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

Xiangdong LIU; Zeyu MI; Huida CHEN

doi:10.21078/JSSI-2020-017-16

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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.

Key words

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:

d S_{t} = μ d t + σ d B_{t}

in 1973. Next, Vasicek^[2] used Ornstein-Uhlenback process describing short-term interest rate in 1977, which is called Vasicek model:

d S_{t} = (μ + λ S_{t}) d t + σ d B_{t}

. Cox, Ingersoll and Ross^[3] proposed the well known CIR model:

d S_{t} = (μ + λ S_{t}) d t + σ \sqrt{S_{t}} d B_{t}

in 1980 and derived pricing formulas for bonds and bond's options. Chan, Karolyi, Longstaff and Sander^[4] proposed CKLS model:

d S_{t} = α (μ - S_{t}) d t + σ S_{t}^{γ} d B_{t}

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:

d S_{t} = (μ + λ S_{t}) d t + σ {\sqrt{S}}_{t} d B_{t} + δ S_{t} d N_{t}

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

d S_{t} = α (μ - S_{t}) d t + σ S_{t}^{γ} d B_{t} + δ S_{t} d N_{t}

under 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

d S_{t} = α (μ - S_{t}) d t + σ S_{t}^{γ} d B_{t}

are not constants, rather than, they depend on Markov process, which is model:

d S_{t} = α (m (t)) [μ (m (t)) - S_{t}] d t + σ (m (t)) S_{t}^{γ} d B_{t}

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

d S_{t} = α (m (t)) [μ (m (t)) - S_{t}] d t + σ (m (t)) S_{t}^{γ} d B_{t}

is close to exact solution with probability 1, under conditions of

\frac{1}{2} \leq γ \leq 1

. Zou, et al.^[13] studied the analytical properties about solution and Euler-Maruyama numerical solutions of Equation

d S_{t} = α (m (t)) [μ (m (t)) - S_{t}] d t + σ (m (t)) S_{t}^{γ} d B_{t}

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 = \frac{3}{2}

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:

d S_{t} = α (m (t)) [μ (m (t)) - S_{t}] d t + β (m (t)) S_{t} d B_{t} + δ (m (t)) S_{t} d N_{t},

(1)

which has a positive initial value, namely,

S_{0} > 0

and

B_{t}

is a scalar Wiener process,

N_{t}

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, \dots, M}

B_{t}, N_{t}, m (t)

are located within the scope of

(Ω, G, {G_{t}}_{t \geq 0}, P)

within whole our article. Let

m (t), B_{t} a n d N_{t}

be independent.

α (\cdot), μ (\cdot), β (\cdot), δ (\cdot)

are the mapping of

C \to R^{+}

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

S_{t}

to Equation (1) will always be positive, up to zero. Theorem 1 ensures that the solution

S_{t}

to Equation (1) has some properties including existence, uniqueness, non-negative and global nature.

Theorem 1 For any given initial value $S_{0} = s_{0} > 0, m (0) = i_{0} \in C, α (\cdot), μ (\cdot), β (\cdot), δ (\cdot) > 0$ , the solution $S_{t}$ to Equation $(1)$ has some properties including existence, uniqueness, non-negative and global nature on $t \geq 0$ .

Proof We will prove existence, uniqueness and global nature of the solution

S_{t}

firstly. Let

φ_{k}

be the point where jumping for

k

th time,

φ_{k}

satisfies

0 = φ_{0} < φ_{1} < φ_{2} \dots < φ_{k} < φ_{k + 1} \to \infty

, define

m (t) = m (φ_{k}), t \in [φ_{k}, φ_{k + 1})

, namely,

m (t) = \sum_{k = 0}^{\infty} m (φ_{k}) 1_{[φ_{k}, φ_{k + 1})} (t), t \geq 0

When

t \in [φ_{0}, φ_{1}) = [0, φ_{1})

, Equation (1) becomes into

d S_{t} = α (i_{0}) [μ (i_{0}) - S_{t}] d t + β (i_{0}) S_{t} d B_{t} + δ (i_{0}) S_{t} d N_{t} .

(2)

Obviously, we have a constant

K_{N_{0}}

K_{N_{0}}

only depends on

N_{0}

α (i_{0}), β (i_{0}), δ (i_{0})

of Equation (2) satisfying

| α (i_{0}) (x) - α (i_{0}) (y) | \leq K_{N_{0}} | x - y |, | β (i_{0}) (x) - β (i_{0}) (y) | \leq K_{N_{0}} | x - y |

and

| δ (i_{0}) (x) - δ (i_{0}) (y) | \leq K_{N_{0}} | x - y |, x, y \in R^{+}

. For

\forall k > 0, t \in [φ_{k}, φ_{k + 1})

, we have a constant

K_{N_{k}}

K_{N_{k}}

only depends on

N_{k}

α (m (t)), β (m (t)), δ (m (t))

of Equation (1) satisfying

| α (m (t)) (x) - α (m (t)) (y) | \leq K_{N_{k}} | x - y |, | β (m (t)) (x) - β (m (t)) (y) | \leq K_{N_{k}} | x - y |

and

| δ (m (t)) (x) - δ (m (t)) (y) | \leq K_{N_{k}} | x - y |, x, y \in R^{+}

. Then, for

t \geq 0

, we have a constant

K_{N}

K_{N}

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) | \leq K_{N} | x - y |, | β (m (t)) (x) - β (m (t)) (y) | \leq K_{N} | x - y |

and

| δ (m (t)) (x) - δ (m (t)) (y) | \leq K_{N} | x - y |, x, y \in R^{+}

. So, for

s_{0} > 0

, which is the first value of

S_{t}

, then the solution

S_{t}

to Equation (1) has some properties including existence, uniqueness and local nature, namely, for

t \in [0, τ_{e}), S_{t}

is the solution with nature of uniqueness to Equation (1),

τ_{e}

is a point where the solution of Equation (1) will explode.

For proving

S_{t}

is a global solution, we certify

τ_{e} = \infty a l m o s t s u r e

firstly. For

\forall k > 0

, satisfying

\frac{1}{k} < s_{0} < k

, construct a stopping time

τ_{k} = inf {t \in [0, τ_{e}) : S_{t} \notin (\frac{1}{k}, k)} .

