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February 15th, 2017, 11:31 AM   #1
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Brownian motion and stochastic process

Can some one explain this introduction for me, more specifically the last sentence, what does it mean that both processes are adapted to the filtration $\mathcal{F}_t$? What does a filtered probability space mean?

"We begin our analysis in the standard Black-Scholes world consisting of a bank account process of price denoted by $B_t$, and a risky stock process $S_t$, both defined on a filtered probability space ($\Omega,\mathcal{F}\!,\mathcal{F}_t,\mathcal{P} $), with the filtration $\mathcal{F}_t$ generated by the standard Brownian motion $W_t$. We assume the bank account process $B_t$ is risk free, and the stock process $S_t $ with non-zero volatility at all times evolves according to a geometric Brownian motion. Both asset processes are therefore adapted to the filtration $\mathcal{F}_t$, with local dynamics shown below."
$$d{S_t} = \mu {S_t}dt + \sigma {S_t}d{W_t},$$
$$d{B_t} = r{B_t}dt.$$

Last edited by skipjack; February 15th, 2017 at 03:05 PM.
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