# Wiener process > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Wiener_process > Markdown URL: https://mediated.wiki/source/Wiener_process.md > Source: https://en.wikipedia.org/wiki/Wiener_process > Source revision: 1356254098 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) Stochastic process generalizing Brownian motion Wiener Process Probability density function Mean 0 Variance σ 2 t {\displaystyle \sigma ^{2}t} A single realization of a one-dimensional Wiener process A single realization of a three-dimensional Wiener process In [mathematics](/source/Mathematics), the **Wiener process** (or **Brownian motion**, due to its historical connection with [the physical process of the same name](/source/Brownian_motion)) is a real-valued [continuous-time](/source/Continuous-time) [stochastic process](/source/Stochastic_process) named after [Norbert Wiener](/source/Norbert_Wiener).[1][2] It is one of the best known [Lévy processes](/source/L%C3%A9vy_process) ([càdlàg](/source/C%C3%A0dl%C3%A0g) stochastic processes with [stationary](/source/Stationary_increments) [independent increments](/source/Independent_increments)). It occurs frequently in pure and [applied mathematics](/source/Applied_mathematics), [economics](/source/Economy), [quantitative finance](/source/Quantitative_finance), [evolutionary biology](/source/Evolutionary_biology), and [physics](/source/Physics). The Wiener process plays an important role in both pure and applied mathematics. In pure mathematics, the Wiener process gave rise to the study of continuous time [martingales](/source/Martingale_(probability_theory)). It is a key process in terms of which more complicated stochastic processes can be described. As such, it plays a vital role in [stochastic calculus](/source/Stochastic_calculus), [diffusion processes](/source/Diffusion_process) and even [potential theory](/source/Potential_theory). It is the driving process of [Schramm–Loewner evolution](/source/Schramm%E2%80%93Loewner_evolution). In [applied mathematics](/source/Applied_mathematics), the Wiener process is used to represent the integral of a [white noise](/source/White_noise) [Gaussian process](/source/Gaussian_process), and so is useful as a model of noise in [electronics engineering](/source/Electronics_engineering) (see [Brownian noise](/source/Brownian_noise)), instrument errors in [filtering theory](/source/Filter_(signal_processing)) and disturbances in [control theory](/source/Control_theory). The Wiener process has applications throughout the mathematical sciences. In physics, researchers use it to model Brownian motion and other types of diffusion, often through the [Fokker–Planck](/source/Fokker%E2%80%93Planck_equation) and [Langevin equations](/source/Langevin_equation), which describe how random motion evolves over time. It also underpins the rigorous [path integral formulation](/source/Path_integral_formulation) of [quantum mechanics](/source/Quantum_mechanics): by the [Feynman–Kac formula](/source/Feynman%E2%80%93Kac_formula), one can represent solutions to the [Schrödinger equation](/source/Schr%C3%B6dinger_equation) in terms of the Wiener process.[3] In [physical cosmology](/source/Physical_cosmology), it also appears in models of [eternal inflation](/source/Eternal_inflation). The Wiener process is prominent in the [mathematical theory of finance](/source/Mathematical_finance) as well, in particular the [Black–Scholes](/source/Black%E2%80%93Scholes) option pricing model.[4] ## Characterisations of the Wiener process The Wiener process *Wt* is characterised by the following properties:[5] 1. *W*0 = 0 [almost surely](/source/Almost_surely). 1. W has [independent increments](/source/Independent_increments): for every *t* > 0, the future increments W t + u − W t , u ≥ 0 , {\textstyle W_{t+u}-W_{t},\,u\geq 0,} are independent of the past values *Ws*, *s* < *t*. 