Hubbry Logo
Jump processJump processMain
Open search
Jump process
Community hub
Jump process
logo
7 pages, 0 posts
0 subscribers
Be the first to start a discussion here.
Be the first to start a discussion here.
Jump process
Jump process
Jump process
View on Wikipedia
from Wikipedia

A jump process is a type of stochastic process that has discrete movements, called jumps, with random arrival times, rather than continuous movement. It is typically modelled as a simple or compound Poisson process.[1]

In finance, various stochastic models are used to model the price movements of financial instruments; for example the Black–Scholes model for pricing options assumes that the underlying instrument follows a traditional diffusion process, with continuous, random movements at all scales, no matter how small. John Carrington Cox and Stephen Ross[2]: 145–166  proposed that prices actually follow a 'jump process'.

Robert C. Merton extended this approach to a hybrid model known as jump diffusion, which states that the prices have large jumps interspersed with small continuous movements.[3]

See also

[edit]

References

[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Jump process

View on Grokipedia
from Grokipedia
A jump process is a stochastic process whose sample paths feature discontinuities, known as jumps, that occur at random times, distinguishing it from continuous processes like . These processes are typically defined on a and exhibit right-continuous paths with left limits ( paths), allowing them to model abrupt changes in systems evolving over time. Pure jump processes represent a specific subclass where sample paths are right-continuous and piecewise constant, with jumps of finite number in any finite time interval and all changes occurring solely through these discontinuities rather than continuous variation. A fundamental example is the Poisson process, a counting process that increments by 1 at each jump, with inter-arrival times following independent exponential distributions at a constant λ, resulting in the number of jumps up to time t following a with parameter λt. More generally, the extends this by assigning random sizes to each jump, drawn from an independent distribution, leading to processes with independent and stationary increments that are infinitely divisible. Jump processes underpin much of the stochastic calculus for discontinuous paths, including extensions of Itô's formula and Girsanov's theorem to handle jumps, which facilitate the solution of stochastic differential equations (SDEs) of the form dX_t = μ dt + σ dW_t + ∫ jump terms. In applications, they are essential in for jump-diffusion models, such as Merton's 1976 framework, which combines with Poisson jumps to better capture large, sudden price movements in asset returns and improve option pricing accuracy. Beyond , jump processes model event arrivals in , neuronal firing patterns in via integrate-and-fire models, and risk events in and .

Overview and Fundamentals

Definition

A jump process is a stochastic process {Xt}t≥0\{X_t\}_{t \geq 0} whose sample paths exhibit discontinuities, known as jumps, occurring at random times, such that Xt≠Xt−X_t \neq X_{t-} for some t>0t > 0, where Xt−=lim⁡s↑tXsX_{t-} = \lim_{s \uparrow t} X_s denotes the left-hand limit of the process at time tt. The magnitude of each jump is quantified by ΔXt=Xt−Xt−\Delta X_t = X_t - X_{t-}. By convention, jump processes are modeled with (right-continuous with left limits) sample paths, ensuring that the process value is well-defined immediately after each jump while preserving information about the pre-jump state. Unlike diffusion processes, such as , which feature continuous paths with no abrupt changes, jump processes capture discrete, instantaneous shifts in value, making them suitable for modeling phenomena with sudden events. A basic illustrative example is a process starting at 0 that remains constant until a random time TT with exponential distribution, at which point it jumps to 1 and stays there thereafter, demonstrating a single discontinuity. Lévy processes, which require stationary and independent increments among other properties, constitute an important subclass of jump processes, extending the framework to include both continuous and discontinuous components.

