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# Statistics
## Stochastic processes
### (Chapter 7)

### Christian Vedel,<br>Department of Economics<br>University of Southern Denmark

### Email: [christian-vs@sam.sdu.dk](mailto:christian-vs@sam.sdu.dk)

### Updated 2026-04-06

---
class: middle
# Today's lecture
.pull-left-wide[
**Modelling random phenomena that evolve over time**

- **Section 1:** Stochastic processes — definition and sample paths
- **Section 2:** Transformations — partial sums and first differences
- **Section 3:** States and waiting times
- **Section 4:** Bernoulli and binomial processes
- **Section 5:** Poisson process
- **Section 6:** Gaussian random walk
- **Section 7:** Wiener process (Brownian motion)
]

.pull-right-narrow[
![Trees](Figures/Trees1.jpg)
]

---
# Why stochastic processes?

.pull-left-wide[
- Economic variables evolve over time: GDP growth, interest rates, exchange rates, asset prices
- **Time series econometrics** builds formal probability models for these sequences
- The processes covered today are the *building blocks* for some of the most important models in economics and finance
]

.pull-left-wide[
- The next two slides show exactly where this leads
]

.pull-left-wide[
> .orange[**Heads up:** the mathematics on the next slides will look unfamiliar — that is intentional. You are **not** expected to understand it now. Return to it after the course and notice how much has clicked into place.]
]

---
# What you are building towards: time series

.pull-left-wide[
> .orange[**Preview only — no need to understand this now**]
]

.pull-left-wide[
The workhorse model of time series econometrics is the **AR(1)**:
`$$Y_t = \mu + \rho \, Y_{t-1} + \varepsilon_t, \qquad \varepsilon_t \overset{iid}{\sim} \mathcal{N}(0,\sigma^2)$$`
]

.pull-left-wide[
The parameter `$\rho$` determines everything:

| `$\rho$` | Process | Consequence |
|:---:|:---|:---|
| `$abs(\rho) < 1$` | Stationary — mean-reverting | Standard regression is valid |
| `$\rho = 1$` | **Gaussian random walk** with drift | Spurious regression — `$R^2 \to 1$` even for unrelated series |
| `$abs(\rho) > 1$` | Explosive | Variance grows without bound |
]

.pull-right-narrow[
- Deciding which case applies requires knowing what a random walk *is* — and how to test for one (**unit root tests**, e.g., Augmented Dickey–Fuller)
- You will encounter all of this in later courses
]

---
# What you are building towards: finance

.pull-left-wide[
> .orange[**Preview only — no need to understand this now**]
]

.pull-left-wide[
Stock prices are modelled as **geometric Brownian motion**:
`$$dS_t = \mu \, S_t \, dt + \sigma \, S_t \, dW_t$$`
where `$W_t$` is a **Wiener process** — the continuous-time process you will meet today
]

.pull-left-wide[
- The **Black–Scholes** option pricing formula is a direct consequence of this model
- Every modern derivatives textbook starts from `$dW_t$`
]

---
# What you are building towards: Markov chains

.pull-left-wide[
> .orange[**Preview only — no need to understand this now**]
]

.pull-left-wide[
Today's lecture also lays the groundwork for **Markov chains** — processes where the future depends *only* on the current state, not the full history:
`$$P(Z_{t+1} = j \mid Z_t = i, Z_{t-1}, \ldots) = P(Z_{t+1} = j \mid Z_t = i)$$`
]

.pull-left-wide[
- Markov chains are used in macroeconomic models of recessions, credit-rating transitions, and job search theory
- They also underlie many machine learning algorithms and reinforcement learning
]

---
class: inverse, middle, center
# Stochastic processes

---
# Time dimension

.pull-left-wide[
- Often, uncertain situations involve a time dimension (e.g., the evolution of a stock price over time)
]

.pull-left-wide[
- Let us consider a simple example: every morning, we toss a fair coin
]

---

# Tossing a coin

.pull-left-wide[
- Let `$Z_t$` be the random variable indicating the outcome of the `$t$`-th coin toss:
`$$Z_t = \begin{cases} 0, & \text{tails on toss } t \\ 1, & \text{heads on toss } t \end{cases}$$`
]

