜇+𝜌𝑏)2/𝜌𝜎2},𝐴𝐷−𝐵𝐶 (4.18) where 𝜇𝐴=²𝜌𝜎^21/2−12𝜇²𝜌𝜎^2−1/2𝑀𝜇1−𝑎,2−𝑐;²𝜌𝜎²−1−𝑎𝜇2−𝑐𝜌𝜎^221/2𝑀𝜇2−𝑎,3−𝑐;²𝜌𝜎²,𝐵=𝜇+𝜌𝑏²𝜌𝜎^21/2−12𝜇+𝜌𝑏²𝜌𝜎^2−1/2𝑀1−𝑎,2−𝑐;𝜇+𝜌𝑏²𝜌𝜎²−1−𝑎2−𝑐𝜇+𝜌𝑏²𝜌𝜎^21/2𝑀2−𝑎,3−𝑐;𝜇+𝜌𝑏²𝜌𝜎²,𝜇𝐶=𝑈𝑐−𝑎,𝑐;²𝜌𝜎²+()𝑈𝜇𝑐−𝑎𝑐−𝑎+1,𝑐+1;²𝜌𝜎²,𝐷=𝑈𝑐−𝑎,𝑐;𝜇+𝜌𝑏²𝜌𝜎²+(𝑐−𝑎)𝑈𝑐−𝑎+1,𝑐+1;𝜇+𝜌𝑏²𝜌𝜎². (4.19)

Acknowledgments

The authors thank the reviewer for valuable insights and suggestions that largely contributed to the improvement of the paper. The research was supported by the National Natural Science Foundation of China (no. 11171179) and the Research Fund for the Doctoral Program of Higher Education of China (no. 20093705110002).