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Let $X_t$ be a continuous-time stochastic process on the state space {'off', 'on'}. Let $U$ be the cumulative duration spent 'on' during a time interval $[0,\tau]$ (that is, $U$ is the uptime during that interval).

Suppose that we simulate $U$ by sampling on an equispaced grid, as in:

  • Choose $\delta t > 0$ such that $m = \tau / \delta t$ is an integer.
  • Put $t_k = (k-1) \cdot \delta t$ for $k = 1 \dots m$.
  • Put $Y_k = X_{t_k}$ for $k = 1 \dots m$.
  • For any realization of $X_t$, put $V = (Y_1 + Y_2 + \dots Y_m) \cdot \delta t$.

Let $F(u) = Pr(U \leq u)$ be the cumulative distribution function for $U$ (intuitively, the process's cdf), and $G(u) = Pr(V \leq u)$ be the cumulative distribution function for $V$ (intuitively, the simulation's cdf).

When does $G$ converge (pointwise/uniformly) to $F$ as $\delta t \rightarrow 0$?

My attempt at an answer: A realization of $X_t$ will be misclassified if it has sojourns 'on' that are of duration less than $\delta t$. As $\delta t \rightarrow 0$, the probability of obtaining a misclassified realization converges to zero.

Many thanks in advance.

PtH
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1 Answers1

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After some pondering, I think I have an answer for pointwise convergence.

Proposition: For any $ u \in [0,\tau] $ and $ \epsilon > 0 $, there exists $ \delta t' > 0 $ such that if $ \delta t < \delta t' $ then $ \lvert G(u) - F(u) \rvert < \epsilon $.

Proof: For any $x_t : [0,\tau] \rightarrow {0,1}$ put \begin{align*} \lambda(x_t) &= \int_{0}^{\tau} x_t \, dt \\ \mu(x_t; \delta t) &= \sum_{k=0}^{\lfloor \tau/\delta t \rfloor} x_{t_k} \delta t \ \text{where}\ t_k = k \cdot \delta t \end{align*} Put \begin{align*} \chi(u; \delta t) &= \{ x_t : \lambda(x_t) = u \ \text{but}\ \mu(x_t; \delta t) \neq u \, \,\text{or}\, \, \lambda(x_t) \neq u \ \text{but}\ \mu(x_t; \delta t) =u \} \\ \xi(u; \delta t) &= Pr\bigl( X_t \in \chi(u; \delta t) \bigr) \end{align*} For any $ u \in [0,\tau] $, $ 0 \leq \xi(u; \delta t) \leq 1 $ (by definition of probabilities) and $ \xi(u; \delta t) \rightarrow 0 $ (pointwise) as $ \delta t \rightarrow 0 $. By the bounded convergence theorem, there exists $ \delta t' $ such that if $ \delta t < \delta t' $ then $$ \int_{[0,\tau]} \xi(u; \delta t) \, du < \epsilon $$ Hence if $ \delta t < \delta t' $ then $$ \lvert G(u) - F(u) \rvert = \int_{0}^{u} \xi(u; \delta t) \, du < \epsilon $$

PtH
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