Wiener process

The unconditional probability density function follows a normal distribution with mean = 0 and variance = t, at a fixed time t:

$${\displaystyle f_{W_{t}}(x)={\frac {1}{\sqrt {2\pi t}}}e^{-x^{2}/(2t)}.}$$

The expectation is zero:

$${\displaystyle \operatorname {E} [W_{t}]=0.}$$

The variance, using the computational formula, is t:

$${\displaystyle \operatorname {Var} (W_{t})=t.}$$

These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus

$${\displaystyle W_{t}=W_{t}-W_{0}\sim N(0,t).}$$

The covariance and correlation (where ${\displaystyle s\leq t}$):

$${\displaystyle {\begin{aligned}\operatorname {cov} (W_{s},W_{t})&=s,\\\operatorname {corr} (W_{s},W_{t})&={\frac {\operatorname {cov} (W_{s},W_{t})}{\sigma _{W_{s}}\sigma _{W_{t}}}}={\frac {s}{\sqrt {st}}}={\sqrt {\frac {s}{t}}}.\end{aligned}}}$$

These results follow from the definition that non-overlapping increments are independent, of which only the property that they are uncorrelated is used. Suppose that ${\displaystyle t_{1}\leq t_{2}}$.

$${\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[(W_{t_{1}}-\operatorname {E} [W_{t_{1}}])\cdot (W_{t_{2}}-\operatorname {E} [W_{t_{2}}])\right]=\operatorname {E} \left[W_{t_{1}}\cdot W_{t_{2}}\right].}$$

Substituting

$${\displaystyle W_{t_{2}}=(W_{t_{2}}-W_{t_{1}})+W_{t_{1}}}$$

we arrive at:

$${\displaystyle {\begin{aligned}\operatorname {E} [W_{t_{1}}\cdot W_{t_{2}}]&=\operatorname {E} \left[W_{t_{1}}\cdot ((W_{t_{2}}-W_{t_{1}})+W_{t_{1}})\right]\\&=\operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]+\operatorname {E} \left[W_{t_{1}}^{2}\right].\end{aligned}}}$$

Since ${\displaystyle W_{t_{1}}=W_{t_{1}}-W_{t_{0}}}$ and ${\displaystyle W_{t_{2}}-W_{t_{1}}}$ are independent,

$${\displaystyle \operatorname {E} \left[W_{t_{1}}\cdot (W_{t_{2}}-W_{t_{1}})\right]=\operatorname {E} [W_{t_{1}}]\cdot \operatorname {E} [W_{t_{2}}-W_{t_{1}}]=0.}$$

Thus

$${\displaystyle \operatorname {cov} (W_{t_{1}},W_{t_{2}})=\operatorname {E} \left[W_{t_{1}}^{2}\right]=t_{1}.}$$

A corollary useful for simulation is that we can write, for t₁ < t₂:

$${\displaystyle W_{t_{2}}=W_{t_{1}}+{\sqrt {t_{2}-t_{1}}}\cdot Z}$$

where Z is an independent standard normal variable.

Wiener (1923) also gave a representation of a Brownian path in terms of a random Fourier series. If ${\displaystyle \xi _{n}}$ are independent Gaussian variables with mean zero and variance one, then

$${\displaystyle W_{t}=\xi _{0}t+{\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \pi nt}{\pi n}}}$$

and

$${\displaystyle W_{t}={\sqrt {2}}\sum _{n=1}^{\infty }\xi _{n}{\frac {\sin \left(\left(n-{\frac {1}{2}}\right)\pi t\right)}{\left(n-{\frac {1}{2}}\right)\pi }}}$$

represent a Brownian motion on ${\displaystyle [0,1]}$. The scaled process

$${\displaystyle {\sqrt {c}}\,W\left({\frac {t}{c}}\right)}$$

is a Brownian motion on ${\displaystyle [0,c]}$ (cf. Karhunen–Loève theorem).

