JumpProcesses.jl: Stochastic Simulation Algorithms for Jump Processes, Jump-ODEs, and Jump-Diffusions
JumpProcesses.jl provides methods for simulating jump and point processes. Across different fields of science, such methods are also known as stochastic simulation algorithms (SSAs), Doob's method, Gillespie methods, Kinetic Monte Carlo's methods, thinning method, and Ogata's method. It also enables the incorporation of jump processes into hybrid jump-ODE models, including piecewise deterministic Markov processes, and into hybrid jump-SDE models, including jump diffusions. It is a component package in the SciML ecosystem, and one of the core solver libraries included in DifferentialEquations.jl.
Historically, jump processes have been developed in the context of dynamical systems to describe dynamics with discontinuous changes — the jumps — in a system's value at random times. In contrast, the development of point processes has been more focused on describing the occurrence of random events — the points — over a support. However, both jump and point processes share many things in common which make JumpProcesses ideal for their study.
As jump and point processes are often considered from a variety of perspectives across different fields, JumpProcesses provides three tutorials on using the package for those with different backgrounds:
- Simulating basic Poisson processes
- Simulating jump processes via SSAs (i.e., Gillespie methods)
- Simulating temporal point processes (TPPs)
These tutorials also explain the types of jump/point processes that can be mathematically modelled with JumpProcesses.jl. For more complicated models that couple ODEs and/or SDEs with continuous noise to jump processes, we provide a tutorial on
For jump processes that involve spatial transport on a graph/mesh, such as Reaction-Diffusion Master Equation models, we provide a tutorial on
Finally, we provide application tutorials which are more extensive tutorials that interface with other libraries going deeper into a topic.
We provide a mathematical overview of the library below, but note users may also skip to the appropriate tutorial listed above to get started with using JumpProcesses.
Mathematical Overview
Let $dN_i(t)$ be a stochastic process such that $dN_i(t) = 1$ with some probability and $0$ otherwise. That is, $dN_i(t)$ encodes that a "jump" in the value of $N_i(t)$ by one occurs at time $t$. Denote the rate, i.e. probability per time, that such jumps occur by the intensity function, $\lambda_i(u(t), p, t)$. Here $u(t)$ represents a vector of dynamic state variables, that may change when $N_i(t)$ jumps. For example, these could be the size of a population changing due to births or deaths, or the number of mRNAs and proteins within a cell (which jump when a gene is transcribed or an mRNA is translated). $p$ represents parameters the intensity may depend on.
In different fields $\lambda_i$ can also be called a propensity, transition rate function, or a hazard function. Note, if we denote $N(t) \equiv N[0, t)$ as the number of points since the start of time until $t$, exclusive of $t$, then $N(t)$ is a stochastic random integer measure. In other words, we have a temporal point process (TPP).
In JumpProcesses.jl's language, we call $\lambda_i$ a rate function, and expect users to provide a function, rate(u,p,t), that returns its value at time t. Given a collection of rates $\{\lambda_i\}_{i=1}^I$, JumpProcesses can then generate exact realizations of pure jump processes of the form
\[du = \sum_{i=1}^I h_i(u,p,t) \, dN_i(t),\]
where $h_i(u,p,t)$ represents the amount that $u(t)$ changes when $N_i(t)$ jumps. JumpProcesses encodes such changes via a user-provided affect! function, which allows even more general changes to the state, $u(t)$, when $N_i(t)$ jumps than just incrementing it by $h_i(u,p,t)$. For example, such changes can themselves be random, allowing for the calculation of marks. In the special case of just one jump, $I = 1$, with $h_1 = 1$, we recover the temporal point process mentioned above.
JumpProcesses provides a variety of algorithms, called aggregators, for determining the next time that a jump occurs, and which $N_i(t)$ jumps at that time. Many of these are optimized for contexts in which each $N_i(t)$ only changes the values of a few components in $u(t)$, as common in many applications such as stochastic chemical kinetics (where each $N_i$ corresponds to a different reaction, and each component of $u$ a different species). To simulate $u(t)$ users must then specify both an aggregator algorithm to determine the time and type of jump that occurs, and a time-stepping method to advance the state from jump to jump. See the tutorials listed above, and reference links below, for more details and examples.
