How do we go about calculating the variance of $$B_{t}^{2}$$ where $$B_{t}$$ is standard Brownian Motion.

Using $$ B_{t}^{2} - {2t} =2\int_0^t B_{s} dB_s$$

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$$\text{Var}(B_t^2)$$

$$=E[(B_t^2-E[B_t^2])^2] $$ This you can evaluate.

From here you can use Ito Isometry - Chapter 7 in Fin Eco workbook (Stochastic Calculus).

In particular: $$E[(\int_0^tB_s dBs)^2] = \int_0^tE[B_s^2]ds $$

An alternative approach to this problem is to transform:

$$B_t = \sqrt{t} \cdot Z $$ where $$Z \sim N(0,1)$$ and then consider moments of the standard normal random variable.

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