docs for exogenous processes

docs/source/How-To/specify_exogenous_processes.md

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+# Specifying Exogenous Processes in the DCEGM Module
+
+## Overview
+
+Exogenous processes in the DCEGM module define how state variables evolve over time based on transition probabilities. These transition probabilities must be properly specified to ensure consistency with the model structure. This document outlines how to define exogenous processes and specify them within the model's options.
+
+## Defining an Exogenous Process Function
+
+An exogenous process function takes relevant state variables and model parameters as inputs and returns an ordered list of transition probabilities for each possible next state. Each transition probability must meet the following requirements:
+
+- It must be a float (dtype `float64`).
+- It must be non-negative.
+- It must sum to 1 across all possible next states.
+
+### Example 1: Health Transition Process
+
+```python
+import jax.numpy as jnp
+
+def prob_exog_health_mother(health_father, params):
+    prob_good_health = (health_father == 0) * 0.7 + (health_father == 1) * 0.3
+    prob_medium_health = (health_father == 0) * 0.2 + (health_father == 1) * 0.5
+    prob_bad_health = (health_father == 0) * 0.1 + (health_father == 1) * 0.2
+
+    return jnp.array([prob_good_health, prob_medium_health, prob_bad_health])
+```
+
+## Specifying Exogenous Processes in Model Options
+
+Each exogenous process should be defined in the `state_space` section of the model options, mapping process names to transition functions and defining possible states.
+
+### Example 2: Specifying Exogenous Processes in `options`
+
+```python
+options = {
+    "state_space": {
+        "exogenous_processes": {
+            "health_mother": {
+                "transition": prob_exog_health_mother,
+                "states": [0, 1, 2],
+            },
+            "job_offer": {
+                "transition": job_offer_process_transition,
+                "states": [0, 1],
+            },
+        },
+    },
+}
+```
+
+## Validating Exogenous Processes
+
+To ensure correctness, the validation function can be used as implemented in the DCEGM module:
+
+```python
+from dcegm.interface import validate_exogenous_processes
+
+validate_exogenous_processes(model, params)
+```
+
+This function checks for compliance with probability constraints and verifies that transition matrices align with the model's state space. Possible errors and their respective messages include:
+
+- **Non-float transition probabilities**: *"Exogenous process {exog_name} does not return float transition probabilities."*
+- **Negative transition probabilities**: *"Exogenous process {exog_name} does not return non-negative transition probabilities."*
+- **Probabilities greater than 1**: *"Exogenous process {exog_name} does not return transition probabilities less or equal to 1."*
+- **Incorrect number of transitions**: *"Exogenous process {exog_name} does not return the correct number of transitions. Expected {expected}, got {actual}."*
+- **Probabilities not summing to 1**: *"Exogenous process {exog_name} transition probabilities do not sum to 1."*
+
+By following this specification, exogenous processes can be correctly integrated into the DCEGM framework, ensuring a consistent and reliable dynamic model setup.