We formulate

inf \emptyset = \infty

in this whole article. Apparently,

τ_{k_{1}} > τ_{k_{2}}

when

k_{1} > k_{2}

. Let

τ_{\infty} = lim_{k \to \infty} τ_{k}

, by definition of

τ_{k}

, we have

τ_{k} \leq t < τ_{e}

, so,

τ_{\infty} \leq τ_{e} a l m o s t s u r e

. If

τ_{k} \to \infty a l m o s t s u r e

when

k \to \infty

, so,

τ_{e} = \infty a l m o s t s u r e

meanwhile

S_{t} > 0 a l m o s t s u r e

t \geq 0

. To prove

τ_{\infty} = \infty a l m o s t s u r e

is equal to proving

P (τ_{\infty} = \infty) = 1

, for convenience, we choose to prove

P (τ_{\infty} = \infty) = 1

Next, we will construct a non-negative nature of the solution by using Lyapunov function method. When

θ \in (0, 1)

, define a function

V : (0, + \infty) \to [0, + \infty),

V (x) = x^{θ} - 1 - θ \log x .

Obviously,

V (x) > 0

for all

x > 0

and

V (x) \to \infty

when

x \to \infty

x \to 0

. Using the Jump-diffusion Itô formula, we have,

d V (S_{t}) = L_{0} V (S_{t}) d t + L_{1} V (S_{t}) d B_{t} + L_{2} V (S_{t}) d N_{t},

(3)

where

\begin{aligned} L_{0} V (S_{t}) = & θ {α (m (t)) μ (m (t)) (S_{t}^{θ - 1} - S_{t}^{- 1}) - α (m (t)) (S_{t}^{θ} - 1) + \frac{β^{2} (m (t))}{2} [(θ - 1) S_{t}^{θ} + 1]}, \\ L_{1} V (S_{t}) = & β (m (t)) θ (S_{t} - 1), \\ L_{2} V (S_{t}) = & {[1 + δ (m (t))]}^{θ} S_{t}^{θ} - θ \log [(1 + δ (m (t))) S_{t}] - S_{t}^{θ} + θ \log S_{t} . \end{aligned}

Then,

\begin{aligned} V (S_{t \land τ_{k}}) = V (s_{0}) + \int_{0}^{t \land τ_{k}} L_{0} V (S_{s}) d s + \int_{0}^{t \land τ_{k}} L_{1} V (S_{s}) d B_{s} + \int_{0}^{t \land τ_{k}} L_{2} V (S_{s}) d N_{s} . \end{aligned}

Hence,

\begin{aligned} E V (S_{t \land τ_{k}}) = V (s_{0}) + E \int_{0}^{t \land τ_{k}} L_{0} V (S_{s}) d s + E \int_{0}^{t \land τ_{k}} L_{1} V (S_{s}) d B_{s} + E \int_{0}^{t \land τ_{k}} L_{2} V (S_{s}) d N_{s} . \end{aligned}

(4)

Let

\bar{α} = max_{i \in C} α (i), \bar{μ} = max_{i \in C} μ (i), \bar{β} = max_{i \in C} β (i), \bar{δ} = max_{i \in C} δ (i), \tilde{α} = min_{i \in C} α (i),

\tilde{μ} = min_{i \in C} μ (i), \tilde{β} = min_{i \in C} β (i), \tilde{δ} = min_{i \in C} δ (i)

, there is,

\begin{aligned} L_{0} V (S_{s}) \leq & θ {\bar{α} \bar{μ} (S_{t}^{θ - 1} - S_{t}^{- 1}) - \tilde{α} (S_{t}^{θ} - 1) + \frac{{\bar{β}}^{2}}{2} [(θ - 1) S_{t}^{θ} + 1]}, \end{aligned}

Due to boundedness of polynomial, we have a constant

K_{1}

satisfying

L_{0} V (X_{s}) \leq K_{1}

. On the other hand, because compensation Poisson process is a martingale, so,

\begin{aligned} E \int_{0}^{t \land τ_{k}} L_{2} V (S_{s}) d N_{s} = E \int_{0}^{t \land τ_{k}} L_{2} V (S_{s}) d (\tilde{N_{s}} + λ s) = E \int_{0}^{t \land τ_{k}} λ L_{2} V (S_{s}) d s . \end{aligned}

Because of

λ L_{2} V (S_{t}) \leq λ {{(1 + \bar{δ})}^{θ} S_{t}^{θ} - θ \log [(1 + \tilde{δ}) S_{t}] - S_{t}^{θ} + θ \log S_{t}},

together with boundedness of polynomial, we have a constant

K_{2}

satisfying

λ L_{2} V (X_{s}) \leq K_{2}

In summary,

E V (S_{t \land τ_{k}}) \leq V (s_{0}) + (K_{1} + K_{2}) T

holds, with regard to

\forall t \in [0, T]

, due to

E V (S_{t \land τ_{k}}) \geq P (τ_{k} \leq T) [V (\frac{1}{k}) \land V (k)]

, therefore,

P (τ_{k} \leq T) \leq \frac{E V (S_{t \land τ_{k}})}{V (\frac{1}{k}) \land V (k)} \leq \frac{V (s_{0}) + (K_{1} + K_{2}) T}{V (\frac{1}{k}) \land V (k)} .

(5)

Because of

V (\frac{1}{k}) \land V (k) \to \infty a s k \to \infty

, so,

P (τ_{\infty} = \infty) = 1

holds.

3 Boundedness of First Moment

We have proved that the solution

S_{t}

to Equation (1) has some properties including existence, uniqueness, non-negative and global nature. This section will devote to boundedness of first moment of

S_{t}

. Theorem 2 establishes boundedness of first moment.

Theorem 2 For $S_{0} = s_{0} > 0$ , which is the first value of $S_{t}$ , $\tilde{α} > \bar{δ} λ$ , the solution $S_{t}$ of Equation $(1)$ satisfies

E S_{t} \leq \frac{\bar{α} \bar{μ}}{\tilde{α} - \bar{δ} λ} + s_{0} \hat{=} κ,

(6)

where $\bar{α} = max_{i \in C} α (i), \bar{μ} = max_{i \in C} μ (i), \bar{δ} = max_{i \in C} δ (i), \tilde{α} = min_{i \in C} α (i)$ .

