I am reading a book on PDE, by Rubenstine and Pinchonver. There is this problem very early in the book, I attempted to work it out, but the problem I have is that the explanation of what brownian motion is, did not click with me. Please look at my solution and comment, also few words of explanation of what brownian motion assumption is will be helpful.
This is the problem
Assume that a broker buys a stock at a certain price. He decides in advance to sell it if its price reaches an upper bound $m_2$ (in order to cash in her profit) or a lower bound $m_1$ (to minimize losses in case the stock dives). How much time on average will the broker hold the stock, assuming that the stock price performs a Brownian motion? The equation and the associated boundary conditions
$ku′′(m) = −1$, $u(m_1) = u(m_2) = 0$
Solve and analyze the results in terms of financial interpretation.
My solution is as follows:
Because we have only one variable we can treat this as an ODE. $ku''(m)=-1 \iff u''(m)=-\frac{1}{k}$ now we can let $\lambda = \frac{1}{k}$
Now we solve the equation $u''(m)=-\lambda $ by integrating both sides with respect to $m$. Thus
$u'(m)=-\lambda m +c_1$
$u(m)= -\frac{\lambda}{2}m^2+c_1m+c_2$
Now using the initial conditions we get
$0=u(m_1)= -\frac{\lambda}{2}{m_1}^2+c_1m_1+c_2$
$0=u(m_2)= -\frac{\lambda}{2}{m_2}^2+c_1m_2+c_2$
Now we subtract the two equations and obtain
$0= -\frac{\lambda}{2}(m_1^2-m_2^2)+c_1(m_1-m_2)$
Thus we get either $m_1 =m_2$ or $\frac{\lambda}{2}=\frac{c_1(m_1-m_2)}{(m_1^2-m_2^2)}$. I am a bit confused here if this is correct because I never used this condition for Brownian motion and I guess that does not feel right.