3.2 Itô Existence and Uniqueness

Definition 3.2.1 (Lipschitz Coefficient).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$, then $\sigma$ is Lipschitz if there exists $C \ge 0$ such that for any $\theta, \eta \in C([0, \infty); \real^{d})$,

\[\norm{\sigma(t, \theta) - \sigma(t, \eta)}_{\real^n}\le C\norm{\theta - \eta}_{u, [0, t]}\]

Lemma 3.2.2.label Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space, $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion, $\sigma$ be a $\bracs{\mathcal{F}_t}$-previsible $L(\real^{d}; \real^{n})$-valued process, $b$ be a $\bracs{\mathcal{F}_t}$-previsible $\real^{n}$-valued process, $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, and

\[X_{t} = \xi + \int_{0}^{t} \sigma_{s} dB_{s} + \int_{0}^{t} b_{s} ds\]

then for any $T > 0$ and $p \ge 2$, there exists $C_{T, p, n}\ge 0$ such that for all $0 \le t \le T$,

\[\ev\braks{\norm{X}_{u, [0, t]}^p}\le C_{T, p, n}\braks{\ev(\norm{\xi}_{\real^n}^p) + \ev\paren{\int_0^t \norm{\sigma_s}_{\real^n}^p + \norm{b_s}_{\real^n}^p ds }}\]

Proof, [Theorem 11.5, RW89]. Assume without loss of generality that $\xi = 0$, then

\[\norm{X}_{u, [0, t]}^{p} \le C_{p}\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^{p} + \paren{\int_0^t \norm{b_s}_{\real^n}ds}^{p}\]

where by Jensen’s inequality,

\[\paren{\int_0^t \norm{b_s}ds}^{p} \le C_{t, p}\int_{0}^{t} \norm{b_s}_{\real^n}^{p} ds\]

and by the BDG inequality and Jensen’s inequality,

\[\ev\braks{\paren{\sup_{0 \le s \le t}\int_0^s \sigma_rdB_r}^p}\le C_{p} \ev\braks{\paren{\int_0^t \norm{\sigma_s}_{\real^d}^2 ds}^{p/2}}\le C_{p} \int_{0}^{t} \norm{\sigma_s}_{\real^d}^{p} ds\]

$\square$

Lemma 3.2.3.label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the Lipschitz condition with constant $K$.

Let $(\Omega, \bracs{\cf_t|t \ge 0}, \bp)$ be a filtered probability space and $B$ be a $\bracs{\mathcal{F}_t}$-adapted standard Brownian motion. For any $\bracs{\mathcal{F}_t}$-adapted process $X: \Omega \to C([0, \infty); \real^{d})$ with continuous sample paths and $\xi \in L^{p}(\Omega, \cf_{0}; \real^{n})$, let

\[P(X, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]

then for any $\bracs{\mathcal{F}_t}$-adapted process $Y: \Omega \to C([0, \infty); \real^{d})$, $\eta \in L^{p}(\Omega, \cf_{0}; \real^{n})$, $T > 0$, and $0 \le t \le T$,

\begin{align*}&\ev\braks{\norm{P(X, \xi) - P(Y, \eta)}_{u, [0, t]}^p}\\&\le C_{K, n, T, p}\braks{\norm{\xi - \eta}_{L^p(\Omega; \real^n)}^p + \ev\paren{\int_0^t \norm{X - Y}_{u, [0, s]}^p ds}}\end{align*}

Theorem 3.2.4 (Itô).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the Lipschitz condition. If for each $T \ge 0$,

\[\sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)}+ \norm{b(s, 0)}_{\real^n}< \infty\]

then the SDE

\[X_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]

is exact and pathwise unique.