Proof Equation (1) can be rewritten into integral form

\begin{aligned} S_{t} = s_{0} + \int_{0}^{t} α (m (s)) [μ (m (s)) - S_{s}] d s + \int_{0}^{t} β (m (s)) S_{s} d B_{s} + \int_{0}^{t} δ (m (s)) S_{s} d N_{s} . \end{aligned}

(7)

For

\forall n > 0

, construct a stopping time

τ_{n} = inf {t : S_{t} > n},

then, we have

\begin{aligned} E S_{t \land τ_{n}} = & E s_{0} + E \int_{0}^{t \land τ_{n}} α (m (s)) [μ (m (s)) - S_{s}] d s + E \int_{0}^{t \land τ_{n}} σ (m (s)) S_{s} d B_{s} \\ + E \int_{0}^{t \land τ_{n}} δ (m (s)) S_{s} d {\tilde{N}}_{s} + E \int_{0}^{t \land τ_{n}} λ δ (m (s)) S_{s} d s \\ = & E s_{0} + E \int_{0}^{t \land τ_{n}} {α (m (s)) μ (m (s)) - [α (m (s)) - λ δ (m (s))]} S_{s} d s \\ \leq & E s_{0} + \int_{0}^{t \land τ_{n}} [\bar{α} \bar{μ} - (\tilde{α} - λ \bar{δ}) E S_{s}] d s \end{aligned}

therefore,

d E S_{t \land τ_{n}} \leq [\bar{α} \bar{μ} - (\tilde{α} - λ \bar{δ}) E S_{t \land τ_{n}}] d t

. Choosing

f (t, x) = e^{(\tilde{α} - λ \bar{δ}) t} x

, so

\frac{\partial f}{\partial t} = (\tilde{α} - λ \bar{δ}) e^{(\tilde{α} - λ \bar{δ}) t} x, \frac{\partial f}{\partial x} = e^{(\tilde{α} - λ \bar{δ}) t}, \frac{\partial^{2} f}{\partial x^{2}} = 0,

applying Itô formula yields,

\begin{aligned} e^{(\tilde{α} - λ \bar{δ}) (t \land τ_{n})} E S_{t \land τ_{n}} \leq & E s_{0} + \int_{0}^{t \land τ_{n}} (\tilde{α} - λ \bar{δ}) e^{(\tilde{α} - λ \bar{δ}) (s \land τ_{n})} E S_{s \land τ_{n}} d s \\ + \int_{0}^{t \land τ_{n}} e^{(\tilde{α} - λ \bar{δ}) (s \land τ_{n})} [\bar{α} \bar{μ} - (\tilde{α} - λ \bar{δ}) E S_{s \land τ_{n}}] d s . \end{aligned}

So,

E S_{t \land τ_{n}} \leq e^{- (\tilde{α} - λ \bar{δ}) (t \land τ_{n})} E s_{0} + \frac{\bar{α} \bar{μ}}{\tilde{α} - λ \bar{δ}} - \frac{\bar{α} \bar{μ}}{\tilde{α} - λ \bar{δ}} e^{- (\tilde{α} - λ \bar{δ}) (t \land τ_{n})} \leq \frac{\bar{α} \bar{μ}}{\tilde{α} - λ \bar{δ}} + s_{0} .

Letting

n \to \infty

, by Fatou lemma, we have

\begin{aligned} E S_{t} = E \underset{n \to \infty}{lim inf} S_{t \land τ_{n}} \leq \underset{n \to \infty}{lim inf} E S_{t \land τ_{n}} \leq \underset{n \to \infty}{lim inf} [\frac{\bar{α} \bar{μ}}{\tilde{α} - λ \bar{δ}} + s_{0}] = \frac{\bar{α} \bar{μ}}{\tilde{α} - λ \bar{δ}} + s_{0}, \end{aligned}

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 = (κ_{i j})_{M \times M}

as follows

\begin{array}{rcl} P (m (t + Δ) = j | m (t) = i) = {\begin{cases} κ_{i j} Δ + o (Δ), & i \neq j, \\ 1 + κ_{i j} Δ + o (Δ), & i = j, \end{cases} \end{array}

(8)

\begin{array}{rcl} κ_{i i} = - \sum_{i \neq j} κ_{i j} . \end{array}

(9)

There is an important definition about

m (t)

. Let

\bar{m} (t) = m_{k}^{Δ} = m (k Δ)

, where

Δ > 0

is given in advance,

t \in [t_{k}, t_{k + 1}), k = 0, 1, 2, \dots

. There is the following important lemma.

Lemma 1 Under Markovian switching, for any $t \in [0, T],$ any small $Δ \in (0, 1),$ we have,

\begin{aligned} E \int_{0}^{t} {[α (m (s)) μ (m (s)) - α (\bar{m} (s)) μ (\bar{m} (s))]}^{2} d s \leq 4 {\bar{α}}^{2} {\bar{μ}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ), \end{aligned}

(10)

where $\bar{α} = max_{i \in C} α (i), \bar{μ} = max_{i \in C} μ (i), κ_{i i}$ is gived by the operator $Γ = (κ_{i j})_{M \times M}$ of $m (t)$ .

Proof For any

t \in [0, T],

any small

Δ \in (0, 1),

we have

\begin{aligned} E \int_{0}^{t} {[α (m (s)) μ (m (s)) - α (\bar{m} (s)) μ (\bar{m} (s))]}^{2} d s \\ \leq & E \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} 4 {\bar{α}}^{2} {\bar{μ}}^{2} (1_{m (s) \neq m (t_{d})}) d s \\ = & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} E (1_{m (s) \neq m (t_{d})}) d s \\ = & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} P (m (s) \neq m (t_{d})) d s \\ = & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} \sum_{i \in C} P (m (t_{d}) = i) P (m (s) \neq i | m (t_{d}) = i) d s . \end{aligned}

By (8) and (9), we have,

\begin{aligned} E \int_{0}^{t} {[α (m (s)) μ (m (s)) - α (\bar{m} (s)) μ (\bar{m} (s))]}^{2} d s \\ \leq & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} \sum_{i \in C} P (m (t_{d}) = i) P (m (s) \neq i | m (t_{d}) = i) d s \\ \leq & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} \sum_{i \in C} P (m (t_{d}) = i) [\sum_{i \neq j} κ_{i j} (s - t_{d}) + o (s - t_{d})] d s \\ \leq & 4 {\bar{α}}^{2} {\bar{μ}}^{2} T \sum_{d = 0}^{[\frac{T}{Δ}]} \int_{t_{d}}^{t_{d + 1}} \sum_{i \in C} P (m (t_{d}) = i) [max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ)] d s \\ \leq & 4 {\bar{α}}^{2} {\bar{μ}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) . \end{aligned}

Therefore, (10) holds.

