A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions

Xiangdong LIU, Zeyu MI, Huida CHEN

Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (1) : 17-32.

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Journal of Systems Science and Information ›› 2020, Vol. 8 ›› Issue (1) : 17-32. DOI: 10.21078/JSSI-2020-017-16
 

A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions

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Abstract

Our article discusses a class of Jump-diffusion stochastic differential system under Markovian switching (JD-SDS-MS). This model is generated by introducing Poisson process and Markovian switching based on a normal stochastic differential equation. Our work dedicates to analytical properties of solutions to this model. First, we give some properties of the solution, including existence, uniqueness, non-negative and global nature. Next, boundedness of first moment of the solution to this model is considered. Third, properties about coefficients of JD-SDS-MS is proved by using a right continuous markov chain. Last, we study the convergence of Euler-Maruyama numerical solutions and apply it to pricing bonds.

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stachastic differential system / Markovian switching / jump-diffusion / analytical properties / numerical solutions

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Xiangdong LIU , Zeyu MI , Huida CHEN. A Class of Jump-Diffusion Stochastic Differential System Under Markovian Switching and Analytical Properties of Solutions. Journal of Systems Science and Information, 2020, 8(1): 17-32 https://doi.org/10.21078/JSSI-2020-017-16

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: dSt=μdt+σdBt in 1973. Next, Vasicek[2] used Ornstein-Uhlenback process describing short-term interest rate in 1977, which is called Vasicek model: dSt=(μ+λSt)dt+σdBt. Cox, Ingersoll and Ross[3] proposed the well known CIR model: dSt=(μ+λSt)dt+σStdBt in 1980 and derived pricing formulas for bonds and bond's options. Chan, Karolyi, Longstaff and Sander[4] proposed CKLS model: dSt=α(μSt)dt+σStγdBt 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: dSt=(μ+λSt)dt+σStdBt+δStdNt 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 formdSt=α(μSt)dt+σStγdBt+δStdNtunder the conditions of γ>1.
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 α, μ, σ of Equation dSt=α(μSt)dt+σStγdBt are not constants, rather than, they depend on Markov process, which is model: dSt=α(m(t))[μ(m(t))St]dt+σ(m(t))StγdBt.
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 EquationdSt=α(m(t))[μ(m(t))St]dt+σ(m(t))StγdBt is close to exact solution with probability 1, under conditions of 12γ1. Zou, et al.[13] studied the analytical properties about solution and Euler-Maruyama numerical solutions of Equation dSt=α(m(t))[μ(m(t))St]dt+σ(m(t))StγdBt approach to exact solution with probability 1, under conditions of γ>1.
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 γ=32 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:
dSt=α(m(t))[μ(m(t))St]dt+β(m(t))StdBt+δ(m(t))StdNt,
(1)
which has a positive initial value, namely, S0>0 and Bt is a scalar Wiener process, Nt is a pure scalar Poisson process whose jumping size is expressed by λ, m(t) is a right continuous Markov chain taking values in a finite state space C={1,2,,M}. Bt, Nt, m(t) are located within the scope of (Ω,G,{Gt}t0,P) within whole our article. Let m(t), Bt and Nt be independent. α(), μ(), β(), δ() are the mapping of CR+.