.pull-left-wide[
- We may be interested in several aspects of this:
  - how many times did heads appear in a year, on average?
  - what is the probability of getting heads (or tails) for a whole week in a row?
  - how many tosses can we expect between two heads?
]

---
# Stochastic process

.pull-left-wide[
> A **stochastic process** `$\{Z_t, t \in T\}$` is a collection of random variables `$Z_t$` indexed by time `$t$`, which belongs to the index set `$T$`.
>
> The outcome of a stochastic process, `$\{z_t, t \in T\}$`, is called a **realization** or a **sample path**.
>
> The outcome `$z_t$` of one of the random variables `$Z_t$` in a stochastic process is called an **incident**.
]

.pull-left-wide[
- Like random variables, stochastic processes are *functions* that assign a random variable `$Z_t$` to a point in time `$t$`
]

.pull-left-wide[
- Same as with random variables, stochastic processes can be of two types:
  - in *discrete time*, if the index set `$T$` is countable (finite or infinite)
  - in *continuous time*, if the index set `$T$` is uncountable (infinite)
]

---
# Stochastic process

.pull-left-wide[
- Suppose that we follow our couple for a week, so `$T = \{ 1, 2, \ldots, 7 \}$`
- A realization may be:
`$$z = \{1, 0, 0, 1, 1, 1, 0 \}$$`
- That is, `$z_1 = 1$`, `$z_2 = 0$`, `$z_3 = 0$`, ..., `$z_7 = 0$`
]

.pull-left-wide[
- Same as we could assign probabilities to random variables, we can also assign probabilities to stochastic processes
- In this case, it is the probability of a certain realization (sample path)
]

.pull-left-wide[
- In our example, if the coin tosses are independent, the probability of each sample path is `$0.5^7 = 0.0078$` (because there are `$2^7$` possible paths, all with equal probability)
]

---
# .red[Practice 1: Stochastic processes]

.pull-left-wide[
**1.** Consider the coin-toss example with `$T = \{1, 2, 3\}$` and `$p = 0.5$`.

a) How many possible sample paths are there?

b) What is the probability of the sample path `$z = \{1, 1, 0\}$`?

**2.** A student records whether it rains (`$Z_t = 1$`) or not (`$Z_t = 0$`) each day for 5 days. Assume each day is independent with `$p = 0.3$`.

a) What is the probability of the sample path `$z = \{0, 0, 0, 0, 0\}$`?
]

---
class: inverse, middle, center
# Transformations of a stochastic process

---
# Partial sum

.pull-left-wide[
- One advantage to working with stochastic processes is that we can use them to model real-life events
- Transformations make our job even easier
]

.pull-left-wide[
- For example, suppose you want to count how many times heads appeared until a certain toss
- We can define the random variable `$X_t$` as:
`$$X_t = Z_1 + Z_2 + \ldots + Z_t = \sum_{s=1}^t Z_s$$`
]

.pull-left-wide[
- The stochastic process `$\{ X_t, t \in T \}$` is a **partial sum transformation** of the initial stochastic process `$\{Z_t, t \in T\}$`
- In our example, we would have `$x = \{ 1, 1, 1, 2, 3, 4, 4 \}$`
]

---
# Other transformations

.pull-left-wide[
- Another useful transformation is the **first-difference**, which shows the change in a stochastic process from one period to the next (e.g., change in income)
]

.pull-left-wide[
- This can be done in absolute terms:
`$$X_t = \Delta Z_t = Z_t - Z_{t-1}$$`
]

.pull-left-wide[
- It can also be done in relative terms (e.g., inflation rate):
`$$X_t = \frac{Z_t - Z_{t-1}}{Z_{t-1}}$$`
]

.pull-left-wide[
- Note that in both of these cases, the transformed stochastic process is "shorter" by one period (`$t$` starts at 2!)
]

---
# .red[Practice 2: Transformations]

.pull-left-wide[
Consider the realization `$z = \{2, 5, 3, 8, 6\}$` for `$t = 1, 2, 3, 4, 5$`.