Notation Under Markovian switching, similarly with the derivation process of (10), we have

\begin{aligned} E \int_{0}^{t} {[α (m (s)) - α (\bar{m} (s))]}^{2} d s \leq {\bar{α}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ), \end{aligned}

(11)

\begin{aligned} E \int_{0}^{t} {[β (m (s)) - β (\bar{m} (s))]}^{2} \\ d s \leq {\bar{β}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ), \end{aligned}

(12)

\begin{aligned} E \int_{0}^{t} {[δ (m (s)) - δ (\bar{m} (s))]}^{2} d s \leq {\bar{δ}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) . \end{aligned}

(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

\bar{m} (t) = m_{k}^{Δ} = m (k Δ)

, we define the discrete Euler-Maruyama numerical solution of Equation (1).

Given fixed timestep

Δ \in (0, 1)

and

S_{0} = s_{0}

s_{k + 1} = s_{k} + α (m_{k}^{Δ}) [μ (m_{k}^{Δ}) - s_{k}] Δ + β (m_{k}^{Δ}) | s_{k} | Δ B_{k} + δ (m_{k}^{Δ}) s_{k} Δ N_{k},

(14)

where

Δ B_{k} = B_{t_{k + 1}} - B_{t_{k}}

is a Wiener process increment and

Δ N_{k} = N_{t_{k + 1}} - N_{t_{k}}

is a Poisson process increment,

k = 0, 1, 2, \dots, t_{k} = k Δ, m_{k}^{Δ} = m (k Δ)

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)

\begin{aligned} s_{t} = s_{0} + \int_{0}^{t} α (\bar{m} (s)) [μ (\bar{m} (s)) - {\bar{s}}_{s}] d s + \int_{0}^{t} β (\bar{m} (s)) | {\bar{s}}_{s} | d B_{s} + \int_{0}^{t} δ (\bar{m} (s)) {\bar{s}}_{s} d N_{s}, \end{aligned}

(15)

where

{\bar{s}}_{t} = s_{k}, \bar{m} (t) = m_{k}^{Δ}, t \in [t_{k}, t_{k + 1}), k = 0, 1, 2, \dots

. Obviously,

{\bar{s}}_{t}

and

\bar{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 $Δ \to 0$ , for any $ε, ξ > 0, T > 0$ , we have

P (sup_{0 \leq t \leq T} {| S_{t} - s_{t} |}^{2} \geq ξ) < ε .

(16)

Proof Step 1. Recalling that

τ_{k} = inf {t \in [0, τ_{e}) : S_{t} \notin (\frac{1}{k}, k)}

, which has been given in proof of Theorem 1. By (5), then

P (τ_{k} \leq T) \leq \frac{V (s_{0}) + (K_{1} + K_{2}) T}{V (\frac{1}{k}) \land V (k)} .

Step 2. For

\forall k > 0

, construct a stopping time

ρ_{k} = inf {t \in [0, T] : s_{t} \notin [\frac{1}{k}, k]} .

When it comes to the property,

ρ_{k}

is equal to

τ_{k}

. Define a function

V_{1} : (0, + \infty) \to [0, + \infty),

V_{1} (x) = x^{\frac{1}{2}} - 1 - \frac{1}{2} \log x .

For (15), applying Jump-diffusion Itô formula yields,

d V_{1} (s_{t}) = L_{0} V_{1} (s_{t}) d t + L_{1} V_{1} (s_{t}) d B_{t} + L_{2} V_{1} (s_{t}) d N_{t},

where

\begin{aligned} L_{0} V (S_{t}) = & \frac{1}{2} {α (m (t)) μ (m (t)) (S_{t}^{- \frac{1}{2}} - S_{t}^{- 1}) - α (m (t)) (S_{t}^{\frac{1}{2}} - 1) + \frac{β^{2} (m (t))}{2} [(- \frac{1}{2}) S_{t}^{\frac{1}{2}} + 1]}, \\ L_{1} V (S_{t}) = & \frac{1}{2} β (m (t)) (S_{t} - 1), \\ L_{2} V (S_{t}) = & {[1 + δ (m (t))]}^{\frac{1}{2}} S_{t}^{\frac{1}{2}} - \frac{1}{2} \log [(1 + δ (m (t))) S_{t}] - S_{t}^{\frac{1}{2}} + \frac{1}{2} \log S_{t}, \end{aligned}

then

\begin{array}{rcl} d V_{1} (s_{t}) & = & \frac{1}{2} {α (m (t)) μ (m (t)) (S_{t}^{- \frac{1}{2}} - S_{t}^{- 1}) - α (m (t)) (S_{t}^{\frac{1}{2}} - 1) \\ + \frac{β^{2} (m (t))}{2} [(- \frac{1}{2}) S_{t}^{\frac{1}{2}} + 1]} d t + \frac{1}{2} β (m (t)) (S_{t} - 1) d B_{t} \\ + {[1 + δ (m (t))]}^{\frac{1}{2}} S_{t}^{\frac{1}{2}} - \frac{1}{2} \log [(1 + δ (m (t))) S_{t}] - S_{t}^{\frac{1}{2}} + \frac{1}{2} \log S_{t} d N_{t} . \end{array}

(17)

s_{s} \in [\frac{1}{k}, k]

implies

{\bar{s}}_{s} \in [\frac{1}{k}, k]