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 St to Equation (1) will always be positive, up to zero. Theorem 1 ensures that the solution St to Equation (1) has some properties including existence, uniqueness, non-negative and global nature.
Theorem 1 For any given initial value S0=s0>0, m(0)=i0C, α(), μ(), β(), δ()>0, the solution St to Equation (1) has some properties including existence, uniqueness, non-negative and global nature on t0.
Proof We will prove existence, uniqueness and global nature of the solution St firstly. Let φk be the point where jumping for kth time, φk satisfies 0=φ0<φ1<φ2<φk<φk+1, define m(t)=m(φk), t[φk,φk+1), namely, m(t)=k=0m(φk)1[φk,φk+1)(t), t0.
When t[φ0,φ1)=[0,φ1), Equation (1) becomes into
dSt=α(i0)[μ(i0)St]dt+β(i0)StdBt+δ(i0)StdNt.
(2)
Obviously, we have a constant KN0, KN0 only depends on N0, α(i0), β(i0), δ(i0) of Equation (2) satisfying |α(i0)(x)α(i0)(y)|KN0|xy|, |β(i0)(x)β(i0)(y)|KN0|xy| and |δ(i0)(x)δ(i0)(y)|KN0|xy|, x, yR+. For  k>0, t[φk,φk+1), we have a constant KNk, KNk only depends on Nk, α(m(t)), β(m(t)), δ(m(t)) of Equation (1) satisfying |α(m(t))(x)α(m(t))(y)|KNk|xy|, |β(m(t))(x)β(m(t))(y)|KNk|xy| and |δ(m(t))(x)δ(m(t))(y)|KNk|xy|, x, yR+. Then, for t0, we have a constant KN, KN only depends on N, α(m(t)), β(m(t)), δ(m(t)) of Equation (1) satisfying |α(m(t))(x)α(m(t))(y)|KN|xy|, |β(m(t))(x)β(m(t))(y)|KN|xy| and |δ(m(t))(x)δ(m(t))(y)|KN|xy|, x, yR+. So, for s0>0, which is the first value of St, then the solution St to Equation (1) has some properties including existence, uniqueness and local nature, namely, for t[0,τe), St is the solution with nature of uniqueness to Equation (1), τe is a point where the solution of Equation (1) will explode.
For proving St is a global solution, we certify τe= almost sure firstly. For  k>0, satisfying 1k<s0<k, construct a stopping time
τk=inf{t[0,τe):St(1k,k)}.
We formulate inf= in this whole article. Apparently, τk1>τk2 when k1>k2. Let τ=limkτk, by definition of τk, we have τkt<τe, so, ττe almost sure. If τk almost sure when k, so, τe= almost sure meanwhile St>0 almost sure on t0. To prove τ= almost sure is equal to proving P(τ=)=1, for convenience, we choose to prove P(τ=)=1.
Next, we will construct a non-negative nature of the solution by using Lyapunov function method. When θ(0,1), define a function V:(0,+)[0,+),
V(x)=xθ1θlogx.
Obviously, V(x)>0 for all x>0 and V(x) when x or x0. Using the Jump-diffusion Itô formula, we have,
dV(St)=L0V(St)dt+L1V(St)dBt+L2V(St)dNt,
(3)
where
L0V(St)= θ{α(m(t))μ(m(t))(Stθ1St1)α(m(t))(Stθ1)+β2(m(t))2[(θ1)Stθ+1]},L1V(St)= β(m(t))θ(St1),L2V(St)= [1+δ(m(t))]θStθθlog[(1+δ(m(t)))St]Stθ+θlogSt.
Then,
V(Stτk)=V(s0)+0tτkL0V(Ss)ds+0tτkL1V(Ss)dBs+0tτkL2V(Ss)dNs.
Hence,
EV(Stτk)=V(s0)+E0tτkL0V(Ss)ds+E0tτkL1V(Ss)dBs+E0tτkL2V(Ss)dNs.
(4)
Let α¯=maxiCα(i), μ¯=maxiCμ(i), β¯=maxiCβ(i), δ¯=maxiCδ(i), α~=miniCα(i), μ~=miniCμ(i), β~=miniCβ(i), δ~=miniCδ(i), there is,
L0V(Ss) θ{α¯μ¯(Stθ1St1)α~(Stθ1)+β¯22[(θ1)Stθ+1]},
Due to boundedness of polynomial, we have a constant K1 satisfying L0V(Xs)K1. On the other hand, because compensation Poisson process is a martingale, so,
E0tτkL2V(Ss)dNs=E0tτkL2V(Ss)d(Ns~+λs)=E0tτkλL2V(Ss)ds.
Because of
λL2V(St)λ{(1+δ¯)θStθθlog[(1+δ~)St]Stθ+θlogSt},
together with boundedness of polynomial, we have a constant K2 satisfying λL2V(Xs)K2.
In summary, EV(Stτk)V(s0)+(K1+K2)T holds, with regard to  t[0,T], due to  EV(Stτk)P(τkT)[V(1k)V(k)], therefore,
P(τkT)EV(Stτk)V(1k)V(k)V(s0)+(K1+K2)TV(1k)V(k).
(5)
Because of V(1k)V(k) as k, so, P(τ=)=1 holds.

