When a Unit-Volatility SDE Is Brownian Motion in Law

Summary

The document considers a stochastic differential equation whose two diffusion coefficients depend on scaled Brownian inputs and the current state. The coefficients are constructed so their squared values sum to one. The question asks whether the process has an explicit solution in terms of the driving Brownian motions, or whether the equation can be transformed into a linear one.

The answer gives a distributional characterization rather than an explicit pathwise formula. It argues that the process is a continuous local martingale and, using the coefficient identity and Itô isometry, has second moment equal to elapsed time. It then invokes Lévy’s characterization to conclude that the process is Brownian motion in law. This identifies the marginal process behavior but does not express its paths as a function of the two inputs. The conclusion depends on the stated independence and regularity assumptions needed for the martingale and characterization arguments; the document does not develop those conditions or solve the equation explicitly.

Key ideas

• A continuous local martingale with the appropriate quadratic variation can be characterized as Brownian motion.
• The diffusion coefficients are designed so their squared values sum to one.
• The answer establishes a law-level result rather than an explicit pathwise solution.
• The argument relies on the driving Brownian motions being independent.

Tags

# Explicit solution SDE


# Explicit solution SDE












I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$

where $W_{t}^{1}$ and $W_{t}^{1}$ are two independent brownian motions and $$A(x,y)=a\frac{\Phi(y)\Phi(-y)e^{0.5(y^2-x^2)}+\Phi(x)\Phi(-x)e^{0.5(x^2-y^2)}}{1+a(1-2\Phi(x))(1-2\Phi(y))}, \ \ \ \ \ \ \ \ B=\sqrt{1-A^2}$$

It seems unlikely, but I was wondering if there may be an explicit solution to this stochastic differential equation in terms of $W_{t}^{1}$ and $W_{t}^{2}$? Perhaps it can be reduced to a linear SDE with a suitable function?

Any help would be greatly appreciated.

## Answer by Sergio Almada (score 1)

https://quant.stackexchange.com/a/14584

When $W_1$ and $W_2$ are independent, $Y$ is equal in law to a Brownian Motion:

- It is obviously a local martingale under its natural filtration, and

- From Ito isometry, $$\mathbf{E}Y_t^2 = \mathbf{E} \int_0^t A(...)^2 + B(...)^2 ds = t.$$ Then, from Levy's theorem (http://almostsure.wordpress.com/2010/04/13/levys-characterization-of-brownian-motion/), these two conditions imply that $Y$ is a Brownian Motion.

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