. Then we have a constant

C_{1}

making

\begin{array}{rcl} \frac{α (\bar{m} (s)) μ (\bar{m} (s))}{2} s_{s}^{- \frac{1}{2}} - 2 α (\bar{m} (s)) μ (\bar{m} (s)) s_{s}^{- 3} \\ + λ [(s_{s} + δ (\bar{m} (s)) {\bar{s}}_{s})^{\frac{1}{2}} + (s_{s} + δ (\bar{m} (s)) {\bar{s}}_{s})^{- 2} - s_{s}^{\frac{1}{2}} - s_{s}^{- 2}] \\ - \frac{α (\bar{m} (s)) μ (\bar{m} (s))}{2} s_{s}^{\frac{1}{2}} + 2 α (\bar{m} (s)) μ (\bar{m} (s)) s_{s}^{- 2} - \frac{σ^{2} (\bar{m} (s))}{8} s_{s}^{\frac{1}{2}} + 3 β^{2} (\bar{m} (s)) s_{s}^{- 2} \\ \leq & \frac{\bar{α} \bar{μ}}{2} s_{s}^{- \frac{1}{2}} - 2 \tilde{α} \tilde{μ} s_{s}^{- 3} + λ [(s_{s} + \bar{δ} {\bar{s}}_{s})^{\frac{1}{2}} + (s_{s} + \bar{δ} {\bar{s}}_{s})^{- 2} - s_{s}^{\frac{1}{2}} - s_{s}^{- 2}] \\ - \frac{\tilde{α} \tilde{μ}}{2} s_{s}^{\frac{1}{2}} + 2 \bar{α} \bar{μ} s_{s}^{- 2} - \frac{{\tilde{β}}^{2}}{8} s_{s}^{\frac{1}{2}} + 3 {\tilde{β}}^{2} s_{s}^{- 2} \\ \leq & C_{1} \end{array}

(18)

holds. By local Lipschitz conditions, we have a constant

C_{2} (k), C_{2} (k)

only depends on

k

, satisfying

\begin{aligned} | s_{s}^{2} - {\bar{s}}_{s}^{2} | \leq C_{2} (k) | s_{s} - {\bar{s}}_{s} | . \end{aligned}

By (17) and (18), we have a constant

C_{3} (k), C_{3} (k)

only depends on

k

, satisfying

\begin{aligned} E V_{1} (s_{t \land ρ_{k}}) \leq V_{1} (s_{0}) + C_{1} T + C_{3} (k) E \int_{0}^{t \land ρ_{k}} | s_{s} - {\bar{s}}_{s} | d s . \end{aligned}

(19)

Let

s \in [0, t \land ρ_{k}]

, by (15),

\begin{array}{rcl} s_{s} - {\bar{s}}_{s} = s_{s} - s_{k} & = & s_{s} - s_{[\frac{s}{Δ}]} \\ = & α (\bar{m} (s)) (μ (\bar{m} (s)) - s_{[\frac{s}{Δ}]}) (s - [\frac{s}{Δ}]) \\ + β (\bar{m} (s)) | s_{[\frac{s}{Δ}]} | (B_{s} - B_{[\frac{s}{Δ}] Δ}) + δ (\bar{m} (s)) s_{[\frac{s}{Δ}]} (N_{s} - N_{[\frac{s}{Δ}] Δ}) \\ \leq & \bar{α} (\bar{μ} + k) (s - [\frac{s}{Δ}] Δ) + \bar{β} k (B_{s} - B_{[\frac{s}{Δ}] Δ}) + \bar{δ} k (N_{s} - N_{[\frac{s}{Δ}] Δ}) . \end{array}

(20)

For

Δ \in (0, 1)

, we have a constant

C_{4}

satisfying

\begin{array}{rcl} E \int_{0}^{t \land ρ_{k}} | s_{s} - {\bar{s}}_{s} | d s & \leq & \bar{α} (\bar{μ} + k) T Δ + \bar{β} k E \int_{0}^{t \land ρ_{k}} | B_{s} - B_{[\frac{s}{Δ}] Δ} | d s \\ + \bar{δ} k E \int_{0}^{t \land ρ_{k}} | N_{s} - N_{[\frac{s}{Δ}] Δ} | d s \\ \leq & \bar{α} (\bar{μ} + k) T Δ + \bar{β} k \int_{0}^{T} E | B_{s} - B_{[\frac{s}{Δ}] Δ} | d s + \bar{β} k \int_{0}^{T} E | N_{s} - N_{[\frac{s}{Δ}] Δ} | d s \\ \leq & \bar{α} (\bar{μ} + k) T Δ + \bar{β} k Δ^{\frac{1}{2}} + \bar{δ} k \sqrt{λ} Δ^{\frac{1}{2}} \\ \leq & [\bar{α} (\bar{μ} + k) T + \bar{β} k + \bar{δ} k \sqrt{λ}] Δ^{\frac{1}{2}} \\ \hat{=} & C_{4} Δ^{\frac{1}{2}} . \end{array}

(21)

By (19) and (21), we have

E V_{1} (s_{t \land ρ_{k}}) \leq V_{1} (s_{0}) + C_{1} T + C_{3} (k) C_{4} Δ^{\frac{1}{2}} .

(22)

E V_{1} (s_{t \land ρ_{k}}) \geq P (ρ_{k} \leq T) [V_{1} (\frac{1}{k}) \land V_{1} (k)]

, therefore,

P (ρ_{k} \leq T) \leq \frac{E V_{1} (s_{t \land ρ_{k}})}{V_{1} (\frac{1}{k}) \land V_{1} (k)} \leq \frac{V_{1} (s_{0}) + C_{1} T + C_{3} (k) C_{4} Δ^{\frac{1}{2}}}{V_{1} (\frac{1}{k}) \land V_{1} (k)} .

(23)

Step 3. Let

θ_{k} = τ_{k} \land ρ_{k}

. This step will prove that we have a constant

D (k), D (k)

only depends on

k

, satisfying

E (sup_{0 \leq t \leq θ_{k} \land T} {| S_{t} - s_{t} |}^{2}) \leq D (k) Δ .

