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

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

Login

+3 votes

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

$$=E[(B_t^2-E[B_t^2])^2] $$ 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.

- All categories
- BUS 1003H - Introduction to Financial Risk (43)
- BUS 2016H - Financial Mathematics (53)
- BUS 3018F - Models (69)
- BUS 3024S - Contingencies (61)
- BUS 4028F - Financial Economics (20)
- BUS 4027W - Actuarial Risk Management (46)
- BUS 4029H - Research Project (5)
- Mphil (1)
- Calculus and Pure Mathematics (3)
- Statistics (16)

...