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I am analysing a problem where I have two correlated stocks described by Brownian motions $$ \frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$ \frac{dS^{2}_{t}}{S^{2}_{t}}=\mu_{2} dt + \sigma_{2} dW^{2}_{t} \quad \quad (2)$$

where the $$ Y_{t} = 2 S^{1}_{t} S^{2}_{t} \quad \quad (3)$$

I am looking for SDE to this equation.

I know that given the below two geometric Brownian motions $$ dX^{1}_{t} =z_{1} dt + Y_{1} dBt \quad \quad (4)$$ $$ dX^{2}_{t} =z_{2} dt + Y_{2} dBt\quad \quad (5)$$

the below equality can be derived

$$d(x^{1}_{t} x^{2}_{t})=x^{1}_{t} dx^{2}_{t} + x^{2}_{t} dx^{1}_{t} + Y_1 Y_2 dt \quad \quad (6)$$

I know as well that the two Brownian motions can be presented as $$ W_t = \rho W^{1}_{t} + \sqrt{1-\rho^2} W^{2}_{t} \quad \quad (7)$$

However when I try to apply the properties to the above problem I am getting confused.

The manual says that $$\frac{dY_t}{Y_t} = 2 \big{(} \frac{dS^{1}_{t}}{S^{1}_{t}} + \frac{dS^{2}_{t}}{S^{2}_{t}} + \frac{ <S^{1} S^{2}>_t}{S^{1}_{t} S^{1}_{t}} \big{)} \quad \quad (8)$$


Can anybody clarify how the $<S^{1} S^{2}>_t$ is derived?
how this relates to the $Y_1 Y_2 dt$ from the eq.(6) ? why sigmas were not used analogically ?
and what it should be substituted for in the further calculation?

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  • $\begingroup$ Looks to me as a straightforward application of Ito's lemma to the multivariate function $f (t,x_1,x_2) = 2x_1x_2$. The term you refer to is called a quadratic (co-)variation. If you are not familiar with any of those two things, I suggest you first open a reference book because it would require a few pages of answer to cover this topic correctly. $\endgroup$ – Quantuple Apr 19 '16 at 6:47
  • $\begingroup$ could you clarify (4),(5) and (6) ? $\endgroup$ – MJ73550 Apr 19 '16 at 8:00
  • $\begingroup$ @MJ73550, IMHO (6) is simply Ito's lemma applied to the product of 2 arithmetic Brownian motions driven by a common Wiener process (hence perfectly correlated) defined in (4) and (5), with unfortunate choices of notation (e.g. the diffusion coefficients $Y_i, i=1,2$ have nothing to do with the process $Y_t $ defined earlier). It therefore seems like the OP's question is: how to obtain the same kind of SDE as (6) for the product of 2 correlated geometric Brownian motions defined as (3) knwoing (1) and (2). $\endgroup$ – Quantuple Apr 19 '16 at 21:42
  • $\begingroup$ I know Ito formula, it is just that (4) and (5) are not GBM, that (6) is the generic formula assuming $z_i,Y_i$ are adapted process with the good integrability condition. My point was to force Michal to be precise. Since his question is just a direct application of Ito $\endgroup$ – MJ73550 Apr 20 '16 at 12:58
  • $\begingroup$ my understanding is that (4) and (5) are two Ito processes, which are perfectly correlated (therefore the Bt is the same for both), so that (5) can be proven. Now I want to apply the property to the problem where two correlated stocks are given and express the final SDE using only one Wt. comparing the third terms of the eq. (6) and (8) it confused me why in the (6) are used Y1Y2 as opposed to S1S2 in the (8). $\endgroup$ – Michal Apr 20 '16 at 13:30

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