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Variance of dividend value
I have the following problem:S follows the usual Brownian motion (dS/S=mu*dt+sigma*dWt)Assume d=constant is a dividend yield.Now, what's the variance (or second moment is fine as well) of int_0^infty exp(-r*t)*d*S_t*dt. (variance of values with dividend method)So in other words, I needE[(int_0^infty exp(-r*t)*d*S_t*dt)^2]I try to post it in LaTeX, doesn't work so far. Help on the formula would be very much appreciated.
Variance of dividend value
I must say, I am not sure if it is visible, here the LaTeX code:E\left[\left(\int_0^{\infty}e^{-rt}dS_tdt\right)^2\right]
Variance of dividend value
because of, let's say, moderate response rate here a clarification of the problem: the dS_t expression might be misleading, it is of course d*S_t. So it writes:We can take d out of the integral and expectation:So, you see, the actual problem is that I want an integral of S_t.
Variance of dividend value
It looks like that dS(t) dt contains two terms. The first one has order (dt)^2 the second dw(t) dt has the order (dt)^(3/2). Thus the the infinite integral in parentesis looks be equal to 0.
Variance of dividend value
No, it's certainly not zero, the integral describes d*the area under the curve in a stock price chart.
Variance of dividend value
Sorry list, my formula is a bit misleading as stated in my comment of Dec 28.dS_t is NOT a differential, it means d*S_t with d=dividend. It would have been better to write "D" instead of "d".Nevertheless, S_t is stochastic. That's the problem here.
Variance of dividend value
You might agree that it looks difficult to expect a short formula. If you think to get a mathematical result about second moment then you can write a parabolic equation for such functional. In order to get this PDE consider needed integral as a function of initial data u ( t , x ) where ( t , x ) is stems from S ( * ; t , x ) but you also need to consider integral from 't' not from 0. Then u ( t , x ) defined for non negative t , x calculate for t = 0. For applcations you might get upper or lower estimates/ Its all depends on what is your final goal.