Historical Development

The theory of jump processes traces its roots to early studies of rare events and point processes in probability. Siméon Denis Poisson's 1837 derivation of the , which models the number of events occurring in fixed intervals, laid foundational groundwork for understanding discontinuous phenomena in settings. This was further advanced in 1898 when Ladislaus von Bortkiewicz applied the distribution to empirical data on infrequent occurrences, such as deaths from horse kicks in the , effectively illustrating the mechanics of point processes with random arrivals. In the 1930s, the formalization of Markov processes by and provided an early framework for processes with discrete jumps, emphasizing state transitions at random times within probabilistic measure theory. By the mid-20th century, Joseph L. Doob's 1953 monograph Stochastic Processes extended martingale theory to encompass discontinuous paths, incorporating jumps as integral components of general evolution and establishing rigorous decomposition techniques. This work bridged earlier probability foundations with dynamic processes exhibiting abrupt changes. Advancements in the included Anatolii Skorokhod's formulation of the embedding problem, which sought stopping times for to match target distributions, later influencing solutions for non-continuous martingales and extensions to jump processes. Concurrently, Hiroshi Kunita and Shinzo Watanabe's 1967 theorem generalized Itô's formula to square-integrable martingales with jumps, using Lévy systems to handle quadratic variations in settings. Modern developments solidified jump processes within broader stochastic frameworks. Jean Jacod's 1979 book Calcul Stochastique et Problèmes de Martingales provided a comprehensive treatment of integration for processes with jumps, detailing martingale problems and random measures for discontinuity . Ken-iti Sato's 1999 text Lévy Processes and Infinitely Divisible Distributions integrated jump mechanisms into Lévy theory, emphasizing infinitely divisible laws and their role in generating discontinuous trajectories. In the post-1970s era, jump processes gained prominence in to capture sudden market shocks.

Mathematical Framework

Jump Measures and Intensity

In jump processes, the jumps are formally captured by the random counting measure N(dt,dx)N(dt, dx), defined as N(dt,dx)=∑s≤t1{ΔXs≠0}δ(s,ΔXs)(dt,dx),N(dt, dx) = \sum_{s \leq t} \mathbf{1}_{\{\Delta X_s \neq 0\}} \delta_{(s, \Delta X_s)}(dt, dx), where ΔXs=Xs−Xs−\Delta X_s = X_s - X_{s-} denotes the jump size at time ss, and δ\delta is the . This measure records the occurrence, timing, and magnitude of all jumps up to time tt, providing a complete description of the discontinuous component of the process XX. The predictable compensator ν(dt,dx)\nu(dt, dx) of the jump measure NN is a unique (up to indistinguishability) predictable random measure such that the compensated process N−νN - \nu is a martingale with respect to the underlying . This compensator encodes the predictable part of the jump activity, allowing for the decomposition of NN into a martingale component and a compensator that reflects the expected jump behavior conditional on the past information. In many models, the compensator admits a representation ν(dt,dx)=λ(t,x) dt μ(dx)\nu(dt, dx) = \lambda(t, x) \, dt \, \mu(dx), where λ(t,x)\lambda(t, x) is the intensity kernel specifying the instantaneous rate of jumps of size xx at time tt, and μ\mu is a reference measure on the jump sizes (often ). The intensity kernel λ(t,x)\lambda(t, x) thus quantifies both the and distribution of jumps, enabling the analysis of jump dynamics in non-homogeneous settings. For stationary jump processes, such as Lévy processes, the compensator takes a time-homogeneous form ν(dt,dx)=dt Π(dx)\nu(dt, dx) = dt \, \Pi(dx), where Π(dx)\Pi(dx) is the Lévy measure on R∖{0}\mathbb{R} \setminus \{0\}. The Lévy measure Π\Pi has no atom at zero and satisfies the integrability condition ∫{∣x∣>1}x2 Π(dx)<∞\int_{\{|x| > 1\}} x^2 \, \Pi(dx) < \infty, ensuring the existence of a well-defined drift term in the characteristic exponent without the need for additional truncation functions for large jumps. This condition, combined with ∫{∣x∣≤1}x2 Π(dx)<∞\int_{\{|x| \leq 1\}} x^2 \, \Pi(dx) < \infty for small jumps, guarantees that the process has finite second moments. Under finite jump activity, where the total intensity ∫λ(t,x) dx<∞\int \lambda(t, x) \, dx < \infty (or equivalently Π(R∖{0})<∞\Pi(\mathbb{R} \setminus \{0\}) < \infty for Lévy processes), the expected contribution from jumps over an infinitesimal interval is captured by the formula E[ΔXt]=∫x λ(t,x) dtE[\Delta X_t] = \int x \, \lambda(t, x) \, dt, reflecting the deterministic drift induced by the average jump size weighted by the intensity kernel. This finite activity assumption implies only finitely many jumps occur almost surely over any finite time horizon, simplifying simulations and analytical tractability while preserving the martingale property of the compensated measure.

