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Stochastic Processes Final Exam, Key Concepts
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1. Brownian Mo- Brownian motion describes the random, continuous movement of particles sus-
tion and Related pended in a fluid. In mathematics, it is a fundamental model for continuous-time
Processes stochastic processes and is used to represent random fluctuations in various
fields, such as physics and finance.
2. Brownian Motion Brownian motion with drift is a variation of standard Brownian motion where a
with Drift constant directional trend, called drift, is added. This means the process not only
fluctuates randomly but also tends to move steadily in one direction over time.
3. Brownian Mo- Brownian motion with constant drift dynamics describes a process where random
tion with Con- Brownian motion is combined with a steady, non-random change in value over
stant Drift Dy- time, such as moving steadily upward or downward.
namics
4. Variance Linear Variance linear in time means that the variability or spread of the Brownian motion
in Time process increases proportionally with time, so the variance at time t is directly
proportional to t.
5. Extremes and Extremes and sample path properties refer to the study of the highest and
Sample Path lowest values a stochastic process, like Brownian motion, can reach, as well as
Properties the detailed behavior of its paths over time. This includes properties such as
continuity, the likelihood of reaching certain levels, and how often the process
crosses those levels.
6. Hitting Times Hitting times refer to the moments when a stochastic process, such as Brownian
and Maximum motion, first reaches a particular value or set. The maximum variable is the highest
Variable value that the process attains within a given time interval. These concepts help
analyze when and how often extremes occur in random processes.
7. Boundary-Cross- Boundary-crossing probability for Brownian motion is the chance that a Brownian
ing Probability motion path crosses a specified level or boundary within a given time period.
(BM)
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8. Level-Crossing A level-crossing time is the instant at which a stochastic process first becomes
Time equal to or crosses a unique specified value.
9. Maximum The maximum process is a new process that records the highest value reached by
Process the original process, such as Brownian motion, up to each point in time.
10. The Maximum The maximum variable is the largest value reached by Brownian motion over
Variable and the a fixed interval. The reflection principle is a mathematical technique used to
Reflection Princi- calculate probabilities involving the maximum by relating paths that exceed a
ple certain threshold to those that do not, effectively 'reflecting' the process at the
boundary.
11. Maximum Distri- Maximum distribution is the probability distribution of the highest value attained
bution by a stochastic process, such as Brownian motion, over a specified interval. It helps
assess the likelihood of extreme outcomes.
12. Fundamental Brownian motion is a random process that models the unpredictable movement
Concepts of of particles suspended in a fluid. It is a continuous-time stochastic process with
Brownian stationary and independent increments, and its changes over time are normally
Motion and distributed. Gaussian processes are a broader class of stochastic processes where
Gaussian any set of random variables has a joint normal distribution, and Brownian motion
Processes is a specific example of a Gaussian process.
13. Gaussian Incre- Gaussian increments refer to the property where the differences between values
ments of a process at different times are normally distributed. In Brownian motion, the
increments over non-overlapping time intervals are independent and each follows
a normal distribution.
14. Quadratic Varia- The quadratic variation of Brownian motion measures the accumulated squared
tion of Brownian changes of the process over a time interval and equals the length of the interval.
Motion
15.
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Stochastic Processes Final Exam, Key Concepts
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Standard Brown- A standard Brownian motion process is a continuous-time stochastic process that
ian Motion starts at zero, has independent and Gaussian increments, stationary increments,
Process and normally distributed changes over any time interval with variance equal to
the length of the interval.
16. Stationary and Stationary processes are stochastic processes whose statistical properties, such as
Related Process- mean and variance, do not change over time. Related processes include those that
es have similar time-invariant properties or are derived from stationary processes,
making them useful for modeling systems with consistent behavior over time.
17. Stationary Stationary processes are stochastic processes whose statistical properties, such
Processes as mean and variance, do not change over time. This means the probability
distributions related to the process remain constant as time shifts.
18. Autocovariance The autocovariance function of a stationary Gaussian process measures how much
Function of a two values of the process at different times vary together based only on the time
Stationary difference between them, not their location in time.
Gaussian Process
19. Stationary Incre- The stationary increments property of Brownian motion means that the distrib-
ments Property ution of the change in the process over any time interval depends only on the
of Brownian Mo- length of the interval, not on its starting point.
tion
20. The Ornstein-Uh- The Ornstein-Uhlenbeck Process is a type of continuous-time stochastic process
lenbeck Process that describes how a system tends to drift towards its long-term mean while expe-
riencing random fluctuations. It is often used to model mean-reverting behaviors
in physics and finance.
21. Equilibrium Dis- The equilibrium distribution is the probability distribution to which a stochastic
tribution process settles as time goes to infinity. For the Ornstein-Uhlenbeck process, this
is often a normal distribution.