**1.** Compute the partial sum process `$x_t = \sum_{s=1}^t z_s$` for `$t = 1, \ldots, 5$`.

**2.** Compute the absolute first-difference `$x_t = z_t - z_{t-1}$` for `$t = 2, \ldots, 5$`.

**3.** Compute the relative first-difference `$x_t = \frac{z_t - z_{t-1}}{z_{t-1}}$` for `$t = 2, \ldots, 5$`.

**4.** Why does the first-difference process have one fewer observation than the original?
]

---
# .red[Raise your hand 1: Transformations]

.pull-left-wide[
**Q1.** Consider `$z = \{3, 1, 4, 1\}$`. What is the partial sum `$x_3$`?

- **A)** 4 — the value of `$z_3$`
- **B)** 8 — the sum of the first 3 values
- **C)** 9 — the sum of all 4 values
]

.pull-left-wide[
**Q2.** A realization is `$z = \{4, 2, 8, 5\}$`. What is the absolute first-difference `$\Delta z_4 = z_4 - z_3$`?

- **A)** 5 — the level at `$t = 4$`
- **B)** `$-3$` — the change from `$t = 3$` to `$t = 4$`
- **C)** 3 — the absolute value of the change
]

---
class: inverse, middle, center
# States and waiting times

---
# States

.pull-left-wide[
- One aspect of a stochastic process that is particularly interesting is its state at a certain point in time
- The **state** of a stochastic process at time `$t$` is given by the random variable representing the value taken at time `$t$`
]

.pull-left-wide[
- In our example, consider the partial sum transformation `$\{ X_t, t \in T \}$`
- The state of this process at time `$t_0 = 2$` is simply `$X_2$`
]

---
# State distribution

.pull-left-wide[
- Note that `$X_2$` (the sum of `$Z_1$` and `$Z_2$`) can take three values: 0, 1, or 2, with probabilities:
`$$f(0) = P(Z_1 = 0, Z_2 = 0) = 0.25$$`
`$$f(1) = P(Z_1 = 0, Z_2 = 1) + P(Z_1 = 1, Z_2 = 0) = 0.5$$`
`$$f(2) = P(Z_1 = 1, Z_2 = 1) = 0.25$$`
]

.pull-left-wide[
- The distribution of `$X_2$` is called the **state distribution** of the stochastic process at time 2
]

.pull-left-wide[
- This also shows the usefulness of stochastic processes: instead of directly specifying the distribution of the number of times heads appeared, we can use a stochastic process
- This approach is more intuitive and often easier
]

---
# Waiting time

.pull-left-wide[
- Another interesting aspect of a stochastic process is the **waiting time**, i.e., the time it will take until a particular state is reached (or passed) for the first time
]

.pull-left-wide[
- For example, we may want to know how long it takes until heads appears twice
- Since the stochastic process consists of random variables, the time until one particular state is realized is also random
]

.pull-left-wide[
- Let `$Y_2$` be the random variable indicating how long it takes until heads appears twice
]

---
# Waiting-time distribution

.pull-left-wide[
- The distribution of the random variable `$Y_2$` is called a **waiting-time distribution**
- In our example, `$Y_2$` can take the values 2, 3, ..., 7
]

.pull-left-wide[
- We can then calculate probabilities for these values. For example:
  - `$f(2)$`: `$Y_2 = 2$` if `$Z_1 = 1$` and `$Z_2 = 1$`, which can happen with probability 0.25
  - `$f(3)$`: `$Y_2 = 3$` if:
    - `$Z_1 = 1$`, `$Z_2 = 0$`, and `$Z_3 = 1$`, which occurs with probability 0.125
    - `$Z_1 = 0$`, `$Z_2 = 1$`, and `$Z_3 = 1$`, which occurs with probability 0.125
  - `$\Rightarrow$` `$f(3) = 2 \cdot 0.125 = 0.25$`
]

.pull-left-wide[
- Note that the stochastic process may not be "long enough" for us to observe the desired value
- In this case, we can calculate `$1 - F(7)$` and/or `$1 - F(6)$`
]

---
# Types of processes

.pull-left-wide[
.small123[
| | Discrete State | Continuous State |
|---|:---:|:---:|
| **Discrete Time** | Bernoulli, Binomial | Gaussian random walk |
| **Continuous Time** | Poisson | Wiener |
]
]

---
# .red[Practice 3: States and waiting times]

.pull-left-wide[
A quality inspector checks items on a production line. Each item is defective with probability `$p = 0.3$`, independently. Let `$X_t$` be the number of defective items found by inspection `$t$`.