(24)

For

\forall t_{1}, t \in [0, T]

, by Hölder inequality, gives

\begin{array}{rcl} E (sup_{0 \leq t_{1} \leq t} {| S_{t_{1} \land θ_{k}} - s_{t_{1} \land θ_{k}} |}^{2}) & \leq & 4 T E \int_{0}^{t \land θ_{k}} {[α (m (s)) μ (m (s)) - α (\bar{m} (s)) μ (\bar{m} (s))]}^{2} d s \\ + 4 T E \int_{0}^{t \land θ_{k}} {[α (m (s)) S_{s} - α (\bar{m} (s)) {\bar{s}}_{s}]}^{2} d s \\ + 4 E sup_{0 \leq t_{1} \leq t} {(\int_{0}^{t_{1} \land θ_{k}} [β (m (s)) S_{s} - β (\bar{m} (s)) | {\bar{s}}_{s} |] d B_{s})}^{2} \\ + 4 E sup_{0 \leq t_{1} \leq t} {(\int_{0}^{t_{1} \land θ_{k}} [δ (m (s)) S_{s} - δ (\bar{m} (s)) {\bar{s}}_{s}] d N_{s})}^{2} . \end{array}

(25)

Let

\begin{aligned} A (t) = & 4 T E \int_{0}^{t \land θ_{k}} {[α (m (s)) μ (m (s)) - α (\bar{m} (s)) μ (\bar{m} (s))]}^{2} d s . \\ B (t) = & 4 T E \int_{0}^{t \land θ_{k}} {[α (m (s)) S_{s} - α (\bar{m} (s)) {\bar{s}}_{s}]}^{2} d s . \\ C (t) = & 4 E sup_{0 \leq t_{1} \leq t} {(\int_{0}^{t_{1} \land θ_{k}} [σ (m (s)) S_{s} - σ (\bar{m} (s)) | {\bar{s}}_{s} |] d B_{s})}^{2} . \\ D (t) = & 4 E sup_{0 \leq t_{1} \leq t} {(\int_{0}^{t_{1} \land θ_{k}} [δ (m (s)) S_{s} - δ (\bar{m} (s)) {\bar{s}}_{s}] d N_{s})}^{2} . \end{aligned}

By (10), we have a constant

C_{5} (k)

satisfying

\begin{aligned} A (t) \leq 16 T {\bar{α}}^{2} {\bar{μ}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) \leq C_{5} (k) Δ + o (Δ) . \end{aligned}

(26)

By triangle inequality,

\begin{array}{rcl} B (t) & = & 4 T E \int_{0}^{t \land θ_{k}} {[α (m (s)) S_{s} - α (\bar{m} (s)) S_{s} + α (\bar{m} (s)) S_{s} - α (\bar{m} (s)) {\bar{s}}_{s}]}^{2} d s \\ \leq & 8 T E \int_{0}^{t \land θ_{k}} {([α (m (s)) - α (\bar{m} (s))] S_{s})}^{2} d s + 8 T E \int_{0}^{t \land θ_{k}} [α (\bar{m} (s)) (S_{s} - {\bar{s}}_{s})]^{2} d s \\ \leq & 8 k^{2} T E \int_{0}^{t \land θ_{k}} {[α (m (s)) - α (\bar{m} (s))]}^{2} d s + 16 {\bar{α}}^{2} T E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s \\ + 16 {\bar{α}}^{2} T E \int_{0}^{t \land θ_{k}} (s_{s} - {\bar{s}}_{s})^{2} d s . \end{array}

(27)

From (11), we have a constant

C_{6} (k)

, satisfying

\begin{aligned} E \int_{0}^{t \land θ_{k}} {[α (m (s)) - α (\bar{m} (s))]}^{2} d s \leq {\bar{α}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) \leq C_{6} (k) Δ + o (Δ) . \end{aligned}

(28)

By (21), (27) and (28), we have constants

C_{7} (k)

and

C_{8} (k)

satisfying,

\begin{array}{rcl} B (t) \leq C_{7} (k) Δ + o (Δ) + 16 {\bar{α}}^{2} T C_{8} (k) Δ + 16 {\bar{α}}^{2} T E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s . \end{array}

(29)

By Burkholder-Davis-Gundy inequality,

\begin{array}{rcl} C (t) & \leq & 16 E \int_{0}^{t \land θ_{k}} {[β (m (s)) S_{s} - β (\bar{m} (s)) {\bar{s}}_{s}]}^{2} d s \\ \leq & 32 k^{2} E \int_{0}^{t \land θ_{k}} [β (m (s)) - β (\bar{m} (s))]^{2} d s + 64 {\bar{β}}^{2} E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s \\ + 64 {\bar{β}}^{2} E \int_{0}^{t \land θ_{k}} (s_{s} - {\bar{s}}_{s})^{2} d s . \end{array}

(30)

From (12), we have a constant

C_{9} (k)

, satisfying

\begin{aligned} E \int_{0}^{t \land θ_{k}} [β (m (s)) - β (\bar{m} (s))]^{2} d s \leq {\bar{β}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) \leq C_{9} (k) Δ + o (Δ) . \end{aligned}

(31)

By local Lipschitz conditions, we have constants

C_{10} (k), C_{11} (k)

satisfying

\begin{aligned} | S_{s} - s_{s} |^{2} \leq C_{10} (k) | S_{s} - s_{s} |^{2}, | s_{s} - {\bar{s}}_{s} |^{2} \leq C_{11} (k) | s_{s} - {\bar{s}}_{s} |^{2} . \end{aligned}

By (21), (30) and (31), we have constants

C_{12} (k), C_{13} (k)

and

C_{14} (k)

, satisfying

\begin{aligned} C (t) \leq C_{12} (k) Δ + o (Δ) + 64 \bar{β} C_{13} (k) Δ + 64 \bar{β} C_{14} (k) E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s . \end{aligned}

(32)

Similarly,

\begin{array}{rcl} D (t) & \leq & 64 λ k^{2} E \int_{0}^{t \land θ_{k}} [δ (m (s)) - δ (\bar{m} (s))]^{2} d s + 128 λ {\bar{δ}}^{2} E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s \\ + 128 λ {\bar{δ}}^{2} E \int_{0}^{t \land θ_{k}} (s_{s} - {\bar{s}}_{s})^{2} d s + 16 λ^{2} k^{2} T E \int_{0}^{t \land θ_{k}} [δ (m (s)) - δ (\bar{m} (s))]^{2} d s \\ + 32 λ^{2} {\bar{δ}}^{2} T E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s + 32 λ^{2} {\bar{δ}}^{2} T E \int_{0}^{t \land θ_{k}} (s_{s} - {\bar{s}}_{s})^{2} d s . \end{array}

