Random + Probability Time-Series

Overview

The MCH TimeSeries module provides tools to simulate stochastic systems and probability-driven processes. It supports three main modeling patterns:

• Geometric Brownian Motion
• Martingales
• Markov Chains

These examples mirror the stories and models from Decision Superhero Vol. 2.

Geometric Brownian Motion (GBM)

Now we move from the general setup to a concrete continuous price process.

GBM is commonly used to simulate continuous price paths such as stock prices or any variable that evolves with compounding random variation.

GBMMfit(HistoricalData, PeriodsToForecast; rng="none")

using MCHammer, Random, Distributions
rng = MersenneTwister(1)
Random.seed!(1)

historical = rand(Normal(10,2.5),1000)
GBMMfit(historical, 12; rng=rng)

GBMM(LastValue, ReturnsMean, ReturnsStd, PeriodsToForecast; rng::Any="none")

using MCHammer, Random, Distributions
rng = MersenneTwister(1)
Random.seed!(1)
GBMM(100000, 0.05,0.05,12, rng=rng)

GBMA_d(price_0, t, rf, exp_vol; rng="none")

**price_0**: Stock Price at period 0
**t**: Number of days out to forecast
**rf**: Risk free rate (usually the country's 25 or 30 bond)
**exp_vol**: Expected volatility is the annual volatility
**rng**: Used seed the results to make the forecast reproducible. Set to none by default which means fully random  but can use any of Julia's rngs to seed results. e.g. *GBMA_d(price_0, t, rf, exp_vol; rng=MersenneTwister(1))*

using MCHammer, Random
rng = MersenneTwister(1)
Random.seed!(1)
GBMA_d(100, 504,0.03,.3, rng=rng)

GBMM_Sim(LastValue, ReturnsMean, ReturnsStd, PeriodsToForecast; trials=1)

Example: Profit Random Walk

Now we combine several GBM paths into a simple business story.

using Dates, MCHammer, Random, Distributions
Random.seed!(1)

ts_trials = []

for i = 1:1000
     Monthly_Sales = GBMM(100000, 0.05,0.05,12)
     Monthly_Expenses = GBMM(50000, 0.03,0.02,12)
     MonthlyCOGS = Monthly_Sales .* 0.3
     MonthlyProfit = Monthly_Sales - Monthly_Expenses - MonthlyCOGS
     push!(ts_trials, MonthlyProfit)
end

dr = collect(Date(2025,1,01):Dates.Month(1):Date(2025,12,31))
trend_chrt(ts_trials,dr)

Martingales

Now we switch to processes that are fair in expectation.

A martingale describes a process where the expected value of the next step equals the current value. This creates a fair game in expectation, with no drift up or down.

marty(Wager, GamesPlayed; GameWinProb = 0.5, CashInHand = Wager)

Example: Simple Gambler

Here is the most compact form of the martingale idea in practice.

marty(50,10)

10-element Vector{Any}:
   0
 100
 150
 100
   0
 100
  50
 -50
  50
   0

Example: Adjusting for Known Odds

Now we introduce biased odds and extra cash to see how the behavior changes.

println(marty(50,10; GameWinProb=0.45, CashInHand=400))

Any[450, 500, 550, 500, 400, 500, 550, 500, 600, 550]

Comparing Different Win Probabilities

Next we compare different win probabilities side by side.

using Plots

Exp11 = scatter(marty(5, 100, GameWinProb = 0.25, CashInHand = 400), title="GameWinProb = 0.25", label="Bets")
Exp12 = scatter(marty(5, 100, GameWinProb = 0.33, CashInHand = 400), title="GameWinProb = 0.33", label="Bets")
Exp13 = scatter(marty(5, 100, GameWinProb = 0.5,  CashInHand = 400), title="GameWinProb = 0.5",  label="Bets")
Exp14 = scatter(marty(5, 100, GameWinProb = 0.55, CashInHand = 400), title="GameWinProb = 0.55", label="Bets")

combined = Plots.plot(Exp11, Exp12, Exp13, Exp14, layout=(2,2), legend=:topleft)

Markov Chain Modeling Functions

Now we move to systems that evolve step by step between discrete states.

