1 Introduction
The development of financial quantities, financial engineering, operations research, communications engineering, computer profession, microelectronics and many other fields couldn't be separated from stochastic differential equations. Therefore, stochastic differential equations and their related properties of solutions have always been the focus of scholars' research.
Without loss of generality, we learn the development and application of stochastic differential equations through as model to interest rate.
Merton
[1] put forward first short-term interest rate model:
in 1973. Next, Vasicek
[2] used Ornstein-Uhlenback process describing short-term interest rate in 1977, which is called Vasicek model:
. Cox, Ingersoll and Ross
[3] proposed the well known CIR model:
in 1980 and derived pricing formulas for bonds and bond's options. Chan, Karolyi, Longstaff and Sander
[4] proposed CKLS model:
in 1992. By a way, the above models may be nested within CKLS model by simply the suitable adjustment on parameters.
The path of Wiener process is continuous with probability 1. Then, the diffusion process driven by Wiener process can only characterize continuous volatility of interest rate. However, in real financial markets, there exists jumping volatility of interest rate because of unexpected events, information disclosure, arbitrage behavior and so on. So some scholars proposed Jump-diffusion short-term interest rate model. The continuous-time model driven by Wiener process and Poisson process is called Jump-diffusion process. Such as Wu, Mao and Chen
[5] proved that Jump-diffusion square root process:
exists a unique solution and its Euler-Maruyama numerical solution converge to true solution. Jiang
[6] constructed Jump-diffusion CKLS short-term interest rate model of the form
under the conditions of
.
The continuous model and Jump-diffusion model mentioned above both assume that first term is a linear structure of interest rates and the coefficient of second term is a constant. While, as early as 1996, Alt-Sahalia
[7] proved drift term of short-term interest rate model is a nonlinear function of the interest rate using non-parametric methods. Buffington et al
[8], Yin, et al.
[9] and Liu, et al.
[10] proved that coefficients
of Equation
are not constants, rather than, they depend on Markov process, which is model:
.
In a sense, Markovian switching can be considered as a way extending a linear model to nonlinearly. On the other hand, Markovian switching deals with the influence of economic cycles (such as bull market and bear market) on the model parameters, and Diebold
[11] showed that Markovian switching has a sustained dependence on economic cycles. Mao, et al
[12] proved that Euler-Maruyama numerical solutions of Equation
is close to exact solution with probability 1, under conditions of
. Zou, et al.
[13] studied the analytical properties about solution and Euler-Maruyama numerical solutions of Equation
approach to exact solution with probability 1, under conditions of
.
In empirical studies, Ma, et al.
[14] introduced Poisson process and Markovian switching into volatility model of interest rate, empirical results showed that the volatility model based on Poisson process and Markovian switching is better than traditional volatility models in 2016. Hsu, et al.
[15] studied a model that relies on risk of jumping. This model could not only describe the conventional time series patten, but also describe the abnormal stochastic patten, which shows Markovian switching patten with risk-dependent size is superior to other models in 2015. Lee
[16] dealt with pricing of options which depends on regime-switching extenting fixed time interval to random time interval under jump-diffusions in 2014. Song, et al.[
17] discussed the pricing of American options under stochastic volatility jump model and stochastic volatility related jump model. Liu, et al.
[18] get conclusion that the corresponding stochastic volatility equation has the best ability to fit VIX index under when
by using empirical test method. Khaled, et al.
[19] solved the delay problem of regime switching via numerical method and mixing method and verified the robustness of a patten under regime switching.
In other fields, stochastic differential systems are also widely used. Such as Sheng, et al.
[20] constructed Hamilton-Jacobi-Bellman (HJB) equations and applied HJB equations research Optimal insurance portfolio. A, et al.
[21] used stochastic diffusion equation studied investment issues under the worst market conditions.
Empirical analysis and application are usually at the forefront of theoretical analysis. As mentioned above, many scholars have studied the stochastic differential equations driven by Poisson process and Markovian switching in the field of empirical analysis. However, in theoretical analysis, the research is still blank.
