Jump-Diffusion Processes

Motivation: why this matters in quant finance

Pure geometric Brownian motion assumes prices move continuously. That assumption is useful for Black-Scholes, but it misses earnings gaps, default announcements, depeg events, exchange halts, and operational-loss arrivals. Those events are not "large Brownian moves"; they are discontinuities.

Lawler's jump-process chapter makes the modelling fork precise. If a process keeps independent and stationary increments but drops continuous paths, the basic new building blocks are Poisson processes and compound Poisson processes. A jump-diffusion is the finance version of combining a Brownian diffusion with one of those jump components.

The payoff is not a catalogue of named models. It is the algebra of jumps: jump arrival times, jump-size distributions, characteristic exponents, generators, compensators, and quadratic variation. Those are the objects that later become jump-risk premia, incomplete-market hedging errors, and integro-differential pricing equations.

The informal idea

A diffusion moves all the time in small rough increments. A jump process waits, then moves suddenly. A compound Poisson process separates those two questions:

When does a jump arrive? A Poisson process $N_t$ counts arrivals at rate $\lambda$ .
How big is the jump? Independent jump sizes $Y_1,Y_2,\ldots$ specify the marks.

The cumulative jump process is

J_t=\sum_{i=1}^{N_t}Y_i.

Adding a Brownian component gives the simplest jump-diffusion log-return model:

X_t=mt+\sigma W_t+J_t.

The Brownian part captures ordinary diffusive noise. The compound Poisson part captures a finite number of discontinuous shocks over finite horizons.

Formal definitions

Let

W_t

be Brownian motion, let

N_t

be a Poisson process with rate

\lambda

, and let

Y_1,Y_2,\ldots

be i.i.d. jump sizes independent of both

W

and

N

. A basic jump-diffusion log process is

X_t=mt+\sigma W_t+\sum_{i=1}^{N_t}Y_i.

An exponential asset-price version is

S_t=S_0e^{X_t}.

The jump part is a compound Poisson process. Lawler parameterises it through a finite measure

\mu=\lambda \mu^\#,

where $\mu^\#$ is the jump-size distribution. The total mass $\mu(\mathbb{R})=\lambda$ is the jump rate, and $\mu(A)$ is the first-order rate of jumps whose sizes fall in $A$ .

Key properties

Characteristic exponents add

For independent Lévy components, characteristic exponents add. A Brownian motion with drift $m$ and variance parameter $\sigma^2$ has exponent

ims-\frac{1}{2}\sigma^2s^2.

A Poisson process with rate $\lambda$ has exponent

\lambda(e^{is}-1).

So the sum of Brownian motion and a unit-jump Poisson process has exponent

\Psi(s)=ims-\frac{1}{2}\sigma^2s^2+\lambda(e^{is}-1).

For a compound Poisson process with Lévy measure $\mu$ ,

\Psi(s)=\int_{\mathbb{R}}\left(e^{isy}-1\right)\,d\mu(y).

The generator has a diffusion part and a jump part

For a unit-jump Poisson process, the generator is

Lf(x)=\lambda[f(x+1)-f(x)].

For a Brownian motion with drift and an independent unit-jump Poisson component,

Lf(x)=mf'(x)+\frac{1}{2}\sigma^2f''(x)+\lambda[f(x+1)-f(x)].

For a compound Poisson component,

Lf(x)=\int_{\mathbb{R}}[f(x+y)-f(x)]\,d\mu(y).

This operator is the jump-process analogue of the diffusion generator.

Compensation removes predictable jump drift

m_J=\int_{\mathbb{R}}y\,d\mu(y)

exists, then the compensated process

M_t=J_t-tm_J

is a martingale. In finance, compensation is the jump analogue of subtracting drift: it separates expected arrival severity from surprise jump risk.

Jump quadratic variation is a sum of squared jumps

For a compound Poisson process,

\langle J\rangle_t=\sum_{s\le t}(\Delta J_s)^2.

Unlike Brownian quadratic variation, this is random. It depends on which jumps actually occurred and how large they were.

Worked examples

Example 1: one-day jump probability

If jump arrivals have annual rate $\lambda=2$ , then over one trading day $\Delta t=1/252$ ,

\mathbb{P}(N_{\Delta t}\ge 1)=1-e^{-2/252}\approx 0.0079.

The chance is small on any one day, but over a year the probability of at least one jump is

1-e^{-2}\approx 0.865.

This is why finite-activity jumps can be almost invisible at daily horizons yet dominate tail scenarios over longer horizons.

Example 2: generator of a fixed-loss jump model

Suppose a log-price has Brownian noise plus jumps of fixed size $-a$ arriving at rate $\lambda$ :

X_t=mt+\sigma W_t-aN_t.

Its generator is

Lf(x)=mf'(x)+\frac{1}{2}\sigma^2f''(x)+\lambda[f(x-a)-f(x)].

The final term says that, in a small time interval, the only first-order jump event moves the state from $x$ to $x-a$ .

Example 3: compensated jump P&L

Let $J_t=\sum_{i=1}^{N_t}Y_i$ and assume $\mathbb{E}[Y_i]$ is finite. Since $\mathbb{E}[J_t]=\lambda t\mathbb{E}[Y]$ , the compensated process

M_t=J_t-\lambda t\mathbb{E}[Y]

has zero conditional drift. It is the jump-risk part of the process after removing the predictable mean arrival severity.

Common confusions and pitfalls

"A jump is just a very large Brownian move." No. Brownian paths are continuous. A jump process has discontinuities and left limits $X_{t-}$ that matter at jump times.

"The Poisson rate determines jump size." The rate controls arrival frequency. The jump-size distribution controls severity. Compound Poisson models keep these separate.

"Compensating a jump process removes jumps." Compensation removes predictable drift, not discontinuities. $J_t-tm_J$ still jumps.

"The quadratic variation of a jump process is deterministic like Brownian motion." For compound Poisson processes it is the realised sum of squared jumps, hence random.

"Adding jumps preserves complete-market Black-Scholes hedging." The source text does not develop market completeness, but the modelling lesson is already visible: Brownian noise and jump arrivals are different risk sources, and a stock-plus-bond hedge cannot mechanically remove arbitrary jump-size risk.

Where this goes next

Poisson Processes: Builds the arrival-time process from exponential waiting times.
Lévy Processes: Places Brownian, Poisson, and compound Poisson processes in one independent-increment family.
Infinitesimal Generators and Kolmogorov Equations: Interprets the jump generator as the operator behind evolution equations.
Local Martingales and Semimartingales: Gives the broader process class that can carry both finite-variation and martingale parts.
Stochastic Differential Equations: Supplies the continuous diffusion component that jump-diffusions extend.

References

Lawler, G. F. (2023). Stochastic Calculus: An Introduction with Applications. Ch. 6 §6.1 (Lévy processes), §6.2 (Poisson process), §6.3 (Compound Poisson process).