Jump diffusion - Nixtla

A jump-diffusion process is a smooth diffusion (like GBM) punctuated by sudden jumps — the model for prices that mostly drift but occasionally gap on news. It is a realistic test for methods that must distinguish ordinary volatility from discrete shocks.

The model $dS_t = \mu S_t\, dt + \sigma S_t\, dW_t + S_{t^-}\, dJ_t, \qquad N_t \sim \mathrm{Poisson}(\lambda\, dt)$ A continuous diffusion accumulates small changes while a Poisson jump process adds occasional discrete moves. The jump intensity sets how often jumps occur; the jump-size distribution sets how large they are. Between jumps the series behaves like its underlying diffusion.

import polars as pl
import matplotlib.pyplot as plt

from synforecast.generators import JumpDiffusionGenerator

1. Jump intensity

lambda_jump is the expected number of jumps per unit time. Between jumps the path is an ordinary diffusion, so raising it adds discontinuities without changing the underlying drift.

base = {
    "min_length": 300,
    "max_length": 300,
    "freq": "D",
    "mu": 0.05,
    "sigma": 0.15,
    "jump_mean": 0.0,
    "jump_std": 0.05,
    "initial_value": 100.0,
    "seed": 42,
}

fig, ax = plt.subplots(figsize=(12, 4))
for lambda_jump in (0.0, 5.0, 30.0):
    df = JumpDiffusionGenerator(
        engine="polars", lambda_jump=lambda_jump, **base
    ).generate(n_series=1)
    ax.plot(
        df["ds"].to_list(), df["y"].to_list(), label=f"lambda_jump={lambda_jump}", alpha=0.85
    )
ax.set(
    xlabel="Timestamp",
    ylabel="Price",
    title="Jump intensity: lambda_jump=0 is pure diffusion",
)
ax.legend(fontsize=8)
plt.tight_layout()
plt.show()

2. Jump direction

jump_mean is the average log jump size, so its sign decides whether shocks push the price up or down. The drift is not compensated for jumps, which is why a negative mean drags the whole path down.

fig, ax = plt.subplots(figsize=(12, 4))
for jump_mean in (-0.05, 0.0, 0.05):
    df = JumpDiffusionGenerator(
        engine="polars",
        min_length=300,
        max_length=300,
        freq="D",
        mu=0.05,
        sigma=0.15,
        lambda_jump=15.0,
        jump_mean=jump_mean,
        jump_std=0.02,
        initial_value=100.0,
        seed=42,
    ).generate(n_series=1)
    ax.plot(
        df["ds"].to_list(), df["y"].to_list(), label=f"jump_mean={jump_mean}", alpha=0.85
    )
ax.set(xlabel="Timestamp", ylabel="Price", title="Jump direction follows the sign of jump_mean")
ax.legend(fontsize=8)
plt.tight_layout()
plt.show()

3. Multiple series

Generate multiple independent jump diffusion paths.

multi_params = {
    "min_length": 150,
    "max_length": 150,
    "freq": "D",
    "mu": 0.05,
    "sigma": 0.15,
    "lambda_jump": 0.2,
    "jump_mean": 0.0,
    "jump_std": 0.1,
    "initial_value": 100.0,
    "seed": 42,
}

multi_gen = JumpDiffusionGenerator(engine="polars", **multi_params)
multi_df = multi_gen.generate(n_series=3)

print(f"Generated 3 series with {len(multi_df)} total observations")
print(f"Overall Statistics: Mean={multi_df['y'].mean():.4f}, Std={multi_df['y'].std():.4f}")
multi_df.filter(pl.col("unique_id") == "0").head(10)

Generated 3 series with 450 total observations
Overall Statistics: Mean=6662.1054, Std=16489.8470

unique_id	ds	y
cat	datetime[ns]	f64
”0”	2000-01-01 00:00:00	100.0
”0”	2000-01-02 00:00:00	84.240947
”0”	2000-01-03 00:00:00	94.557346
”0”	2000-01-04 00:00:00	107.575297
”0”	2000-01-05 00:00:00	114.688591
”0”	2000-01-06 00:00:00	159.7268
”0”	2000-01-07 00:00:00	146.174789
”0”	2000-01-08 00:00:00	171.596255
”0”	2000-01-09 00:00:00	174.553944
”0”	2000-01-10 00:00:00	168.409026

fig, ax = plt.subplots(figsize=(12, 4))
for uid in multi_df["unique_id"].unique().to_list():
    series = multi_df.filter(pl.col("unique_id") == uid)
    ax.plot(series["ds"].to_list(), series["y"].to_list(), label=uid, alpha=0.8)
ax.set_xlabel("Timestamp")
ax.set_ylabel("Value")
ax.set_title("Multiple jump diffusion series")
ax.legend()
plt.tight_layout()
plt.show()

Related generators

Geometric Brownian motion — the jump-free diffusion.

Hawkes process — when the jumps cluster and self-excite.

Jump intensity and size parameters are in the generator reference.

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