Poisson and Levy processes; Ornstein-Uhlenbeck; geometric Brownian motion; Fokker-Planck; Langevin; branching; renewal; Gaussian process; Markov decision process; spatial point process; stationary processes. Complementary to existing…
Stochastic Processes
Poisson process (counting)
N(t) with rate lambda has independent stationary increments and N(t) - N(s) ~ Poisson(lambda(t-s)); inter-arrival times exponential;…
Levy process (Levy-Ito decomposition)
Stationary independent-increment processes decompose as drift + Brownian motion + compensated Poisson jumps; characteristic function via…
Ornstein-Uhlenbeck process
dX = -theta X dt + sigma dW: mean-reverting Gaussian process; stationary distribution N(0, sigma^2 / 2 theta); models velocity in Langevin…
Geometric Brownian motion
dS = mu S dt + sigma S dW: lognormal-distributed exponential-of-Brownian; foundation of Black-Scholes option pricing and…
Fokker-Planck density evolution
Time-evolution of probability density rho(x,t) for SDE dX = mu(x) dt + sigma(x) dW: partial_t rho = -partial_x(mu rho) + 0.5…
Langevin equation
Langevin 1908: dv/dt = -gamma v + sqrt(2 gamma kT / m) eta(t); stochastic damped Newton equation; recovers Maxwell-Boltzmann at equilibrium…
Branching process (Galton-Watson)
Galton-Watson 1873: each individual produces independent offspring count X with mean m; population dies out almost surely iff m <= 1; basis…
Renewal process
Generalizes Poisson with iid inter-arrival times of arbitrary distribution F; renewal function m(t) = E[N(t)]; key renewal theorem gives…
Gaussian process (GP)
Distribution over functions: any finite collection {f(x_i)} jointly Gaussian with mean m(x) and covariance k(x,x'); foundation of Bayesian…
Markov decision process
(S, A, P, R, gamma) tuple: state-action-transition-reward-discount; Bellman equation V*(s) = max_a [R(s,a) + gamma sum_s' P(s'|s,a)…
Point process (spatial)
Random configurations of points in R^d: Poisson, Cox (doubly-stochastic Poisson), Hawkes (self-exciting), Gibbs (interaction); Campbell…
Stationary process and ergodicity
Strict-stationary: joint law invariant under time-shift; weak-stationary: constant mean and covariance depending only on lag; Birkhoff…
Ito vs Stratonovich SDE
SDE dX = mu dt + sigma dW interpretable two ways: Ito uses left-endpoint (martingale property + non-anticipating) vs Stratonovich uses…
Girsanov theorem (change of measure)
Girsanov 1960 / Cameron-Martin 1944: under change of probability-measure via Radon-Nikodym derivative exp(-int theta dW - 0.5 int theta^2…
Martingale representation theorem
Every L2 martingale adapted to Brownian filtration has stochastic-integral representation M_t = M_0 + int phi_s dW_s; foundation of…
Doob-Meyer decomposition
Doob 1953 / Meyer 1962: any submartingale X uniquely decomposes as X = M + A where M martingale + A predictable increasing; central tool in…
Skorohod embedding
Skorohod 1961: any centered random-variable X embeddable in BM via stopping-time tau s.t. B_tau ~= X; basis of weak-convergence proofs +…
Reflected Brownian motion
BM with absorbing/reflecting boundary modeling queues, financial-default-barriers, semilinear-PDEs; Tanaka equation Y_t = sup_{s <=t}…
Chapman-Kolmogorov (1931)
Chapman-Kolmogorov 1931 forward + backward equations; foundational text in Markov-process theory.
Doob decomposition (1953)
J L Doob 1953 'Stochastic Processes'; martingale-decomposition; modern semimartingale theory + financial-math.
Lévy process (Lévy 1934)
P Lévy 1934 Lévy-Khintchine; modern Lévy-Itô decomposition; basis of jump-diffusion + heavy-tailed financial models.
Brownian motion (Wiener 1923)
N Wiener 1923 mathematical Brownian-motion; modern Wiener measure + path-integrals; foundational stochastic-process.
Kalman filter (1960)
R E Kalman 1960 LQG state-estimator; modern unscented + ensemble + particle-filter generalizations.
Rough paths (Lyons 1998)
T Lyons 1998 rough-path theory; modern Hairer 2014 (Fields 2014) regularity-structures; pathwise stochastic analysis.
Tracy-Widom distribution F_beta for largest eigenvalue of GUE/GOE/GSE
Tracy-Widom distribution (Tracy-Widom 1994 *Comm Math Phys* 159, 151; 1996 *Comm Math Phys* 177, 727) is the limit law of the largest…
Wigner-Dyson level-spacing statistics P_beta(s) for chaotic spectra
Wigner-Dyson level-spacing statistics (Wigner 1955 *Ann Math* 62, 548; Dyson 1962 *J Math Phys* 3, 140-175) describe the universal…
Lai-Robbins asymptotic regret lower bound R_T >= ln(T) sum Delta_i / KL
Lai-Robbins lower bound (Lai-Robbins 1985 *Adv Appl Math* 6, 4) is the asymptotic information-theoretic floor on cumulative regret R_T = T…