Random process not so random after all (deterministic)

I would like to show (demonstrate by simulation) a random process that turns out after $i$ interactions to be deterministic, i.e. ends at predefined value (roughly) known at time $t=1$.

Conditions for the solution:

1) The random process should display randomness visible (graph) = interactive

2) The "convergence" to the end result should be gradual and therefore the end result surprising to the observer.

3) Solution in R (hopefully).

The meaning of surprising in this case: I mean, during the generating process, the observer expectation is not (rational?) .... (Surprise in sense of conviction that random is not random after all).

Sub-question (setup):

Does it matter whether the terminal (final) value is revealed to observer a priori (at the beginning of the generating process series) or after the process ends (to confirm the values match)?

Summary:

1) If solution is not provided, how to approach it? (any hint helps)

2) Which tool (statistical/modeling) should be used? How to set up the process?

EDIT: improved version (of what I'm asking)

Simulate a (large) number of identically distributed but dependent random variables, such that their sum is ideally equal to a prespecified value, or failing that (for continuous RVs, we will hit a single number with probability 0), which is in a given range with a given probability. (with @Stephan kind help.)

• Can we simulate an unconstrained process, then modify the values to achieve your target, finally reveal the modified process? Or does this need to be "online"? Jul 25 '18 at 7:39
• I'm open on how to approach it...I'm not sure until I see the solution that it would be convincing... so at this point I cannot tell... Jul 25 '18 at 7:43
• en.wikipedia.org/wiki/I_know_it_when_I_see_it Jul 25 '18 at 7:44
• well, I do not understand the term online. Do you mean that scenario 1) would be a pre-generated series which is adapted such as to match the predefined value 2) online: The series would be generated "online" while series would slowly converge to predetermined value? (parametric?, starting value random, etc.). In this case the option 2) is what is needed... Jul 25 '18 at 7:48
• I was struggling with what term to use myself. Yes, I meant that modifying a known series is easy, but generating a series and modifying it without "looking ahead" is harder, since the unmodified series might end up all over the place (which can even be formalized). Jul 25 '18 at 7:53

Suppose that after $T$ iterations, your process should end up at a predefined value $m$.

You can first simulate a process $f_t$ with whatever characteristics you want and then modify it as follows:

$$f_t \mapsto f_t + \frac{t}{T}(m-f_T)$$

Note that this requires that we know the end value $f_T$ of the unconstrained process when we modify it at time $t$.

Here is an example with $T=1000$ and $m=2$: target <- 2
n_steps <- 1000

set.seed(1)
process_raw <- cumsum(rnorm(n_steps+1))
process <- process_raw + (0:n_steps)/n_steps * (target-tail(process_raw,1))

plot(process,type="l")
abline(h=target,lty=2)

If you need something that can be calculated at time $t$ without "looking ahead", you can use $f_t$ instead of $f_T$ in the above formula.

$$f_t \mapsto f_t + \frac{t}{T}(m-f_t)$$

However, then your process will have lower and lower variance as $t\to T$: process <- process_raw + (0:n_steps)/n_steps * (target-process_raw)

plot(process,type="l")
abline(h=target,lty=2)

What you are asking for is called a Brownian bridge ; there is an answer elsewhere on CV that asks how to convert a Brownian bridge into a Brownian "excursion": Simulating a Brownian Excursion using a Brownian Bridge? . That question uses the rbridge() function from the e1071 package:

library(e1071)
set.seed(101)
r <- rbridge()  ## default: end time=1, frequency=1000
plot(r)
abline(h=0,col="red") The underlying code for rbridge is pretty simple:

z <- rwiener(end = end, frequency = frequency)
ts(z - time(z) * as.vector(z)[frequency], start = 1/frequency,
frequency = frequency)

Stripped of its extra details, this corresponds to $B(t) = W(t) - t/T \cdot W(T)$, where $W$ is a Wiener process and $T$ is the ending time, as suggested in the Wikipedia article.

This simulates a Brownian bridge that starts and ends at zero; I would suggest adding a linear trend if you want to modify the starting and ending points (as suggested by the "general case" section of the Wikipedia article).

An ensemble:

rr <- replicate(500,rbridge()) A comment on a previous version of this answer argued that the volatility would not be constant; based on a numerical experiment (easier than thinking), I believe this is false - the volatility is constant (we can also see this by looking at the $B(t)$ expression above ...
However, a careful observer should be able to detect that something is going on, because $\Delta x$ will be biased toward the starting point (similar to, but not the same as, an Ornstein-Uhlenbeck process).