Let $\{X(t);t\geq 0\}$ be an IID stochastic process given by: $dX(t) = \mu X (t) dt + \sigma X(t) dW(t)$ where $W(t)\sim N(0,1)$ and $\mu,\sigma>0$ are constants.
I want to calculate: $E[\max(aX(t)+b,0) \mid X(s) = x]$
So I do the following:
$E[\max(aX(t)+b,0)]=E[(aX+b)I\{aX+b\geq0\}]=aE[XI\{X\geq-\frac{b}{a}\}]+bP(X\geq -\frac{b}{a})$, where $I$ is the indicator function.
Under expectation we have $aE[XI\{X\geq-\frac{b}{a}\}] = \int_{-\infty}^{+\infty}xI\{x\geq-\frac{b}{a}\}\phi(x)dx$ where $\phi$ is the pdf of normal distribution. But I don't know how i should handle this.
Any suggestions?