Martingale problem: help writing and equation using the martingale assumptions

[Nyn]

Hi,

I have been given this problem and the solution however, neither make sense to me.

$$x_t=\int exp(t-s)E(k_s|F_t)ds$$

(the integral is from t to infinity)

where

$$k_u=m^d_u-(m^d_u)^*-α(y_u-y_u^*)$$ for all u>0


suppose (m_t, t>0) is a martingale, what is an equation for x_t using the martingale assumption for (m_t, t>0).

the solutions say

$$E_tk_s^2=m_t+E_t(-m_s^*-α(y_s-y_s^*))=m_t+v_s. thus, x_t=m_t+ \int v_s de$$

So, first off I don't know what the martingale assumptions are a simple google search did not yield any useful results. Secondly I can't seem to follow the math (it's been a while since I have done anything other that simple integration).

I am hoping that someone is familiar with the martingale assumptions and can explain this problem to me-or maybe knows about a cite with a good beginners explanation to the topic.

Thanks,

 

[Adrmay]

The martingale assumption in this problem is that the process m_t is a martingale, which means that it is a stochastic process whose expectation at time t is equal to its value at time t. This means that for all times t, E[m_t] = m_t.Using this assumption, we can rewrite the equation for x_t as follows:x_t = \int exp(t-s)E(k_s|F_t)ds = \int exp(t-s)[m_s + E(-m_s^* - α(y_s - y_s^*))]ds = m_t + \int v_s ds Where v_s is defined as E(-m_s^* - α(y_s - y_s^*)). This is because the martingale assumption implies that the expectation of m_s is equal to its value at time s, so we can rewrite the equation as m_t + the expectation of the remaining terms.