**1.** Find the state distribution of `$X_4$` — compute `$f(0)$`, `$f(1)$`, `$f(2)$`.

**2.** Let `$Y_1$` be the waiting time until the first defective item is found. Compute `$f(1)$`, `$f(2)$`, `$f(3)$`.

**3.** What is `$1 - F(3)$` for `$Y_1$`? What does this represent in the context of quality control?
]

---
class: inverse, middle, center
# The Bernoulli process and the binomial process

---
# The Bernoulli process

.pull-left-wide[
> A **Bernoulli process** `$\{Z_t, t \in T \}$` satisfies the following conditions:
>
> 1. Time: `$T = \{ 1, 2, 3, \ldots \}$`
>
> 2. State: `$Z_t \in \{ 0, 1 \}$` for all `$t$`
>
> 3. Distribution of outcomes: each state follows a Bernoulli distribution with parameter `$p$`
>
> 4. Relationship between outcomes: each state is independent of the other states: `$Z_t$` and `$Z_s$` are independent for all `$s \not= t$`
]

---
# The Bernoulli process

.pull-left-wide[
- The Bernoulli process is characterized by the fact that, at each point in time, an incident can either occur (`$Z_t = 1$`) or not (`$Z_t = 0$`)
- This means that an element from a Bernoulli population is drawn at each point in time
]

.pull-left-wide[
- Our previous example (coin toss) is a Bernoulli process with `$p = 0.5$`
]

---
# Example of path of a Bernoulli process (`$p = 0.5$`)

.center[
![Bernoulli path](Figures/7. Bernoulli path.png)
]

---
# The binomial process

.pull-left-wide[
- Sometimes, what is of interest is the number of events in a Bernoulli process
- As before, we can obtain this using the partial sum transformation:
`$$X_t = \sum_{s=1}^t Z_s$$`
]

.pull-left-wide[
- This is a sum of independent Bernoulli-distributed random variables, which yields a binomial process
]

---
# The binomial process

.pull-left-wide[
> A **binomial process** `$\{X_t, t \in T \}$` satisfies the following conditions:
>
> 1. Time: `$T = \{ 1, 2, 3, \ldots \}$`
>
> 2. State: `$X_t \in \{ 0, 1, \ldots, t \}$` for all `$t$`
>
> 3. Distribution of outcomes: states are such that `$(X_{t+n} - X_t)$` follows a binomial distribution with parameters `$n$` and `$p$`, for all `$n$` and `$t$`, and `$X_0 = 0$`
>
> 4. Relationship between outcomes: `$(X_{t_1} - X_{t_2})$` and `$(X_{t_3} - X_{t_4})$` are independent, for all `$t_1 > t_2 > t_3 > t_4$`
]

---
# Example of path of a binomial process (`$p = 0.5$`)

.center[
![Binomial path](Figures/7. Binomial path.png)
]

---
# The binomial process

.pull-left-wide[
- With a binomial process, the probability of a certain number of events in a time interval depends on the *length* of the time interval and not on *when* this time interval occurs
]

.pull-left-wide[
- In addition, the number of events in a time interval is independent of the number of events in any other non-overlapping time interval
]

.pull-left-wide[
- A binomial process is a **counting process**: it models the number of events that occurred up to a certain point in time
]

---
# State distribution of a binomial process

.pull-left-wide[
- Using the definition of the binomial process, we can easily see that:
`$$X_t - X_0 = X_t \sim Bin(t, p)$$`
- This gives us the state distribution of the binomial process
]

.pull-left-wide[
- We can therefore determine the probability function of `$X_t$`:
`$$f(x) = {t \choose x} \cdot p^x \cdot (1 - p)^{t-x}$$`
]

