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  • Let $a,b : [0,T] \times \mathbb{R} \to \mathbb{R}$ be Borel functions with linear growth.
  • Let $X$ be the solution of

$$ X_t= \int_{0}^t a(s,X_s) dW_s + \int_{0}^t b(s,X_s) ds , t \in[0,T]$$

  • $\tau_n := \inf \{ t \geq 0, |X_t| \geq n\}$
  1. Show that $E( \sup_{t \leq T} |{X_t}^{\tau_n} |^2 ) < \infty$
  2. With Gronwall lemma, show the existence of $C$ a constant independant of $n$ such that $E( \sup_{t \leq T} |{X_t}^{\tau_n} |^2 ) < C$
  3. $E( \sup_{t \leq T} |{X_t}|^2 ) < C$
  1. We may use somewhat the Burkholder-Davis-Gundy inequality

\begin{align*} Z_t & ={X_t}^{\tau_n} \\ \mathbb{E} \sup_{t \leq T} |Z_t|^2 &\leq C_1 \mathbb{E}\langle M \rangle_T^{2/2} \\ &= C_1 \int_{0}^T a(s, M_s)^2 ds \\ &\leq C_1 \sup_{t \leq T} (1 + |Z_T|)\\ &\leq C_1 (1+n) \\ &< \infty \\ \end{align*}

  1. We use the following inequality

\begin{align*} Z_t & ={X_t}^{\tau_n} \\ Z_t^2 &\leq 2 x_0^2 + 2| \int_{0}^t a(s,Z_s) dW_s |^2 + 2 | \int_{0}^t b(s,Z_s) ds | ^2 \\ \end{align*}

We evaluate each term separately. \begin{align*} E ( \sup_{t <T} | \int_{0}^t a(s,Z_s) dW_s |^2 ) &\leq C E( | \int_{0}^t |a(s,Z_s)|^2 ds | ) \\ &\leq C E ( \int_{0}^T (1+ |Z_s| )^2 ds) \\ &\leq C' E ( \int_{0}^T ( 1+ \sup_{t <T} |Z_s|^2 ) ds ) \end{align*}

We use the Jensen inequality with expectation with respect to an uniform distribution.

\begin{align*} \left| \int_0^t b(s,X_s) \, ds \right|^2 &= t^2 \left| \int_0^t b(s,X_s) \, \frac{ds}{t} \right|^2\\ &\leq t^{2-1} \int_0^t |b(s,X_s)|^2 \, ds \\ &\leq K^2 T^{2-1} \int_0^t (1+|X_s|^2) \,ds \\ &\leq K^2 T^{2-1} \int_0^t \left( 1+ \sup_{r \leq s} |X_r|^2 \right) \, ds\end{align*}

  • $u(t) := \mathbb{E} \left( \sup_{r \leq t} |Z_r|^2 \right)$
  • $ u(t) \leq c_1 + c_2 \int_0^t u(s)ds$
  • $u(t) \leq c_1 e^{c_2 t} \leq c_1 e^{ c_2 T}$ by Gronwall inequality
  1. Fatou Lemma

$E( \sup_{t \leq T} |{X_t}|^2 ) = E ( \liminf \sup_{t \leq T} |{X_t}^{\tau_n}|^2 ) \leq \liminf E ( \sup_{t \leq T} |{X_t}^{\tau_n}|^2 ) < \infty $

zestiria
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