The joint distribution of the running maximum

$${\displaystyle M_{t}=\max _{0\leq s\leq t}W_{s}}$$

and W_t is

$${\displaystyle f_{M_{t},W_{t}}(m,w)={\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}},\qquad m\geq 0,w\leq m.}$$

To get the unconditional distribution of ${\displaystyle f_{M_{t}}}$, integrate over −∞ < w ≤ m:

$${\displaystyle {\begin{aligned}f_{M_{t}}(m)&=\int _{-\infty }^{m}f_{M_{t},W_{t}}(m,w)\,dw=\int _{-\infty }^{m}{\frac {2(2m-w)}{t{\sqrt {2\pi t}}}}e^{-{\frac {(2m-w)^{2}}{2t}}}\,dw\\[5pt]&={\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}},\qquad m\geq 0,\end{aligned}}}$$

the probability density function of a Half-normal distribution. The expectation[6] is

$${\displaystyle \operatorname {E} [M_{t}]=\int _{0}^{\infty }mf_{M_{t}}(m)\,dm=\int _{0}^{\infty }m{\sqrt {\frac {2}{\pi t}}}e^{-{\frac {m^{2}}{2t}}}\,dm={\sqrt {\frac {2t}{\pi }}}}$$

If at time ${\displaystyle t}$ the Wiener process has a known value ${\displaystyle W_{t}}$, it is possible to calculate the conditional probability distribution of the maximum in interval ${\displaystyle [0,t]}$ (cf. Probability distribution of extreme points of a Wiener stochastic process). The cumulative probability distribution function of the maximum value, conditioned by the known value ${\displaystyle W_{t}}$, is:

$${\displaystyle \,F_{M_{W_{t}}}(m)=\Pr \left(M_{W_{t}}=\max _{0\leq s\leq t}W(s)\leq m\mid W(t)=W_{t}\right)=\ 1-\ e^{-2{\frac {m(m-W_{t})}{t}}}\ \,,\,\ \ m>\max(0,W_{t})}$$

For every c > 0 the process ${\displaystyle V_{t}=(1/{\sqrt {c}})W_{ct}}$ is another Wiener process.

The process ${\displaystyle V_{t}=W_{1}-W_{1-t}}$ for 0 ≤ t ≤ 1 is distributed like W_t for 0 ≤ t ≤ 1.

The process ${\displaystyle V_{t}=tW_{1/t}}$ is another Wiener process.

If a polynomial p(x, t) satisfies the partial differential equation

$${\displaystyle \left({\frac {\partial }{\partial t}}-{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t)=0}$$

then the stochastic process

$${\displaystyle M_{t}=p(W_{t},t)}$$

is a martingale.

Example: ${\displaystyle W_{t}^{2}-t}$ is a martingale, which shows that the quadratic variation of W on [0, t] is equal to t. It follows that the expected time of first exit of W from (−c, c) is equal to c².

More generally, for every polynomial p(x, t) the following stochastic process is a martingale:

$${\displaystyle M_{t}=p(W_{t},t)-\int _{0}^{t}a(W_{s},s)\,\mathrm {d} s,}$$

where a is the polynomial

$${\displaystyle a(x,t)=\left({\frac {\partial }{\partial t}}+{\frac {1}{2}}{\frac {\partial ^{2}}{\partial x^{2}}}\right)p(x,t).}$$

Example: ${\displaystyle p(x,t)=\left(x^{2}-t\right)^{2},}$ ${\displaystyle a(x,t)=4x^{2};}$ the process

$${\displaystyle \left(W_{t}^{2}-t\right)^{2}-4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s}$$

is a martingale, which shows that the quadratic variation of the martingale ${\displaystyle W_{t}^{2}-t}$ on [0, t] is equal to

$${\displaystyle 4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s.}$$

About functions p(xa, t) more general than polynomials, see local martingales.

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.

• 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).
• For any ${\displaystyle \epsilon >0}$, ${\displaystyle w(t)}$ is almost surely not ${\displaystyle ({\tfrac {1}{2}}+\epsilon )}$-Hölder continuous, and almost surely ${\displaystyle ({\tfrac {1}{2}}-\epsilon )}$-Hölder continuous.[7]
• Points of local maximum 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
  $${\displaystyle \lim _{s\to t}{\frac {|w(s)-w(t)|}{|s-t|}}\to \infty .}$$

  
  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 on every interval.
• The quadratic variation of w over [0,t] is t.
• Zeros of the function w are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable).

Local modulus of continuity:

$${\displaystyle \limsup _{\varepsilon \to 0+}{\frac {|w(\varepsilon )|}{\sqrt {2\varepsilon \log \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.}$$

Global modulus of continuity (Lévy):

$${\displaystyle \limsup _{\varepsilon \to 0+}\sup _{0\leq s<t\leq 1,t-s\leq \varepsilon }{\frac {|w(s)-w(t)|}{\sqrt {2\varepsilon \log(1/\varepsilon )}}}=1,\qquad {\text{almost surely}}.}$$

The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely.