3 Boundedness of First Moment

We have proved that the solution St to Equation (1) has some properties including existence, uniqueness, non-negative and global nature. This section will devote to boundedness of first moment of St. Theorem 2 establishes boundedness of first moment.
Theorem 2 For S0=s0>0, which is the first value of St, α~>δ¯λ, the solution St of Equation (1) satisfies
EStα¯μ¯α~δ¯λ+s0=^κ,
(6)
where α¯=maxiCα(i), μ¯=maxiCμ(i), δ¯=maxiCδ(i), α~=miniCα(i).
Proof Equation (1) can be rewritten into integral form
St=s0+0tα(m(s))[μ(m(s))Ss]ds+0tβ(m(s))SsdBs+0tδ(m(s))SsdNs.
(7)
For  n>0, construct a stopping time
τn=inf{t:St>n},
then, we have
EStτn=Es0+E0tτnα(m(s))[μ(m(s))Ss]ds+E0tτnσ(m(s))SsdBs+E0tτnδ(m(s))SsdN~s+E0tτnλδ(m(s))Ssds=Es0+E0tτn{α(m(s))μ(m(s))[α(m(s))λδ(m(s))]}SsdsEs0+0tτn[α¯μ¯(α~λδ¯)ESs]ds
therefore, dEStτn[α¯μ¯(α~λδ¯)EStτn]dt. Choosing f(t,x)=e(α~λδ¯)tx, so ft=(α~λδ¯)e(α~λδ¯)tx, fx=e(α~λδ¯)t, 2fx2=0, applying Itô formula yields,
e(α~λδ¯)(tτn)EStτnEs0+0tτn(α~λδ¯)e(α~λδ¯)(sτn)ESsτnds+0tτne(α~λδ¯)(sτn)[α¯μ¯(α~λδ¯)ESsτn]ds.
So, EStτn e(α~λδ¯)(tτn)Es0+α¯μ¯α~λδ¯α¯μ¯α~λδ¯e(α~λδ¯)(tτn)α¯μ¯α~λδ¯+s0. Letting n, by Fatou lemma, we have
ESt=Elim infnStτnlim infnEStτnlim infn[α¯μ¯α~λδ¯+s0]=α¯μ¯α~λδ¯+s0,
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 m(t), which has been given in Equation (1).
As shown above, m(t) has been given in Equation (1), which has a generator Γ=(κij)M×M as follows
P(m(t+Δ)=j|m(t)=i)={κijΔ+o(Δ),ij,1+κijΔ+o(Δ),i=j,
(8)
κii=ijκij.
(9)
There is an important definition about m(t). Let m¯(t)=mkΔ=m(kΔ), where Δ>0 is given in advance, t[tk,tk+1), k=0,1,2,. There is the following important lemma.
Lemma 1 Under Markovian switching, for any t[0,T], any small Δ(0,1), we have,
E0t[α(m(s))μ(m(s))α(m¯(s))μ(m¯(s))]2ds4α¯2μ¯2max1iM(κii)Δ+o(Δ),
(10)
where α¯=maxiCα(i), μ¯=maxiCμ(i), κii is gived by the operator Γ=(κij)M×M of m(t).
Proof For any t[0,T], any small Δ(0,1), we have
E0t[α(m(s))μ(m(s))α(m¯(s))μ(m¯(s))]2ds Ed=0[TΔ]tdtd+14α¯2μ¯2(1m(s)m(td))ds= 4α¯2μ¯2Td=0[TΔ]tdtd+1E(1m(s)m(td))ds= 4α¯2μ¯2Td=0[TΔ]tdtd+1P(m(s)m(td))ds= 4α¯2μ¯2Td=0[TΔ]tdtd+1iCP(m(td)=i)P(m(s)i|m(td)=i)ds.
