Joint Distributions of Merton Jump Diffusion Processes

Summary

The document asks whether knowing the distribution of a Merton jump diffusion at each individual time is enough to determine the joint distribution of its values across multiple times. It defines log price as a drift and Brownian component plus a compound Poisson sum, with independent jump arrivals and normally distributed jump sizes.

Its focus is the distinction between single-time distributions and the full process law: the latter includes dependence across times. The document poses the question but provides no derivation, answer, numerical example, or empirical evidence. In this model, a joint law can be constructed from the independent increments of the Brownian and Poisson processes, while respecting that earlier jumps remain part of later log prices. The post itself does not explain that construction, specify parameter values, or discuss practical calibration and estimation. It is therefore useful as a prompt about stochastic-process distributions, but offers little guidance for pricing or trading without a supplementary explanation.

Key ideas

• The Merton model combines drift, Brownian variation, and a compound Poisson jump component in log price.
• A distribution at one time does not by itself describe dependence among prices at multiple times.
• The post asks whether the process's finite-dimensional joint distributions can be calculated.
• It provides no solution or evidence for the question it raises.

Tags

# The distribution of the jump diffusion process


# The distribution of the jump diffusion process












In the Merton jump diffusion model the process of the share price can be expressed as $$S_{t}=S_{0}\cdot\exp\left\{ X_{t}\right\} ,$$ where $$X_{t}=\mu t+\sigma W_{t}+\sum_{i=1}^{N_{t}}Y_{i}.$$ Here $\mu$ and $\sigma$ are constants, $W_{t}$ is a Wiener process, $N_{t}$ is a Poisson process, and $Y_{i}$s are $N\left(\mu_{A},\sigma_{A}\right)$ $iid$ variables. $W_{t}$, $N_{t}$ and $Y_{i}$ are independent.

I know that we can express the distribution of $X_{t}$ and hence the distribution of $S_{t}$ as well, so we know the distribution of the process for a given $t$, but can we calculate the distribution of the (full) $X$ process? I mean do we know the joint distribution of $$X_{t_{1}},X_{t_{2}},...,X_{t_{n}}$$ for every $n$ and for every $t_{1},t_{2},...,t_{n}$ instants?

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