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Does the equation below have a name, or is it similar to some other well-known process/ equation?

Equation of interest:

$$S_c = S_{c-1} + S_{c-1}\omega_c\delta_c$$ $\delta\sim\mathcal{N}(0,1)$ is a standard normal shock

$\omega\sim\mathcal{B}(1,~0.5)$ is a coin flip with a 50% chance of being 1 (0 otherwise)

$c=(1,~2,~\dots,~C^{\ast})$ is sequence of coin flips ($C^{\ast}$ total flips)

$S_0\sim\mathcal{N}(0,1)$ initializes the process

$S_{C^{\ast}}$ is the value of $S_c$ at the end of the iterations, and is the value of interest.

Note that the $c^{th}$ coin flip dictates whether or not the $c^{th}$ standard normal shock occurs, and thus whether or not $S_c$ is different from $S_{c-1}$.

Below is some R code that will generate one value of $S_{C^{\ast}}$ when $C^{\ast}$ is 4. It's set up to make the omegas and deltas the same each time, but to allow the order in which they accumulate to be random. Note that the order of accumulation for a given set of omegas and deltas does not affect the value of $S_{C^{\ast}}$.

set.seed(2) # set the RNG seed so that all the shocks and coin flips are the same

Cstar <- 4 # number of coin flips/ number of iterations
S0 <- rnorm(1) # initial standard normal shock
Sc <- c(S0, rep(NA, Cstar)) # vector to hold the shocks as they accumulate

deltas <- rnorm(Cstar) # standard normal shocks
omegas <- rbinom(Cstar, 1, 0.5) # coin tosses

rm(.Random.seed, envir=globalenv()) # remove the non-random seed, so the order in which the shocks/ coin flips are accumulated is random
c.2.Cstar <- sample(1:Cstar, Cstar) # a scrambled sequence from c=1 to C*
ctr <- 1 # the counter used for indexing Sc (but not the omegas and deltas)

for(i in c.2.Cstar){
    ctr <- ctr+1
    Sc[ctr] <- Sc[ctr-1] + Sc[ctr-1]*omegas[i]*deltas[i] 
}

SCstar <- Sc[ctr]
print(c.2.Cstar)
print(SCstar)
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  • $\begingroup$ It is a Markov regime switching time series model, since at the time $t$ the process depends only on process value at time $t$, the regime $w$ and innovation $\delta$. If there is a more specific name, then probably it is known to researchers specializing in that field (regime switching time series models). The multiplication of $S_{t-1}$ and the innovation $\delta$ seems strange for me, since it is hard for me to think of an application where such process might arise. $\endgroup$
    – mpiktas
    Jul 2 '14 at 8:13
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I gather all these r.v.'s are independent.

This is a Markov process, because the past is fully encapsulated in $S_{t-1}$, and so current probabilities conditional in its whole past are equivalent to current probabilities conditional only on the previous period.

It is also a martingale process because a) it is absolutely integrable (its expected value exists) and b),

$$E[S_t \mid t-1]= E(1+\omega_t\delta_t)\cdot S_{t-1} = [1+E(\omega_t)E(\delta_t)]\cdot S_{t-1} = S_{t-1}$$

Finally it is a mean-stationary process that is not second-order stationary. Writing recursively we obtain (using $t$ for indexing)

$$S_t = \left(\prod_{i=1}^t(1+\omega_i\delta_i)\right)S_0$$

$$E[S_t] = E\left(\prod_{i=1}^t(1+\omega_i\delta_i)\right)\cdot E[S_0]=0$$

While $$E[S_t^2] = \left(\prod_{i=1}^tE(1+\omega_i\delta_i)^2\right)\cdot E[S_0^2]$$

$$E(1+\omega_i\delta_i)^2 = E(1+2\omega_i\delta_i+\omega_i^2\delta_i^2) = 1+0+E(\omega_i^2)E(\delta_i^2) = 1+\frac 12 = 3/2$$

So

$$\text{Var}(S_t) = \left (\frac 32\right)^t$$

If we treat the more general case writing the term as $(a+\omega_i\delta_i)$, then the autocovariance function $\gamma_k$ is

$$\gamma_k = \text{Cov}(S_t,S_{t-k}) = E[S_tS_{t-k}] = E\left[\Big(S_{t-k} \cdot \prod_{i=t-k+1}^t(a+\omega_i\delta_i)\Big)\cdot S_{t-k}\right]$$

$$\Rightarrow \gamma_k = a^k\cdot E(S_{t-k}^2) = a^k\left (a^2+\frac 12\right)^{t-k}$$

We see that if $|a|=1/\sqrt2$ the process will be covariance-stationary having variance equal to unity, while if $|a|<1/\sqrt2$ the process will tend to a constant value, as its unconditional variance will tend to zero. This would be worth simulating.

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