By (8) and (9), we have,
E0t[α(m(s))μ(m(s))α(m¯(s))μ(m¯(s))]2ds 4α¯2μ¯2Td=0[TΔ]tdtd+1iCP(m(td)=i)P(m(s)i|m(td)=i)ds 4α¯2μ¯2Td=0[TΔ]tdtd+1iCP(m(td)=i)[ijκij(std)+o(std)]ds 4α¯2μ¯2Td=0[TΔ]tdtd+1iCP(m(td)=i)[max1iM(κii)Δ+o(Δ)]ds 4α¯2μ¯2max1iM(κii)Δ+o(Δ).
Therefore, (10) holds.
Notation Under Markovian switching, similarly with the derivation process of (10), we have
E0t[α(m(s))α(m¯(s))]2dsα¯2max1iM(κii)Δ+o(Δ),
(11)
E0t[β(m(s))β(m¯(s))]2dsβ¯2max1iM(κii)Δ+o(Δ),
(12)
E0t[δ(m(s))δ(m¯(s))]2dsδ¯2max1iM(κii)Δ+o(Δ).
(13)
It is noteworthy that those properties all can be proved by using the property of m(t), 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 m¯(t)=mkΔ=m(kΔ), we define the discrete Euler-Maruyama numerical solution of Equation (1).
Given fixed timestep Δ(0,1) and S0=s0,
sk+1=sk+α(mkΔ)[μ(mkΔ)sk]Δ+β(mkΔ)|sk|ΔBk+δ(mkΔ)skΔNk,
(14)
where ΔBk=Btk+1Btk is a Wiener process increment and ΔNk=Ntk+1Ntk is a Poisson process increment, k=0,1,2,, tk=kΔ, mkΔ=m(kΔ).
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)
st=s0+0tα(m¯(s))[μ(m¯(s))s¯s]ds+0tβ(m¯(s))|s¯s|dBs+0tδ(m¯(s))s¯sdNs,
(15)
where s¯t=sk, m¯(t)=mkΔ, t[tk,tk+1), k=0,1,2,. Obviously, s¯t and m¯(t) are step functions.
Theorem 3 ensures that continuous Euler-Maruyama numerical solution of Equation (1) converges to real solution in probability.
Theorem 3 Letting Δ0, for any ε, ξ>0, T>0, we have
P(sup0tT|Stst|2ξ)<ε.
(16)
Proof Step 1. Recalling that τk=inf{t[0,τe):St(1k,k)}, which has been given in proof of Theorem 1. By (5), then
P(τkT)V(s0)+(K1+K2)TV(1k)V(k).
Step 2. For  k>0, construct a stopping time
ρk=inf{t[0,T]:st[1k,k]}.
When it comes to the property, ρk is equal to τk. Define a function V1:(0,+)[0,+),
V1(x)=x12112logx.
For (15), applying Jump-diffusion Itô formula yields,
dV1(st)=L0V1(st)dt+L1V1(st)dBt+L2V1(st)dNt,
where
L0V(St)= 12{α(m(t))μ(m(t))(St12St1)α(m(t))(St121)+β2(m(t))2[(12)St12+1]},L1V(St)= 12β(m(t))(St1),L2V(St)= [1+δ(m(t))]12St1212log[(1+δ(m(t)))St]St12+12logSt,
then
dV1(st)=12{α(m(t))μ(m(t))(St12St1)α(m(t))(St121)+β2(m(t))2[(12)St12+1]}dt+12β(m(t))(St1)dBt+[1+δ(m(t))]12St1212log[(1+δ(m(t)))St]St12+12logStdNt.
(17)
ss[1k,k] implies s¯s[1k,k]. Then we have a constant C1 making