Semimartingale Representation

Jump processes, as càdlàg adapted processes with jumps, fit naturally into the semimartingale framework, which enables the development of stochastic integration and calculus for such paths. A jump process XX is a semimartingale if it admits a decomposition of the form Xt=X0+At+MtX_t = X_0 + A_t + M_t, where AA is an adapted càdlàg process of finite variation (including continuous drift and the compensator of jumps), and MM is a local martingale (including any continuous martingale part and the compensated jumps). This decomposition separates the predictable finite variation and martingale components, allowing jump processes to be treated as limits of processes with finitely many jumps or via random measures. For pure jump processes without a diffusion component, the continuous martingale part vanishes, and the process is given by the integral with respect to the jump measure and its compensator. The Itô formula extends to jump semimartingales, providing a chain rule that accounts for jumps explicitly. For a twice continuously differentiable function FF applied to a jump semimartingale XX, the formula takes the form dF(Xt)=F′(Xt−) dXtc+12F′′(Xt−) d[Xc,Xc]t+∑s≤t[F(Xs)−F(Xs−)−F′(Xs−)ΔXs],dF(X_t) = F'(X_{t-}) \, dX_t^c + \frac{1}{2} F''(X_{t-}) \, d[X^c, X^c]_t + \sum_{s \leq t} \left[ F(X_s) - F(X_{s-}) - F'(X_{s-}) \Delta X_s \right], where XcX^c denotes the continuous part of XX, and the sum corrects for the jump discontinuities by including higher-order Taylor terms implicitly through the difference F(Xs)−F(Xs−)F(X_s) - F(X_{s-}). This extension ensures that the formula holds pathwise for càdlàg trajectories, facilitating applications in stochastic differential equations driven by jumps. The jump correction term is crucial for processes with infinite activity, where uncompensated small jumps would otherwise diverge. The quadratic variation of a jump semimartingale incorporates both continuous and jump contributions, defined as [X,X]t=∑s≤t(ΔXs)2+[Xc,Xc]t[X, X]_t = \sum_{s \leq t} (\Delta X_s)^2 + [X^c, X^c]_t, where the first term sums the squared jumps and the second is the quadratic variation of the continuous part (often ⟨Xc,Xc⟩t\langle X^c, X^c \rangle_t for the predictable version). This structure highlights how jumps contribute discretely to the total variation, distinguishing jump processes from purely continuous semimartingales like diffusions. For a jump process to qualify as a semimartingale, its jumps must satisfy certain integrability conditions, such as finite variation (finitely many jumps or summable jump sizes) or, for infinite activity, the existence of a compensator ensuring that the compensated jump measure has finite second moments locally, i.e., ∫(∣x∣2∧1)ν(dt,dx)<∞\int (|x|^2 \wedge 1) \nu(dt, dx) < \infty, where ν\nu is the intensity measure of the jumps. This compensation via the predictable jump measure (as detailed in related sections on jump measures) renders the process integrable against bounded predictable processes. Without such conditions, the process may exhibit infinite variation and fail to support stochastic integration.

Types and Examples

Pure Jump Processes

A pure jump process is a type of stochastic process characterized by the absence of any continuous component in its paths, meaning all changes occur through discrete jumps. Within the framework of Lévy processes, it is defined by a Lévy triplet of the form (0,0,ν)(0, 0, \nu), where ν\nu is the Lévy measure that governs the size and frequency of jumps, with no Gaussian variance or Brownian motion term. The paths of such a process can be represented explicitly as Xt=∑s≤tΔXsX_t = \sum_{s \leq t} \Delta X_s, where ΔXs=Xs−Xs−\Delta X_s = X_s - X_{s-} denotes the jump at time ss. The distinguishing feature of pure jump processes is their purely discontinuous sample paths, which lack any smooth, continuous evolution akin to that in diffusion processes. For processes with finite jump activity, the paths resemble step functions, with a finite number of jumps over any finite time interval. In contrast, processes with infinite activity exhibit paths accumulating infinitely many small jumps, leading to more irregular but still discontinuous trajectories, without any underlying Brownian component. The activity level is determined by the Lévy measure: finite activity occurs when ∫∣x∣<1ν(dx)<∞\int_{|x|<1} \nu(dx) < \infty, while infinite activity arises otherwise, allowing for clusters of small jumps that contribute significantly to the overall variation. Representative examples of pure jump processes include subordinators, which are non-decreasing Lévy processes consisting solely of positive jumps and no negative movements, often used to model accumulation phenomena. Another class comprises stable processes with jumps only, such as α\alpha-stable Lévy processes for 0<α<20 < \alpha < 2, where the Lévy measure takes the form ν(dx)=c∣x∣−1−αdx\nu(dx) = c |x|^{-1-\alpha} dx (with c>0c > 0), resulting in heavy-tailed jump distributions and infinite activity. The variance gamma process serves as a prominent example of an infinite-activity pure jump process, constructed as a with drift subordinated by a , yielding paths with infinitely many jumps and finite variation.