.pull-left-wide[
- The expected value and variance of `$X_t$` also come from the binomial distribution:
`$$E(X_t) = t \cdot p \qquad Var(X_t) = t \cdot p \cdot (1 - p)$$`
]

.pull-right-narrow[
*Coin toss:* The number of heads after `$t$` fair coin tosses follows `$Bin(t,\; 0.5)$`.
]

---
# Waiting-time distribution of a binomial process

.pull-left-wide[
- As before, let `$Y_{x_0}$` be the time until `$X_t = x_0$` for the first time
- The distribution of `$Y_{x_0}$` is called the **negative binomial distribution**:
`$$Y_{x_0} \sim Negbin(x_0, p)$$`
]

.pull-left-wide[
- The probability function of `$Y_{x_0}$` is:
`$$f(y) = {{y - 1} \choose {x_0 - 1}} \cdot p^{x_0} \cdot (1 - p)^{y - x_0}$$`
]

.pull-left-wide[
- The expected value and variance of `$Y_{x_0}$` are:
`$$E(Y_{x_0}) = \frac{x_0}{p} \qquad Var(Y_{x_0}) = x_0 \cdot \frac{1 - p}{p^2}$$`
]

.pull-right-narrow[
*Coin toss:* The number of tosses needed to get `$x_0$` heads follows `$Negbin(x_0,\; 0.5)$`.
]

---
# Waiting-time distribution of a binomial process

.pull-left-wide[
- We can also ask: what is the time until the first event occurs?
- This is a negative binomial distribution with parameters 1 and `$p$`
]

.pull-left-wide[
- This distribution is also very useful and has its own name: the **geometric distribution**:
`$$Y_{1} \sim Negbin(1, p) = Geo(p)$$`
]

.pull-left-wide[
- The probability function for the geometric distribution is:
`$$f(y) = p \cdot (1 - p)^{y - 1}$$`
- This distribution also represents the distribution of the waiting time until the next event occurs
]

.pull-right-narrow[
*Coin toss:* The number of fair coin tosses until the first head follows `$Geo(0.5)$`.
]

---
# .red[Practice 4: Bernoulli and binomial processes]

.pull-left-wide[
A website receives visits according to a Bernoulli process with `$p = 0.4$` per minute.

**1.** What is `$f(3)$` — the probability of exactly 3 visits in the first 8 minutes?

**2.** What are `$E(X_8)$` and `$Var(X_8)$`?

**3.** What is the probability that the first visit occurs on minute 4? Use the geometric distribution.

**4.** What is the expected waiting time until the 2nd visit?
]

---
# .red[Raise your hand 2: Bernoulli and binomial processes]

.pull-left-wide[
**Q1.** A binomial process has `$p = 0.3$` and `$t = 10$`. What is `$E(X_{10})$`?

- **A)** 0.3 — the probability of success
- **B)** 3 — expected count of events
- **C)** 7 — expected number of non-events
]

.pull-left-wide[
**Q2.** The waiting time until the *first* event in a Bernoulli process follows which distribution?

- **A)** Binomial — counts the total number of events in a fixed number of trials
- **B)** Geometric — special case of negative binomial with `$x_0 = 1$`
- **C)** Negative binomial — the general distribution for waiting times in a binomial process
]

---
class: inverse, middle, center
# The Poisson process

---
# The Poisson process

.pull-left-wide[
> A **Poisson process** `$\{X_t, t \in T \}$` satisfies the following conditions:
>
> 1. Time: `$T = (0, \infty)$`
>
> 2. State: `$X_t \in \{ 0, 1, 2, \ldots \}$` for all `$t$`
>
> 3. Distribution of outcomes: states are such that `$(X_{t + \tau} - X_t)$` follows a Poisson distribution with parameter `$(\lambda \cdot \tau)$`, for all `$\tau$` and `$t$`, and `$X_0 = 0$`
>
> 4. Relationship between outcomes: `$(X_{t_1} - X_{t_2})$` and `$(X_{t_3} - X_{t_4})$` are independent, for all `$t_1 > t_2 > t_3 > t_4$`
]

---
# The Poisson process

.pull-left-wide[
- The Poisson process gives the number of events that have occurred up to a given point in time
- Therefore, it is also a counting process, but in continuous time
]

.pull-left-wide[
- Note that the number of events occurring in any time interval of length `$\tau$` follows a Poisson distribution with parameter `$\lambda \cdot \tau$`, meaning that only the *length* of the interval matters, not *when* the time interval starts
]

---
# State distribution of a Poisson process

.pull-left-wide[
- Using the definition of the Poisson process, we can easily see that:
`$$X_t - X_0 = X_t \sim Poisson(\lambda \cdot t)$$`
- This gives us the state distribution of the Poisson process as a Poisson distribution
]

.pull-left-wide[
- We can therefore determine the probability function of `$X_t$`:
`$$f(x) = \frac{(\lambda \cdot t)^x}{x!} \cdot e^{-(\lambda \cdot t)}$$`
]

.pull-left-wide[
- The expected value and variance of `$X_t$` also come from the Poisson distribution:
`$$E(X_t) = \lambda \cdot t \qquad Var(X_t) = \lambda \cdot t$$`
]

---
# Probability updating in a Poisson process

.pull-left-wide[
- If we know that `$b$` events occurred until time `$s$`, we can update the probability that `$x$` events will occur until time `$t$`:
`$$f(x \mid X_s = b) = P(X_t - X_s = x - b)$$`
]

.pull-left-wide[
- But we know that `$X_t - X_s$` follows a Poisson distribution with parameter `$\lambda \cdot (t - s)$`
- Therefore, the conditional probability function becomes:
`$$f(x \mid X_s = b) = \frac{[\lambda \cdot (t - s)]^{x - b}}{(x - b)!} \cdot e^{-[\lambda \cdot (t-s)]}$$`
]

---
# Waiting-time distribution of a Poisson process

.pull-left-wide[
- As before, let `$Y_{x_0}$` be the time until `$X_t = x_0$` for the first time
- The distribution of `$Y_{x_0}$` is called the **Erlang distribution**:
`$$Y_{x_0} \sim Erlang(\lambda, x_0)$$`
]

.pull-left-wide[
- The probability function of `$Y_{x_0}$` is:
`$$f(y) = \frac{\lambda^{x_0} \cdot y^{x_0 - 1}}{(x_0 - 1)!} \cdot e^{-\lambda \cdot y}$$`
]

.pull-left-wide[
- The expected value and variance of `$Y_{x_0}$` are:
`$$E(Y_{x_0}) = \frac{x_0}{\lambda} \qquad Var(Y_{x_0}) = \frac{x_0}{\lambda^2}$$`
]

.footnote[
  .small123[After Danish statistician Agner Krarup Erlang: https://en.wikipedia.org/wiki/Agner_Krarup_Erlang]
]

---
# Waiting-time distribution of a Poisson process

.pull-left-wide[
- We can also ask: what is the time until the first (or the next) event occurs?
- This is an Erlang distribution with parameters `$\lambda$` and 1
]

.pull-left-wide[
- This distribution is also quite useful and has its own name: the **exponential distribution**:
`$$Y_{1} \sim Erlang(1, \lambda) = Exp(\lambda)$$`
]

.pull-left-wide[
- The probability function for the exponential distribution is:
`$$f(y) = \lambda \cdot e^{-\lambda \cdot y}$$`
]

---
# .red[Practice 5: Poisson process]

.pull-left-wide[
Customers arrive at a shop according to a Poisson process with rate `$\lambda = 3$` per hour.

**1.** What is `$f(5)$` — the probability that exactly 5 customers arrive in the first 2 hours?

**2.** What are `$E(X_2)$` and `$Var(X_2)$`?

**3.** If 4 customers arrived by hour 1, what is `$f(x \mid X_1 = 4)$` for the number arriving by hour 2?

**4.** What is the expected waiting time until the 3rd customer arrives?
]

---
# .red[Raise your hand 3: Poisson process]

.pull-left-wide[
**Q1.** A Poisson process has `$\lambda = 4$` per hour. What is `$Var(X_3)$`?