1. W has Gaussian increments: for all u , t ≥ 0 {\textstyle u,t\geq 0} , W t + u − W t ∼ N ( 0 , u ) . {\textstyle W_{t+u}-W_{t}\sim {\mathcal {N}}(0,u).} That is, a time step u results in an increment that is [normally distributed](/source/Normal_distribution) with mean 0 and variance u. 1. W has almost surely continuous paths: *Wt* is almost surely continuous in t. That the process has independent increments means that if 0 ≤ *s*1 < *t*1 ≤ *s*2 < *t*2 then *W**t*1 − *W**s*1 and *W**t*2 − *W**s*2 are independent random variables, and the similar condition holds for n increments. Condition 2 can equivalently be formulated: For every *t* > 0 and u ≥ 0 {\textstyle u\geq 0} , the increment W t + u − W t {\textstyle W_{t+u}-W_{t}} is independent of the sigma-algebra F t B = σ ( W s : 0 ≤ s ≤ t ) . {\textstyle {\mathcal {F}}_{t}^{B}=\sigma (W_{s}:0\leq s\leq t).} . An alternative characterisation of the Wiener process is the so-called *Lévy characterisation* that says that the Wiener process is an almost surely continuous [martingale](/source/Martingale_(probability_theory)) with *W*0 = 0 and [quadratic variation](/source/Quadratic_variation) [*W**t*, *W**t*] = *t* (which means that *W**t*2 − *t* is also a martingale). A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent *N*(0, 1) random variables. This representation can be obtained using the [Karhunen–Loève theorem](/source/Karhunen%E2%80%93Lo%C3%A8ve_theorem). Another characterisation of a Wiener process is the [definite integral](/source/Definite_integral) (from time zero to time t) of a zero mean, unit variance, delta correlated ("white") [Gaussian process](/source/Gaussian_process).[6] The Wiener process can be constructed as the [scaling limit](/source/Scaling_limit) of a [random walk](/source/Random_walk), or other discrete-time stochastic processes with stationary independent increments. This is known as [Donsker's theorem](/source/Donsker's_theorem). Like the random walk, the Wiener process is recurrent in one or two dimensions (meaning that it returns almost surely to any fixed [neighborhood](/source/Neighborhood_(mathematics)) of the origin infinitely often) whereas it is not recurrent in dimensions three and higher (where a multidimensional Wiener process is a process such that its coordinates are independent Wiener processes).[7] Unlike the random walk, it is [scale invariant](/source/Scale_invariance), meaning that α − 1 W α 2 t {\displaystyle \alpha ^{-1}W_{\alpha ^{2}t}} is a Wiener process for any nonzero constant α. The **Wiener measure** is the [probability law](/source/Law_(stochastic_processes)) on the space of [continuous functions](/source/Continuous_function) *g*, with *g*(0) = 0, induced by the Wiener process. An [integral](/source/Integral) based on Wiener measure may be called a **Wiener integral**. ## Wiener process as a limit of random walk Let ξ 1 , ξ 2 , … {\textstyle \xi _{1},\xi _{2},\ldots } be [i.i.d.](/source/Independent_and_identically_distributed_random_variables) random variables with mean 0 and variance 1. For each n, define a continuous time stochastic process W n ( t ) = 1 n ∑ 1 ≤ k ≤ ⌊ n t ⌋ ξ k , t ∈ [ 0 , 1 ] . {\displaystyle W_{n}(t)={\frac {1}{\sqrt {n}}}\sum \limits _{1\leq k\leq \lfloor nt\rfloor }\xi _{k},\qquad t\in [0,1].} This is a random step function. Increments of *Wn* are independent because the ξ k {\textstyle \xi _{k}} are independent. For large n, W n ( t ) − W n ( s ) {\textstyle W_{n}(t)-W_{n}(s)} is close to N ( 0 , t − s ) {\textstyle N(0,t-s)} by the central limit theorem. [Donsker's theorem](/source/Donsker's_theorem) asserts that as n → ∞ {\textstyle n\to \infty } , *Wn* approaches a Wiener process, which mathematically explains the ubiquity of Brownian motion in natural phenomena.