I am new to this.
If variance of Brownian motion b is t, what is the variance of db?
db is delta of b
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1 Answer
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Let $(B_t)$ be a standard Brownian motion. Then, $B_t\sim N(0,t)$ and $B_t-B_s\sim N(0,t-s)$.
Informally, you can say $\mathrm{d}B_t\sim N(0,\mathrm{d}t)$ where $\mathrm{d}B_t=B_{t+\mathrm{d}t}-B_t$ is an infinitesimal increment.
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$\begingroup$ I see. What about var(sigma x db)? $\endgroup$ Commented Jan 19, 2020 at 9:15
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$\begingroup$ @InfoLearner good question. A constant can be pulled out of the variance if you remember to square it, i.e. $\sigma \mathrm{d}B_t\sim N(0,\sigma^2\mathrm{d}t)$. $\endgroup$– KevinCommented Jan 19, 2020 at 10:46
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$\begingroup$ Do you recommend any book or link or online video that explains this concept? $\endgroup$ Commented Jan 19, 2020 at 12:58
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1$\begingroup$ Itô Integration is covered in every text book on stochastic calculus. Shreve‘s book on StoCal for finance (Part II) would be a good book to start with. You could also google StoCal Lecture notes and I am sure you’ll find many resources introducing the Itô integral. $\endgroup$– KevinCommented Jan 19, 2020 at 13:05