Fixing an Outcome Gives a Path
A stochastic process is a family of random variables indexed by time. Fixing the outcome of the experiment removes the randomness from the values along that family: what remains is an ordinary function of time. Studying this function helps distinguish a property of one realized trajectory from a statement about the process as a whole. It also makes an important point about definitions: a path can only be called bounded or monotone when the state space has the relevant structure.
Let \((\Omega,\mathcal{F},\mathbb{P})\) be a probability space, let \(T\) be a time index set, and let \(S\) be a state space. A process \((X_t)_{t\in T}\) is a family of random variables \(X_t:\Omega\to S\). For each fixed \(\omega\in\Omega\), the function \(t\mapsto X_t(\omega)\) is the sample path associated with \(\omega\). It has domain \(T\) and takes values in \(S\). In discrete time, for example, \(T=\{0,1,2,\ldots\}\), so a path is a sequence. In continuous time, \(T\) may be an interval, so a path is a function defined at every time in that interval.
Some path properties need more structure than just a state space. If \(S\) is a metric space with distance \(d\), a path \(f:T\to S\) is bounded when there is an \(x_0\in S\) and a finite \(C\) such that \(d(f(t),x_0)\le C\) for every \(t\in T\). If \(S\) is a subset of the real numbers, this is equivalent to the usual boundedness of the real-valued function. Monotonicity, however, needs an order: for a real-valued path, nondecreasing means \(f(s)\le f(t)\) whenever \(s\le t\), and nonincreasing means \(f(s)\ge f(t)\) whenever \(s\le t\). These comparisons are not defined for a general state space.
Continuity also requires a notion of distance in the state space and a suitable notion of closeness in the time index. For a real-valued path on an interval, continuity at \(t\) means that \(f(s)\) approaches \(f(t)\) as \(s\) approaches \(t\) within the domain. In discrete time, the index set alone does not provide the usual idea of a path being continuous between successive integer times. Instead, it is often useful to examine the step-to-step increments \(f(n+1)-f(n)\). A jump in continuous time is a change visible through one-sided limits; for real-valued \(f\), if the left limit \(f(t-)=\lim_{s\uparrow t}f(s)\) exists, the signed jump at \(t\) is \(f(t)-f(t-)\). This definition applies at interior times with a left-hand approach, not at the left endpoint of the time domain.
What Can Be Said for Every Path?
Proof. Write the finite set as \(T=\{t_1,\ldots,t_m\}\), where \(m\ge1\). The finitely many real numbers \(f(t_1),\ldots,f(t_m)\) have a largest and a smallest value: comparing the first two produces their larger and smaller values, and comparing each resulting value with the next one gives a largest and smallest value after finitely many comparisons. Thus there are indices \(r,q\in\{1,\ldots,m\}\) such that \(f(t_r)\le f(t_j)\le f(t_q)\) for every \(j\). Setting \(C=\max\{|f(t_r)|,|f(t_q)|\}\), we have \(|f(t_j)|\le C\) for every \(j\), so the path is bounded and attains both extrema. \(\square\)
This is a deterministic fact about finite collections of values, so it holds for every outcome of a process with real-valued states observed at finitely many times. It does not claim that a path on an infinite time set is bounded. Nor does it say that a path with values in an arbitrary state space has a maximum or minimum: those notions require an order.
Worked Example: An Alternating Discrete-Time Path
Take \(\Omega=\{-1,1\}\), with probability \(1/2\) assigned to each outcome, time set \(T=\{0,1,2,\ldots\}\), and real-valued process \(X_n(\omega)=\omega(-1)^n\). Fix \(\omega=1\). Its path begins \(1,-1,1,-1,\ldots\). For each integer \(n\ge0\), the path value is either \(1\) or \(-1\), so \(|X_n(1)|=1\) and this path is bounded by \(1\).