Based on the above analysis, to make up for the gap in theoretical analysis, our article discusses a class of Jump-diffusion stochastic differential system under Markovian switching:
which has a positive initial value, namely, and is a scalar Wiener process, is a pure scalar Poisson process whose jumping size is expressed by , is a right continuous Markov chain taking values in a finite state space . are located within the scope of within whole our article. Let be independent. are the mapping of .
For convenience of expression, Equation (1) will be recorded as JD-SDS-MS.
Our article is divided into five parts to elaborate specifically as below. In Section 2, we give properties of the solution to model JD-SDS-MS, including existence, uniqueness, non-negative and global nature by using Lyapunov function method. In Section 3, boundedness of first moment of the solution to this model is considered. In Section 4, properties about coefficients of this model is proved by using a right continuous Markov chain. In Section 5, our article defines approximate Euler-Maruyama numerical solutions of JD-SDS-MS by using Euler-Maruyama method. Further more, we prove that approximate Euler-Maruyama numerical solutions are close to exact solution with probability 1. In Section 6, the convergence of Euler-Maruyama numerical solutions is applied to financial quantities, taking the pricing of a bond as an example.
2 Solution: Existence, Uniqueness, Non-Negative and Global Nature
When using Equation (1) to describe variables such as stock prices in financial markets or other actual situations, the value of the solution to Equation (1) will always be positive, up to zero. Theorem 1 ensures that the solution to Equation (1) has some properties including existence, uniqueness, non-negative and global nature.
Theorem 1 For any given initial value , the solution to Equation has some properties including existence, uniqueness, non-negative and global nature on .
Proof We will prove existence, uniqueness and global nature of the solution firstly. Let be the point where jumping for th time, satisfies , define , namely, .
When , Equation (1) becomes into
Obviously, we have a constant , only depends on , of Equation (2) satisfying and . For , we have a constant , only depends on , of Equation (1) satisfying and . Then, for , we have a constant , only depends on , of Equation (1) satisfying and . So, for , which is the first value of , then the solution to Equation (1) has some properties including existence, uniqueness and local nature, namely, for is the solution with nature of uniqueness to Equation (1), is a point where the solution of Equation (1) will explode.
For proving is a global solution, we certify firstly. For , satisfying , construct a stopping time
We formulate in this whole article. Apparently, when . Let , by definition of , we have , so, . If when , so, meanwhile on . To prove is equal to proving , for convenience, we choose to prove .
Next, we will construct a non-negative nature of the solution by using Lyapunov function method. When , define a function
Obviously, for all and when or . Using the Jump-diffusion Itô formula, we have,
where
Then,
Hence,
Let , there is,
Due to boundedness of polynomial, we have a constant satisfying . On the other hand, because compensation Poisson process is a martingale, so,
Because of
together with boundedness of polynomial, we have a constant satisfying .
In summary, holds, with regard to , due to , therefore,
Because of , so, holds.
3 Boundedness of First Moment
We have proved that the solution to Equation (1) has some properties including existence, uniqueness, non-negative and global nature. This section will devote to boundedness of first moment of . Theorem 2 establishes boundedness of first moment.
Theorem 2 For , which is the first value of , , the solution of Equation satisfies
where .
Proof Equation (1) can be rewritten into integral form
For , construct a stopping time
then, we have
therefore, . Choosing , so applying Itô formula yields,
So, Letting , by Fatou lemma, we have
so, (1) holds.
4 Properties About Coefficients of JD-SDS-MS
In this section, we will prove some properties about coefficients of JD-SDS-MS by using the property of , which has been given in Equation (1).
As shown above, has been given in Equation (1), which has a generator as follows
There is an important definition about . Let , where is given in advance, . There is the following important lemma.
Lemma 1 Under Markovian switching, for any any small we have,
where is gived by the operator of .
Proof For any any small we have
By (8) and (9), we have,
Therefore, (10) holds.
