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Path densities and finite jump counts for bounded finite-state rates

Statement

Let $X_{t}$ be a finite-state continuous-time Markov jump process on $[0, T]$ under the first-jump rate convention, with initial law $ρ$ and bounded continuous rates $κ_{x y} (t)$ , where $T \geq 0$ . Put $M : = \sup_{x \in E, 0 \leq t \leq T} λ_{x} (t) < \infty$ . For $n \geq 0$ , let $x_{0}, \dots, x_{n} \in E$ with $x_{j - 1} \neq x_{j}$ for $1 \leq j \leq n$ , let $0 < t_{1} < \dots < t_{n} < T$ , and set $t_{0} : = 0$ , $t_{n + 1} : = T$ . Relative to counting measure on state sequences and Lebesgue measure on the ordered time simplex, the probability density of exactly this state sequence and these jump times is

$ρ (x_{0}) (\prod_{j = 1}^{n} κ_{x_{j - 1} x_{j}} (t_{j})) \exp (- \sum_{j = 0}^{n} \int_{t_{j}}^{t_{j + 1}} λ_{x_{j}} (s) d s) .$

Here an empty product is $1$ and for $n = 0$ the formula is $ρ (x_{0}) \exp (- \int_{0}^{T} λ_{x_{0}} (s) d s)$ ; if $T = 0$ , this is the only path case. For any subinterval $J = [s, t]$ of length $δ$ with $M δ < 1$ , let $N_{J}$ count its jumps. Then for every integer $n \geq 1$ ,

$P (N_{J} \geq n) \leq (M δ)^{n}, E [N_{J}] \leq \frac{M δ}{1 - M δ} < \infty .$

If $M = 0$ there are no jumps. If $M > 0$ , divide $[0, T]$ into finitely many intervals with $M δ < 1$ ; the preceding bound then gives $E [N_{T}] < \infty$ . Thus bounded finite-state rates imply nonexplosion on every finite horizon.

Statement references

Finite-state continuous-time Markov jump processes with bounded rates

The Axiom of Countable Choice ( ${A C}_{ω}$ )

Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$

Facts & Assumptions

Given: A finite nonempty state set, a finite $T > 0$ , bounded continuous rates, an initial probability law, and the process rate convention in the Statement.

[F1]

Finite-state continuous-time Markov jump processes with bounded rates: $E$ is finite, the rates $κ_{x y} (t)$ are bounded and continuous, and conditional on state $x$ at time $s$ the next-jump survival is $H_{x} (s, t) = \exp (- \int_{s}^{t} λ_{x} (u) d u)$ with next-jump density $H_{x} (s, t) κ_{x y} (t)$ to state $y \neq x$ .

[A1]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): every countable family of nonempty sets has a choice function. This is the exact axiom used to treat the restricted Lebesgue outer measure as a complete measure on the finite-dimensional jump-time simplices.

[F2]

Lebesgue measurable sets, the family $L (R^{n})$ , and the restricted set function $λ_{n}$ and Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume: time-simplex densities use Lebesgue measure, and a box of side lengths $δ_{1}, \dots, δ_{n}$ has volume $\prod_{j} δ_{j}$ .

[F3]

Every complete ordered field is Archimedean: for every real $r$ there is a positive integer $m > r$ .

[F4]

Expectation of a nonnegative or integrable random variable, Monotone convergence for the integral, and The Lebesgue integral is linear on $L^{1} (μ)$ : expectation is integration with respect to the probability measure, increasing nonnegative measurable functions have integrals increasing to the integral of their pointwise limit, and the integral is linear on integrable functions.

Proof

Given: The process, rates, horizon, initial law, and Countable Choice as stated.

1.1F1constructalgebra

Conditional on state $x_{0}$ at time $0$ , the first-jump convention in [F1] gives the survival factor $H_{x_{0}} (0, t_{1})$ and jump density $H_{x_{0}} (0, t_{1}) κ_{x_{0} x_{1}} (t_{1})$ . After this jump, the Markov rule restarts at $x_{1}$ . Multiplying the successive conditional survival and jump factors through state $x_{n}$ , and then the final no-jump survival on $[t_{n}, T]$ , gives exactly the stated path density. If $n = 0$ , the same convention gives $ρ (x_{0}) H_{x_{0}} (0, T)$ , which is the stated boundary formula.

1.2A1F1F2F4algebra

Let $J = [s, t]$ have length $δ$ with $M δ < 1$ . The density of the first $n$ ordered jumps, with survival factors removed, is at most the product of the $n$ rates. Summing these products over each successive state gives at most $M^{n}$ . The ordered time simplex is contained in the cube $J^{n}$ , whose Lebesgue volume is $δ^{n}$ by [F2]. Hence $P (N_{J} \geq n) \leq (M δ)^{n}$ . Pointwise $N_{J} = \sum_{n \geq 1} 1_{{N_{J} \geq n}}$ ; [F4] gives $E [N_{J}] \leq \sum_{n \geq 1} (M δ)^{n} = M δ / (1 - M δ)$ .

2.1F1F3F4step 1.1step 1.2algebra∎

If $T = 0$ or $M = 0$ , there are no jumps. Otherwise choose a positive integer $m > M T$ by [F3] and partition $[0, T]$ into $m$ intervals of length $δ = T / m$ , so $M δ < 1$ . Jump times have densities by step 1.1, hence no jump occurs at a fixed partition endpoint with positive probability. The total count is the sum of the interval counts almost surely; linearity of expectation [F4] and step 1.2 give $E [N_{T}] \leq m M δ / (1 - M δ) < \infty$ . If $P (N_{T} = \infty) > 0$ , then $E [N_{T}] \geq n P (N_{T} = \infty)$ for every $n$ , contradicting finiteness. Thus the process is nonexplosive and the countable path-density collection covers the path law.

Path densities and finite jump counts for bounded finite-state rates

Statement

Statement references

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources