How to solve Stochastic differential equation?

Question

I do not have a clue on how to solve out this type of question, and how to deal with integration with a combination of brownian motion and linear function. Can anyone help me out please?

Answer 1

HINT:

$$\int_0^t\int_0^{s}\frac{B_s'}{(1-s')^2}ds'ds=\int_0^t\int_s'^t \frac{B_s'}{(1-s')^2}ds'ds \tag 1$$

and

$$\int_0^t \frac{B_s'(t-s')}{(1-s')^2}ds'=\int_0^t \frac{B_s'(t-1+1-s')}{(1-s'^2)}ds' \tag 2$$

Can you complete?

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We are given

$$X_t=xt+B_t-(1-t)\int_0^t \frac{B_s}{(1-s)^2}ds \tag 3$$

and asked to show that this $X_t$ is a solution of the following

$$X_t=B_t+\int_0^t \frac{x-X_s}{1-s}ds \tag 4$$

Let's substitute $(3)$ into $(4)$ to obtain

$$\begin{align} X_t&=xt+B_t-(1-t)\int_0^t \frac{B_s}{(1-s)^2}ds\\\\ &=B_t+\int_0^t \frac{x-(xs+B_s-(1-s)\int_0^t \frac{B_s'}{(1-s')^2}ds')}{(1-s)}ds\\\\ &=B_t+xt-\int_0^s \frac{B_s}{(1-s)}ds+\int_0^t \int_0^s \frac{B_s'}{(1-s')^2}ds'ds \tag 5 \end{align}$$

Using Hints $(1)$ and $(2)$,

$$\int_0^t \int_0^s \frac{B_s'}{(1-s')^2}ds'ds=\int_0^t \frac{B_s}{1-s}ds-(1-t)\int_0^t \frac{B_s}{(1-s)^2}ds$$

and we are done!