Page 3 of 26 3/31/2026
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Stochastic Processes Final Exam, Key Concepts
Study online at https://quizlet.com/_igev3b
1. Brownian Mo- Brownian motion describes the random, continuous movement of particles sus-
tion and Related pended in a fluid. In mathematics, it is a fundamental model for continuous-time
Processes stochastic processes and is used to represent random fluctuations in various
fields, such as physics and finance.
2. Brownian Motion Brownian motion with drift is a variation of standard Brownian motion where a
with Drift constant directional trend, called drift, is added. This means the process not only
fluctuates randomly but also tends to move steadily in one direction over time.
3. Brownian Mo- Brownian motion with constant drift dynamics describes a process where random
tion with Con- Brownian motion is combined with a steady, non-random change in value over
stant Drift Dy- time, such as moving steadily upward or downward.
namics
4. Variance Linear Variance linear in time means that the variability or spread of the Brownian motion
in Time process increases proportionally with time, so the variance at time t is directly
proportional to t.
5. Extremes and Extremes and sample path properties refer to the study of the highest and
Sample Path lowest values a stochastic process, like Brownian motion, can reach, as well as
Properties the detailed behavior of its paths over time. This includes properties such as
continuity, the likelihood of reaching certain levels, and how often the process
crosses those levels.
6. Hitting Times Hitting times refer to the moments when a stochastic process, such as Brownian
and Maximum motion, first reaches a particular value or set. The maximum variable is the highest
Variable value that the process attains within a given time interval. These concepts help
analyze when and how often extremes occur in random processes.
7. Boundary-Cross- Boundary-crossing probability for Brownian motion is the chance that a Brownian
ing Probability motion path crosses a specified level or boundary within a given time period.
(BM)
Page 1 of 26 3/31/2026
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Stochastic Processes Final Exam, Key Concepts
Study online at https://quizlet.com/_igev3b
8. Level-Crossing A level-crossing time is the instant at which a stochastic process first becomes
Time equal to or crosses a unique specified value.
9. Maximum The maximum process is a new process that records the highest value reached by
Process the original process, such as Brownian motion, up to each point in time.
10. The Maximum The maximum variable is the largest value reached by Brownian motion over
Variable and the a fixed interval. The reflection principle is a mathematical technique used to
Reflection Princi- calculate probabilities involving the maximum by relating paths that exceed a
ple certain threshold to those that do not, effectively 'reflecting' the process at the
boundary.
11. Maximum Distri- Maximum distribution is the probability distribution of the highest value attained
bution by a stochastic process, such as Brownian motion, over a specified interval. It helps
assess the likelihood of extreme outcomes.
12. Fundamental Brownian motion is a random process that models the unpredictable movement
Concepts of of particles suspended in a fluid. It is a continuous-time stochastic process with
Brownian stationary and independent increments, and its changes over time are normally
Motion and distributed. Gaussian processes are a broader class of stochastic processes where
Gaussian any set of random variables has a joint normal distribution, and Brownian motion
Processes is a specific example of a Gaussian process.
13. Gaussian Incre- Gaussian increments refer to the property where the differences between values
ments of a process at different times are normally distributed. In Brownian motion, the
increments over non-overlapping time intervals are independent and each follows
a normal distribution.
14. Quadratic Varia- The quadratic variation of Brownian motion measures the accumulated squared
tion of Brownian changes of the process over a time interval and equals the length of the interval.
Motion
15.
Page 2 of 26 3/31/2026
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Stochastic Processes Final Exam, Key Concepts
Study online at https://quizlet.com/_igev3b
Standard Brown- A standard Brownian motion process is a continuous-time stochastic process that
ian Motion starts at zero, has independent and Gaussian increments, stationary increments,
Process and normally distributed changes over any time interval with variance equal to
the length of the interval.
16. Stationary and Stationary processes are stochastic processes whose statistical properties, such as
Related Process- mean and variance, do not change over time. Related processes include those that
es have similar time-invariant properties or are derived from stationary processes,
making them useful for modeling systems with consistent behavior over time.
17. Stationary Stationary processes are stochastic processes whose statistical properties, such
Processes as mean and variance, do not change over time. This means the probability
distributions related to the process remain constant as time shifts.
18. Autocovariance The autocovariance function of a stationary Gaussian process measures how much
Function of a two values of the process at different times vary together based only on the time
Stationary difference between them, not their location in time.
Gaussian Process
19. Stationary Incre- The stationary increments property of Brownian motion means that the distrib-
ments Property ution of the change in the process over any time interval depends only on the
of Brownian Mo- length of the interval, not on its starting point.
tion
20. The Ornstein-Uh- The Ornstein-Uhlenbeck Process is a type of continuous-time stochastic process
lenbeck Process that describes how a system tends to drift towards its long-term mean while expe-
riencing random fluctuations. It is often used to model mean-reverting behaviors
in physics and finance.
21. Equilibrium Dis- The equilibrium distribution is the probability distribution to which a stochastic
tribution process settles as time goes to infinity. For the Ornstein-Uhlenbeck process, this
is often a normal distribution.
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