1

The fact that $(dB_t)^2=dt$ is it a convention notation, or it can be proved rigorously ? Or more generaly, if $$dY_t=a(t)dt+b(t)dB_t,$$ does the equatity $$(dY_t)^2=b(t)^2dt$$ can by justify rigorously ?

If yes, how ? because, I'm not so sure how to do it.

Todd
  • 574

1 Answers1

1

The notation $(dB_t)^2 = dt$ is just a convenient [abuse of] notation. Formally, the notation $(dB_t)^2$ is the [forward] differential of the quadratic variation of $B_t.$

All the formal derivation, therefore, must be done in terms of the quadratic variation. In this context, it certainly is possible to rigorously justify $$(dB_t)^2 := d[B]_t = dt \\ (dY_t)^2 := d[Y]_t = b(t)^2dt$$

It's a happy coincidence that for certain stochastic differential equations, the quadratic variation may be determined via squaring the entire SDE and applying some rules on the product of differentials. This happy coincidence is why we use this particular abuse of notation.

Brian Moehring
  • 21,088
  • Do you have a reference for $(dB_t)^2$ being the quadratic variation ? This look strange to me... – Todd Jan 08 '20 at 17:54
  • @Todd Compare the "informal derivation" and "Itô drift-diffusion processes" sections of the Itô's lemma wikipedia page. I don't personally have a book showing $(dY_t)^2$ is defined as the Itô differential of the quadratic variation, because my only text that talks about $(dY_t)^2$ is an intro financial math text that never mentions quadratic variation, while my stochastic calculus text uses quadratic variation but never uses the notation $(dY_t)^2$. Both mentions are conveniently in the same place in the derivation of Itô's lemma, though. – Brian Moehring Jan 08 '20 at 22:09