JumpProcesses also allows such jumps to be coupled into ODE models (i.e. piecewise deterministic Markov Processes), or continuous-noise SDE models (i.e. jump-diffusions). For example, a jump-diffusion JumpProcesses can simulate would be
\[du = f(u,p,t)dt + \sum_{i}g_i(u,t)dW_i(t) + \sum_{j}h_j(u,p,t)dN_j(t)\]
where $f$ encodes the drift of the process, each $g_i$ the strength of a diffusion component of the process, and each $W_i(t)$ denotes an independent standard Brownian Motion.
JumpProcesses is designed to simulate all the types of jumps described above.
Reference Documentation
In addition to the tutorials linked above, the documentation contains
- References on the types of jumps and available simulation methods
- References on jump time stepping methods
- An FAQ with information on changing parameters between simulations and using callbacks
- API documentation
Installation
There are two ways to install JumpProcesses.jl. First, users may install the meta DifferentialEquations.jl package, which installs and wraps OrdinaryDiffEq.jl for solving ODEs, StochasticDiffEq.jl for solving SDEs, and JumpProcesses.jl, along with a number of other useful packages for solving models involving ODEs, SDEs and/or jump process. This single install will provide the user with all the facilities for developing and solving Jump problems.
To install the DifferentialEquations.jl package, refer to the following link for complete installation details.
If the user wishes to install the JumpProcesses.jl library separately, which is a lighter dependency than DifferentialEquations.jl, then the following code will install JumpProcesses.jl using the Julia package manager:
using Pkg
Pkg.add("JumpProcesses")Contributing and Getting Help
Please refer to the SciML ColPrac: Contributor's Guide on Collaborative Practices for Community Packages for guidance on PRs, issues, and other matters relating to contributing to SciML.
See the SciML Style Guide for common coding practices and other style decisions.
There are a few community forums for getting help and asking questions:
- The #diffeq-bridged and #sciml-bridged channels in the Julia Slack
- The #diffeq-bridged and #sciml-bridged channels in the Julia Zulip
- The Julia Discourse forums
- See also the SciML Community page
Reproducibility
The documentation of this SciML package was built using these direct dependencies,
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Julia Version 1.11.6
Commit 9615af0f269 (2025-07-09 12:58 UTC)
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Official https://julialang.org/ release
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[efe28fd5] OpenSpecFun_jll v0.5.6+0
[91d4177d] Opus_jll v1.5.2+0
[36c8627f] Pango_jll v1.56.3+0
⌅ [30392449] Pixman_jll v0.44.2+0
[c0090381] Qt6Base_jll v6.8.2+1
[629bc702] Qt6Declarative_jll v6.8.2+1
[ce943373] Qt6ShaderTools_jll v6.8.2+1
[e99dba38] Qt6Wayland_jll v6.8.2+1
[f50d1b31] Rmath_jll v0.5.1+0
⌅ [fb77eaff] Sundials_jll v5.2.3+0
[a44049a8] Vulkan_Loader_jll v1.3.243+0
[a2964d1f] Wayland_jll v1.24.0+0
[ffd25f8a] XZ_jll v5.8.1+0
[f67eecfb] Xorg_libICE_jll v1.1.2+0
[c834827a] Xorg_libSM_jll v1.2.6+0
