\hspace{0.1cm}(\text{since} \hspace{0.2cm} W_s \hspace{0.1cm} is \hspace{0.1cm} \mathcal{F}_s-\text{measurable  })\\
      &=&\mathbb{E}((W_t-W_s)^2)+ W_s\mathbb{E}(W_t-W_s)+W_s\mathbb{E}(W_t/\mathcal{F}_s)-t\\
      & &(\text{since the increments are independents})\\
      &=&t-s+0+W_s^2-t\hspace{0.2cm} \text{since} \hspace{0.2cm} W_t-W_s\rightsquigarrow\mathcal{N}(0, t-s)\\
      &=&W_s^2-s.
      \end{aligned}$$

Using the same argument as above, we have : $$\begin{aligned}
  \mathbb{E}(e^{\gamma W_t}/\mathcal{F}_s)&=& e^{\gamma W_s}\mathbb{E}(e^{\gamma(W_t-W_s)}/\mathcal{F}_s)\\
  &=& e^{\gamma W_s}\mathbb{E}(e^{\gamma W_{t-s}})\\
  &=&e^{\gamma W_s}\int_{-\infty}^{+\infty}\dfrac{e^{-x^2/2(t-s)}}{\sqrt{2\pi(t-s)}}dx\\
  &=&e^{\gamma W_s}e^{\gamma^2(t-s)/2}=e^{\gamma W_s+\gamma^2(t-s)/2}.
  \end{aligned}$$ Therefore, $$\begin{aligned}
  \mathbb{E}\left(exp\left(\gamma W_t-\gamma^2\dfrac{t}{2}\right)/\mathcal{F}_s\right)&=&\mathbb{E}(e^{\gamma W_t}/\mathcal{F}_s)e^{-\gamma^2t/2}\\
  &=&e^{\gamma W_s+\gamma^2(t-s)/2}e^{-\gamma^2t/2}\\
  &=& \exp(\gamma W_s-{\gamma^2 s}/2).
  \end{aligned}$$

The following proposition is from [@Oks].

Almost all sample paths of a Brownian motion are nowhere differentiable.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\mathcal{F}_t)$ a filtration on this space. Let $(S_k)_{k\geq 1}$ be an $\mathcal{F}_t$-adapted stochastic process on $(\Omega, \mathcal{F}, \mathbb{P})$ with $0\leq S_1(\omega)\leq S_2(\omega)\leq ...$ for all $k\geq 1$ and $\omega \in \Omega$. The $\mathcal{F}_t$- adapted process $N=(N_t)_{t\geq 0}$ defined by : $$\begin{aligned}
N_t:=\sum_{k\geq 1}\mathbf{1}_{\{S_k\leq k\}}\end{aligned}$$ is called counting process with jump times $S_k$.

Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space and $(\mathcal{F}_t)$ a filtration on this space. A counting process $(N)_t$, $\mathcal{F}_t$- adapted is called poisson process of intensity $\lambda >0$ if :

1.  $N_0=0$.

2.  $\forall\; 0\leq t_0<t_1<...<t_n$, the random variables $\{N_{t_j}-N_{t_{j-1}} \hspace{0.3cm} 1\leq j\leq n\}$ are independent.

3.  For $0\leq s\leq t$, $N_t-N_s\approx N_{t-s}$, where $\approx$ stand for the equality in probability law.