(33)

From (13), we have a constant

C_{15} (k)

, satisfying

\begin{aligned} E \int_{0}^{t \land θ_{k}} {[δ (m (s)) - δ (\bar{m} (s))]}^{2} d s \leq {\bar{δ}}^{2} max_{1 \leq i \leq M} (- κ_{i i}) Δ + o (Δ) \leq C_{15} (k) Δ + o (Δ) . \end{aligned}

(34)

By (21), (33) and (34), we have constants

C_{16} (k)

and

C_{17} (k)

, satisfying

\begin{aligned} D (t) \leq & C_{16} (k) Δ + o (Δ) + (128 λ {\bar{δ}}^{2} + 32 λ^{2} {\bar{δ}}^{2} T) C_{17} (k) Δ \\ + (128 λ {\bar{δ}}^{2} + 32 λ^{2} {\bar{δ}}^{2} T) E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s . \end{aligned}

(35)

By (26), (29), (32) and (35), we have a constant

C_{18} (k)

satisfying

\begin{aligned} E (sup_{0 \leq t_{1} \leq t} {| S_{t_{1} \land θ_{k}} - s_{t_{1} \land θ_{k}} |}^{2}) \leq & C_{18} (k) Δ + o (Δ) + (16 {\bar{α}}^{2} T + 64 {\bar{β}}^{2} C_{14} (k) \\ + 128 λ {\bar{δ}}^{2} + 32 λ^{2} {\bar{δ}}^{2} T) E \int_{0}^{t \land θ_{k}} (S_{s} - s_{s})^{2} d s . \end{aligned}

(36)

By Gronwall inequality, (24) holds. Namely, we have a constant

D (k)

satisfying

E (sup_{0 \leq t \leq θ_{k} \land T} {| S_{t} - s_{t} |}^{2}) \leq D (k) Δ .

Step 4. For any small

ε, ξ \in (0, 1)

, set

\bar{Ω} = {ω : sup_{0 \leq t \leq T} {| S_{t} - s_{t} |}^{2} \geq ξ} .

By (24),

\begin{aligned} ξ P (\bar{Ω} \cap {θ_{k} \geq T}) = E {1_{{θ_{k} \geq T}} 1_{{\bar{Ω}}} ξ} \leq E {sup_{0 \leq t \leq T} {| S_{t} - s_{t} |}^{2}} \leq D (k) Δ . \end{aligned}

By (5) and (23), we have,

\begin{aligned} P (\bar{Ω}) \leq & P (\bar{Ω} \cap {θ_{k} \geq T}) + P (θ_{k} \leq T) \\ \leq & P (\bar{Ω} \cap {θ_{k} \geq T}) + P (τ_{k} \leq T) + P (ρ_{k} \leq T) \\ \leq & \frac{D (k)}{ξ} Δ + \frac{V (s_{0}) + (K_{1} + K_{2}) T}{V (\frac{1}{k}) \land V (k)} + \frac{V_{1} (s_{0}) + C_{1} T + C_{3} (k) C_{4} Δ^{\frac{1}{2}}}{V_{1} (\frac{1}{k}) \land V_{1} (k)} . \end{aligned}

When

k \to \infty

V (\frac{1}{k}) \land V (k) \to \infty

and

V_{1} (\frac{1}{k}) \land V_{1} (k) \to \infty

hold. Choosing a sufficiently large

k

satisfying

\frac{V (s_{0}) + (K_{1} + K_{2}) T}{V (\frac{1}{k}) \land V (k)} + \frac{V_{1} (s_{0}) + C_{1} T}{V_{1} (\frac{1}{k}) \land V_{1} (k)} < \frac{ε}{2},

fix above

k

, choosing a sufficiently small

Δ

satisfying

\frac{D (k)}{ξ} Δ + \frac{C_{3} (k) C_{4} Δ^{\frac{1}{2}}}{V_{1} (\frac{1}{k}) \land V_{1} (k)} < \frac{ε}{2} .

so, we have

P (\bar{Ω}) = P (sup_{0 \leq t \leq T} {| S_{t} - s_{t} |}^{2} \geq ξ) < 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 $Δ \to 0$ , for any $ε, ξ > 0, T > 0$ , we have

P (sup_{0 \leq t \leq T} {| s_{t} - {\bar{s}}_{t} |}^{2} \geq ξ) < ε .

(37)

Proof By (20),

\begin{aligned} E (sup_{0 \leq t \leq T} {| s_{t} - {\bar{s}}_{t} |}^{2}) \leq & 3 {\bar{α}}^{2} (\bar{μ} + k)^{2} Δ^{2} + 3 {\bar{β}}^{2} k^{2} E (sup_{0 \leq t \leq T} {| B_{t} - B_{[\frac{t}{Δ}] Δ} |}^{2}) \\ + 3 {\bar{δ}}^{2} k^{2} E (sup_{0 \leq t \leq T} | N_{t} - N_{[\frac{t}{Δ}] Δ} |^{2}) . \end{aligned}

(38)

Due to Doob-martingale inequality, then

\begin{aligned} E (sup_{0 \leq t \leq T} | B_{t} - B_{[\frac{t}{Δ}] Δ} |^{4}) \leq {(\frac{4}{3})}^{4} \sum_{k = 0}^{[\frac{T}{Δ}] - 1} E | B_{(k + 1) Δ} - B_{k Δ} |^{4} \leq \frac{4^{4}}{3^{3}} T Δ, \end{aligned}

Applying Lyapunov inequality again, yields

E (sup_{0 \leq t \leq T} | B_{t} - B_{[\frac{t}{Δ}] Δ} |^{2}) \leq {(E (sup_{0 \leq t \leq T} | B_{t} - B_{[\frac{t}{Δ}] Δ} |^{4}))}^{\frac{2}{4}} \leq \frac{16}{9 \sqrt{3}} T^{\frac{1}{2}} Δ^{\frac{1}{2}} .

