Deriving the Martingale PDE from an Itô Diffusion

Summary

The document presents a method for finding a partial differential equation that makes a function of a stochastic process a martingale. It starts with a square-root diffusion whose drift pulls the process toward a constant level and whose noise scales with the square root of the state. Applying Itô’s lemma to the function of time and state produces a drift term; setting that term to zero gives the martingale condition.

The attempted solution correctly identifies the general generator-based structure, with a time derivative, a drift contribution, and a second-derivative term weighted by half the local variance. Its final expression, however, uses the process value in places where the PDE should be written for a generic state variable, and should make clear that the diffusion coefficient is squared. The document offers no boundary or regularity conditions, so the equation alone does not specify a unique solution.

Key ideas

• Itô’s lemma gives the drift of a function evaluated along a diffusion.
• A zero drift term is the condition used to obtain the martingale PDE.
• The PDE combines time change, state drift, and half the local variance times the second state derivative.
• The equation should be expressed using a generic state variable and the squared diffusion coefficient.
• Boundary conditions and regularity assumptions are not supplied.

Tags

# Find the PDE for a function that makes it a martingale


# Find the PDE for a function that makes it a martingale












> Given the SDE, find the PDE for the function $V(t,x)$ such that $V(t,S_t)$ is a martingale. $dS_t = \kappa(m - S_t)dt + \sigma\sqrt{S_t}dB_t$ where $\kappa$,$m$, and $\sigma$ are constants.

Attempted solution: The stochastic differential equation or SDE can be written in differential form such that $$dS_t = \mu(t,S_t)dt + \sigma(t,S_t)dB_t$$ thus from c.) we have $$\mu(t,S_t) = \kappa(m-S_t) \ \ \ \text{and} \ \ \ \sigma(t,S_t) = \sigma\sqrt{S_t}$$ $V(t,S_t)$ is martingale if and only if $V(t,x)$ satisfies $$\partial_t V(t,x) + \partial_x V(t,x)\mu(t,x) + \frac{1}{2}\partial_{xx}V(t,x)\sigma^2(t,x) = 0$$ therefore the PDE required to satisfy this condition is $$\partial_t V(t,x) + \partial_x V(t,x)\kappa(m - S_t) + \frac{1}{2}\partial_{xx}V(t,x)\sigma^2 S_t = 0$$

Not sure if this is correct any suggestions is greatly appreciated.

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