[8] ## Properties of a one-dimensional Wiener process Five sampled processes, with expected standard deviation in gray ### Basic properties The unconditional [probability density function](/source/Probability_density_function) follows a [normal distribution](/source/Normal_distribution) with mean = 0 and variance = t, at a fixed time t: f W t ( x ) = 1 2 π t e − x 2 / ( 2 t ) . {\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.} The [expectation](/source/Expected_value) is zero: E ⁡ [ W t ] = 0. {\displaystyle \operatorname {E} [W_{t}]=0.} The [variance](/source/Variance), using the computational formula, is t: Var ⁡ ( W t ) = t . {\displaystyle \operatorname {Var} (W_{t})=t.} These results follow immediately from the definition that increments have a [normal distribution](/source/Normal_distribution), centered at zero. Thus W t = W t − W 0 ∼ N ( 0 , t ) . {\displaystyle W_{t}=W_{t}-W_{0}\sim N(0,t).} A useful decomposition for proving martingale properties also called *Brownian increment decomposition* is W t = W s + ( W t − W s ) , s ≤ t {\displaystyle W_{t}=W_{s}+(W_{t}-W_{s}),\;s\leq t} ### Covariance and correlation The [covariance](/source/Covariance_function) and [correlation](/source/Correlation_function) (where s ≤ t {\textstyle s\leq t} ): cov ⁡ ( W s , W t ) = s , corr ⁡ ( W s , W t ) = cov ⁡ ( W s , W t ) σ W s σ W t = s s t = s t . {\displaystyle {\begin{aligned}\operatorname {cov} (W_{s},W_{t})&=s,\\\operatorname {corr} (W_{s},W_{t})&={\frac {\operatorname {cov} (W_{s},W_{t})}{\sigma _{W_{s}}\sigma _{W_{t}}}}={\frac {s}{\sqrt {st}}}={\sqrt {\frac {s}{t}}}.\end{aligned}}} These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used.[9] Suppose that t 1 ≤ t 2 {\textstyle t_{1}\leq t_{2}} . cov ⁡ ( W t 1 , W t 2 ) = E ⁡ [ ( W t 1 − E ⁡ [ W t 1 ] ) ⋅ ( W t 2 − E ⁡ [ W t 2 ] ) ] = E ⁡ [ W t 1 ⋅ W t 2 ] . {\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[(W_{t_{1}}-\operatorname {E} [W_{t_{1}}])\cdot (W_{t_{2}}-\operatorname {E} [W_{t_{2}}])\right]=\operatorname {E} \left[W_{t_{1}}\cdot W_{t_{2}}\right].} Substituting W t 2 = ( W t 2 − W t 1 ) + W t 1 {\displaystyle W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}} we arrive at: E ⁡ [ W t 1 ⋅ W t 2 ] = E ⁡ [ W t 1 ⋅ ( ( W t 2 − W t 1 ) + W t 1 ) ] = E ⁡ [ W t 1 ⋅ ( W t 2 − W t 1 ) ] + E ⁡ [ W t 1 2 ] . {\displaystyle {\begin{aligned}\operatorname {E} [W_{t_{1}}\cdot W_{t_{2}}]&=\operatorname {E} \left[W_{t_{1}}\cdot ((W_{t_{2}}-W_{t_{1}})+W_{t_{1}})\right]\\&=\operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]+\operatorname {E} \left[W_{t_{1}}^{2}\right].\end{aligned}}} Since W t 1 = W t 1 − W t 0 {\textstyle W_{t_{1}}=W_{t_{1}}-W_{t_{0}}} and W t 2 − W t 1 {\textstyle W_{t_{2}}-W_{t_{1}}} are independent, E ⁡ [ W t 1 ⋅ ( W t 2 − W t 1 ) ] = E ⁡ [ W t 1 ] ⋅ E ⁡ [ W t 2 − W t 1 ] = 0. {\displaystyle \operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]=\operatorname {E} [W_{t_{1}}]\cdot \operatorname {E} [W_{t_{2}}-W_{t_{1}}]=0.} Thus cov ⁡ ( W t 1 , W t 2 ) = E ⁡ [ W t 1 2 ] = t 1 . {\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[W_{t_{1}}^{2}\right]=t_{1}.} A corollary useful for simulation is that we can write, for *t*1 < *t*2: W t 2 = W t 1 + t 2 − t 1 ⋅ Z {\displaystyle W_{t_{2}}=W_{t_{1}}+{\sqrt {t_{2}-t_{1}}}\cdot Z} where Z is an independent standard normal variable. ### Wiener representation Wiener (1923) also gave a representation of a Brownian path in terms of a random [Fourier series](/source/Fourier_series). If ξ n {\textstyle \xi _{n}} are independent Gaussian variables with mean zero and variance one, then W t = ξ 0 t + 2 ∑ n = 1 ∞ ξ n sin ⁡ π n t π n {\displaystyle W_{t}=\xi _{0}t+{\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \pi nt}{\pi n}}} and W t = 2 ∑ n = 1 ∞ ξ n sin ⁡ ( ( n − 1 2 ) π t ) ( n − 1 2 ) π {\displaystyle W_{t}={\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \left(\left(n-{\frac {1}{2}}\right)\pi t\right)}{\left(n-{\frac {1}{2}}\right)\pi }}} represent a Brownian motion on [ 0 , 1 ] {\textstyle [0,1]} . The scaled process c W ( t c ) {\displaystyle {\sqrt {c}}\,W\left({\frac {t}{c}}\right)} is a Brownian motion on [ 0 , c ] {\textstyle [0,c]} (cf. [Karhunen–Loève theorem](/source/Karhunen%E2%80%93Lo%C3%A8ve_theorem)). ### Running maximum The joint distribution of the running maximum M t = max 0 ≤ s ≤ t W s {\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}} and *Wt* is f M t , W t ( m , w ) = 2 ( 2 m − w ) t 2 π t e − ( 2 m − w ) 2 2 t , m ≥ 0 , w ≤ m . {\displaystyle f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},\qquad m\geq 0,w\leq m.} To get the unconditional distribution of f M t {\textstyle f_{M_{t}}} , integrate over −∞ < *w* ≤ *m*: f M t ( m ) = ∫ − ∞ m f M t , W t ( m , w ) d w = ∫ − ∞ m 2 ( 2 m − w ) t 2 π t e − ( 2 m − w ) 2 2 t d w = 2 π t e − m 2 2 t , m ≥ 0 , {\displaystyle {\begin{aligned}f_{M_{t}}(m)&=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw\\[5pt]&={\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}},\qquad m\geq 0,\end{aligned}}} the probability density function of a [Half-normal distribution](/source/Half-normal_distribution). The expectation[10] is E ⁡ [ M t ] = ∫ 0 ∞ m f M t ( m ) d m = ∫ 0 ∞ m 2 π t e − m 2 2 t d m = 2 t π {\displaystyle \operatorname {E} [M_{t}]=\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}}\,dm={\sqrt {\frac {2t}{\pi }}}} If at time t the Wiener process has a known value W t {\textstyle W_{t}} , it is possible to calculate the conditional probability distribution of the maximum in interval [ 0 , t ] {\textstyle [0,t]} (cf. [Probability distribution of extreme points of a Wiener stochastic process](/source/Probability_distribution_of_extreme_points_of_a_Wiener_stochastic_process)). The [cumulative probability distribution function](/source/Cumulative_probability_distribution_function) of the maximum value, [conditioned](/source/Conditional_probability) by the known value W t {\textstyle W_{t}} , is: F M W t ( m ) = Pr ( M W t = max 0 ≤ s ≤ t W ( s ) ≤ m ∣ W ( t ) = W t ) = 1 − e − 2 m ( m − W t ) t , m > max ( 0 , W t ) {\displaystyle \,F_{M_{W_{t}}}(m)=\Pr \left(M_{W_{t}}=\max _{0\leq s\leq t}W(s)\leq m\mid W(t)=W_{t}\right)=\ 1-\ e^{-2{\frac {m(m-W_{t})}{t}}}\ \,,\,\ \ m>\max(0,W_{t})} ### Self-similarity A demonstration of Brownian scaling, showing V t = ( 1 / c ) W c t {\textstyle V_{t}=(1/{\sqrt {c}})W_{ct}} for decreasing c. Note that the average features of the function do not change while zooming in, and note that it zooms in quadratically faster horizontally than vertically. #### Brownian scaling For every *c* > 0 the process V t = ( 1 / c ) W c t {\textstyle V_{t}=(1/{\sqrt {c}})W_{ct}} is another Wiener process. #### Time reversal The process V t = W 1 − t − W 1 {\textstyle V_{t}=W_{1-t}-W_{1}} for 0 ≤ *t* ≤ 1 is distributed like *Wt* for 0 ≤ *t* ≤ 1. #### Time inversion The process V t = t W 1 / t {\textstyle V_{t}=tW_{1/t}} is another Wiener process. #### Projective invariance Consider a Wiener process W ( t ) {\textstyle W(t)} , t ∈ R {\textstyle t\in \mathbb {R} } , conditioned so that lim t → ± ∞ t W ( t ) = 0 {\textstyle \lim _{t\to \pm \infty }tW(t)=0} (which holds almost surely) and as usual W ( 0 ) = 0 {\textstyle W(0)=0} . Then the following are all Wiener processes:[11] W 1 , s ( t ) = W ( t + s ) − W ( s ) , s ∈ R W 2 , σ ( t ) = σ − 1 / 2 W ( σ t ) , σ > 0 W 3 ( t ) = t W ( − 1 / t ) . {\displaystyle {\begin{array}{rcl}W_{1,s}(t)&=&W(t+s)-W(s),\quad s\in \mathbb {R} \\W_{2,\sigma }(t)&=&\sigma ^{-1/2}W(\sigma t),\quad \sigma >0\\W_{3}(t)&=&tW(-1/t).