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# 数学代写|随机过程Stochastic Porcesses代考|AMATH562 Examples of Strongly Semimartingale-Consistent Multivariate Special Semimartingales

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## 数学代写|随机过程代写Stochastic Porcesses代考|Examples of Strongly Semimartingale-Consistent Multivariate Special Semimartingales

Example 5.4 Let $\mathcal{X}=\mathbb{R} \times \mathbb{R}$ and consider a Poisson process $X=\left(X^1, X^2\right)$. We will investigate the strong semimartingale consistency of $X$. There is a one-to-one correspondence between any time-homogeneous Poisson process with values in $\mathcal{X}=$ $\mathbb{R} \times \mathbb{R}$ and a homogeneous Poisson measure, say $\mu$, on $E:={0,1}^2 \backslash{(0,0)}$. See for instance the discussion in Bielecki et al. (2008b).

Let $v$ denote the $\mathbb{F}$-dual predictable projection of $\mu$. The measure $v$ is a measure on a finite set, so it is uniquely determined by its values on the atoms in $E$. Therefore $X=\left(X^1, X^2\right)$ is uniquely determined by
$$v(d t,{(1,0)})=\lambda_{1,0} d t, \quad v(d t,{(0,1)})=\lambda_{0,1} d t, \quad v(d t,{(1,1)})=\lambda_{1,1} d t$$
for some positive constants $\lambda_{10}, \lambda_{01}$, and $\lambda_{11}$. A time homogeneous Poisson process with values in $\mathcal{X}=\mathbb{R} \times \mathbb{R}$ is a special semimartingale, and the $\mathbb{F}$-characteristic triple of $X$ is $(B, 0, v)$, where
$$B_t=\left[\begin{array}{l} B_t^1 \ B_t^2 \end{array}\right]:=\left[\begin{array}{l} \left(\lambda_{10}+\lambda_{11}\right) t \ \left(\lambda_{01}+\lambda_{11}\right) t \end{array}\right] .$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Examples of Multivariate Special Semimartingales That Are Not Strongly Semimartingale Consistent

Example 5.7 Consider the canonical stochastic basis $(\Omega, \mathcal{F}, \mathbb{F}, \mathbb{P})$, with $\mathcal{X}=\mathbb{R} \times$ $\mathbb{R}^N$. Let $X^2$ be a Markov chain that takes values in $\left{e_1, \ldots, e_N\right}$, where $\left(e_i\right)_{i=1}^N$ is the canonical basis in $\mathbb{R}^N$. We assume that $X^2$ admits a constant generator matrix $\Lambda$. Consider the special semimartingale $X=\left(X^1, X^2\right)$, with $X^1$ given by
$$d X_t^1=b^{\top} X_t^2 d t+\sigma d W_t, \quad X_0^1=x^1 \in \mathbb{R},$$

where $W$ is a real-valued standard Brownian motion, $\sigma \in \mathbb{R}{+}$, and $b \in \mathbb{R}^N, b \neq 0$. The $\mathbb{F}$-characteristics of $X^1$ are $$\left(\left(\int_0^t b^{\top} X_u^2 d u\right){t \geq 0},\left(\sigma^2 t\right){t \geq 0}, 0\right) .$$ According to Stricker’s theorem, $X^1$ is a special semimartingale on $\left(\Omega, \mathcal{F}, \mathbb{F}^1, \mathbb{P}\right)$. Clearly, however, the $\mathbb{F}$-characteristics of $X^1$ do not coincide with its $\mathbb{F}^1$-characteristics. Indeed, Theorem $5.1$ implies that the $\mathbb{F}^1$-characteristics of $X^1$ are $$\left(\left(\int_0^t b^{\top} \mathbb{E}\left(X_u^2 \mid \mathcal{F}_u^1\right) d u\right){t \geq 0},\left(\sigma^2 t\right)_{t \geq 0}, 0\right) .$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Examples of Strongly Semimartingale-Consistent Multivariate Special Semimartingales

$$v(d t,(1,0))=\lambda_{1,0} d t, \quad v(d t,(0,1))=\lambda_{0,1} d t, \quad v(d t,(1,1))=\lambda_{1,1} d t$$

$$B_t=\left[B_t^1 B_t^2\right]:=\left[\left(\lambda_{10}+\lambda_{11}\right) t\left(\lambda_{01}+\lambda_{11}\right) t\right] .$$

## 数学代写|随机过程代写Stochastic Porcesses代考|Examples of Multivariate Special Semimartingales That Are Not Strongly Semimartingale Consistent

$$d X_t^1=b^{\top} X_t^2 d t+\sigma d W_t, \quad X_0^1=x^1 \in \mathbb{R},$$

$$\left(\left(\int_0^t b^{\top} X_u^2 d u\right) t \geq 0,\left(\sigma^2 t\right) t \geq 0,0\right) .$$

$$\left(\left(\int_0^t b^{\top} \mathbb{E}\left(X_u^2 \mid \mathcal{F}u^1\right) d u\right) t \geq 0,\left(\sigma^2 t\right){t \geq 0}, 0\right)$$

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