- **A)** 4 — the rate parameter `$\lambda$`
- **B)** 12 — `$\lambda \cdot t$`
- **C)** 16 — `$\lambda^2$`
]

.pull-left-wide[
**Q2.** 10 customers arrived in the first 2 hours of a Poisson process with `$\lambda = 3$`. How are the arrivals in the *next* hour distributed?

- **A)** Poisson(3) — non-overlapping intervals are independent
- **B)** Poisson(13) — adding the 10 existing arrivals to `$\lambda$`
- **C)** Poisson(5) — because 10 customers arrived in 2 hours, suggesting a rate of 5 per hour
]

---
class: inverse, middle, center
# The Gaussian random walk

---
# The Gaussian random walk

.pull-left-wide[
> A **Gaussian random walk** `$\{X_t, t \in T \}$` satisfies the following conditions:
>
> 1. Time: `$T = \{ 1, 2, 3, \ldots \}$`
>
> 2. State: `$X_t \in (-\infty, \infty)$` for all `$t$`
>
> 3. Distribution of outcomes: states are such that `$(X_{t + n} - X_t)$` follows a Normal distribution with mean `$n \cdot \mu$` and variance `$n \cdot \sigma^2$`, for all `$n$` and `$t$`; and `$X_0 = 0$`
>
> 4. Relationship between outcomes: `$(X_{t_1} - X_{t_2})$` and `$(X_{t_3} - X_{t_4})$` are independent, for all `$t_1 > t_2 > t_3 > t_4$`
]

.pull-left-wide[
- `$\mu$` is called the **drift** parameter and `$\sigma^2$` the **diffusion** parameter
]

---
# The Gaussian random walk

.pull-left-wide[
- A random walk is a partial sum of independent and identically distributed random variables (e.g., the binomial process is a random walk)
- The Gaussian random walk is a continuous-state discrete-time process
]

.pull-left-wide[
- It represents the "drunken walk:" each step can occur in a random direction, either forwards or backwards
- The change in this process, at any point in time, is normally distributed with mean and variance that depend on the *length* of the time interval but not on *when* the interval starts
]

---
# State distribution of a Gaussian random walk

.pull-left-wide[
- Using the definition of the Gaussian random walk, we can easily see that:
`$$X_t - X_0 = X_t \sim \mathcal{N}(t \cdot \mu, t \cdot \sigma^2)$$`
- This gives us the state distribution of the Gaussian random walk as a normal distribution
]

.pull-left-wide[
- We can therefore determine the probability density function of `$X_t$`:
`$$f(x) = \frac{1}{ \sigma \cdot \sqrt{2 \cdot \pi \cdot t }} \cdot e^{- \frac{(x - t \cdot \mu)^2}{2 \cdot t \cdot \sigma^2}}$$`
]

.pull-left-wide[
- The expected value and variance of `$X_t$` also come from the normal distribution:
`$$E(X_t) = t \cdot \mu \qquad Var(X_t) = t \cdot \sigma^2$$`
]

---
# Probability updating in a Gaussian random walk

.pull-left-wide[
- Same as in the case of a Poisson process, we can calculate the conditional CDF of `$X_t$` given that `$X_s = b$`:
`$$F(x \mid X_s = b) = P(X_t - X_s \leq x - b)$$`
]

.pull-left-wide[
- But we know that `$X_t - X_s$` follows a normal distribution with mean `$(t - s) \cdot \mu$` and variance `$(t - s) \cdot \sigma^2$`
- Therefore, the conditional CDF can be derived from the normal distribution `$\mathcal{N}((t - s) \cdot \mu, (t - s) \cdot \sigma^2)$`:
`$$F(x \mid X_s = b) = \Phi \left( \frac{(x - b) - (t - s) \cdot \mu}{\sqrt{(t - s) \cdot \sigma^2}} \right)$$`
]