\end{array}}} Thus the Wiener process is invariant under the projective group [PSL(2,R)](/source/PSL(2%2CR)), being invariant under the generators of the group. The action of an element g = [ a b c d ] {\textstyle g={\begin{bmatrix}a&b\\c&d\end{bmatrix}}} is W g ( t ) = ( c t + d ) W ( a t + b c t + d ) − c t W ( a c ) − d W ( b d ) , {\displaystyle W_{g}(t)=(ct+d)W\left({\frac {at+b}{ct+d}}\right)-ctW\left({\frac {a}{c}}\right)-dW\left({\frac {b}{d}}\right),} which defines a [group action](/source/Group_action), in the sense that ( W g ) h = W g h . {\textstyle (W_{g})_{h}=W_{gh}.} #### Conformal invariance in two dimensions Let W ( t ) {\textstyle W(t)} be a two-dimensional Wiener process, regarded as a complex-valued process with W ( 0 ) = 0 ∈ C {\textstyle W(0)=0\in \mathbb {C} } . Let D ⊂ C {\textstyle D\subset \mathbb {C} } be an open set containing 0, and τ D {\textstyle \tau _{D}} be associated Markov time: τ D = inf { t ≥ 0 | W ( t ) ∉ D } . {\displaystyle \tau _{D}=\inf\{t\geq 0|W(t)\not \in D\}.} If f : D → C {\textstyle f:D\to \mathbb {C} } is a [holomorphic function](/source/Holomorphic_function) which is not constant, such that f ( 0 ) = 0 {\textstyle f(0)=0} , then f ( W t ) {\textstyle f(W_{t})} is a time-changed Wiener process in f ( D ) {\textstyle f(D)} .[12] More precisely, the process Y ( t ) {\textstyle Y(t)} is Wiener in D with the Markov time S ( t ) {\textstyle S(t)} where Y ( t ) = f ( W ( σ ( t ) ) ) {\displaystyle Y(t)=f(W(\sigma (t)))} S ( t ) = ∫ 0 t | f ′ ( W ( s ) ) | 2 d s {\displaystyle S(t)=\int _{0}^{t}|f'(W(s))|^{2}\,ds} σ ( t ) = S − 1 ( t ) : t = ∫ 0 σ ( t ) | f ′ ( W ( s ) ) | 2 d s . {\displaystyle \sigma (t)=S^{-1}(t):\quad t=\int _{0}^{\sigma (t)}|f'(W(s))|^{2}\,ds.} ### A class of Brownian martingales If a [polynomial](/source/Polynomial) *p*(*x*, *t*) satisfies the [partial differential equation](/source/Partial_differential_equation) ( ∂ ∂ t + 1 2 ∂ 2 ∂ x 2 ) p ( x , t ) = 0 {\displaystyle \left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t)=0} then the stochastic process M t = p ( W t , t ) {\displaystyle M_{t}=p(W_{t},t)} is a [martingale](/source/Martingale_(probability_theory)). **Example:** W t 2 − t {\textstyle W_{t}^{2}-t} is a martingale, which shows that the [quadratic variation](/source/Quadratic_variation) of W on [0, *t*] is equal to t. It follows that the expected [time of first exit](/source/First_exit_time) of W from (−*c*, *c*) is equal to *c*2. More generally, for every polynomial *p*(*x*, *t*) the following stochastic process is a martingale: M t = p ( W t , t ) − ∫ 0 t a ( W s , s ) d s , {\displaystyle M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s,} where a is the polynomial a ( x , t ) = ( ∂ ∂ t + 1 2 ∂ 2 ∂ x 2 ) p ( x , t ) . {\displaystyle a(x,t)=\left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t).} **Example:** p ( x , t ) = ( x 2 − t ) 2 , {\textstyle p(x,t)=\left(x^{2}-t\right)^{2},} a ( x , t ) = 4 x 2 ; {\textstyle a(x,t)=4x^{2};} the process ( W t 2 − t ) 2 − 4 ∫ 0 t W s 2 d s {\displaystyle \left(W_{t}^{2}-t\right)^{2}-4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s} is a martingale, which shows that the quadratic variation of the martingale W t 2 − t {\textstyle W_{t}^{2}-t} on [0, *t*] is equal to 4 ∫ 0 t W s 2 d s . {\displaystyle 4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s.