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Here is a concise roadmap of the key ideas:
- (1) Non‑existence (explosion) for
$$ dX_t = X_t^3dt+X_t^2dW_t.$$
One shows that any candidate solution must “blow up” in finite time with probability 1, using either the Feller test for explosions or an explicit change of variable that sends the equation to one with singular drift at zero. - (2) Non‑uniqueness for
$$ dX_t = 3X_t^{1/3}dt+3X_t^{2/3}dW_t,\quad X_0=0.$$
Here the coefficients fail the usual Hölder/Lipschitz conditions at 0, and one constructs infinitely many strong solutions by observing that any cube of a Brownian motion (plus an arbitrary constant) also solves the SDE (Itô–Watanabe construction).
1. Preliminaries: Feller’s test and non‑Lipschitz examples
1.1 Feller’s test for explosion
For a one–dimensional Itô SDE
$$ dX_t = b(X_t)dt +\sigma(X_t)dW_t,$$
the Feller test gives necessary and sufficient criteria, in terms of the scale function$\displaystyle s(x)=\int^x\exp\bigl[-2!\int^y\frac{b(z)}{\sigma^2(z)}dz\bigr]dy$ and speed measure, for whether$\tau_{\rm explode}<\infty$ with positive probability (mate.dm.uba.ar).
1.2 Non‑Lipschitz coefficients and non‑uniqueness
If $b$ or $\sigma$ fail to be (locally) Hölder‑$1/2$ in $x$, one may lose pathwise uniqueness. A classical Itô–Watanabe example:
$$ dX_t = 3X_t^{1/3}dt + 3X_t^{2/3}dW_t,\quad X_0=0,$$
admits both the trivial solution $X_t\equiv0$ and the nontrivial one $X_t = \bigl(W_t\bigr)^3$ (Department of Statistics, personal.math.ubc.ca).
2. Part (1): Explosion of $dX_t=X_t^3dt + X_t^2dW_t$
2.1 Heuristic divergence and change of variable
Set $Y_t=1/X_t$. By Itô’s formula,
$$ dY_t = -X_t^{-2}dX_t + \tfrac12\cdot2X_t^{-3}(X_t^2)^2dt = -X_t^{-2}(X_t^3dt + X_t^2dW_t) + X_tdt = -X_tdt - X_t^{-1}dW_t + X_tdt = -X_t^{-1}dW_t.$$
Thus$\displaystyle dY_t = -\frac{1}{X_t}dW_t = -Y_tdW_t$,so $Y$ formally solves a nondegenerate SDE that cannot start at $Y_0=\infty$. Hence no continuous solution $X_t$ can exist from a finite initial condition without blowing up in finite time.
2.2 Feller test verificationHere$\displaystyle b(x)=x^3,\ \sigma(x)=x^2$. The integrals in the scale function near infinity diverge in such a way that the explosion time $\tau<\infty$ a.s. (mate.dm.uba.ar, mate.dm.uba.ar).
3. Part (2): Infinitely many solutions for $dX_t=3X_t^{1/3}dt + 3X_t^{2/3}dW_t$, $X_0=0$
3.1 Two explicit solutions3.2 Family of solutions
More generally, for any constant $c\in\Bbb R$, one checks that
$$ X_t = \bigl(W_t + c\bigr)^3 \quad\Longrightarrow\quad dX_t = 3(W_t + c)^{1/3}d\bigl(W_t+c\bigr) + \tfrac12\cdot3\cdot2(W_t+c)^{-2/3}dt =3X_t^{1/3}dW_t + 3X_t^{2/3}dt,$$
so each choice of $c$ gives a distinct strong solution started from $0$. Hence infinitely many solutions. (Department of Statistics, Mathematics Stack Exchange).
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hbghlyj
Posted 2025-5-19 05:05
