Research Article | Open Access
A Multivariate Stochastic Hybrid Model with Switching Coefficients and Jumps: Solution and Distribution
In this work, a class of multidimensional stochastic hybrid dynamic models is studied. The system under investigation is a first-order linear nonhomogeneous system of Itô-Doob type stochastic differential equations with switching coefficients. The switching of the system is governed by a discrete dynamic which is monitored by a non-homogeneous Poisson process. Closed-form solutions of the systems are obtained. Furthermore, the major part of the work is devoted to finding closed-form probability density functions of the solution processes of linear homogeneous and Ornstein-Uhlenbeck type systems with jumps.
The study of stochastic hybrid systems exhibiting both continuous and discrete dynamics has been an area of great interest over the years. The properties of various types of stochastic hybrid systems have been studied extensively. Davis [1, 2] introduced a piecewise-deterministic Markov process, where transitions between discrete modes are triggered by random events and deterministic conditions for hitting the boundary, while the continuous-state process between jumps for the model is governed by a deterministic differential equation. Hespanha  proposed a model where transitions between modes are triggered by stochastic events much like transitions between states of a continuous-time Markov chains. Hu et al.  proposed a stochastic hybrid system where the deterministic differential equations for the evolution of the continuous-state process are replaced by Itô-Doob type stochastic differential equations [5, 6]. However, in this proposed model, the transitions are only triggered by hitting the boundaries. Siu and Ladde  studied a stochastic hybrid dynamic process where the transitions of its discrete time state are governed by either a non-homogeneous Poisson process or triggered by hitting the boundaries, while the continuous state is governed by a stochastic diffusion. A study of a stochastic hybrid system whose continuous time component is stochastic and altered by transitions of a finite-state Markov chain can be found in Chandra and Ladde , Korzeniowski and Ladde , and Ladde . Yin et al.  provided an algorithm of the numerical solutions for a class of jump diffusions with regime switching. Mao and Yuan  and Yin and Zhu  summarize a wide range of properties of stochastic differential equations with Markovian switching coefficients. In view of applications, stochastic hybrid systems have been employed to diverse fields of studies, such as communication networks [3, 14], air traffic management [15, 16], and insurance risk models .
In this work, we attempt to solve two fundamental problems in the stochastic modeling of dynamic processes described by Itô-Doob type stochastic differential equations with jumps. First, we find closed-form solutions of the stochastic hybrid systems. By using the closed-form solutions, we determine the closed-form probability density functions of solution processes of special cases of the general systems. The presented method provides an accessible way of obtaining the probability density functions without solving the Fokker-Planck partial differential equations  or approximating their solutions.
The rest of the paper is organized as follows. In Section 2, closed-form solution processes of the multidimensional systems are obtained through utilizing the result of A. G. Ladde and G. S. Ladde  piecewisely on the intervals between jumps. The problem of finding the closed-form probability density functions is investigated in a systematic and coherent way. In Section 3 the probability density function of the solution process of one-dimensional linear homogeneous system of Itô-Doob type stochastic differential equations is derived. This is an extension of the geometric Brownian motion processes . Then, by using the concept of modal matrix, the probability density function of the solution process of -dimensional linear homogeneous systems is obtained in Section 4. The probability distribution of the solution process of the system with continuous dynamic consisting of only drift part and additive noise, namely, a Ornstein-Uhlenbeck system , is extended to hybrid system in Section 5. Some concluding remarks are given in Section 6.
2. Model Formulation
In this section, we develop a conceptual stochastic model for dynamic processes in chemical, biological, engineering, medical, physical, and social science [19, 21, 22] that are under the influence of discrete time events. The continuous time dynamic of the stochastic model between jumps follows a first-order linear non-homogeneous system of Itô-Doob type stochastic differential equations. At random times, governed by a non-homogeneous Poisson process, the coefficients of the continuous time dynamic are switched, and the process is multiplied by a random factor which results in a discontinuous jump.
Let be a real -dimensional process. and are matrices for any and . Let be -dimensional vectors for any and . Let the continuous dynamic of the process be determined by the following system of stochastic differential equations: where and are independent -dimensional and -dimensional standard Wiener processes, and is a non-homogeneous Poisson process with intensity . Here, we denote as for all . When for all and , system (2.1) reduces to a first-order linear homogeneous system of Itô-Doob type stochastic differential equations, given below Here, and are such that solution process of (2.2) is nonnegative.
