Real-valued continuous-time [[Stochastic Processes|stochastic process]], often referred to as Brownian motion. The process is a [[Markov Decision Process|Markov process]] characterized by the following properties: >[!info] Definition Wiener Process >A real-valued [[Stochastic Processes|stochastic process]], more precisely an [[Itô Diffusion]] $\{ W_{t} \colon t \leq 0 \}$ defined on a [[Probability Space|probability space]] $(\Sigma, \mathcal{A}, \mu)$ is a standard Wiener process, if: >- The initial value is zero with probability 1. >- The increment $W_{t}-W{s}$ is independent of the past $W_{u}$ with $0 \leq u \leq s$ (uncorrelated for different timesteps). >- The increment $W_{t}-W_{s}$ is a [[Gaussian Distribution|normal]] variable with [[Expectations|mean]] $0$ and [[Covariance and Variance|variance]] $t-s$:$W_t-W_s\sim \mathcal{N}(0,t-s).$ >- For $0 \leq u \leq s$ the increment $W_{t}-W_{s}$ is equal in [[Probability Distribution|distribution]] to $W_{t-s}$. ![[Pasted image 20250602115551.png|center|400]] >[!success] Properties >- [[Continuity for Functions - Pointwise, Uniform, Absolute and Lipschitz|Continuous]] everywhere, [[Derivative, Gradient, Jacobian and Hessian|differentiable]] nowhere >- Unbounded at any finite interval