Wiener process

The Wiener process is characterised by the following properties: • almost surely. • has independent increments: for every , the future increments W_{t+u} - W_t,\, u \ge 0, are independent of the past values , • has Gaussian increments: for all u, t \ge 0, W_{t+u} - W_t \sim \mathcal N(0,u). That is, a time step results in an increment that is normally distributed with mean 0 and variance . • has almost surely continuous paths: is almost surely continuous in . That the process has independent increments means that if then and are independent random variables, and the similar condition holds for increments. Condition 2 can equivalently be formulated: For every and u \ge 0, the increment W_{t+u} - W_t is independent of the sigma-algebra \mathcal{F}_t^B = \sigma(W_s : 0 \le s \le 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 with and quadratic variation (which means that is also a martingale). A third characterisation is that the Wiener process has a spectral representation as a sine series whose coefficients are independent random variables. This representation can be obtained using the Karhunen–Loève theorem. Another characterisation of a Wiener process is the definite integral (from time zero to time ) of a zero mean, unit variance, delta correlated ("white") Gaussian process. The Wiener process can be constructed as the scaling limit of a random walk, or other discrete-time stochastic processes with stationary independent increments. This is known as 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 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). Unlike the random walk, it is scale invariant, meaning that \alpha^{-1} W_{\alpha^2 t} is a Wiener process for any nonzero constant . The Wiener measure is the probability law on the space of continuous functions , with , induced by the Wiener process. An integral based on Wiener measure may be called a Wiener integral. ==Wiener process as a limit of random walk==

Let \xi_1, \xi_2, \ldots be i.i.d. random variables with mean 0 and variance 1. For each , define a continuous time stochastic process 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 are independent because the \xi_k are independent. For large , W_n(t)-W_n(s) is close to N(0,t-s) by the central limit theorem. Donsker's theorem asserts that as n \to \infty, approaches a Wiener process, which mathematically explains the ubiquity of Brownian motion in natural phenomena. == Properties of a one-dimensional Wiener process ==

Basic properties The unconditional probability density function follows a normal distribution with mean = 0 and variance = , at a fixed time : f_{W_t}(x) = \frac{1}{\sqrt{2 \pi t}} e^{-x^2/(2t)}. The expectation is zero: \operatorname E[W_t] = 0. The variance, using the computational formula, is : \operatorname{Var}(W_t) = t. These results follow immediately from the definition that increments have a normal distribution, centered at zero. Thus 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 \le t Covariance and correlation The covariance and correlation (where s \leq t): \begin{align} \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{align} 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 t_1\leq t_2. \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} we arrive at: \begin{align} \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{align} Since W_{t_1}=W_{t_1} - W_{t_0} and W_{t_2} - W_{t_1} are independent, \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 \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 : W_{t_2} = W_{t_1}+\sqrt{t_2-t_1}\cdot Z where 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. If \xi_n are independent Gaussian variables with mean zero and variance one, then W_t = \xi_0 t+ \sqrt{2}\sum_{n=1}^\infty \xi_n\frac{\sin \pi n t}{\pi n} and 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]. The scaled process \sqrt{c}\, W\left(\frac{t}{c}\right) is a Brownian motion on [0,c] (cf. Karhunen–Loève theorem). Running maximum The joint distribution of the running maximum M_t = \max_{0 \leq s \leq t} W_s and is f_{M_t,W_t}(m,w) = \frac{2(2m - w)}{t\sqrt{2 \pi t}} e^{-\frac{(2m-w)^2}{2t}}, \qquad m \ge 0, w \leq m. To get the unconditional distribution of f_{M_t}, integrate over : \begin{align} 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 \ge 0, \end{align} the probability density function of a Half-normal distribution. The expectation is \operatorname{E}[M_t] = \int_0^\infty m f_{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 the Wiener process has a known value W_{t}, it is possible to calculate the conditional probability distribution of the maximum in interval [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 W_t, is: \, 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 Brownian scaling For every the process V_t = (1 / \sqrt c) W_{ct} is another Wiener process. Time reversal The process V_t = W_{1-t} - W_{1} for is distributed like for . Time inversion The process V_t = t W_{1/t} is another Wiener process. Projective invariance Consider a Wiener process W(t), t\in\mathbb R, conditioned so that \lim_{t\to\pm\infty}tW(t)=0 (which holds almost surely) and as usual W(0)=0. Then the following are all Wiener processes: \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), being invariant under the generators of the group. The action of an element g = \begin{bmatrix}a&b\\c&d\end{bmatrix} is 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, in the sense that (W_g)_h = W_{gh}. Conformal invariance in two dimensions Let W(t) be a two-dimensional Wiener process, regarded as a complex-valued process with W(0)=0\in\mathbb C. Let D\subset\mathbb C be an open set containing 0, and \tau_D be associated Markov time: \tau_D = \inf \{ t\ge 0 |W(t)\not\in D\}. If f:D\to \mathbb C is a holomorphic function which is not constant, such that f(0)=0, then f(W_t) is a time-changed Wiener process in f(D). More precisely, the process Y(t) is Wiener in with the Markov time S(t) where Y(t) = f(W(\sigma(t))) S(t) = \int_0^t|f'(W(s))|^2\,ds \sigma(t) = S^{-1}(t):\quad t = \int_0^{\sigma(t)}|f'(W(s))|^2\,ds. A class of Brownian martingales If a polynomial satisfies the partial differential equation \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 ) is a martingale. Example: W_t^2 - t is a martingale, which shows that the quadratic variation of on is equal to . It follows that the expected time of first exit of from (−c, c) is equal to . More generally, for every polynomial the following stochastic process is a martingale: M_t = p ( W_t, t ) - \int_0^t a(W_s,s) \, \mathrm{d}s, where is the polynomial 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) = \left(x^2 - t\right)^2, a(x,t) = 4x^2; the process \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 on [0, t] is equal to 4 \int_0^t W_s^2 \, \mathrm{d}s. About functions more general than polynomials, see local martingales. Some properties of sample paths The set of all functions 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 takes both (strictly) positive and (strictly) negative values on (0, ε). • The function is continuous everywhere but differentiable nowhere (like the Weierstrass function). • For any \epsilon > 0, w(t) is almost surely not (\tfrac 1 2 + \epsilon)-Hölder continuous, and almost surely (\tfrac 1 2 - \epsilon)-Hölder continuous. • Points of local maximum of the function are a dense countable set; the maximum values are pairwise different; each local maximum is sharp in the following sense: if has a local maximum at then \lim_{s \to t} \frac \to \infty. The same holds for local minima. • The function has no points of local increase, that is, no satisfies the following for some ε in (0, t): first, for all in (t − ε, t), and second, for all in (t, t + ε). (Local increase is a weaker condition than that is increasing on (t − ε, t + ε).) The same holds for local decrease. • The function is of unbounded variation on every interval. • The quadratic variation of over [0,t] is . • Zeros of the function are a nowhere dense perfect set of Lebesgue measure 0 and Hausdorff dimension 1/2 (therefore, uncountable). Quantitative properties ===== Law of the iterated logarithm ===== \limsup_{t\to+\infty} \frac{ |w(t)| }{ \sqrt{ 2t \log\log t } } = 1, \quad \text{almost surely}. ===== Modulus of continuity ===== Local modulus of continuity: \limsup_{\varepsilon \to 0+} \frac{ |w(\varepsilon)| }{ \sqrt{ 2\varepsilon \log\log(1/\varepsilon) } } = 1, \qquad \text{almost surely}. Global modulus of continuity (Lévy): \limsup_{\varepsilon\to0+} \sup_{0\le s ===== Dimension doubling theorem ===== The dimension doubling theorems say that the Hausdorff dimension of a set under a Brownian motion doubles almost surely. Local time The image of the Lebesgue measure on [0, t] under the map (the pushforward measure) has a density . Thus, \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 (namely: all continuous functions; all locally integrable functions; all non-negative measurable functions). The density is (more exactly, can and will be chosen to be) continuous. The number is called the local time at of on [0, t]. It is strictly positive for all of the interval (a, b) where a and b are the least and the greatest value of on [0, t], respectively. (For outside this interval the local time evidently vanishes.) Treated as a function of two variables and , the local time is still continuous. Treated as a function of (while is fixed), the local time is a singular function corresponding to a nonatomic measure on the set of zeros of . 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. Information rate The information rate of the Wiener process with respect to the squared error distance, i.e. its quadratic rate-distortion function, is given by R(D) = \frac{2}{\pi^2 D \ln 2} \approx 0.29D^{-1}. Therefore, it is impossible to encode \{w_t \}_{t \in [0,T]} using a binary code of less than T R(D) bits and recover it with expected mean squared error less than . On the other hand, for any \varepsilon>0, there exists large enough and a binary code of no more than 2^{TR(D)} distinct elements such that the expected mean squared error in recovering \{w_t \}_{t \in [0,T]} from this code is at most 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 T_s before applying a binary code to represent these samples, the optimal trade-off between code rate R(T_s,D) and expected mean square error (in estimating the continuous-time Wiener process) follows the parametric representation 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_\theta = \frac{T_s}{6} + T_s\int_0^1 \min\left\{S(\varphi)-\frac{1}{6},\theta \right\} d\varphi, where S(\varphi) = (2 \sin(\pi \varphi /2))^{-2} and \log^+[x] = \max\{0,\log(x)\}. In particular, T_s/6 is the mean squared error associated only with the sampling operation (without encoding). == Related processes ==

