ODE that depends nonlinearly on a jump process

[cumberworth]

I need to solve a differential equation that has a nonlinear dependence on a jump process, whose rate is itself dependent on the solution of the ODE. This does seem to be possible with this package; it seems only possible to add a jump process to an ODE. Is there any plan to allow for a more general dependence of an ODE on a jump process than what is currently available?

[ChrisRackauckas]

Have you seen https://diffeq.sciml.ai/stable/tutorials/discrete_stochastic_example/ and https://diffeq.sciml.ai/stable/tutorials/jump_diffusion/? The rate is just a function of (u,p,t) so you can make the jump rate use u, which is from the combined system.

[cumberworth]

I have seen that, but it doesn't allow for the ODE to use the jump process. What I want is to solve something of the form

$$ \frac{\mathrm{d} u}{\mathrm{d} t} = f(u, p, N, t) $$

where $p$ are parameters, and $N$ is a Poisson process with rate $\lambda(u, p, N, t)$. From the documentation, it is only possible to handle problems of the form

$$ \frac{\mathrm{d} u}{\mathrm{d} t} = f(u, p, t) + c(u, p, t) \mathrm{d} N $$

where $c(u, p, t)$ sets the jump size and the Poisson process has rate $\lambda(u, p, t)$.

[ChrisRackauckas]

Oh yes, this would require using the random ordinary differential equation (RODE) solvers with a Poisson process.

[cumberworth]

I thought about this as well, but according to the documentation, the RODE problems seem to be restricted to the form

$$ \frac{\mathrm{d} u}{\mathrm{d} t} ​= f(u, p, t, W(t)) $$

where $W(t)$ is the noise/stochastic process. Reading through the documentation for noise processes, I cannot see that it is possible to define a noise process that has a distribution that depends on the solution to the differential equation that it is being used in, or even on it's own solution. Again, I would like to have a Poisson process who's rate depends on both the current value of the solution to the ODE and on the current solution of the Poisson process.

[isaacsas]

Apologies if I've misunderstood, but if you just want to have a counting process as an input in the ODE, why not make it an extra state-variable within u? i.e. below u[1] is the ODE component and u[2] is the counting process

using DifferentialEquations
function f!(du,u,p,t)
    du[1] = 10 - u[2]*u[1]
    du[2] = 0
    nothing
end
# rate the counting process fires at 
rate(u,p,t) = u[1]*u[2] + 1
function affect!(integrator) 
    # when a jump occurs we increment that value of the counting process by 1
    integrator.u[2] += 1   
    nothing
end
vrj = VariableRateJump(rate, affect!)
oprob = ODEProblem(f!, [100.0, 0.0], (0.0, 10.0))
jprob = JumpProblem(oprob, Direct(), vrj)
sol = solve(jprob, Tsit5())

[isaacsas]

Also, doesn't your general mathematical problem still fit within the jump-ODE form you wrote down above, i.e. just take your combined equation to be

$$ du(t) = \begin{pmatrix} du_1(t) \\ du_2(t) \end{pmatrix} = \begin{pmatrix} f(u,p,t) \\ 0 \end{pmatrix} + \begin{pmatrix} 0 \\ 1 \end{pmatrix} dN(t) $$

choosing $N(t)$ to have whatever intensity function you want via the rate(u,p,t) function.

[cumberworth]

Ah, yes, you are right. Thank you both for helping to clarify this.