A Markov chain models a system moving step by step between states where the probability of the next state depends only on the current one.

What you can predict:

1. Long-run equilibrium
2. State distribution at each step
3. Growth or decline of states
4. Speed of convergence
5. Absorbing outcomes

Core Markov Functions

cmatrix(dimensions::Int64)

markov_solve(T; state_names=nothing)

State | Probability | Type="Ergodic"

State | <one column per absorbing state> | ExpectedSteps | Type

markov_ts(T, start_vec; trials=1, state_names=nothing)

dist₁ = start_vec * T
dist₂ = dist₁    * T
...
distₖ = distₖ₋₁  * T

markov_ts_plot(ms; title="State Probabilities Over Time",
                  xlabel="Time Step",
                  ylabel="Probability",
                  states=nothing)

State | Type | 1 | 2 | … | trials

markov_state_graph(T, ms; state_names, title, size)

Example A — Soft Drink Brand Switching (Ergodic Chain)

Now we see how these tools behave in a real consumer switching problem.

People switch cola brands all the time, and it feels random when you look at just one shopper, but across millions of decisions the switching pattern becomes predictable. A Markov chain captures those movements by tracking the chance that a Classic drinker moves to Zero, or a DietCola fan slides to CherryCola, or a CaffeineFree loyalist quietly disappears. Run those transitions forward and you can see where the market is heading, which brands gain ground, which ones fade and what the long run looks like. In short, if you want to understand how consumers drift between choices over time, a Markov chain gives you a simple way to model the flow and forecast the future.

Why this matters: Brand switching looks chaotic, but the long term trend is predictable. This model shows who wins and how fast.

A1. Analytic Solve

We start by understanding the long-run behavior directly from the transition matrix.

using MCHammer, DataFrames

drink_names  = ["CherryCola", "DietCola", "CaffeineFree", "Classic", "Zero"]
brand_share  = [0.10, 0.25, 0.05, 0.35, 0.25]

drink_preferences = [
    0.60  0.03  0.15  0.20  0.02;
    0.02  0.40  0.30  0.20  0.08;
    0.15  0.25  0.30  0.25  0.05;
    0.15  0.02  0.10  0.70  0.03;
    0.15  0.30  0.05  0.05  0.45
]

markov_solve(drink_preferences; state_names = drink_names)

A2. Time-Series Forecast

Next we push the chain forward in time to see how shares evolve.

using MCHammer, Plots

drink_periods = 10

ms_drink = markov_ts(drink_preferences, brand_share;
                     trials      = drink_periods,
                     state_names = drink_names)

A2.1 Plot

Now we convert the forecast into a simple time-series chart.

plt_drink_ts = markov_ts_plot(
    ms_drink;
    title  = "Brand Share Over Time",
    ylabel = "Market Share",
    xlabel = "Years Out"
)
plt_drink_ts

A3. State Graph

Finally we switch from time-series to structure and visualize the full chain.

plt_drink_state = markov_state_graph(
    drink_preferences, ms_drink;
    state_names = drink_names,
    title       = "Brand Switching\n(Node Size = Year 10 Share)",
    size        = (700, 700)
)
plt_drink_state

Example B — Marital Status Over Time (6-State Ergodic)

Now we move from brand choices to life events that evolve more slowly.

The time series plot for the marital example shows how each relationship category rises or falls over the years. Instead of only seeing the final state, the chart reveals the path the population takes to get there. You can watch Single shrink or grow, Married stabilize or decline, Divorced expand at a steady pace and Widowed increase slowly as the population ages. Each line tracks one state across every forecast step, making it easy to see momentum, turning points and long term trends. It is the unfolding story of the system, not just the ending.

