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I have been reading Chapter 4 of Shreve's Stochastic Calculus for Finance II. It is easy to understand the simple process, $\Delta(t)$, defined on Page 126, which is just a constant inside a given subinterval.

Later in the Exercise 4.2 and 4.3, it is mentioned again. The process $\Delta(t)$ is simple and nonrandom in 4.2 while it is simple but random in 4.3.

How should I understand the randomness of such process?

I am just a beginner of financial math. Thanks in advance! Good reference will also be appreciated!

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closed as off-topic by Gordon, LocalVolatility, lehalle, vanguard2k, Quantuple Feb 16 '17 at 13:21

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  • "Basic financial questions are off-topic as they are assumed to be common knowledge for those studying or working in the field of quantitative finance." – Gordon, LocalVolatility, Quantuple
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  • $\begingroup$ To make this question more generally applicable, could you quote the relevant definitions, ideally to the point where potential answerers don't need a copy of Shreve's to answer? $\endgroup$ – barrycarter Feb 26 '16 at 16:16
  • $\begingroup$ Yes please recall the exact context of yoir question $\endgroup$ – lehalle Feb 15 '17 at 23:50
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Thanks for everybody answering my question!

Here is my understanding. If a process $\Delta(t)$ is nonrandom, then one could tell what the values will be for all time $t$ when one is standing at $t=0$. On the other hand, if such process is random, then one stands at $t=0$ he cannot see anything in the future.

Moreover, the randomness of a simple process are crucial when one takes expectation on it. Say, a simple process $\Delta(t)$ is nonrandom, then one could take out of what is known $E(\Delta(t))=\Delta(t)$. The rule fails if such process is random.

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$\Delta(t)$ simply represents change in process value over the interval $t$ to $t+1$. If change in process value over this interval is deterministic, then you can call process $\Delta(t)$ as nonrandom. Let's suppose, the change in bond price $(B_t)$ governed by following equation: $$dB_t=rB_t dt$$ If $r$ is constant, then there is no uncertainty in the change in bond price.

Now, suppose stock price $(S_t)$ follows geometric Brownian motion and satisfy following SDE: $$dS_t = \mu S_t dt + \sigma S_t dW_t$$ where, $W_t$ is Wiener process. Here, change in stock price is not constant but a random unit.

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To put it simply, $\Delta(t)$ is the change in your stock price in a very small interval. Say, if you measure your stock every minute. The value of your stock is a random variable because you don't know where it will end up with. The randomness every minute you can observe would be $\Delta(t)$.

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