Compound Poisson Processes

A is a defined as Xt=∑i=1NtYiX_t = \sum_{i=1}^{N_t} Y_i for t≥0t \geq 0, where NtN_t is a Poisson process with intensity λ>0\lambda > 0, and the YiY_i are independent and identically distributed random variables representing jump sizes, independent of NtN_t. This construction models scenarios where events occur at Poisson-distributed times, each contributing a random increment YiY_i drawn from a fixed distribution FYF_Y. The moments of XtX_t follow directly from the independence and the properties of the Poisson process. The expected value is E[Xt]=λtE[Y]\mathbb{E}[X_t] = \lambda t \mathbb{E}[Y], assuming E[∣Y∣]<∞\mathbb{E}[|Y|] < \infty, while the variance is Var(Xt)=λtE[Y2]\mathrm{Var}(X_t) = \lambda t \mathbb{E}[Y^2], provided E[Y2]<∞\mathbb{E}[Y^2] < \infty. For common jump distributions, explicit forms exist; for instance, if the YiY_i are exponentially distributed with rate β>0\beta > 0, then conditionally on Nt=kN_t = k, XtX_t follows a Gamma distribution with shape kk and rate β\beta, so the unconditional distribution is a Poisson-weighted mixture of these Gammas: P(Xt∈dx)=∑k=0∞e−λt(λt)kk!fΓ(k,β)(x)dxP(X_t \in dx) = \sum_{k=0}^\infty e^{-\lambda t} \frac{(\lambda t)^k}{k!} f_{\Gamma(k, \beta)}(x) dx, where fΓ(k,β)f_{\Gamma(k, \beta)} is the Gamma density (degenerate at 0 for k=0k=0). In the framework of Lévy processes, the Lévy measure of a compound Poisson process is Π(dx)=λFY(dx)\Pi(dx) = \lambda F_Y(dx), which fully characterizes the jump structure since the process has finite activity (expected number of jumps in [0,t][0,t] is λt<∞\lambda t < \infty). This measure is finite, Π(R)=λ<∞\Pi(\mathbb{R}) = \lambda < \infty, distinguishing it from infinite-activity cases. To simulate paths of a compound Poisson process over [0,T][0,T], one standard method generates the number of jumps NT∼Poisson(λT)N_T \sim \mathrm{Poisson}(\lambda T), then samples NTN_T i.i.d. interarrival times from the exponential distribution with rate λ\lambda to obtain jump times 0<T1<⋯<TNT≤T0 < T_1 < \cdots < T_{N_T} \leq T, and adds i.i.d. jumps Yi∼FYY_i \sim F_Y at those times, setting Xt=∑Ti≤tYiX_t = \sum_{T_i \leq t} Y_i. This thinning-based approach leverages the memoryless property of the Poisson process for efficient computation.

