Definition

A standard (one-dimensional) Wiener process (also called Brownian motion) is a stochastic process {Wt}t≥0+\{W_t\}_{t\geq 0+} indexed by t∈ℝ+t \in \mathbb{R}^+ with properties:

  1. W0=0W_0 = 0
  2. With probability 11, function t→Wtt \to W_t continuous in tt
  3. stationary, independent increments for process {Wt}t≥0\{W_t\}_{t\geq 0}
  4. normal distribution (0,t)(0,t) for increment Wt+s−WsW_{t+s} - W_s

dd-dimensional

(Standard) Brownian motion is a stochastic process (Bt)t≥0(B_t)_{t \geq 0} in ℝd\mathbb{R}^d that satisfies the following properties:

  1. B0=0B_0=0
  2. (independence of increments) For all 0<t1<...<tk0 < t_1 < ... < t_k, random variables (Bt1,Bt2−Bt1,...,Btk−Btk−1)(B_{t_1}, B_{t_2}-B_{t_1},...,B_{t_k}-B_{t_{k-1}}) are mutually independent.
  3. (law of the increments) For all 0≤s<t<∞0 \leq s < t < \infty,B_t - B_s \sim \mathcal{N}(0, (t-s) I_d)$$ ([[Gaussian distribution|normal distribution]]) 4. (_continuity of the paths_) Almost surely, $t \mapsto B_t$ is [[continuous function|continuous]] ## MATLAB realization ```matlab %BPATH1 Brownian path simulation randn('state',100) % set the state of randn T = 1; N = 500; dt = T/N; dW = zeros(1,N); % preallocate arrays ... W = zeros(1,N); % for efficiency dW(1) = sqrt(dt)*randn; % first approximation outside the loop ... W(1) = dW(1); % since W(0) = 0 is not allowed for j = 2:N dW(j) = sqrt(dt)*randn; % general increment W(j) = W(j-1) + dW(j); end plot([0:dt:T],[0,W],'r-') % plot W against t xlabel('t','FontSize',16) ylabel('W(t)','FontSize',16,'Rotation',0) ``` ## See also - [[Euler-Maruyama method]] --- ## References 1. https://galton.uchicago.edu/~lalley/Courses/313/BrownianMotionCurrent.pdf 2. https://sites.me.ucsb.edu/~moehlis/APC591/tutorials/tutorial7/node2.html 3. https://en.wikipedia.org/wiki/Wiener_process 4. https://www.physik.uni-augsburg.de/theo1/hanggi/History/Langevin1908.pdf 5. https://chewisinho.github.io/main.pdf, p. 5 ParseError: Can't use function '$' in math mode at position 41: …}(0, (t-s) I_d)$̲$ ([[Gaussian d…