} About functions *p*(*xa*, *t*) more general than polynomials, see [local martingales](/source/Local_martingale#Martingales_via_local_martingales). ### Some properties of sample paths The set of all functions w with these properties is of full Wiener measure. That is, a path (sample function) of the Wiener process has all these properties almost surely: #### Qualitative properties - For every ε > 0, the function w takes both (strictly) positive and (strictly) negative values on (0, ε). - The function w is continuous everywhere but differentiable nowhere (like the [Weierstrass function](/source/Weierstrass_function)). - For any ϵ > 0 {\textstyle \epsilon >0} , w ( t ) {\textstyle w(t)} is almost surely not ( 1 2 + ϵ ) {\textstyle ({\tfrac {1}{2}}+\epsilon )} -[Hölder continuous](/source/H%C3%B6lder_continuous), and almost surely ( 1 2 − ϵ ) {\textstyle ({\tfrac {1}{2}}-\epsilon )} -Hölder continuous.[13] - Points of [local maximum](/source/Maxima_and_minima) of the function w are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if w has a local maximum at t then lim s → t | w ( s ) − w ( t ) | | s − t | → ∞ . {\displaystyle \lim _{s\to t}{\frac {|w(s)-w(t)|}{|s-t|}}\to \infty .} The same holds for local minima. - The function w has no points of local increase, that is, no *t* > 0 satisfies the following for some ε in (0, *t*): first, *w*(*s*) ≤ *w*(*t*) for all s in (*t* − ε, *t*), and second, *w*(*s*) ≥ *w*(*t*) for all s in (*t*, *t* + ε). (Local increase is a weaker condition than that w is increasing on (*t* − *ε*, *t* + *ε*).) The same holds for local decrease. - The function w is of [unbounded variation](/source/Bounded_variation) on every interval. - The [quadratic variation](/source/Quadratic_variation) of w over [0,*t*] is t. - [Zeros](/source/Root_of_a_function) of the function w are a [nowhere dense](/source/Nowhere_dense_set) [perfect set](/source/Perfect_set) of Lebesgue measure 0 and [Hausdorff dimension](/source/Hausdorff_dimension) 1/2 (therefore, uncountable). #### Quantitative properties #### [Law of the iterated logarithm](/source/Law_of_the_iterated_logarithm) lim sup t → + ∞ | w ( t ) | 2 t log ⁡ log ⁡ t = 1 , almost surely . {\displaystyle \limsup _{t\to +\infty }{\frac {|w(t)|}{\sqrt {2t\log \log t}}}=1,\quad {\text{almost surely}}.} #### [Modulus of continuity](/source/Modulus_of_continuity) Local modulus of continuity: lim sup ε → 0 + | w ( ε ) | 2 ε log ⁡ log ⁡ ( 1 / ε ) = 1 , almost surely . {\displaystyle \limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon )|}{\sqrt {2\varepsilon \log \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.} [Global modulus of continuity](/source/L%C3%A9vy's_modulus_of_continuity_theorem) (Lévy): lim sup ε → 0 + sup 0 ≤ s < t ≤ 1 , t − s ≤ ε | w ( s ) − w ( t ) | 2 ε log ⁡ ( 1 / ε ) = 1 , almost surely . {\displaystyle \limsup _{\varepsilon \to 0+}\sup _{0\leq s 0 {\textstyle \varepsilon >0} , there exists T large enough and a [binary code](/source/Binary_code) of no more than 2 T R ( D ) {\textstyle 2^{TR(D)}} distinct elements such that the expected [mean squared error](/source/Mean_squared_error) in recovering { w t } t ∈ [ 0 , T ] {\textstyle \{w_{t}\}_{t\in [0,T]}} from this code is at most D − ε {\textstyle D-\varepsilon } . In many cases, it is impossible to [encode](/source/Binary_code) the Wiener process without [sampling](/source/Sampling_(signal_processing)) it first. When the Wiener process is sampled at intervals T s {\textstyle T_{s}} before applying a binary code to represent these samples, the optimal trade-off between [code rate](/source/Code_rate) R ( T s , D ) {\textstyle R(T_{s},D)} and expected [mean square error](/source/Mean_square_error) D (in estimating the continuous-time Wiener process) follows the parametric representation [15] R ( T s , D θ ) = T s 2 ∫ 0 1 log 2 + ⁡ [ S ( φ ) − 1 6 θ ] d φ , {\displaystyle R(T_{s},D_{\theta })={\frac {T_{s}}{2}}\int _{0}^{1}\log _{2}^{+}\left[{\frac {S(\varphi )-{\frac {1}{6}}}{\theta }}\right]d\varphi ,} D θ = T s 6 + T s ∫ 0 1 min { S ( φ ) − 1 6 , θ } d φ , {\displaystyle D_{\theta }={\frac {T_{s}}{6}}+T_{s}\int _{0}^{1}\min \left\{S(\varphi )-{\frac {1}{6}},\theta \right\}d\varphi ,} where S ( φ ) = ( 2 sin ⁡ ( π φ / 2 ) ) − 2 {\textstyle