Let $X_{t_{(t\geq 0)}}$ be stochastic process in continuous time (eg. a Gemetric Brownian Motion). $X_0 = x_0$ is a known constant and the distribution of $X_t$ is known.
Let $C$ be a positive scalar and $p_1<p_2<p_3$ also be 3 positive scalars. Now consider $Y$ defined as
$$ Y=\left\{\begin{matrix} p_1 & \text{if} & X_{T_1}>C \\ p_2 & \text{if} & X_{T_2}>C \text{ and } X_{T_1}<C \\ p_3 & \text{if} & X_{T_3}>C \text{ and } \{X_{T_1}<C,X_{T_2}<C\} \\ 0 & \text{else} & \end{matrix}\right. $$ for $0<T_1<T_2<T_3$
What is the expected value of $Y$?
When I look at this I think we can easily use the conditional expectation because the the distribution of $X_t$ is known. Let's consider each of the outcomes of $Y$.
- Event 1 happens if $X_{T_1}>C$ and the probability of that is $P(X_{T_1}>C_t)=:Q_1$
- Event 2, here comes the conditional: the probability is $P(X_{T_2}>C|X_{T_1}<C)=:Q_2$
- Event 3: the probability is here $P(X_{T_3}>C|X_{T_1}<C,X_{T_2}<C)0;Q_3$
I am in a position where I can compute all the conditional expectations mentioned aboove in my problem. Can I conclude that $$ E[Y|X_0=x_0]=p_1Q_1+p_2Q_2+p_3Q_3 $$ I know that this is very simple probability but somehow I can't decide whether my approach is correct. I simply don't recall whether we can sum up the events when they are all conditioned on each other.
No, your equation is not valid. What you have is $E[Y|X_0=x_0]=p_1Q_1+p_2Q_2P(X_{T_1} <C)+p_3Q_3P(X_{T_1}<C, X_{T_2} <C)$.