α(m¯(s))μ(m¯(s))2ss122α(m¯(s))μ(m¯(s))ss3+λ[(ss+δ(m¯(s))s¯s)12+(ss+δ(m¯(s))s¯s)2ss12ss2]α(m¯(s))μ(m¯(s))2ss12+2α(m¯(s))μ(m¯(s))ss2σ2(m¯(s))8ss12+3β2(m¯(s))ss2α¯μ¯2ss122α~μ~ss3+λ[(ss+δ¯s¯s)12+(ss+δ¯s¯s)2ss12ss2]α~μ~2ss12+2α¯μ¯ss2β~28ss12+3β~2ss2C1
(18)
holds. By local Lipschitz conditions, we have a constant C2(k), C2(k) only depends on k, satisfying
|ss2s¯s2|C2(k)|sss¯s|.
By (17) and (18), we have a constant C3(k), C3(k) only depends on k, satisfying
EV1(stρk)V1(s0)+C1T+C3(k)E0tρk|sss¯s|ds.
(19)
Let s[0,tρk], by (15),
sss¯s=sssk=sss[sΔ]=α(m¯(s))(μ(m¯(s))s[sΔ])(s[sΔ])+β(m¯(s))|s[sΔ]|(BsB[sΔ]Δ)+δ(m¯(s))s[sΔ](NsN[sΔ]Δ)α¯(μ¯+k)(s[sΔ]Δ)+β¯k(BsB[sΔ]Δ)+δ¯k(NsN[sΔ]Δ).
(20)
For Δ(0,1), we have a constant C4 satisfying
E0tρk|sss¯s|dsα¯(μ¯+k)TΔ+β¯kE0tρk|BsB[sΔ]Δ|ds+δ¯kE0tρk|NsN[sΔ]Δ|ds α¯(μ¯+k)TΔ+β¯k0TE|BsB[sΔ]Δ|ds+β¯k0TE|NsN[sΔ]Δ|ds α¯(μ¯+k)TΔ+β¯kΔ12+δ¯kλΔ12[α¯(μ¯+k)T+β¯k+δ¯kλ]Δ12=^ C4Δ12.
(21)
By (19) and (21), we have
EV1(stρk)V1(s0)+C1T+C3(k)C4Δ12.
(22)
By EV1(stρk)P(ρkT)[V1(1k)V1(k)], therefore,
P(ρkT)EV1(stρk)V1(1k)V1(k)V1(s0)+C1T+C3(k)C4Δ12V1(1k)V1(k).
(23)
Step 3. Let θk=τkρk. This step will prove that we have a constant D(k), D(k) only depends on k, satisfying
E(sup0tθkT|Stst|2)D(k)Δ.
(24)
For  t1, t[0,T], by Hölder inequality, gives
E(sup0t1t|St1θkst1θk|2)4TE0tθk[α(m(s))μ(m(s))α(m¯(s))μ(m¯(s))]2ds+4TE0tθk[α(m(s))Ssα(m¯(s))s¯s]2ds+4Esup0t1t(0t1θk[β(m(s))Ssβ(m¯(s))|s¯s|]dBs)2+4Esup0t1t(0t1θk[δ(m(s))Ssδ(m¯(s))s¯s]dNs)2.
(25)
Let
A(t)= 4TE0tθk[α(m(s))μ(m(s))α(m¯(s))μ(m¯(s))]2ds.B(t)= 4TE0tθk[α(m(s))Ssα(m¯(s))s¯s]2ds.C(t)= 4Esup0t1t(0t1θk[σ(m(s))Ssσ(m¯(s))|s¯s|]dBs)2.D(t)= 4Esup0t1t(0t1θk[δ(m(s))Ssδ(m¯(s))s¯s]dNs)2.
By (10), we have a constant C5(k) satisfying
A(t)16Tα¯2μ¯2max1iM(κii)Δ+o(Δ)C5(k)Δ+o(Δ).
(26)
By triangle inequality,
B(t)=4TE0tθk[α(m(s))Ssα(m¯(s))Ss+α(m¯(s))Ssα(m¯(s))s¯s]2ds8TE0tθk([α(m(s))α(m¯(s))]Ss)2ds+8TE0tθk[α(m¯(s))(Sss¯s)]2ds8k2TE0tθk[α(m(s))α(m¯(s))]2ds+16α¯2TE0tθk(Ssss)2ds+16α¯2TE0tθk(sss¯s)2ds.
(27)
From (11), we have a constant C6(k), satisfying
E0tθk[α(m(s))α(m¯(s))]2dsα¯2max1iM(κii)Δ+o(Δ)C6(k)Δ+o(Δ).
(28)
By (21), (27) and (28), we have constants C7(k) and C8(k) satisfying,