The image of the Lebesgue measure on [0, t] under the map w (the pushforward measure) has a density L_t. Thus,

$${\displaystyle \int _{0}^{t}f(w(s))\,\mathrm {d} s=\int _{-\infty }^{+\infty }f(x)L_{t}(x)\,\mathrm {d} x}$$

for a wide class of functions f (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density L_t is (more exactly, can and will be chosen to be) continuous. The number L_t(x) is called the local time at x of w on [0, t]. It is strictly positive for all x of the interval (a, b) where a and b are the least and the greatest value of w on [0, t], respectively. (For x outside this interval the local time evidently vanishes.) Treated as a function of two variables x and t, the local time is still continuous. Treated as a function of t (while x is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of w.

These continuity properties are fairly non-trivial. Consider that the local time can also be defined (as the density of the pushforward measure) for a smooth function. Then, however, the density is discontinuous, unless the given function is monotone. In other words, there is a conflict between good behavior of a function and good behavior of its local time. In this sense, the continuity of the local time of the Wiener process is another manifestation of non-smoothness of the trajectory.

The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by [8]

$${\displaystyle R(D)={\frac {2}{\pi ^{2}\ln 2D}}\approx 0.29D^{-1}.}$$

Therefore, it is impossible to encode ${\displaystyle \{w_{t}\}_{t\in [0,T]}}$ using a binary code of less than ${\displaystyle TR(D)}$ bits and recover it with expected mean squared error less than ${\displaystyle D}$. On the other hand, for any ${\displaystyle \varepsilon >0}$, there exists ${\displaystyle T}$ large enough and a binary code of no more than ${\displaystyle 2^{TR(D)}}$ distinct elements such that the expected mean squared error in recovering ${\displaystyle \{w_{t}\}_{t\in [0,T]}}$ from this code is at most ${\displaystyle D-\varepsilon }$.

In many cases, it is impossible to encode the Wiener process without sampling it first. When the Wiener process is sampled at intervals ${\displaystyle T_{s}}$ before applying a binary code to represent these samples, the optimal trade-off between code rate ${\displaystyle R(T_{s},D)}$ and expected mean square error ${\displaystyle D}$ (in estimating the continuous-time Wiener process) follows the parametric representation [9]

$${\displaystyle R(T_{s},D_{\theta })={\frac {T_{s}}{2}}\int _{0}^{1}\log _{2}^{+}\left[{\frac {S(\varphi )-{\frac {1}{6}}}{\theta }}\right]d\varphi ,}$$

$${\displaystyle D_{\theta }={\frac {T_{s}}{6}}+T_{s}\int _{0}^{1}\min \left\{S(\varphi )-{\frac {1}{6}},\theta \right\}d\varphi ,}$$

where ${\displaystyle S(\varphi )=(2\sin(\pi \varphi /2))^{-2}}$ and ${\displaystyle \log ^{+}[x]=\max\{0,\log(x)\}}$. In particular, ${\displaystyle T_{s}/6}$ is the mean squared error associated only with the sampling operation (without encoding).

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 X_t 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 X_t 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 adapted to the Brownian filtration; and the Brownian filtration is, by definition, the filtration generated by the Wiener process.

The time-integral of the Wiener process

$${\displaystyle W^{(-1)}(t):=\int _{0}^{t}W(s)\,ds}$$

is called integrated Brownian motion or integrated Wiener process. It arises in many applications and can be shown to have the distribution N(0, t³/3),[11] calculated using the fact that the covariance of the Wiener process is ${\displaystyle t\wedge s=\min(t,s)}$.[12]

For the general case of the process defined by

$${\displaystyle V_{f}(t)=\int _{0}^{t}f'(s)W(s)\,ds=\int _{0}^{t}(f(t)-f(s))\,dW_{s}}$$

Then, for ${\displaystyle a>0}$,

$${\displaystyle \operatorname {Var} (V_{f}(t))=\int _{0}^{t}(f(t)-f(s))^{2}\,ds}$$

$${\displaystyle \operatorname {cov} (V_{f}(t+a),V_{f}(t))=\int _{0}^{t}(f(t+a)-f(s))(f(t)-f(s))\,ds}$$