[4f6342f7] Xorg_libX11_jll v1.8.12+0
[0c0b7dd1] Xorg_libXau_jll v1.0.13+0
[935fb764] Xorg_libXcursor_jll v1.2.4+0
[a3789734] Xorg_libXdmcp_jll v1.1.6+0
[1082639a] Xorg_libXext_jll v1.3.7+0
[d091e8ba] Xorg_libXfixes_jll v6.0.1+0
[a51aa0fd] Xorg_libXi_jll v1.8.3+0
[d1454406] Xorg_libXinerama_jll v1.1.6+0
[ec84b674] Xorg_libXrandr_jll v1.5.5+0
[ea2f1a96] Xorg_libXrender_jll v0.9.12+0
[c7cfdc94] Xorg_libxcb_jll v1.17.1+0
[cc61e674] Xorg_libxkbfile_jll v1.1.3+0
[e920d4aa] Xorg_xcb_util_cursor_jll v0.1.5+0
[12413925] Xorg_xcb_util_image_jll v0.4.1+0
[2def613f] Xorg_xcb_util_jll v0.4.1+0
[975044d2] Xorg_xcb_util_keysyms_jll v0.4.1+0
[0d47668e] Xorg_xcb_util_renderutil_jll v0.3.10+0
[c22f9ab0] Xorg_xcb_util_wm_jll v0.4.2+0
[35661453] Xorg_xkbcomp_jll v1.4.7+0
[33bec58e] Xorg_xkeyboard_config_jll v2.44.0+0
[c5fb5394] Xorg_xtrans_jll v1.6.0+0
[3161d3a3] Zstd_jll v1.5.7+1
[35ca27e7] eudev_jll v3.2.14+0
[214eeab7] fzf_jll v0.61.1+0
[a4ae2306] libaom_jll v3.12.1+0
[0ac62f75] libass_jll v0.17.4+0
[1183f4f0] libdecor_jll v0.2.2+0
[2db6ffa8] libevdev_jll v1.13.4+0
[f638f0a6] libfdk_aac_jll v2.0.4+0
[36db933b] libinput_jll v1.28.1+0
[b53b4c65] libpng_jll v1.6.50+0
[f27f6e37] libvorbis_jll v1.3.8+0
[f8abcde7] micromamba_jll v1.5.12+0
[009596ad] mtdev_jll v1.1.7+0
[1317d2d5] oneTBB_jll v2022.0.0+0
[4d7b5844] pixi_jll v0.41.3+0
[1270edf5] x264_jll v10164.0.1+0
[dfaa095f] x265_jll v4.1.0+0
[d8fb68d0] xkbcommon_jll v1.9.2+0
[0dad84c5] ArgTools v1.1.2
[56f22d72] Artifacts v1.11.0
[2a0f44e3] Base64 v1.11.0
[ade2ca70] Dates v1.11.0
[8ba89e20] Distributed v1.11.0
[f43a241f] Downloads v1.6.0
[7b1f6079] FileWatching v1.11.0
[9fa8497b] Future v1.11.0
[b77e0a4c] InteractiveUtils v1.11.0
[4af54fe1] LazyArtifacts v1.11.0
[b27032c2] LibCURL v0.6.4
[76f85450] LibGit2 v1.11.0
[8f399da3] Libdl v1.11.0
[37e2e46d] LinearAlgebra v1.11.0
[56ddb016] Logging v1.11.0
[d6f4376e] Markdown v1.11.0
[a63ad114] Mmap v1.11.0
[ca575930] NetworkOptions v1.2.0
[44cfe95a] Pkg v1.11.0
[de0858da] Printf v1.11.0
[3fa0cd96] REPL v1.11.0
[9a3f8284] Random v1.11.0
[ea8e919c] SHA v0.7.0
[9e88b42a] Serialization v1.11.0
[1a1011a3] SharedArrays v1.11.0
[6462fe0b] Sockets v1.11.0
[2f01184e] SparseArrays v1.11.0
[f489334b] StyledStrings v1.11.0
[4607b0f0] SuiteSparse
[fa267f1f] TOML v1.0.3
[a4e569a6] Tar v1.10.0
[8dfed614] Test v1.11.0
[cf7118a7] UUIDs v1.11.0
[4ec0a83e] Unicode v1.11.0
[e66e0078] CompilerSupportLibraries_jll v1.1.1+0
[deac9b47] LibCURL_jll v8.6.0+0
[e37daf67] LibGit2_jll v1.7.2+0
[29816b5a] LibSSH2_jll v1.11.0+1
[c8ffd9c3] MbedTLS_jll v2.28.6+0
[14a3606d] MozillaCACerts_jll v2023.12.12
[4536629a] OpenBLAS_jll v0.3.27+1
[05823500] OpenLibm_jll v0.8.5+0
[efcefdf7] PCRE2_jll v10.42.0+1
[bea87d4a] SuiteSparse_jll v7.7.0+0
[83775a58] Zlib_jll v1.2.13+1
[8e850b90] libblastrampoline_jll v5.11.0+0
[8e850ede] nghttp2_jll v1.59.0+0
[3f19e933] p7zip_jll v17.4.0+2
Info Packages marked with ⌃ and ⌅ have new versions available. Those with ⌃ may be upgradable, but those with ⌅ are restricted by compatibility constraints from upgrading. To see why use `status --outdated -m`You can also download the manifest file and the project file.