Risk-Neutral Measure for a Drifted Compound Poisson Asset

Summary

The document poses a change-of-measure problem for an asset driven by a compound Poisson process with drift and no Brownian component. Jump sizes are independent and identically distributed with a finite mean, while the jump intensity is constant. The price dynamics use a state-dependent Lipschitz function to scale compensated jumps, and the author asks how to choose an equivalent risk-neutral measure so the drift is removed and the jumps are compensated under the new measure.

The text supplies the model assumptions and identifies that additional restrictions on the scaling function may be needed. It does not provide a derivation, a Radon–Nikodym density, or a resulting pricing method, so the measure change and conditions for its validity remain unresolved. Readers should treat it as a statement of a modeling question rather than a worked solution; further assumptions about admissible measure changes and the jump distribution would be needed to establish a particular risk-neutral model.

Key ideas

• The asset is modeled with drift and compensated compound Poisson jumps, without a Brownian term.
• Jump sizes have a common distribution and finite mean, and arrivals have constant intensity.
• A state-dependent Lipschitz function scales the jump component.
• The document asks how to change measure to remove drift but does not give a solution.
• Additional restrictions on the scaling function may be needed for a risk-neutral measure.

Tags

# Change of Measure for Jump Process with Drift and no Brownian motion


# Change of Measure for Jump Process with Drift and no Brownian motion












If on $(\Omega, \mathcal{F},\mathbb{P})$, $r>0$ is a constant and $Z_t =\sum_{i=1}^{N_t} Y_i$ where $Y_i$ are i.i.d with $E[Y_i]=L$ denotes the size of the jump and can have distributions like normal, exponential, etc; $N_t$ is the Poisson process with intensity $\lambda$. If $X_t$ is given by $$dX_t = X_t(rdt -g(X(t))d(Z_t-\lambda Lt))$$

where $g:\mathbb{R}\rightarrow \mathbb{R}-0$ is Lipschitz continuous function.

My question is how do we get a risk-neutral measure $\mathbb{Q}$ and the Radon Nikodym derivative so that we have a form like $$dX_t = X_t(\theta(t)d\tilde{Z}_t)$$ under the new measure where $\tilde{Z}$ is the compensated Poisson process under $\mathbb{Q}$?

In order to get a risk-neutral measure, we can further impose some restrictions on $g$.

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