In fact, ${\displaystyle V_{f}(t)}$ is always a zero mean normal random variable. This allows for simulation of ${\displaystyle V_{f}(t+a)}$ given ${\displaystyle V_{f}(t)}$ by taking

$${\displaystyle V_{f}(t+a)=A\cdot V_{f}(t)+B\cdot Z}$$

where Z is a standard normal variable and

$${\displaystyle A={\frac {\operatorname {cov} (V_{f}(t+a),V_{f}(t))}{\operatorname {Var} (V_{f}(t))}}}$$

$${\displaystyle B^{2}=\operatorname {Var} (V_{f}(t+a))-A^{2}\operatorname {Var} (V_{f}(t))}$$

The case of ${\displaystyle V_{f}(t)=W^{(-1)}(t)}$ corresponds to ${\displaystyle f(t)=t}$. All these results can be seen as direct consequences of Itô isometry. The n-times-integrated Wiener process is a zero-mean normal variable with variance ${\displaystyle {\frac {t}{2n+1}}\left({\frac {t^{n}}{n!}}\right)^{2}}$. This is given by the Cauchy formula for repeated integration.

Every continuous martingale (starting at the origin) is a time changed Wiener process.

Example: 2W_t = V(4t) where V is another Wiener process (different from W but distributed like W).

Example. ${\displaystyle W_{t}^{2}-t=V_{A(t)}}$ where ${\displaystyle A(t)=4\int _{0}^{t}W_{s}^{2}\,\mathrm {d} s}$ and V is another Wiener process.

In general, if M is a continuous martingale then ${\displaystyle M_{t}-M_{0}=V_{A(t)}}$ where A(t) is the quadratic variation of M on [0, t], and V is a Wiener process.

Corollary. (See also Doob's martingale convergence theorems) Let M_t be a continuous martingale, and

$${\displaystyle M_{\infty }^{-}=\liminf _{t\to \infty }M_{t},}$$

$${\displaystyle M_{\infty }^{+}=\limsup _{t\to \infty }M_{t}.}$$

Then only the following two cases are possible:

$${\displaystyle -\infty <M_{\infty }^{-}=M_{\infty }^{+}<+\infty ,}$$

$${\displaystyle -\infty =M_{\infty }^{-}<M_{\infty }^{+}=+\infty$$ ;}

other cases (such as ${\displaystyle M_{\infty }^{-}=M_{\infty }^{+}=+\infty ,}$   ${\displaystyle M_{\infty }^{-}<M_{\infty }^{+}<+\infty }$ etc.) are of probability 0.

Especially, a nonnegative continuous martingale has a finite limit (as t → ∞) almost surely.

All stated (in this subsection) for martingales holds also for local martingales.

A wide class of continuous semimartingales (especially, of diffusion processes) is related to the Wiener process via a combination of time change and change of measure.

Using this fact, the qualitative properties stated above for the Wiener process can be generalized to a wide class of continuous semimartingales.[13][14]

The complex-valued Wiener process may be defined as a complex-valued random process of the form ${\displaystyle Z_{t}=X_{t}+iY_{t}}$ where ${\displaystyle X_{t}}$ and ${\displaystyle Y_{t}}$ are independent Wiener processes (real-valued).[15]

Brownian scaling, time reversal, time inversion: the same as in the real-valued case.

Rotation invariance: for every complex number ${\displaystyle c}$ such that ${\displaystyle |c|=1}$ the process ${\displaystyle c\cdot Z_{t}}$ is another complex-valued Wiener process.

If ${\displaystyle f}$ is an entire function then the process ${\displaystyle f(Z_{t})-f(0)}$ is a time-changed complex-valued Wiener process.

Example: ${\displaystyle Z_{t}^{2}=\left(X_{t}^{2}-Y_{t}^{2}\right)+2X_{t}Y_{t}i=U_{A(t)}}$ where

$${\displaystyle A(t)=4\int _{0}^{t}|Z_{s}|^{2}\,\mathrm {d} s}$$

and ${\displaystyle U}$ is another complex-valued Wiener process.

In contrast to the real-valued case, a complex-valued martingale is generally not a time-changed complex-valued Wiener process. For example, the martingale ${\displaystyle 2X_{t}+iY_{t}}$ is not (here ${\displaystyle X_{t}}$ and ${\displaystyle Y_{t}}$ are independent Wiener processes, as before).

The Brownian sheet is a multiparamateric generalization. The definition varies from authors, some define the Brownian sheet to have specifically a two-dimensional time parameter ${\displaystyle t}$ while others define it for general dimensions.