B(t)C7(k)Δ+o(Δ)+16α¯2TC8(k)Δ+16α¯2TE0tθk(Ssss)2ds.
(29)
By Burkholder-Davis-Gundy inequality,
C(t)16E0tθk[β(m(s))Ssβ(m¯(s))s¯s]2ds32k2E0tθk[β(m(s))β(m¯(s))]2ds+64β¯2E0tθk(Ssss)2ds+64β¯2E0tθk(sss¯s)2ds.
(30)
From (12), we have a constant C9(k), satisfying
E0tθk[β(m(s))β(m¯(s))]2dsβ¯2max1iM(κii)Δ+o(Δ)C9(k)Δ+o(Δ).
(31)
By local Lipschitz conditions, we have constants C10(k), C11(k) satisfying
|Ssss|2C10(k)|Ssss|2, |sss¯s|2C11(k)|sss¯s|2.
By (21), (30) and (31), we have constants C12(k), C13(k) and C14(k), satisfying
C(t)C12(k)Δ+o(Δ)+64β¯C13(k)Δ+64β¯C14(k)E0tθk(Ssss)2ds.
(32)
Similarly,
D(t) 64λk2E0tθk[δ(m(s))δ(m¯(s))]2ds+128λδ¯2E0tθk(Ssss)2ds+128λδ¯2E0tθk(sss¯s)2ds+16λ2k2TE0tθk[δ(m(s))δ(m¯(s))]2ds+32λ2δ¯2TE0tθk(Ssss)2ds+32λ2δ¯2TE0tθk(sss¯s)2ds.
(33)
From (13), we have a constant C15(k), satisfying
E0tθk[δ(m(s))δ(m¯(s))]2dsδ¯2max1iM(κii)Δ+o(Δ)C15(k)Δ+o(Δ).
(34)
By (21), (33) and (34), we have constants C16(k) and C17(k), satisfying
D(t) C16(k)Δ+o(Δ)+(128λδ¯2+32λ2δ¯2T)C17(k)Δ +(128λδ¯2+32λ2δ¯2T)E0tθk(Ssss)2ds.
(35)
By (26), (29), (32) and (35), we have a constant C18(k) satisfying
E(sup0t1t|St1θkst1θk|2) C18(k)Δ+o(Δ)+(16α¯2T+64β¯2C14(k) +128λδ¯2+32λ2δ¯2T)E0tθk(Ssss)2ds.
(36)
By Gronwall inequality, (24) holds. Namely, we have a constant D(k) satisfying
E(sup0tθkT|Stst|2)D(k)Δ.
Step 4. For any small ε, ξ(0,1), set
Ω¯={ω:sup0tT|Stst|2ξ}.
By (24),
ξP(Ω¯{θkT})=E{1{θkT} 1{Ω¯} ξ}E{sup0tT|Stst|2}D(k)Δ.
By (5) and (23), we have,
P(Ω¯)P(Ω¯{θkT})+P(θkT)P(Ω¯{θkT})+P(τkT)+P(ρkT)D(k)ξΔ+V(s0)+(K1+K2)TV(1k)V(k)+V1(s0)+C1T+C3(k)C4Δ12V1(1k)V1(k).
When k,  V(1k)V(k) and V1(1k)V1(k) hold. Choosing a sufficiently large k satisfying
V(s0)+(K1+K2)TV(1k)V(k)+V1(s0)+C1TV1(1k)V1(k)<ε2,
fix above k, choosing a sufficiently small Δ satisfying
D(k)ξΔ+C3(k)C4Δ12V1(1k)V1(k)<ε2.
so, we have
P(Ω¯)=P(sup0tT|Stst|2ξ)<ε,
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 Δ0, for any ε, ξ>0, T>0, we have
P(sup0tT|sts¯t|2ξ)<ε.
(37)
Proof By (20),
E(sup0tT|sts¯t|2) 3α¯2(μ¯+k)2Δ2+3β¯2k2E(sup0tT|BtB[tΔ]Δ|2) +3δ¯2k2E(sup0tT|NtN[tΔ]Δ|2).
(38)
Due to Doob-martingale inequality, then
E(sup0tT|BtB[tΔ]Δ|4)(43)4k=0[TΔ]1E|B(k+1)ΔBkΔ|44433TΔ,