Proof of existence, [Theorem 11.2, RW89]. For each $t \ge 0$, let $\mathcal{G}_{t}^{\circ} = \sigma(\bracs{B_s|1 \le s \le t}\cup \bracs{\xi})$, $\mathcal{G}^{\circ} = \sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, $\mathcal{N}$ be the collection of $\bp$-null sets in the completion of $\sigma(\bracsn{\mathcal{G}_t^\circ|t \ge 0})$, and $\mathcal{G}_{t} = \sigma(\mathcal{G}_{t}^{\circ} \cup \mathcal{N})$.

Let $\xi \in L^{\infty}(\Omega, \cf_{0}; \real^{n})$ and $X^{(0)}= \xi$. For each $m \in \natz$, define inductively

\[X^{(m+1)}= P(X^{(m)}, \xi)_{t} = \xi + \int_{0}^{t} \sigma(s, X^{(m)})dB_{s} + \int_{0}^{t} b(s, X^{(m)})ds\]

then $X^{(m)}$ has continuous sample paths, and for each $T > 0$, $\ev[{\normn{X^{(m)}}_{u, [0, s]}^2}] < \infty$. By Lemma 3.2.3,

\[\ev\braks{\norm{X^{(m+2)} - X^{(m+1)}}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, t]}^2}dt\]

Therefore

\[\ev\braks{\norm{X^{(m+1)} - X^{(m)}}_{u, [0, T]}^2}\le C_{0} C \frac{T^{n}}{n!}\]

and $X^{(m)}$ converges uniformly in $L^{2}$ to a limiting process $X$, which satisfies the given equation.$\square$

Proof of uniqueness. Let $X$ and $Y$ be solutions of the SDE with the same setup, then by Lemma 3.2.3,

\[\ev\braks{\norm{X - Y}_{u, [0, T]}^2}\le C \int_{0}^{t} \ev\braks{\norm{X - Y}_{u, [0, t]}^2}dt\]

which implies that $X|_{[0, T]}= Y|_{[0, T]}$ almost surely.$\square$

Lemma 3.2.5 (Gronwall).label Let $f \in C([0, T]; \real)$, and $c, K > 0$ such that

\[f(t) \le c + K\int_{0}^{t} f(s)ds\]

for all $t \in [0, T]$, then

\[\frac{d}{dt}\braks{e^{-Kt}\int_0^t f(s)ds}\le ce^{-Kt}\]

and $f(t) \le ce^{Kt}$.

Theorem 3.2.6 (Blagoveshchenskii-Blagoveshchensk).label Let $\sigma: [0, \infty) \times C([0, \infty); \real^{d}) \to L(\real^{d}; \real^{n})$ and $b: [0, \infty) \times C([0, \infty); \real^{d}) \to \real^{n}$ be previsible path functionals satisfying the Lipschitz condition. If for each $T \ge 0$,

\[\sup_{0 \le s \le T}\norm{\sigma(s, 0)}_{L(\real^d; \real^n)}+ \norm{b(s, 0)}_{\real^n}< \infty\]

then the SDE

\[X_{t} = \xi + \int_{0}^{t} \sigma(s, X)dB_{s} + \int_{0}^{t} b(s, X)ds\]

admits a strong solution

\[F: \real^{n} \times C([0, \infty); \real^{d}) \to C([0, \infty); \real^{n})\]

such that:

(1)
For each $\theta \in C([0, \infty); \real^{d})$, $F(\cdot, \theta): \real^{d} \to C([0, \infty); \real^{n})$ is continuous.
(2)
For each solution $X^{y}$ initial condition $y \in \real^{n}$ and $s \ge 0$,
\[X_{s + t}^{y} = F(X_{s}^{y}, \tau_{-s}B)_{t} \quad \forall t \ge 0\]

almost surely, where $(\tau_{-s}B)_{r} = B_{r+s}$.
(3)
For each $y \in \real^{n}$, $\bracs{X_t^y|t \ge 0}$ is a Markov process.

Unnamed Website

3.2 Itô Existence and Uniqueness

Direct References

Unnamed Website

Direct References