B1. Time-Series Forecast

We start by forecasting how the distribution across marital states changes over time.

using MCHammer, Plots

marital_names = ["Single", "CommonLaw", "Married",
                 "Separated", "Divorced", "Widowed"]

initial_marital = [0.291, 0.126, 0.443, 0.024, 0.062, 0.054]

marital_T = [
    0.65  0.15  0.12  0.03  0.01  0.04;
    0.00  0.50  0.33  0.08  0.04  0.05;
    0.00  0.00  0.13  0.40  0.42  0.05;
    0.00  0.20  0.09  0.02  0.65  0.04;
    0.00  0.15  0.10  0.00  0.70  0.05;
    0.00  0.25  0.25  0.00  0.00  0.50
]

ms_marital = markov_ts(marital_T, initial_marital;
                       trials      = 25,
                       state_names = marital_names)

B1.1 Plot

Next we plot how each marital state moves over the forecast horizon.

plt_marital_ts = markov_ts_plot(
    ms_marital;
    title  = "Marital Status Over Time",
    ylabel = "Proportion of Population",
    xlabel = "Years Out"
)
plt_marital_ts

B2. State Graph

Finally we look at the structure of transitions and the long-run distribution.

plt_marital_state = markov_state_graph(
    marital_T, ms_marital;
    state_names = marital_names,
    title       = "Marital Status Transitions\n(Node Size = Year 25 Share)",
    size        = (700, 700)
)
plt_marital_state

Example C — Accounts Receivable & Default (Absorbing Chain)

Now we close with a financial example that has clear end states.

Receivables slide through aging buckets just like shoppers drift between cola brands, and even though each invoice feels like a one-off headache, the overall flow settles into a predictable pattern. A Markov chain captures how money moves from 30 days to 60, then 90, then either paid or default. Once you know those transition probabilities, you can jump straight to the end of the story. The model tells you the final state without tracking every month: how much cash will eventually be collected, how much will be lost and how long it takes for the portfolio to clear. If you want a simple shortcut to the endgame of your AR pipeline, a Markov chain gives you the final answer.

C1. Analytic Absorbing Solution

We begin by solving the absorbing chain to understand paid versus default outcomes.

using MCHammer, Plots

ar_names   = ["30days", "30_60days", "60_90days",
              "90daysPlus", "default", "paid"]

ar_current = [0.50, 0.30, 0.15, 0.05, 0.0, 0.0]

ar_rates = [
    0.50  0.10  0.10  0.05  0.05  0.20;
    0.00  0.30  0.15  0.25  0.075 0.225;
    0.00  0.00  0.25  0.25  0.04  0.46;
    0.00  0.00  0.00  0.30  0.20  0.50;
    0.00  0.00  0.00  0.00  1.00  0.00;
    0.00  0.00  0.00  0.00  0.00  1.00
]

ar_solve = markov_solve(ar_rates; state_names = ar_names)

C2. Time-Series Forecast

Next we track how the portfolio migrates between buckets over time.

ms_ar = markov_ts(ar_rates, ar_current;
                  trials      = 12,
                  state_names = ar_names)

C2.1 Plot

Now we plot how much of the portfolio sits in each bucket at each period.

plt_ar_ts = markov_ts_plot(
    ms_ar;
    title  = "Payment Status Over Time",
    ylabel = "Proportion of Receivables",
    xlabel = "Months Out"
)
plt_ar_ts

C3. State Graph

Finally we summarize the structure and final distribution in a single picture.

plt_ar_state = markov_state_graph(
    ar_rates, ms_ar;
    state_names = ar_names,
    title       = "Receivables Transitions\n(Node Size = Month 12 Share)",
    size        = (700, 700)
)
plt_ar_state

Sources & References

• Eric Torkia, Decision Superhero Vol. 2, chapter 10 : Beyond Monte Carlo — When Uncertainty HAS Memory, Technics Publishing, 2026
• Available on Amazon : https://a.co/d/4YlJFzY . Volumes 2 and 3 to be released in Fall 2025 and Summer 2026.
• Eric Torkia, Decision Superhero Vol. 3, Chapter 5