Stochastic differential equation problem for $e^{2 \int W_t}$.

Question

We have some process with $X_{T} = \displaystyle \exp(2 \int_0^T W_t dt)$.

We want to determine:

1) $X_0$ and process $Y_t$ : $X_T = X_0 + \displaystyle \int_0^T X_t Y_t dW_t$

2) $var(X_T)$.

The last problem is quite easy to solve. First of all $\displaystyle \mathbb{P}(X_T \le t) = \mathbb{P}(\xi \le 0.5\log(t))$. Hence we have distribution and it's not hard to compute following integral.

But how to deal with first problem? As I understand we need to use Ito's theorem. But I guess it should be used in inverse direction. Maybe it's quite easy too, but I newbie in this problem. Also as far as I understand $X_0 = 1$. Any hints?

Answer 1

We have some process with $X_{T} = \displaystyle \exp(2 \int_0^T W_t dt)$. We want to determine:

1. $X_0$ and process $Y_t$ : $X_T = X_0 + \displaystyle \int_0^T X_t Y_t dW_t$

The process $X_{t}$ is actually differentiable and so we have the ODE

$$\frac{d}{dt}X_{t}=X_{t}2W_{t},$$

which has the integral form

$$X_{t}=X_{0}+2\int X_{s}W_{s}ds.$$

So now it remains to do a stochastic integration by parts. Let $F_{s}:=\int_{0}^{s}2X_{r}dr$

$$=X_{0}+F_{t}W_{t}-\int_{0}^{t}F_{s}dW_{s}.$$

And so we let

$$Y_{s}:=\frac{F_{t}-F_{s}}{X_{s}}.$$