Properties and Analysis

Strong Markov Property

The strong Markov property is a fundamental characteristic of many jump processes, extending the ordinary Markov property to hold at arbitrary stopping times. Specifically, a jump process {Xt}t≥0\{X_t\}_{t \geq 0} satisfies the strong Markov property if, for any stopping time τ\tau, the post-τ\tau process {Xτ+t}t≥0\{X_{\tau + t}\}_{t \geq 0} (on the event {τ<∞}\{\tau < \infty\}), conditional on Fτ\mathcal{F}_\tau, has the same distribution as the original process started from XτX_\tau, and is independent of Fτ\mathcal{F}_\tau given XτX_\tau. This property ensures that the process "restarts" in a Markovian fashion at unpredictable times, preserving the conditional independence of future increments given the current state. For the Poisson process, a canonical example of a jump process, the strong Markov property holds due to the memoryless nature of its exponential interarrival times. Consider a Poisson process N(t)N(t) with rate λ>0\lambda > 0; the interarrival times are i.i.d. exponential random variables with parameter λ\lambda. At a stopping time τ\tau, the residual time until the next jump is again exponentially distributed with rate λ\lambda, independent of the history up to τ\tau, because P(A(τ)>t∣Fτ)=e−λtP(A(\tau) > t \mid \mathcal{F}_\tau) = e^{-\lambda t} for the forward recurrence time A(τ)A(\tau). This memoryless property implies that the process after τ\tau behaves identically to a fresh Poisson process starting from N(τ)N(\tau), confirming the strong Markov property. The strong facilitates the analysis of jump processes by allowing their embedding into constructions or frameworks, where evolution is governed by transition operators. For pure jump processes, this is reflected in the infinitesimal generator L\mathcal{L}, which for a with jump measure λ(dy)\lambda(dy) takes the form Lf(x)=∫[f(x+y)−f(x)] λ(dy),\mathcal{L} f(x) = \int [f(x + y) - f(x)] \, \lambda(dy), capturing the expected change due to jumps. This generator enables solving and studying long-run behavior within Feller s. Not all processes exhibiting jumps satisfy the strong , as it requires the underlying dynamics to be Markovian. A is a deterministic jump process that increments by at each time t=1,2,3,…t = 1, 2, 3, \dots, regardless of the current state; here, the timing of future jumps depends explicitly on the absolute time rather than solely on the present value, violating the and thus the strong version.

Stationarity and Ergodicity

A Markov jump process, modeled as a continuous-time Markov chain (CTMC), is stationary if its finite-dimensional distributions are invariant under time shifts, meaning the statistical properties do not change over time when the process starts from the stationary distribution. For such processes, the stationary distribution π\pi satisfies the global balance equation πQ=0\pi Q = 0, where QQ is the infinitesimal generator matrix with off-diagonal entries qij≥0q_{ij} \geq 0 representing jump rates from state ii to jj (for i≠ji \neq j) and diagonal entries qii=−∑j≠iqijq_{ii} = -\sum_{j \neq i} q_{ij} ensuring row sums of zero. The vector π\pi is normalized such that ∑iπi=1\sum_i \pi_i = 1, representing the long-run proportion of time spent in each state under equilibrium. This equation equates the total rate of probability flow into each state with the flow out, establishing steady-state conditions. The existence and uniqueness of π\pi depend on the chain's structure. For finite-state CTMCs that are irreducible—meaning every state is reachable from every other—a unique exists, as the generator QQ has a one-dimensional kernel, allowing a positive solution to πQ=0\pi Q = 0 normalized to a . In infinite-state spaces, existence requires additional conditions like positive recurrence, where the time to any state is finite, ensuring πi=1/mi\pi_i = 1 / m_i with mi<∞m_i < \infty the mean return time to state ii. Positive recurrence can be verified using Foster-Lyapunov criteria, which involve a non-negative function VV (Lyapunov function) satisfying a drift condition: outside a finite set CC, the expected change AV(x)≤−ϵV(x)+K\mathcal{A}V(x) \leq -\epsilon V(x) + K for some ϵ>0\epsilon > 0, K<∞K < \infty, where A\mathcal{A} is the generator applied to VV, implying the chain drifts toward CC and admits a unique π\pi. These criteria extend to general state spaces and confirm positive recurrence under irreducibility. Ergodicity ensures that the long-run behavior of the process aligns with the stationary distribution, allowing time averages to converge to ensemble averages. A CTMC is ergodic if it is irreducible and positive recurrent, in which case the transition probabilities satisfy Pij(t)→πjP_{ij}(t) \to \pi_j as t→∞t \to \infty for all i,ji, j, independently of the initial state. For finite-state irreducible CTMCs, this convergence holds without further aperiodicity requirements, as the continuous-time embedding avoids periodicity issues inherent in discrete-time chains. The ergodic theorem states that for any bounded measurable function ff, the time average (1/T)∫0Tf(Xs) ds→∫f dπ(1/T) \int_0^T f(X_s) \, ds \to \int f \, d\pi almost surely as T→∞T \to \infty, where the convergence relies on the mixing properties induced by positive recurrence. In infinite-state cases, Foster-Lyapunov conditions guarantee geometric ergodicity, with explicit rates of convergence to π\pi. A representative example is the birth-death process, a one-dimensional jump process on non-negative integers with jumps only to adjacent states via birth rates λn>0\lambda_n > 0 (upward) and death rates μn>0\mu_n > 0 (downward). Under irreducibility, the process is recurrent if ∑n=1∞∏k=1nμkλk=∞\sum_{n=1}^\infty \prod_{k=1}^n \frac{\mu_k}{\lambda_k} = \infty, and positive recurrent if additionally ∑n=1∞∏k=1nλk−1μk<∞\sum_{n=1}^\infty \prod_{k=1}^n \frac{\lambda_{k-1}}{\mu_k} < \infty—the stationary distribution π\pi satisfies detailed balance: πnλn=πn+1μn+1\pi_n \lambda_n = \pi_{n+1} \mu_{n+1}, yielding πn=π0∏k=1n(λk−1/μk)\pi_n = \pi_0 \prod_{k=1}^n (\lambda_{k-1} / \mu_k), normalized appropriately. This π\pi gives the long-run distribution of the state (e.g., population size), with ergodicity implying that sample path averages converge to expectations under π\pi.

