{"id":328,"date":"2025-10-07T17:55:31","date_gmt":"2025-10-07T17:55:31","guid":{"rendered":"https:\/\/mrivas.su.domains\/gbe\/?p=328"},"modified":"2025-10-13T20:22:05","modified_gmt":"2025-10-13T20:22:05","slug":"brownian-motion-and-martingales-and-their-relations","status":"publish","type":"post","link":"https:\/\/mrivas.su.domains\/gbe\/uncategorized\/brownian-motion-and-martingales-and-their-relations\/","title":{"rendered":"Brownian motion and martingales and their relations"},"content":{"rendered":"\n

Brownian motion<\/h2>\n\n\n\n

Brownian motion is defined as a stochastic process \"\{X_t\}_{t \geq 0}\" which is Gaussian, i.e. such that for any \"0 \leq t_0 < t_1 < \cdots < t_n\", \"X_{t_j}\" for \"j = 0, 1, \cdots, n\" have a jointly Gaussian (normal distribution), \"\mathbb{E}X_t = 0\" for all \"t\", and the covariance is \"\mathbb{E}X_sX_t = \min(s,t)\" for all \"s, t \geq 0\". A normal distribution is uniquely determined by its mean vector \"\mu\" and covariance matrix \"C\". Each \"X_t\" has a normal distribution \"\mathcal{N}(0, t)\". <\/p>\n\n\n\n

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Brownian motion – multiple sample paths from t = 0 to t = 5. <\/figcaption><\/figure>\n\n\n\n
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Density at t = 2.5. <\/figcaption><\/figure>\n\n\n\n

It follows from the definition that Brownian motion has independent increments<\/em>, in other words whenever \"0 \leq s < t \leq u < v\", \"X_t - X_s\" and \"X_v - X_u\" are independent. <\/p>\n\n\n\n

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A consequence of the definition is that for any \"c > 0\", the process \"X_{c^2t}/ c\" is also a Brownian motion. <\/p>\n\n\n\n

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By a therorem of Norbert Wiener, Brownian motion can be chosen such that with probability 1 (or for all \"w\"), \"X_t(w)\" is continuous as a function of \"t\". a Brownian motion with this property is called the standard <\/em> Brownian motion.<\/p>\n\n\n\n

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A Markov time <\/em>\"\tau\" for Brownian motion is defined as a random variable \"\tau \geq 0\" such that for each \"t > 0\", the event \"\{ \tau < t\}\" is a function of \"X_s\" for \"0 \leq s \eq t\". <\/p>\n\n\n\n

The strong Markov property <\/em>of Brownian motion says that if \"\tau\" is a Markov time such that \"\math{P}(\tau < \infty) > 0\", then conditional on \"\tau < \infty\", the process \"V_{t} \def X_{\tau + t} - X_{\tau}\" for \"t \geq 0\" is itself a Brownian motion and is independent of \"X_s\" for \"s \leq \tau\". <\/p>\n\n\n\n

Using the strong Markov property one can prove the reflection principle<\/a><\/strong><\/em> for Brownian motion: for any \"b > 0\" and \"t > 0\", the probability that \"X_s \geq b\" for some \"s\" with \"0 < s \leq t\" equals \"2\mathbb{P}\left(X_t \geq b\right)\". <\/p>\n\n\n\n

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What may be called “Wald’s identity<\/a>“<\/em> for Brownian motion says that if \"\tau\" is a Markov time with \"\mathbb{E}\left[\tau\right] < \infty\", and so \"\tau < \infty\" with probability 1, \"\mathbb{E}\left[X_{\tau}\right] = 0\" and \"\mathbb{E}\left[X_{\tau}^2\right] = \mathbb{E}\left[\tau\right]\". <\/p>\n\n\n\n

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The Brownian Bridge <\/em>is a Gaussian process \"Y_t\" defined for \"0 \leq t \leq 1\" with \"\mathbb{E}\left[Y_t\right] \equiv 0\" and covariance \"\mathbb{E}\left[Y_SY_t\right] = s\left(1 - t\right)\" for \"0 \leq s \leq t \leq 1\". <\/p>\n\n\n\n

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The path starts and ends at 0 (a.s.).
The dashed curves plot the \"\pm 2\sqrt{t(1-t)}\" band (two standard deviations), and the dotted line marks the zero mean.
Note that \"Y_0=Y_1=0\" a.s., and the fluctuations are largest near \"t=\tfrac12\".<\/p>\n\n\n\n

Relations between the Brownian motion and Brownian bridge:<\/p>\n\n\n\n