(39)

Similarly,

\begin{array}{lll} E (sup_{0 \leq t \leq T} | N_{t} - N_{[\frac{t}{Δ}] Δ} |^{2}) \\ \leq & \sum_{k = 0}^{[\frac{T}{Δ}] - 1} E (sup_{k Δ \leq t \leq (k + 1) Δ} | N_{t} - N_{[\frac{t}{Δ}] Δ} |^{2}) \\ \leq & 2 \sum_{k = 0}^{[\frac{T}{Δ}] - 1} E (sup_{k Δ \leq t \leq (k + 1) Δ} {| {\tilde{N}}_{t} - {\tilde{N}}_{[\frac{t}{Δ}] Δ} |}^{2} + sup_{k Δ \leq t \leq (k + 1) Δ} {| λ (t - [\frac{t}{Δ}] Δ) |}^{2}) \\ \leq & \sum_{k = 0}^{[\frac{T}{Δ}] - 1} (8 E | {\tilde{N}}_{(k + 1) Δ} - {\tilde{N}}_{k Δ} |^{2} + 2 E | λ Δ |^{2}) \\ \leq & T (8 λ Δ + 2 λ^{2} Δ^{2}) . \end{array}

(40)

For

Δ \in (0, 1)

, by (38), (39) and (40), we have a constant

D_{1}

, satisfying

E (sup_{0 \leq t \leq T} | s_{t} - {\bar{s}}_{t} |^{2}) \leq D_{1} Δ^{\frac{1}{2}} .

(41)

For any small

ε, ξ \in (0, 1)

, set

\tilde{Ω} = {ω : sup_{0 \leq t \leq T} {| s_{t} - {\bar{s}}_{t} |}^{2} \geq ξ} .

By (41),

ξ P (\tilde{Ω} \cap {ρ_{k} \geq T}) = E {1_{{ρ_{k} \geq T}} 1_{{\tilde{Ω}}} ξ} \leq E {sup_{0 \leq t \leq T} {| s_{t} - {\bar{s}}_{t} |}^{2}} \leq D_{1} Δ^{\frac{1}{2}} .

(42)

By (23) and (42), we have

\begin{array}{rcl} P (\tilde{Ω}) & \leq & P (\tilde{Ω} \cap {ρ_{k} \geq T}) + P (ρ_{k} \leq T) \\ \leq & \frac{D_{1}}{ξ} Δ^{\frac{1}{2}} + \frac{V_{1} (s_{0}) + C_{1} T + C_{3} (k) C_{4} Δ^{\frac{1}{2}}}{V_{1} (\frac{1}{k}) \land V_{1} (k)} . \end{array}

When

k \to \infty

V_{1} (\frac{1}{k}) \land V_{1} (k) \to \infty

holds. Choosing a sufficiently large

k

satisfying

\frac{V_{1} (s_{0}) + C_{1} T}{V_{1} (\frac{1}{k}) \land V_{1} (k)} < \frac{ε}{2},

fix above

k

, choosing a sufficiently small

Δ

satisfying

\frac{D_{1}}{ξ} Δ^{\frac{1}{2}} + \frac{C_{3} (k) C_{4} Δ^{\frac{1}{2}}}{V (\frac{1}{k}) \land V (k)} < \frac{ε}{2},

so, we have

P (\tilde{Ω}) = P (sup_{0 \leq t \leq T} {| s_{t} - {\bar{s}}_{t} |}^{2} \geq ξ) < 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 $Δ \to 0$ , for any $ε, ξ > 0, T > 0$ , we have

P (sup_{0 \leq t \leq T} | S_{t} - {\bar{s}}_{t} | \geq ξ) < ε .

(43)

Proof For any

ξ > 0

P (sup_{0 \leq t \leq T} | S_{t} - {\bar{s}}_{t} | \geq ξ) \leq P (sup_{0 \leq t \leq T} | S_{t} - s_{t} | \geq \frac{ξ}{2}) + P (sup_{0 \leq t \leq T} | s_{t} - {\bar{s}}_{t} | \geq \frac{ξ}{2}) .

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,

S_{t}

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

\begin{aligned} B (T) = E [\exp (- \int_{0}^{T} S_{t} d t)] . \end{aligned}

(44)

Using

{\bar{s}}_{t}

, we get an asymptotic value to

B (T)

as follows

\begin{aligned} {\bar{B}}_{Δ} (T) = E [\exp (- \int_{0}^{T} | {\bar{s}}_{t} | d t)] . \end{aligned}

(45)

Theorem 6 ensures that

{\bar{B}}_{Δ} (T)

is close to

B (T)

with probability 1.

Theorem 6 Letting $Δ \to 0$ , for any $ε, ξ > 0$ , we have

P (| B (T) - {\bar{B}}_{Δ} (T) | \geq ξ) < ε .

(46)

Proof For any

ε, ξ > 0

, it is sufficient to prove

P (| B (T) - {\bar{B}}_{Δ} (T) | \geq ξ) < ε

, namely, for any small

ε, ξ > 0

, proving

\begin{aligned} P [| \exp (- \int_{0}^{T} S_{t} d t) - \exp (- \int_{0}^{T} | {\bar{s}}_{t} | d t) | \geq ξ] < ε . \end{aligned}

Due to

\exp (- | x |) - \exp (- | y |) \leq | x - y |

and the non-negative nature about

S_{t}

, we have

\begin{array}{rcl} | \exp (- \int_{0}^{T} S_{t} d t) - \exp (- \int_{0}^{T} | {\bar{s}}_{t} | d t) | & \leq & | \int_{0}^{T} [S_{t} - {\bar{s}}_{t}] d t | \\ \leq & T sup_{0 \leq t \leq T} | S_{t} - {\bar{s}}_{t} | . \end{array}

Due to (43), (46) holds.

References

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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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Abstract
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Cite this article
1 Introduction
2 Solution: Existence, Uniqueness, Non-Negative and Global Nature
3 Boundedness of First Moment
4 Properties About Coefficients of JD-SDS-MS
5 Convergence of Euler-Maruyama Numerical Solutions
6 The Pricing of a Bond
References
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2018-03-06	2018-06-11	2020-02-25
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6 The Pricing of a Bond

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