Applications

Financial Modeling

Jump processes play a crucial role in financial modeling by capturing sudden, discontinuous changes in asset prices, such as those observed during market crashes or news events, which continuous diffusion models like Black-Scholes fail to represent adequately. These models extend the geometric Brownian motion by incorporating a jump component, allowing for the explanation of empirical anomalies including excess kurtosis in return distributions and asymmetric volatility patterns. A foundational approach is the Merton jump-diffusion model, introduced in 1976, which combines a diffusion process with a compound Poisson jump process. The asset price StS_t follows the stochastic differential equation dSt=μSt dt+σSt dWt+St− dJt,dS_t = \mu S_t \, dt + \sigma S_t \, dW_t + S_{t-} \, dJ_t, where WtW_t is a standard Brownian motion, μ\mu and σ\sigma are the drift and volatility parameters, and JtJ_t is a compound Poisson process with intensity λ\lambda and log-normal jump sizes Yi=ln⁡(1+ki)Y_i = \ln(1 + k_i) to model multiplicative jumps. This framework accounts for rare but significant price shocks while preserving the continuous dynamics for smaller fluctuations. To address limitations in capturing return asymmetry, Kou's double exponential jump-diffusion model (2002) modifies the jump size distribution to a double exponential form, with upward jumps following an exponential distribution with parameter η1\eta_1 and downward jumps with η2>η1\eta_2 > \eta_1, enabling better fitting of the skewness observed in equity returns. For derivative pricing under these models, risk-neutral valuation is employed, adjusting the physical measure drift and incorporating a jump to reflect aversion to discontinuous risks. Option prices can be derived via the of the log-price process, often using methods for efficient computation, as detailed in the affine jump-diffusion framework. These techniques yield semi-closed-form solutions for European options by inverting the transform to obtain the risk-neutral density. Empirically, jump-diffusion models provide strong evidence for their relevance in equity and markets, particularly in explaining fat-tailed return distributions and the —the upward-sloping curve for out-of-the-money options—observed persistently after the 1987 . Studies on options post-1987 reveal that incorporating jumps significantly improves model fit by capturing the negative skewness and crash-like events implicit in option prices, reducing pricing errors compared to pure models.