Applying Lyapunov inequality again, yields
E(sup0tT|BtB[tΔ]Δ|2)(E(sup0tT|BtB[tΔ]Δ|4))241693T12Δ12.
(39)
Similarly,
E(sup0tT|NtN[tΔ]Δ|2)k=0[TΔ]1E(supkΔt(k+1)Δ|NtN[tΔ]Δ|2)2k=0[TΔ]1E(supkΔt(k+1)Δ|N~tN~[tΔ]Δ|2+supkΔt(k+1)Δ|λ(t[tΔ]Δ)|2)k=0[TΔ]1(8E|N~(k+1)ΔN~kΔ|2+2E|λΔ|2)T(8λΔ+2λ2Δ2).
(40)
For Δ(0,1), by (38), (39) and (40), we have a constant D1, satisfying
E(sup0tT|sts¯t|2)D1Δ12.
(41)
For any small ε, ξ(0,1), set
Ω~={ω:sup0tT|sts¯t|2ξ}.
By (41),
ξP(Ω~{ρkT})=E{1{ρkT} 1{Ω~} ξ}E{sup0tT|sts¯t|2}D1Δ12.
(42)
By (23) and (42), we have
P(Ω~)P(Ω~{ρkT})+P(ρkT)D1ξΔ12+V1(s0)+C1T+C3(k)C4Δ12V1(1k)V1(k).
When k, V1(1k)V1(k) holds. Choosing a sufficiently large k satisfying
V1(s0)+C1TV1(1k)V1(k)<ε2,
fix above k, choosing a sufficiently small Δ satisfying
D1ξΔ12+C3(k)C4Δ12V(1k)V(k)<ε2,
so, we have
P(Ω~)=P(sup0tT|sts¯t|2ξ)<ε,
namely, (37) holds.
Theorem 5 ensures that discrete Euler-Maruyama Numerical Solution of Equation (1) converges to real solution in probability.
Theorem 5 Letting Δ0, for any ε, ξ>0, T>0, we have
P(sup0tT|Sts¯t|ξ)<ε.
(43)
Proof For any ξ>0,
P(sup0tT|Sts¯t|ξ)P(sup0tT|Stst|ξ2)+P(sup0tT|sts¯t|ξ2).
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, St is the solution of JD-SDS-MS, let T is maturity date of bonds, the price of the bond at T is expressed as follows
B(T)=E[exp(0TStdt)].
(44)
Using s¯t, we get an asymptotic value to B(T) as follows
B¯Δ(T)=E[exp(0T|s¯t|dt)].
(45)
Theorem 6 ensures that B¯Δ(T) is close to B(T) with probability 1.
Theorem 6 Letting Δ0, for any ε, ξ>0, we have
P(|B(T)B¯Δ(T)|ξ)<ε.
(46)
Proof For any ε, ξ>0, it is sufficient to prove P(|B(T)B¯Δ(T)|ξ)<ε, namely, for any small ε, ξ>0, proving
P[|exp(0TStdt)exp(0T|s¯t|dt)|ξ]<ε.
Due to exp(|x|)exp(|y|)|xy| and the non-negative nature about St, we have
|exp(0TStdt)exp(0T|s¯t|dt)||0T[Sts¯t]dt|Tsup0tT|Sts¯t|.
Due to (43), (46) holds.

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Funding

the National Natural Science Foundation of China(71471075)
Fundamental Research Funds for the Central University(19JNLH09)
Humanities and Social Sciences Foundation of Ministry of Education, China(14YJAZH052)
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