Sde $Y_t=Y_0+W_t +\int_{0}^t \gamma(Y_s) ds$

Soient :

+ $T>0$
+ $(W_t)$ si a standard brownian motion adapted to the filtration e
+ $f : [0,T] \to \mathbb{R} $ in $L¹([0,T])$
+ $g: [0,T] \to \mathbb{R}$ in $ L^2([0,T])$
+ $R_t= \int_{0}^t \exp \left( - \int_{0}^{r} f(s) ds \right) g(r) dW_r$
+ $Y_t= \exp \left( \int_{0}^{t} f(s) ds \right)$ $(\star)$

$
\begin{cases}
dX_t &= f(t)X_t dt + g(t) dW_t\\
X_0 &= 0\\
\end{cases}
$

1. Show that $R$ is well-defined
2. Show that $Y$ is a gaussian process
3. Show that $ t \mapsto \int_{0}^t f(s) ds$ has bounded variations
4. Show that if $X$ is defined as $(\star)$, then $X$ solves the SDE
5. let $X$ be a solution of the SDE, show that $X=Y$
6. Can we use the usual theorems of existence and uniqueness ?

_____________________

1. $\int_{0}^{T} g(r) dW_r$ is a gaussian random variable with law $\mathcal{N}(0, {\lVert g \rVert}_2 ^2)$ because $g$ is square integrable.

Let $\phi(r)=\exp \left( - \int_{0}^{r} f(s) ds \right) g(r)$,

we have $\phi(r)^2=\exp( - 2 \int_{0}^r f(s) ds) g(r)^2 \leq \exp( 2 {\lVert f \rVert}_1 ) g(r)^2 $

therefore $\int_0^T \phi(r)^2 dr \leq \exp( 2 {\lVert f \rVert}_1 ) {\lVert g \rVert}_2 ^2$

so $R_t$ is well-defined.

2. ???
3. $ \psi :t \mapsto \int_{0}^t f(s) ds$ has for derivative $f$ and the total variation $TV(\psi)= \int_{0}^T \mid \psi' \mid =\int_{0}^T \mid f \mid$ and we know that $f \in L^1$

4. We use Ito formula

5. ???

6. ????

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