Couldn't understand decomposition inside derivation of option valuation using Martingale method In the derivation of option valuation using Martingale method in continuous time framework of my book named "Brownian Motion Calculus by Ubbo F Wiersema" I have faced some issue. I add the derivation below:
In the continuous time setting, start from the standard SDE for the stock price, $$\frac{dS_t}{S_t}=\mu dt+\sigma dB_t$$
Introduce the discounted stock price, $$S^{\star}_t\stackrel{\text{def}}{=}\frac{S_t}{e^{rt}}$$
Ito's formula for $S^{\star}_t$ as a function of $t$ and $S$ gives, $$\frac{dS^{\star}_t}{S^{\star}_t}=(\mu-r)dt+\sigma dB_t=\sigma\left[ \frac{\mu-r}{\sigma}dt+dB_t \right]=\sigma\left[ \phi dt+dB_t \right]$$
The probability density of $B_t$ at $B_t=x$ is, $$\frac{1}{\sqrt t \sqrt{2\pi}}\exp\left[-\frac12\left(\frac{x}{\sqrt t}\right)^2\right]$$
This can be decomposed into the product of two terms,

$$\frac{1}{\sqrt t \sqrt{2\pi}}\exp\left[-\frac12\left(\frac{\phi t+x}{\sqrt t}\right)^2\right]\exp\left[\frac12 \phi^2 t+\phi x\right]$$

With $y \stackrel{\text { def }}{=} \varphi t+x$ the first term can be written as
$$
\frac{1}{\sqrt{t} \sqrt{2 \pi}} \exp \left[-\frac{1}{2}\left(\frac{y}{\sqrt{t}}\right)^{2}\right]
$$
which is the probability density of another Brownian motion, say $\widehat{B}(t)$, at $\widehat{B}(t)=y$. It defines $\widehat{B}(t) \stackrel{\text { def }}{=} \varphi t+B(t)$, so $\widehat{d} \widehat{B}(t)=\varphi d t+d B(t)$. Substituting the latter into the SDE for $S^{\star}$ gives
$$
\frac{d S^{\star}(t)}{S^{\star}(t)}=\sigma d \widehat{B}(t) \quad \text { and } \quad \frac{d S(t)}{S(t)}=r d t+\sigma d \widehat{B}(t)
$$
This says that under the probability distribution of Brownian motion $\widehat{B}(t)$,
$S^{\star}$ is a martingale.
I couldn't understand how they bring the decomposition like this? Another thing is, "Why we need to sustain martingale?". Does it mean we couldn't predict the future regardless of all prior knowledge, as it introduce arbitrage?

@Ali, give an answer to show the decomposition.
$$
\exp\left[-\frac{1}{2}\left(
\frac{\phi t + x}{\sqrt{t}}\right)^{2}\right]=
\exp\left[-\frac{1}{2}\left(
\phi^{2}t+2\phi x + x^{2}/t\right)\right]=
\exp\left[-\frac{1}{2}
\phi^{2}t-\phi x\right]
\exp\left[-\frac{1}{2t}x^{2}\right]
$$
But, still, I couldn't get any intuition of that decomposition. Like, I want to know why this decomposition was introduced, and its meaning in the context of option valuation.
Does this related with Girsanov transformation?

$$
\exp\left[-\frac{1}{2}\left(
\frac{\phi t + x}{\sqrt{t}}\right)^{2}\right]=
\exp\left[-\frac{1}{2}\left(
\phi^{2}t+2\phi x + x^{2}/t\right)\right]=
\exp\left[-\frac{1}{2}
\phi^{2}t-\phi x\right]
\exp\left[-\frac{1}{2t}x^{2}\right]
$$