of Brownian motion on Riemannian manifolds is times the Laplace–Beltrami operator. The image above shows Brownian motion on the surface of a 2-sphere. The stochastic process defined by 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, 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. Conditioned also to stay positive on (0, 1), the process is called Brownian excursion. 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 can be written 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^{2t}} is distributed like the Ornstein–Uhlenbeck process with parameters \theta = 1, \mu = 0, and \sigma^2 = 2. The time of hitting a single point by the Wiener process is a random variable with the Lévy distribution. The family of these random variables (indexed by all positive numbers ) is a left-continuous modification of a Lévy process. The right-continuous modification of this process is given by times of first exit from closed intervals [0, x]. The local time of a Brownian motion describes the time that the process spends at the point . Formally L^x(t) =\int_0^t \delta(x-B_t)\,ds where δ is the Dirac delta function. The behaviour of the local time is characterised by Ray–Knight theorems. Brownian martingales Let 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 the conditional probability of 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 ). Then the process 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. Also B_t^2 - t and e^{\theta B_t - \tfrac{\theta^2}{2} t} are martingales. Integrated Brownian motion The time-integral of the Wiener process 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 , calculated using the fact that the covariance of the Wiener process is t \wedge s = \min(t, s). For the general case of the process defined by V_f(t) = \int_0^t f'(s)W(s) \,ds=\int_0^t (f(t)-f(s))\,dW_s Then, for a > 0, \operatorname{Var}(V_f(t))=\int_0^t (f(t)-f(s))^2 \,ds \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) is always a zero mean normal random variable. This allows for simulation of V_f(t+a) given V_f(t) by taking V_f(t+a)=A\cdot V_f(t) +B\cdot Z where Z is a standard normal variable and A=\frac{\operatorname{cov}(V_f(t+a),V_f(t))}{\operatorname{Var}(V_f(t))} B^2=\operatorname{Var}(V_f(t+a))-A^2\operatorname{Var}(V_f(t)) The case of V_f(t)=W^{(-1)}(t) corresponds to 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 \frac{t}{2n+1}\left ( \frac{t^n}{n!} \right )^2 . This is given by the Cauchy formula for repeated integration. Time change Every continuous martingale (starting at the origin) is a time changed Wiener process. Example: where is another Wiener process (different from but distributed like ). Example. W_t^2 - t = V_{A(t)} where A(t) = 4 \int_0^t W_s^2 \, \mathrm{d} s and is another Wiener process. In general, if is a continuous martingale then M_t - M_0 = V_{A(t)} where is the quadratic variation of on [0, t], and is a Wiener process. Corollary. (See also Doob's martingale convergence theorems) Let be a continuous martingale, and M^-_\infty = \liminf_{t\to\infty} M_t, M^+_\infty = \limsup_{t\to\infty} M_t. Then only the following two cases are possible: -\infty -\infty = M^-_\infty other cases (such as M^-_\infty = M^+_\infty = +\infty,   M^-_\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. Change of measure 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. Complex-valued Wiener process The complex-valued Wiener process may be defined as a complex-valued random process of the form Z_t = X_t + i Y_t where X_t and Y_t are independent Wiener processes (real-valued). In other words, it is the 2-dimensional Wiener process, where we identify \R^2 with \mathbb C. Self-similarity Brownian scaling, time reversal, time inversion: the same as in the real-valued case. Rotation invariance: for every complex number c such that |c|=1 the process c \cdot Z_t is another complex-valued Wiener process. Time change If f is an entire function then the process f(Z_t) - f(0) is a time-changed complex-valued Wiener process. Example: Z_t^2 = \left(X_t^2 - Y_t^2\right) + 2 X_t Y_t i = U_{A(t)} where A(t) = 4 \int_0^t |Z_s|^2 \, \mathrm{d} s and 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 2 X_t + i Y_t is not (here X_t and Y_t are independent Wiener processes, as before). Brownian sheet 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 t while others define it for general dimensions. == See also ==