# 金融数学代写|Semimartingale decomposition assignment

1. Semimartingale decomposition. Let $W$ be a one-dimensional standard Brownian motion defined on a filtered probability space $(\Omega, \mathbb{F}, \mathbb{P})$.
(a) Show that the process
$$B_t=W_t-t W_1, \quad t \in[0,1],$$
is Gaussian and compute its mean value and the covariance function.
(b) Show that $B_t$ is not $\mathcal{F}t^W$-measurable. Show that the process $B$ is adapted to the filtration $\mathbb{G}=$ $\left(\mathcal{G}_t\right){t \geq 0}$ where $\mathcal{G}t=\sigma\left(\mathcal{F}_t^W, \sigma\left(W_1\right)\right)$.

(c) Use the properties of conditional expectations to show that

$\mathbb{E}_{\mathbb{P}}\left(W_t-W_s \mid \mathcal{G}_s\right)=\frac{t-s}{1-s}\left(W_1-W_s\right), \quad 0 \leq s<t<1 .$

(d) Show that $W$ is not a $\mathbb{G}$-martingale.

(e) It is known that for every $\alpha$ such that $0<\alpha<\frac{1}{2}$ there exists a random variable $H$ such that

$\left|W_t-W_s\right| \leq H|t-s|^\alpha .$

Use this fact to show that the process $\left(X_t\right)_{t \in[0,1]}$, given by the formula

$X_t=\int_0^t \frac{W_1-W_s}{1-s} d s,$

is well defined.

(f) Using (e), (c) and assuming in what follows that expectation and integration with respect to time can be interchanged show that the process

$M_t=W_t-X_t, \quad t \leq 1,$

is a $\mathbb{G}$-martingale.

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