Notation Under Markovian switching, similarly with the derivation process of (10), we have
It is noteworthy that those properties all can be proved by using the property of , which has been given in Equation (1).
5 Convergence of Euler-Maruyama Numerical Solutions
In general, only a few simple stochastic differential equations can find analytical solutions. It is difficult to find analytical solutions to complex stochastic differential equations such as Equation (1). In other words, we can't find the analytical solution of Equation (1). This section defines Euler-Maruyama numerical solutions of JD-SDS-MS by using Euler-Maruyama method. Further more, we prove that the discrete and continuous approximate Euler-Maruyama numerical solutions are close to exact solution with probability 1.
Based on , we define the discrete Euler-Maruyama numerical solution of Equation (1).
Given fixed timestep and ,
where is a Wiener process increment and is a Poisson process increment, .
The continuous-time model is easier to handle than the discrete-time model, so, we construct a continuous Euler-Maruyama numerical solution of Equation (1)
where . Obviously, and are step functions.
Theorem 3 ensures that continuous Euler-Maruyama numerical solution of Equation (1) converges to real solution in probability.
Theorem 3 Letting , for any , we have
Proof Step 1. Recalling that , which has been given in proof of Theorem 1. By (5), then
Step 2. For , construct a stopping time
When it comes to the property, is equal to . Define a function
For (15), applying Jump-diffusion Itô formula yields,
where
then
implies . Then we have a constant making
holds. By local Lipschitz conditions, we have a constant only depends on , satisfying
By (17) and (18), we have a constant only depends on , satisfying
Let , by (15),
For , we have a constant satisfying
By (19) and (21), we have
By , therefore,
Step 3. Let . This step will prove that we have a constant only depends on , satisfying
For , by Hölder inequality, gives
Let
By (10), we have a constant satisfying
By triangle inequality,
From (11), we have a constant , satisfying
By (21), (27) and (28), we have constants and satisfying,
By Burkholder-Davis-Gundy inequality,
From (12), we have a constant , satisfying
By local Lipschitz conditions, we have constants satisfying
By (21), (30) and (31), we have constants and , satisfying
Similarly,
From (13), we have a constant , satisfying
By (21), (33) and (34), we have constants and , satisfying
By (26), (29), (32) and (35), we have a constant satisfying
By Gronwall inequality, (24) holds. Namely, we have a constant satisfying
Step 4. For any small , set
By (24),
By (5) and (23), we have,
When , and hold. Choosing a sufficiently large satisfying
fix above , choosing a sufficiently small satisfying
so, we have
namely, (16) holds.
Theorem 3 has proved that the continuous Euler-Maruyama numerical solution of Equation (1) converges to real solution in probability, for ensuring the discrete Euler-Maruyama numerical solution of Equation (1) converging to real solution in probability, we need to prove Theorem 4 firstly.
Theorem 4 Letting , for any , we have
Proof By (20),
Due to Doob-martingale inequality, then
Applying Lyapunov inequality again, yields
Similarly,
For , by (38), (39) and (40), we have a constant , satisfying
For any small , set
By (41),
By (23) and (42), we have
When , holds. Choosing a sufficiently large satisfying
fix above , choosing a sufficiently small satisfying
so, we have
namely, (37) holds.
Theorem 5 ensures that discrete Euler-Maruyama Numerical Solution of Equation (1) converges to real solution in probability.
Theorem 5 Letting , for any , we have
Proof For any ,
By (16) and (37), (43) holds.
6 The Pricing of a Bond
In this section, the convergence of Euler-Maruyama numerical solutions is applied to financial quantities, taking the pricing of a bond as an example.
As shown above, is the solution of JD-SDS-MS, let is maturity date of bonds, the price of the bond at is expressed as follows
Using , we get an asymptotic value to as follows
Theorem 6 ensures that is close to with probability 1.
Theorem 6 Letting , for any , we have
Proof For any , it is sufficient to prove , namely, for any small , proving
Due to and the non-negative nature about , we have
Due to (43), (46) holds.
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