S(\varphi )=(2\sin(\pi \varphi /2))^{-2}} and log + ⁡ [ x ] = max { 0 , log ⁡ ( x ) } {\textstyle \log ^{+}[x]=\max\{0,\log(x)\}} . In particular, T s / 6 {\textstyle T_{s}/6} is the mean squared error associated only with the sampling operation (without encoding). ## Related processes Wiener processes with drift (blue) and without drift (red) 2D Wiener processes with drift (blue) and without drift (red) The [generator](/source/Infinitesimal_generator_(stochastic_processes)) of Brownian motion on [Riemannian manifolds](/source/Riemannian_manifold) is 1⁄2 times the [Laplace–Beltrami operator](/source/Laplace%E2%80%93Beltrami_operator). The image above shows Brownian motion on the surface of a 2-sphere. The stochastic process defined by X t = μ t + σ W t {\displaystyle X_{t}=\mu t+\sigma W_{t}} is called a **Wiener process with drift μ** and infinitesimal variance σ2. These processes exhaust continuous [Lévy processes](/source/L%C3%A9vy_process), which means that they are the only continuous Lévy processes, as a consequence of the Lévy–Khintchine representation. Two random processes on the time interval [0, 1] appear, roughly speaking, when conditioning the Wiener process to vanish on both ends of [0,1]. With no further conditioning, the process takes both positive and negative values on [0, 1] and is called [Brownian bridge](/source/Brownian_bridge). Conditioned also to stay positive on (0, 1), the process is called [Brownian excursion](/source/Brownian_excursion).[16] In both cases a rigorous treatment involves a limiting procedure, since the formula *P*(*A*|*B*) = *P*(*A* ∩ *B*)/*P*(*B*) does not apply when *P*(*B*) = 0. A [geometric Brownian motion](/source/Geometric_Brownian_motion) can be written e μ t − σ 2 t 2 + σ W t . {\displaystyle e^{\mu t-{\frac {\sigma ^{2}t}{2}}+\sigma W_{t}}.} It is a stochastic process which is used to model processes that can never take on negative values, such as the value of stocks. The stochastic process X t = e − t W e 2 t {\displaystyle X_{t}=e^{-t}W_{e^{2t}}} is distributed like the [Ornstein–Uhlenbeck process](/source/Ornstein%E2%80%93Uhlenbeck_process) with parameters θ = 1 {\textstyle \theta =1} , μ = 0 {\textstyle \mu =0} , and σ 2 = 2 {\textstyle \sigma ^{2}=2} . The [time of hitting](/source/Hitting_time) a single point *x* > 0 by the Wiener process is a random variable with the [Lévy distribution](/source/L%C3%A9vy_distribution). The family of these random variables (indexed by all positive numbers x) is a [left-continuous](/source/Left-continuous) modification of a [Lévy process](/source/L%C3%A9vy_process). The [right-continuous](/source/Right-continuous) [modification](/source/Random_process) of this process is given by times of [first exit](/source/Hitting_time) from closed intervals [0, *x*]. The [local time](/source/Local_time_(mathematics)) *L* = (*Lxt*)*x* ∈ **R**, *t* ≥ 0 of a Brownian motion describes the time that the process spends at the point x. Formally L x ( t ) = ∫ 0 t δ ( x − B t ) d s {\displaystyle L^{x}(t)=\int _{0}^{t}\delta (x-B_{t})\,ds} where *δ* is the [Dirac delta function](/source/Dirac_delta_function). The behaviour of the local time is characterised by [Ray–Knight theorems](/source/Local_time_(mathematics)#Ray-Knight_Theorems). ### Brownian martingales Let A be an event related to the Wiener process (more formally: a set, measurable with respect to the Wiener measure, in the space of functions), and *Xt* the conditional probability of A given the Wiener process on the time interval [0, *t*] (more formally: the Wiener measure of the set of trajectories whose concatenation with the given partial trajectory on [0, *t*] belongs to A). Then the process *Xt* is a continuous martingale. Its martingale property follows immediately from the definitions, but its continuity is a very special fact – a special case of a general theorem stating that all Brownian martingales are continuous. A Brownian martingale is, by definition, a [martingale](/source/Martingale_(probability_theory)) adapted to the Brownian filtration; and the Brownian filtration is, by definition, the [filtration](/source/Filtration_(probability_theory)) generated by the Wiener process. Also B t 2 − t {\textstyle B_{t}^{2}-t} and e θ B t − θ 2 2 t {\textstyle e^{\theta B_{t}-{\tfrac {\theta ^{2}}{2}}t}} are martingales.