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Solving a Nonlinear Stochastic Differential Equation by Guessing a Transform

Article Quant Q&A · Author: MitchCL
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Summary

The document considers a stochastic differential equation with a cubic drift term, a quadratic Brownian diffusion term, and an initial value of one. The proposed approach is to identify a function of time and Brownian motion whose Itô differential matches those coefficients. The answer suggests the reciprocal transform of one minus Brownian motion and recommends starting from the diffusion term when searching for a tractable form.

The response offers a candidate process rather than a detailed verification with Itô’s formula. It also notes that the candidate has a singularity when Brownian motion reaches one, so it can only describe a solution up to that hitting time; the assertion that Brownian motion is nonzero does not address this issue. The exchange gives a useful illustration of guess-and-check reasoning, but it does not discuss uniqueness, continuation, or behavior at the singularity.

Key ideas

  • The proposed equation has nonlinear drift and diffusion terms that invite a change of variables.
  • The answer suggests matching the Brownian-motion term first when guessing a solution.
  • A reciprocal function of Brownian motion is offered as a candidate process.
  • The candidate becomes singular when Brownian motion reaches one, limiting its validity in time.
  • The response does not provide a full Itô verification or discuss uniqueness.

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# Trying to solve an SDE with a power function


# Trying to solve an SDE with a power function












I want to solve an SDE: $$ dX_t=X^3_t\,dt+X^2_t\,dB_t, $$ with $X_0$=1.

In my understanding (I'm currently using J. Michael Steele's textbook "Stochastic calculus and financial applications"), SDEs like this are solved with Ito's formula and I need to find such a function $f$ that $X_t=f(t, B_t)$ and then just apply the formula to it. However, I can't quite find a good function for this SDE, and looking through Steele's examples, I fear that power functions have some caveats that I'm not aware of.

How can I solve this SDE?

## Answer by Rylan (score 1)

https://quant.stackexchange.com/a/81888

It looks like for a standard brownian motion $B_t \neq0$, then the process $X_t = \frac{1}{1-B_t}$ solves the SDE you've proposed.

If I remember my (S)DEs correctly, solving one basically starts with guess-and-check, with some "nice" forms admitting solutions that are easier to find through a procedure. In the case of this SDE, I focused on the $dB_t$ term, which I think is often a good starting point.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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