To obtain solution process of system (2.1), we first consider the solution process of the initial-value system when , , and ’s are fixed over time, that is, the solution process on a subinterval between jumps. Under the condition that the matrices pairwise commute, the solution can be explicitly obtained; see A. G. Ladde and G. S. Ladde  or Movellan . We state the result in the following lemma.
Lemma 2.1. Let be the solution of the following initial value problem (IVP): then the is expressed by provided that and for all .
Let be the solution to system (2.3) with , , , , and . Now, we consider the following system of two interconnected stochastic dynamics: where , , are iid positive random variables with , and . Here, we assume that , , , and are independent.
Proposition 2.2. If and for all and , then the solution to the system (2.5) is given by
Proof. By applying the result of (2.4) on the subintervals , for , and , we have the solution to system (2.5) as the following piecewise function: where is a solution process in (2.4) with , , , , and , then, we have Next, substitute . The term can be replaced by because the solution process is continuous between jumps. Repeating the substitution gives the desired result.
In the following, we present two important particular by-products of Proposition 2.2. When for all and , system (2.5) reduces to the following first-order linear homogeneous system of Itô-Doob type stochastic differential equations with jumps:
The solution of the above system is given in the following corollary. The result follows from Proposition 2.2 by letting be zero for all and .
Corollary 2.3. If and for all and , then the solution of system (2.9) is given by
In the case when are zeros for all and , system (2.5) becomes a linear system with additive noise. The continuous dynamics between jumps are now governed by Ornstein-Uhlenbeck equations as follows: Denote for all , then the above system can be rewritten as where 's are matrices, and is a -dimensional standard Wiener process.
Corollary 2.4. The solution of system (2.11) is given by
3. Probability Distribution of One-Dimensional Linear Homogeneous Models
In this section, we will derive the probability density function of the solution process of the scalar version of system (2.9). Now, takes values in , and in this case, and are scalars for all in the system (2.9) and the solution process (2.10). Some auxiliary results are presented below. The following lemma provides the joint density function of the jump times given the number of jumps due to the non-homogeneous Poisson process . The proof of this result can be found in many textbooks, for example, see Cox and Lewis  or Crowder et al. .
Lemma 3.1. For a non-homogeneous Poisson process with intensity , the joint density function of the jump times conditioned on is given by
Next lemma gives the conditional probability density function of given the number of jumps and the jump times.
Lemma 3.2. Given that , and , has a probability density function as where is the th convolution of the common probability density function of , for , and denotes the normal density function with mean and variance .
Proof. Given that , and , , from (2.10), we have
where we denote and as the sum of last three terms in (3.3).
Since is the common probability density function of , as the sum of iid random variables has the probability density function as the th convolution . We further note that is the sum of independent normal variables due to the independent increment property of Wiener process. Then is normally distributed with mean and variance as Since and are independent, then and are also independent. Using transformation method  on , we can obtain the conditional probability density function of as If follows that
Having obtained the conditional probability density function of , we can derive the marginal probability distribution of the solution process in the one-dimensional case as follows.
Proposition 3.3. The probability density function of the scalar version of the solution process to the system (2.5) is given by
Proof. From (3.1) and (3.2), the joint density function of , , given the condition is given by Integrating with respect to , and gives the probability density function of given that as By multiplying the above density by the probability of jumps, namely, and then taking the summation over , the marginal probability density function of given in (3.7) is established.
4. Probability Distribution of Multivariate Linear Homogeneous Models
In this section, we will derive the probability density function of the solution process for the -dimensional stochastic system (2.9) under the following assumptions. (i)The drift and diffusion coefficients, and for all and , are diagonalizable. (ii)The coefficients in each regime pairwise commute, that is, and for all and . (iii) for , where denotes the set of diagonalizable matrices whose eigenvector matrix, , is such that for any .
We first consider the stochastic system on the interval between jumps. Given that and , , consider the following SDE on , for some : In the following, we provide the necessary background material that will be used, subsequently. The following result provides a way to find a modal matrix that can diagonalize the coefficients in the above system.
Theorem 4.1 (see ). A set of diagonalizable matrices commutes if and only if the set is simultaneously diagonalizable, that is, there exists an invertible matrix that can diagonalize all the matrices simultaneously.