[17] ### Integrated Brownian motion The time-integral of the Wiener process W ( − 1 ) ( t ) := ∫ 0 t W ( s ) d s {\displaystyle W^{(-1)}(t):=\int _{0}^{t}W(s)\,ds} is called **integrated Brownian motion** or **integrated Wiener process**. It arises in many applications and can be shown to have the distribution *N*(0, *t*3/3),[18] calculated using the fact that the covariance of the Wiener process is t ∧ s = min ( t , s ) {\textstyle t\wedge s=\min(t,s)} .[19] For the general case of the process defined by V f ( t ) = ∫ 0 t f ′ ( s ) W ( s ) d s = ∫ 0 t ( f ( t ) − f ( s ) ) d W s {\displaystyle V_{f}(t)=\int _{0}^{t}f'(s)W(s)\,ds=\int _{0}^{t}(f(t)-f(s))\,dW_{s}} Then, for a > 0 {\textstyle a>0} , Var ⁡ ( V f ( t ) ) = ∫ 0 t ( f ( t ) − f ( s ) ) 2 d s {\displaystyle \operatorname {Var} (V_{f}(t))=\int _{0}^{t}(f(t)-f(s))^{2}\,ds} cov ⁡ ( V f ( t + a ) , V f ( t ) ) = ∫ 0 t ( f ( t + a ) − f ( s ) ) ( f ( t ) − f ( s ) ) d s {\displaystyle \operatorname {cov} (V_{f}(t+a),V_{f}(t))=\int _{0}^{t}(f(t+a)-f(s))(f(t)-f(s))\,ds} In fact, V f ( t ) {\textstyle V_{f}(t)} is always a zero mean normal random variable. This allows for simulation of V f ( t + a ) {\textstyle V_{f}(t+a)} given V f ( t ) {\textstyle V_{f}(t)} by taking V f ( t + a ) = A ⋅ V f ( t ) + B ⋅ Z {\displaystyle V_{f}(t+a)=A\cdot V_{f}(t)+B\cdot Z} where *Z* is a standard normal variable and A = cov ⁡ ( V f ( t + a ) , V f ( t ) ) Var ⁡ ( V f ( t ) ) {\displaystyle A={\frac {\operatorname {cov} (V_{f}(t+a),V_{f}(t))}{\operatorname {Var} (V_{f}(t))}}} B 2 = Var ⁡ ( V f ( t + a ) ) − A 2 Var ⁡ ( V f ( t ) ) {\displaystyle B^{2}=\operatorname {Var} (V_{f}(t+a))-A^{2}\operatorname {Var} (V_{f}(t))} The case of V f ( t ) = W ( − 1 ) ( t ) {\textstyle V_{f}(t)=W^{(-1)}(t)} corresponds to f ( t ) = t {\textstyle f(t)=t} . All these results can be seen as direct consequences of [Itô isometry](/source/It%C3%B4_isometry). The *n*-times-integrated Wiener process is a zero-mean normal variable with variance t 2 n + 1 ( t n n ! ) 2 {\textstyle {\frac {t}{2n+1}}\left({\frac {t^{n}}{n!}}\right)^{2}} . This is given by the [Cauchy formula for repeated integration](/source/Cauchy_formula_for_repeated_integration). ### Time change Every continuous martingale (starting at the origin) is a time changed Wiener process. **Example:** 2*W**t* = *V*(4*t*) where V is another Wiener process (different from W but distributed like W). **Example.** W t 2 − t = V A ( t ) {\textstyle W_{t}^{2}-t=V_{A(t)}} where A ( t ) = 4 ∫ 0 t W s 2 d s {\textstyle A(t)=4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s} and V is another Wiener process. In general, if M is a continuous martingale then M t − M 0 = V A ( t ) {\textstyle M_{t}-M_{0}=V_{A(t)}} where *A*(*t*) is the [quadratic variation](/source/Quadratic_variation) of M on [0, *t*], and V is a Wiener process. **Corollary.** (See also [Doob's martingale convergence theorems](/source/Doob's_martingale_convergence_theorems)) Let *Mt* be a continuous martingale, and M ∞ − = lim inf t → ∞ M t , {\displaystyle M_{\infty }^{-}=\liminf _{t\to \infty }M_{t},} M ∞ + = lim sup t → ∞ M t . {\displaystyle M_{\infty }^{+}=\limsup _{t\to \infty }M_{t}.} Then only the following two cases are possible: − ∞ < M ∞ − = M ∞ + < + ∞ , {\displaystyle -\infty