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edited Oct 21 2010 at 13:29 |
A (standard, real-valued) Brownian motion $W = \{W(t): t >=0\}$\geq 0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ s,t \geq 0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost surely, the function $t \mapsto W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, t \geq 0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0 <= \leq s <=t$; \leq t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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edited Oct 21 2010 at 7:56 |
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edited Oct 21 2010 at 7:08 |
A (standard, real-valued) Brownian motion $W = \{W(t): t>=0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost surely, the function $t \mapsto W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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edited Oct 21 2010 at 7:03 |
A (standard, real-valued) Brownian motion $W = \{W(t): t>=0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost surely, the function $t \mapsto W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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edited Oct 21 2010 at 6:47 |
A (standard, real-valued) Brownian motion `$W $W = {W(t): t>=0}$' \{W(t): t>=0\}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ with $s<t$, the increment $W(t) – W(s)$ is normally distributed with mean zero and variance $t-s$, and 4) almost surely, the function $t \mapsto W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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edited Oct 21 2010 at 6:28 |
A (standard, real-valued) Brownian motion `$W = {W(t): {W(t): t>=0}$' is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2)the process has independent increments,
3) for all $s,t>=0$ with s
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4
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edited Oct 21 2010 at 6:19 |
A (standard, real-valued) Brownian motion $W `$W ={W(t): {W(t): t>=0}$ is commonly defined by the
following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments,
3) for all $s,t>=0$ with $s
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3
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edited Oct 21 2010 at 6:13 |
A (standard, real-valued) Brownian motion $W = {W(t): t>=0}$ is commonly defined by the
following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments,
3) for all $s,t>=0$ with $s
W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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2
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edited Oct 21 2010 at 6:07 |
A (standard, real-valued) Brownian motion $W = {W(t): t>=0}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ with $s W(t)$ is continuous.
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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1
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asked Oct 21 2010 at 6:00 |
The conditions in the definition of Brownian motion
A (standard, real-valued) Brownian motion $W = {W(t): t>=0}$ is commonly defined by the following properties: 1) $W(0) = 0$ a.s., 2) the process has independent increments, 3) for all $s,t>=0$ with $s
As is well known, the above set of conditions can be reduced to 2), 3') for all $t>=0$, $W(t)$ has mean zero and variance $t$, and 4). [Note that, in 3'), $W(t)$ is not assumed to be normally distributed.] But what about omitting condition 2)? Can you find an example of a process $W$ satisfying conditions 1), 3), and 4), but not 2)? [Note that such $W$ must have (the Brownian motion) covariance $E[W(s)W(t)] = s$, $0<= s<=t$; hence, it cannot be a Gaussian process, for otherwise it would be a Brownian motion.]
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