Remark 4.2. In fact, the set of diagonalizable and commuting matrices shares the same set of independent eigenvectors. The eigenvector matrix is the one that simultaneously diagonalizes all the matrices in this set, and the resulting diagonal elements are the eigenvalues of each matrix.
We recall  that a random vector is said to have a -dimensional multivariate normal distribution with mean and covariance matrix, and , if its probability density function is given by The following lemma gives the useful fact that the linear transformation of a multivariate normal random vector is again multivariate normally distributed.
Theorem 4.3 (see ). If has a multivariate normal distribution with mean and covariance matrix, and , then as a linear transformation of follows also multivariate normal distribution with mean and covariance matrix, and .
Proof. Since and every linear combination of normal random variables is still normal, then follows a multivariate normal distribution. By the linearity of expectation we have
To find the probability density function of satisfying the SDE (4.1), we need to introduce some notations and definitions that will be used, subsequently. First note that according to Theorem 4.1 and Remark 4.2, s and s have the same eigenvector matrix, denoted by . Moreover, and where and are the sets of eigenvalues of and , respectively, for all . Next, we define a linear transformation , then . Define .
The below proposition gives the probability distribution of solution process on the interval over which the coefficients are constant.
Lemma 4.4. Under assumptions (i)–(iii), the process satisfying the SDE (4.1) has a probability density function given by where
Proof. For , we have , and . By multiplying on both sides of the SDE (4.1), we obtain the transformed SDE as follows:
where and are diagonal matrices as defined before.
From the application of Lemma 2.1 with , the solution process of the transformed system (4.6) is for .
It follows from assumption (iii) that since , then, we can rewrite system (4.7) in the following form: where and are defined above.
Since is a standard Wiener process, then has a multivariate normal distribution with mean zero and covariance matrix , where is the identity matrix. Then, by Theorem 4.3, as a linear transformation of is also multivariate normally distributed with mean and covariance matrix , where and the (u,v)th element of is , for . The probability density function of is We now apply the method of transformation from to , then, where , and is the Jacobian matrix. The Jacobian determinant can be computed as Then, from (4.10) and (4.11), we have the probability density function of as Next step is to find the probability density function of by using method of transformation from to . Since , or , we have where The result follows from combining (4.13) and (4.14).
Now, we are to develop the conditional probability density function of solution process , which is a result parallel to the one-dimensional case in Lemma 3.2.
Lemma 4.5. Under assumptions (i)–(iii) and given that , and , , the solution process has a conditional probability density function as where is the common probability density function of , .
Proof. We will apply the result in Lemma 4.4 piecewisely to the system (2.9) under the conditions and . First, we note that the joint probability density function of can be expressed as then, for , consider that as a product of two random variables where the first one has the probability density function given in (4.4), and is the random jump factor at time . By the independence of and , we then have then (4.17) can be written as The conditional probability density function of given in (4.16) is obtained by integrating (4.19) with respect to , and .
Now, we can derive the unconditional probability distribution of the solution process of the -dimensional system given in the following proposition.
Proposition 4.6. Under assumptions (i)–(iii), the probability density function of the solution process to the n-dimensional system (2.9) is given by
Proof. The proof follows the argument in Proposition 3.3 by incorporating the random jumps.
Remark 4.7. It is obvious that the result in Proposition 4.6 yields the one-dimensional result as a special case. As a result of this, the proof for Proposition 4.6 is considered to be an alternative proof of the one-dimensional result in Proposition 3.3.
5. Probability Distribution of an Ornstein-Uhlenbeck Model with Jumps
In this section, we will derive the probability distribution of an Ornstein-Uhlenbeck model with jumps described by system (2.12). To obtain the desired result, we need the following lemma which gives the probability distribution of an Ornstein-Uhlenbeck process which is the continuous dynamic between jumps in system (2.12).
Suppose that the number of jumps and the jump times are given, then, by applying the above lemma piecewisely, the following result gives the conditional probability density function of .
Lemma 5.2. Under the conditions , and , , the solution process for system (2.12) has a conditional probability density function as where is the common probability density function of , .
Proof. As we noted before that the joint probability density function of , under the conditions and , , can be expressed as then, by applying the result in Lemma 5.1 on each interval between jumps , we have, for , where and . Then, for , consider that as a product of two random variables where the first one has the probability density function given in (5.5), and is the random jump factor at time . By the independence of and , then Then the conditional joint probability density function (5.4) can be written as