Simulating a Hawkes-Driven Jump Process with Euler Steps

Summary

The document sets up a jump process whose value falls multiplicatively at each event. Jump sizes are generated from exponential random variables, while event arrivals follow a Hawkes process: its intensity mean-reverts and increases after each jump. It asks how to simulate sample paths over a finite horizon using a fixed Euler time grid and given starting parameters.

The material provides a model specification and simulation task, but no worked algorithm, code, simulated paths, or validation evidence. A simulation would need to update the intensity over each time step, sample whether an event occurs based on the current intensity, and apply a sampled jump when it does. The document does not discuss discretization error, handling multiple events within a step, or alternative event-time simulation methods, so those implementation choices and their effects remain open.

Key ideas

• The process decreases multiplicatively whenever the counting process records an event.
• The event intensity follows mean-reverting dynamics and rises after events, giving it Hawkes-type clustering.
• The proposed task uses a fixed Euler grid to generate sample paths over a finite time interval.
• The document does not provide a simulation procedure or assess discretization accuracy.

Tags

# Euler discretization with jumps


# Euler discretization with jumps












There is a process

$B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$,

where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$.

${N_t}$ is a counting process with intensity $λ_t$ which solves the stochastic differential equation

$dλ_t = 0.5 (0.3 − λ_t) dt + 0.3dN_t$(a Hawkes Process) and $λ_0 = 0$.

How can I use Euler discretization scheme to generate sample paths of $B_t$ for t ∈ [0, 10]? Assume that the number of Euler steps per unit of time is 100 and $B_0$ = 10, 000.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.