Course → API Map

This page connects the mathematical concepts in the five chapters with the corresponding StochX API components.

Chapter 1 — Discrete-Time Markov Chains

Concept	StochX	Documentation
one-step transition matrix P	MarkovChain	cmt_d.md
transition in n steps P^(n)	n_step_transition	cmt_d.md
law mu_n = mu_0 P^n	state_distribution	cmt_d.md
Chapman–Kolmogorov	chapman_kolmogorov	cmt_d.md
accessibility	accessible	cmt_d.md
communication	communicate	cmt_d.md
communication classes	communicating_classes	cmt_d.md
closed classes	closed_classes	cmt_d.md
recurrent/transient classification	classify_states	cmt_d.md
period	period	cmt_d.md
stationary distribution	stationary_distribution	cmt_d.md
limiting distribution	limiting_distribution	cmt_d.md
absorption probability	absorption_probability	cmt_d.md

Chapter 2 — Poisson Processes

Concept	StochX	Documentation
homogeneous Poisson process	PoissonProcess	poisson.md
Poisson counting law	count_probability	poisson.md
independent increments	increment utilities	poisson.md
inter-arrival times	interarrival_samples	poisson.md
arrival times	arrival_times	poisson.md
simulation	simulate	poisson.md
conditioning on counts	conditional_first_arrival_cdf, conditional_arrival_times	poisson.md
superposition	superpose	poisson.md
thinning	split	poisson.md
non-homogeneous process	NonHomogeneousPoissonProcess	poisson.md

Chapter 3 — Continuous-Time Markov Chains

Concept	StochX	Documentation
infinitesimal generator Q	ContinuousTimeMarkovChain	cmtc.md
transition matrix P(t)	transition_matrix_at	cmtc.md
infinitesimal transition approximation	infinitesimal_transition_matrix	cmtc.md
Chapman–Kolmogorov	chapman_kolmogorov	cmtc.md
Kolmogorov equations	forward_derivative, backward_derivative	cmtc.md
matrix exponential	transition_matrix_at	cmtc.md
stationary law	stationary_distribution	cmtc.md
holding rate / waiting time	holding_rate, holding_time	cmtc.md
embedded jump chain	jump_chain_matrix, jump_chain	cmtc.md
simulated trajectory	CTMCPath, simulate	cmtc.md

Birth-death processes

Concept	StochX	Documentation
birth/death rates	BirthDeathProcess	birth_death.md
finite generator	generator_matrix	birth_death.md
embedded jump chain	jump_chain_matrix	birth_death.md
Kolmogorov evolution	kolmogorov_derivative	birth_death.md
stationary product weights	stationary_weights	birth_death.md
pure birth/death/immigration models	class constructors and formulas	birth_death.md
explosion criterion	pure_birth_reciprocal_rate_sum	birth_death.md

Chapter 4 — Conditional Expectation

Concept	StochX	Documentation
finite probability space	FiniteProbabilitySpace	conditional_expectation.md
random variable	RandomVariable	conditional_expectation.md
conditioning partition	Partition	conditional_expectation.md
E(X | G)	conditional_expectation	conditional_expectation.md
E(X | Y)	conditional_expectation_given	conditional_expectation.md
conditional probability	conditional_probability	conditional_expectation.md
total expectation	total_expectation	conditional_expectation.md
tower property	tower	conditional_expectation.md
conditional variance/covariance	conditional_variance, conditional_covariance	conditional_expectation.md
L^2 projection	l2_projection	conditional_expectation.md

Chapter 5 — Discrete-Time Martingales

Concept	StochX	Documentation
filtration	Filtration	martingales.md
adapted process	is_adapted	martingales.md
martingale	Martingale.is_martingale	martingales.md
submartingale	Martingale.is_submartingale	martingales.md
supermartingale	Martingale.is_supermartingale	martingales.md
Doob martingale	Martingale.doob	martingales.md
stopping time	StoppingTime	martingales.md
stopped process	StoppedProcess	martingales.md