The increment from time \(n\) to \(n+1\) is \(X_{n+1}(1)-X_n(1)=(-1)^{n+1}-(-1)^n=-2(-1)^n\). Thus the increment is \(-2\) when \(n\) is even and \(2\) when \(n\) is odd. The path is not nondecreasing, since \(X_1(1)=-1<1=X_0(1)\); it is not nonincreasing, since \(X_2(1)=1>-1=X_1(1)\). The same boundedness and failure of monotonicity hold for \(\omega=-1\), whose path begins \(-1,1,-1,1,\ldots\).
Check: for either outcome and every \(n\), \(|\omega(-1)^n|=1\), and consecutive values alternate between \(-1\) and \(1\). This confirms the bound and the two inequalities that rule out monotonicity. The increments describe changes between indexed observations; no claim about continuous-time continuity is needed.
Continuous-Time Paths: Smooth Change and a Jump
Worked Example: A Continuous Path Whose Direction Depends on the Outcome
Let \(\Omega=[0,1]\) with its Borel sets and uniform probability, let \(U(\omega)=\omega\), and define the real-valued process \(X_t=(U-\tfrac12)t\) for \(0\le t\le2\). For a fixed outcome with \(U=\tfrac14\), the path is \(f(t)=-\tfrac14t\). It has \(f(0)=0\), \(f(1)=-\tfrac14\), and \(f(2)=-\tfrac12\). For \(0\le s\le t\le2\), \(f(t)-f(s)=-\tfrac14(t-s)\le0\), so this path is nonincreasing. Also, \(|f(t)|=\tfrac14t\le\tfrac12\) on the time interval.
For an outcome with \(U=\tfrac34\), the path is \(g(t)=\tfrac14t\). Now \(g(t)-g(s)=\tfrac14(t-s)\ge0\), so this path is nondecreasing; its values at \(t=0,1,2\) are \(0,\tfrac14,\tfrac12\). More generally, the coefficient \(U-\tfrac12\) determines the direction: the path is nondecreasing when \(U\ge\tfrac12\), nonincreasing when \(U\le\tfrac12\), and constant when \(U=\tfrac12\). Every path in this example is continuous, since for all \(s,t\in[0,2]\), \(|X_t-X_s|=|U-\tfrac12||t-s|\le\tfrac12|t-s|\).
Check: substituting \(t=2\) gives \(X_2=2(U-\tfrac12)\). At \(U=\tfrac14\) this is \(2(-\tfrac14)=-\tfrac12\), and at \(U=\tfrac34\) it is \(2(\tfrac14)=\tfrac12\), matching the stated endpoint values.
A path can also have a sudden change. The next result checks continuity and the jump directly from the definition, including which time is the jump time.
Proof. If \(t<u\), choose a positive distance smaller than \(u-t\). Every sufficiently nearby \(s\in[0,2]\) then satisfies \(s<u\), so \(f_u(s)=0=f_u(t)\); hence \(f_u\) is continuous at \(t\). If \(t>u\), choose a positive distance smaller than \(t-u\). Every sufficiently nearby \(s\in[0,2]\) then satisfies \(s>u\), so \(f_u(s)=1=f_u(t)\); hence the path is continuous at \(t\). At \(u\), the value is \(f_u(u)=1\), and for every \(s>u\) the value is also \(1\), proving right-continuity. For every \(s<u\), \(f_u(s)=0\), so the left limit is \(0\). The jump is therefore \(1-0=1\). Since values immediately to the left of \(u\) equal \(0\), while \(f_u(u)=1\), the function is not continuous at \(u\). \(\square\)
Worked Example: A Random Jump Time
Use \(\Omega=[0,1]\) with uniform probability and \(U(\omega)=\omega\), and set \(X_t=\mathbf{1}_{\{t\ge U\}}\) for \(0\le t\le2\). Fix the outcome \(U=\tfrac34\). The path is \(0\) for \(0\le t<\tfrac34\) and \(1\) for \(\tfrac34\le t\le2\). For example, \(X_{1/2}=0\), \(X_{3/4}=1\), and \(X_1=1\). It is nondecreasing and bounded between \(0\) and \(1\). Its left limit at \(3/4\) is \(0\), its value there is \(1\), and its jump size is \(1-0=1\). It is continuous at times strictly less than \(3/4\) and strictly greater than \(3/4\), but not at the jump time.