---
class: inverse, middle, center
# The Wiener process

---
# The Wiener process

.pull-left-wide[
> A **Wiener process** (**Brownian motion**) `$\{X_t, t \in T \}$` satisfies the following conditions:
>
> 1. Time: `$T = (0, \infty)$`
>
> 2. State: `$X_t \in (-\infty, \infty)$` for all `$t$`
>
> 3. Distribution of outcomes: states are such that `$(X_{t + \tau} - X_t)$` follows a Normal distribution with mean `$\tau \cdot \mu$` and variance `$\tau \cdot \sigma^2$`, for all `$\tau$` and `$t$`; and `$X_0 = 0$`
>
> 4. Relationship between outcomes: `$(X_{t_1} - X_{t_2})$` and `$(X_{t_3} - X_{t_4})$` are independent, for all `$t_1 > t_2 > t_3 > t_4$`
]

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# The Wiener process vs Gaussian random walk

.pull-left-wide[
- The Wiener process is very similar to a Gaussian random walk, except that it takes place in continuous time
]

.pull-left-wide[
- In other words, the Gaussian random walk only changes at certain points in time (positive integers), while the Wiener process changes continuously
]

---
# Example of path of a Gaussian random walk and of a Wiener process (`$\mu = 0$`, `$\sigma^2 = 4$`)

.center[
![Wiener path](Figures/7. Wiener path.png)
]

---
# State distribution of a Wiener process

.pull-left-wide[
- The state distribution of a Wiener process is completely identical to the state distribution of a Gaussian random walk:
`$$X_t - X_0 = X_t \sim \mathcal{N}(t \cdot \mu, t \cdot \sigma^2)$$`
]

.pull-left-wide[
- With probability density function:
`$$f(x) = \frac{1}{ \sigma \cdot \sqrt{2 \cdot \pi \cdot t }} \cdot e^{- \frac{(x - t \cdot \mu)^2}{2 \cdot t \cdot \sigma^2}}$$`
]

.pull-left-wide[
- The expected value and variance of `$X_t$` are:
`$$E(X_t) = t \cdot \mu \qquad Var(X_t) = t \cdot \sigma^2$$`
]

---
# Waiting-time distribution of a Wiener process

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- As before, let `$Y_{x_0}$` be the time until `$X_t = x_0$` for the first time
- The distribution of `$Y_{x_0}$` is called the **inverse Gaussian distribution**, or the **Wald distribution**:
`$$Y_{x_0} \sim IG \left( \frac{x_0}{\mu}, \frac{x_0^2}{\sigma^2} \right)$$`
]

.pull-left-wide[
- The expected value and variance of `$Y_{x_0}$` are:
`$$E(Y_{x_0}) = \frac{x_0}{\mu} \qquad Var(Y_{x_0}) = \frac{x_0}{\mu} \cdot \frac{\sigma^2}{\mu^2}$$`
]

---
# .red[Practice 6: Gaussian random walk and Wiener process]

.pull-left-wide[
A stock price follows a Gaussian random walk with `$\mu = 0.1$` and `$\sigma^2 = 1$`.

**1.** What is the distribution of `$X_{10}$`? Give the mean and variance.

**2.** If `$X_5 = 3$`, what is the conditional distribution of `$X_{10}$`?

**3.** Compute `$F(4 \mid X_5 = 3)$` — the probability that `$X_{10} \leq 4$` given `$X_5 = 3$`.

**4.** How does the Wiener process differ from the Gaussian random walk? What stays the same?
]

---
# .red[Raise your hand 4: Poisson and Gaussian]

.pull-left-wide[
**Q1.** A Poisson process has `$\lambda = 2$` per hour. What is `$E(X_3)$`?

- **A)** 2 — the rate parameter
- **B)** 6 — rate times time
- **C)** 0.5 — the mean waiting time until the first event
]

.pull-left-wide[
**Q2.** A Gaussian random walk has `$\mu = 0$` and `$\sigma^2 = 1$`. What is `$Var(X_4)$`?

- **A)** 1 — the diffusion parameter
- **B)** 2 — square root of 4
- **C)** 4 — variance scales with time
]

---
# Before next time
.pull-left[
- Read the assigned reading
- Next time: The sampling process

.red[We will do things with numbers from the real world!!]

]

.pull-right[
![Trees](Figures/Trees1.jpg)
]