Queueing and Reliability Theory

In , jump processes provide a natural framework for modeling the dynamics of service systems, where customer arrivals and departures induce discrete jumps in the queue length state space. The queue length evolves as a piecewise constant process, with holding times between jumps governed by exponential or general distributions, capturing the nature of system occupancy. This approach is particularly suited to systems with countable states, allowing through the intensity of jumps corresponding to arrival and service completion rates. The M/M/1 queue exemplifies this modeling, represented as a birth-death (CTMC) that is a pure jump process. Arrivals occur as Poisson jumps at rate λ\lambda, increasing the queue length by 1, while service completions are exponential jumps at rate μ\mu (when the queue is nonempty), decreasing it by 1. Under the stability condition ρ=λ/μ<1\rho = \lambda / \mu < 1, the process is ergodic, and the of the queue length NN is geometric: P(N=k)=(1−ρ)ρkP(N = k) = (1 - \rho) \rho^k for k=0,1,2,…k = 0, 1, 2, \dots. Generalizations to the G/G/1 queue extend this framework by allowing general interarrival times (modeled as renewal jumps) and general service times, without assuming Poisson arrivals. The queue length process remains a jump process, but full continuous-time analysis is challenging; instead, embedded Markov chains are constructed at jump epochs, such as departure instants, to study the state evolution discretely. A seminal result in this context is the Pollaczek-Khinchine formula for the M/G/1 special case (Poisson arrivals, general services), which expresses the mean waiting time WW as W=λE[S2]2(1−ρ),W = \frac{\lambda E[S^2]}{2(1 - \rho)}, where SS is the service time and E[S2]E[S^2] accounts for service variability induced by jump timings; this highlights how jump process variability directly impacts queue performance metrics. In reliability theory, jump processes model system degradation and recovery, with failures manifesting as jumps that transition the system from an operational to a failed state. Failure times form a renewal process—a special pure jump process with successive jumps of size 1 at inter-failure epochs following a general distribution—allowing the counting of cumulative failures over time. Repairs are incorporated as regenerative jumps, resetting the system to an state upon completion, forming an alternating renewal process that alternates between up (operational) and down (repair) periods. For systems with exponentially distributed times to (rate λ\lambda) and repairs (rate μ\mu), the long-run —the proportion of time the is operational—is given by A=μλ+μ,A = \frac{\mu}{\lambda + \mu}, derived from the equilibrium of the alternating renewal process under . This underscores the balance between and repair jump rates in determining dependability.

References

  1. https://personal.ntu.edu.sg/nprivault/MA5182/stochastic-calculus-jump-processes.pdf
  2. https://www.ceremade.dauphine.fr/~poisat/files/M2/jump-processes.pdf
  3. https://sites.math.rutgers.edu/~tmp140/courses/Spring2015/622-Lecture1.pdf
  4. https://arxiv.org/pdf/math/0309246
  5. https://math.arizona.edu/~tgk/529/section2.pdf
  6. https://www.math.utah.edu/~davar/math7880/S11/Chapters/Ch6.pdf
  7. http://www.math.ntua.gr/~papapan/teaching/LevyProcesses.pdf
  8. https://bibliotekanauki.pl/articles/2012070.pdf
  9. https://link.springer.com/article/10.1007/s00440-019-00935-z
  10. https://arxiv.org/pdf/math.PR/0401114
  11. https://www.semanticscholar.org/paper/On-Square-Integrable-Martingales-Kunita-Watanabe/9732ae1fd8518337602743e1eb408ecc7f741194
  12. https://link.springer.com/book/10.1007/BFb0064907
  13. https://www.ma.imperial.ac.uk/~ajacquie/IC_AMDP/IC_AMDP_Docs/Literature/Jarrow_Protter_History_Stochastic_Integration.pdf
  14. https://link.springer.com/book/9783540441104
  15. https://almostsuremath.com/2011/02/25/properties-of-levy-processes/
  16. https://people.math.ethz.ch/~jteichma/slides_week1to3.pdf
  17. https://public.econ.duke.edu/~get/browse/courses/883/Spr15/COURSE-MATERIALS/10_PureJumpLevyProcesses.pdf
  18. https://engineering.nyu.edu/sites/default/files/2018-09/CarrEuropeanFinReview1998.pdf
  19. https://www.math.utah.edu/~davar/math7880/S11/Chapters/Ch9.pdf
  20. http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-PP.pdf
  21. https://personal.math.ubc.ca/~holmescerfon/teaching/asa22/handout-Lecture4_2022.pdf
  22. https://mathematicaster.org/teaching/lcs22/norris_markov.pdf
  23. https://probability.ca/MT/BOOK.pdf
  24. https://www.sciencedirect.com/science/article/pii/0304405X76900222
  25. http://www.columbia.edu/~sk75/MagSci02.pdf
  26. https://www.nber.org/system/files/working_papers/w5894/w5894.pdf
  27. http://www.columbia.edu/~ks20/stochastic-I/stochastic-I-CTMC.pdf
  28. https://www.stats.ox.ac.uk/~winkel/bs3a07l13-14.pdf
  29. https://www.netlab.tkk.fi/opetus/s383143/kalvot/E_mg1jono.pdf
  30. https://ocw.mit.edu/courses/6-262-discrete-stochastic-processes-spring-2011/931ffa0940899c27f34b71ad64fd2bb0_MIT6_262S11_chap04.pdf
  31. https://link.springer.com/chapter/10.1007/978-981-4451-51-2_13
Add your contribution
Related Hubs
User Avatar
No comments yet.