The endpoints of the random time range are worth checking. If \(U=0\), then \(X_t=1\) for every \(t\in[0,2]\), so the path is constant and has no jump inside the time domain. If \(U=1\), the path is \(0\) for \(t<1\) and \(1\) for \(t\ge1\), so it jumps at the interior time \(1\). Thus a jump occurs in the domain for \(U\in(0,1]\), while at \(U=0\) the change is at the left endpoint and there is no left-hand limit within the domain. Under the uniform probability on \([0,1]\), the event \(\{U=0\}\) has probability zero; the endpoint distinction still matters when describing individual paths.
Check: for the fixed outcome \(U=\tfrac34\), the defining condition \(t\ge U\) is false at \(t=\tfrac12\), true at \(t=\tfrac34\), and true at \(t=1\), giving the three values stated. The jump-size calculation uses the left limit, not the value just after the jump: \(1-0=1\).
Boundedness Does Not Guarantee Continuity
Worked Example: A Bounded Path Without a Limit at the Origin
Consider the deterministic real-valued process on a one-outcome probability space, with time set \([0,1]\), given by \(X_0=0\) and \(X_t=\sin(1/t)\) for \(t>0\). This is a valid process: at each time its value is a random variable (a constant on the one-outcome space). Its path is bounded because \(|\sin(1/t)|\le1\) for \(t>0\), and \(|X_0|=0\).
To check whether the path is continuous at zero, take \(t_k=(\tfrac{\pi}{2}+2\pi k)^{-1}\) and \(s_k=(\tfrac{3\pi}{2}+2\pi k)^{-1}\), for integers \(k\ge0\). Both sequences are positive and tend to zero because their denominators tend to infinity. Substitution gives \(X_{t_k}=\sin(\tfrac{\pi}{2}+2\pi k)=1\), while \(X_{s_k}=\sin(\tfrac{3\pi}{2}+2\pi k)=-1\). Since values along times approaching zero do not approach the same limit, the path has no limit at zero and is not continuous there.
Check: the two sequences use sine values \(1\) and \(-1\) at every index, respectively. Thus neither can converge to \(X_0=0\), even though all path values lie in \([-1,1]\). This verifies directly that boundedness and continuity are separate properties.
When describing a process, state clearly whether a claim concerns every path, almost every path, or only a particular outcome. Also state the time domain: a path may be bounded on a finite interval but not on an unbounded one, and endpoint behavior can differ from behavior at interior times. Finally, check the structure of the state space before using words such as “increasing,” “distance,” or “jump size.” For real-valued paths these have familiar meanings; for a general state space they require additional definitions.
Common mistake. It is invalid to define a path taking values in an arbitrary state space \(S\) as bounded by requiring \(|X_t(\omega)|\le C\), or as nondecreasing by comparing \(X_s(\omega)\le X_t(\omega)\), unless absolute value and an order are defined on \(S\). For example, if \(S=\{a,b\}\) is just a set of two labels, neither \(|a|\) nor the comparison \(a\le b\) is specified. The correction is to restrict those definitions to real-valued paths, or to state the needed structure explicitly: use a metric for boundedness and an order for monotonicity. A path being a function alone does not supply either structure.
Check Your Understanding
For each question, specify the time domain and state space when they affect the answer.
- For a real-valued path on a finite, nonempty set of times, why must its maximum be attained?
- For the alternating path \(f(n)=(-1)^n\), compute \(f(4)-f(3)\). Is the path nondecreasing, nonincreasing, or neither?
- For \(f_u(t)=\mathbf{1}_{\{t\ge u\}}\) with \(u\in(0,1)\), find its value, left limit, and jump size at \(u\).
- In the random jump-time example, what path results when \(U=0\), and why is it different from the path when \(U=1\)?
- Give a reason that boundedness alone does not imply continuity, using the path \(X_0=0\), \(X_t=\sin(1/t)\) for \(t>0\).