Deterministic and Random Simple Processes in Stochastic Calculus
Summary
The document clarifies the distinction between deterministic and random simple processes in stochastic calculus. A deterministic process has values that are known in advance, while a random process has values that depend on uncertain future outcomes. This distinction matters when taking expectations: a deterministic quantity can be taken outside an expectation, whereas a random one generally cannot.
The answers illustrate the idea with finance examples. A bond whose price follows a differential equation with a constant rate has predictable changes, while a stock following geometric Brownian motion has uncertain increments driven by a Wiener process. Another answer informally describes process increments over short time intervals. The discussion is introductory and does not fully state the formal measurability or information-filtration conditions used to define simple adapted processes in Shreve; its examples are meant as intuition rather than a complete mathematical treatment.
Key ideas
- A deterministic process has future values fixed and knowable in advance.
- A random process depends on outcomes that are uncertain at the current time.
- Deterministic quantities can be taken outside expectations, but random quantities generally cannot.
- A constant-rate bond model gives predictable changes, while geometric Brownian motion models uncertain stock changes.
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# How to understand nonrandom/random process in Shreve book? # How to understand nonrandom/random process in Shreve book? 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! ## Answer by Pandaaaaaaa (score 1) https://quant.stackexchange.com/a/24945 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. ## Answer by Neeraj (score 0) https://quant.stackexchange.com/a/24570 $\Delta(t)$ 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. ## Answer by SmallChess (score 0) https://quant.stackexchange.com/a/24584 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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