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Given a one-dimensional SDE $$ \begin{cases} dX_t &= b(t,X_t)dt+\sigma(t,X_t)dB_t, \\ X_0 &= Z \end{cases} $$ for $t\in[0,T]$ where $Z$ is square integrable: $\mathsf E[Z^2]<\infty$ the sufficient conditions for existence and uniqueness are: the growth condition $$ |b(t,x)|+|\sigma(t,x)|\leq C(1+|x|)\quad\forall x\in \mathbb R, t\in [0,T] $$ and the Lipschitz condition $$ |b(t,x'')-b(t,x')|+|\sigma(t,x'')-\sigma(t,x')|\leq D|x''-x'|\quad\forall x',x''\in \mathbb R. $$ These conditions are stated e.g. in Theorem 5.2.1, "Stochastic Differential Equations" (p. 68 here).

However, in the definition of an Ito diffusion as a process satisfying $$ dX_t = b(X_t)dt+\sigma(X_t)dB_t $$ for $t\geq s$, where $X_s = x$, in the same book (p. 114 on the linked webpage) it is written that conditions simplify to Lipschitz condition. Could you help me to understand the reasoning of that?

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@sos440: thank you - would you put this as an answer? – Ilya Feb 27 '12 at 8:48
up vote 4 down vote accepted

Let's consider a simplest case: Let $f : \mathbb{R} \to \mathbb{R}$ satisfy global Lipschitz condition

$$ |f(x) - f(y)| \leq L |x - y|, \quad x, y \in \mathbb{R}$$

with Lipschitz constant $L$. Then by triangle inequality, we have

$$ |f(x)| \leq |f(x) - f(0)| + |f(0)| \leq L|x| + |f(0)| \leq C(|x| + 1)$$

for large $C \geq \max(L, |f(0)|)$. Same argument applies to this case.

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