Finite-horizon moments for Lipschitz Itô SDEs

If $T=0$, then $\mathrm{\mid }{X}_0{\mathrm{\mid }}^p\le 2^{p\mathrm{/}2}(1+\mathrm{\mid }{X}_0{\mathrm{\mid }}^p)$ and the result follows. For $T>0$, put $V(x)=(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2}$. It is $C^2$ and ${\mathrm{\partial }}_{i}V=p{x}_i(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-1},{\mathrm{\partial }}_{ij}V=p{\delta }_{ij}(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-1}+p(p-2)x_i{x}_j(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-2}\mathrm{.}$ The gradient bound follows from $\mathrm{\mid }x\mathrm{\mid }\le (1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{1\mathrm{/}2}$. The Hessian is a nonnegative multiple of $I$ plus a nonnegative multiple of $x{x}^{\mathsf{T}}$, so it is positive semidefinite. Its eigenvalues orthogonal to $x$ are $p(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-1}$, and the eigenvalue in the $x$ direction is $p(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-2}(1+(p-1)\mathrm{\mid }x{\mathrm{\mid }}^2)$; each is at most $p(p-1)(1+\mathrm{\mid }x{\mathrm{\mid }}^2{)}^{p\mathrm{/}2-1}$ because $p\ge 2$. At $x=0$ all eigenvalues equal $p$, which obeys the same bound.

By [F1], $E{\sup }_{t\le T}\mathrm{\mid }{X}_t{\mathrm{\mid }}^2<\mathrm{\infty }$, so [F2] makes $b(\cdot ,X_{\cdot })$ progressive and $\sigma (\cdot ,X_{\cdot })$ predictable. The growth bound gives $E{\int }_0^T(\mathrm{\mid }b(s,X_s){\mathrm{\mid }}^2+\mathrm{\parallel }\sigma (s,X_s){\mathrm{\parallel }}_F^2)ds\le 2K^{2}T(1+E{\sup }_{s\le T}\mathrm{\mid }{X}_s{\mathrm{\mid }}^2)<\mathrm{\infty }\mathrm{.}$ Thus the drift-square integral and diffusion energy are finite almost surely; pathwise Cauchy–Schwarz gives ${\int }_0^T\mathrm{\mid }b(s,X_s)\mathrm{\mid }ds\le \sqrt{T}({\int }_0^T\mathrm{\mid }b(s,X_s){\mathrm{\mid }}^{2}ds{)}^{1\mathrm{/}2}<\mathrm{\infty }$. Hence $X$ satisfies the continuous Brownian Itô-process hypotheses in [F2].

For $R>0$ define ${\tau }_R:=\inf \{s\in [0,T]:\mathrm{\mid }{X}_s\mathrm{\mid }\ge R\}\wedge T$, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. For $t<T$, $\{{\tau }_R\le t\}=\{{\sup }_{s\le t}\mathrm{\mid }{X}_s\mathrm{\mid }\ge R\}$; by continuity and [F4] that supremum is measurable using the countable rational times in $[0,t]$ and $t$, all of which are ${\mathcal{F}}_t$-measurable. For $t\ge T$ the event is all of $\mathrm{\Omega }$, so ${\tau }_R$ is a stopping time. On $[0,{\tau }_R]$ the process is bounded by $R$ unless $\mathrm{\mid }{X}_0\mathrm{\mid }\ge R$, in which case ${\tau }_R=0$. Hence $1_{[0,{\tau }_R]}\mathrm{\nabla }V(X_s{)}^{\mathsf{T}}\sigma (s,X_s)$ is predictable and has finite energy on $[0,T]$.

Let ${\mathcal{L}}_{s}V(x)=b(s,x)\cdot \mathrm{\nabla }V(x)+\frac{1}{2}\mathrm{tr}(\sigma {\sigma }^{\mathsf{T}}(s,x)D^{2}V(x))$. The inequality $1+\mathrm{\mid }x\mathrm{\mid }\le \sqrt{2(1+\mathrm{\mid }x{\mathrm{\mid }}^2)}$ follows by squaring and using $(\mathrm{\mid }x\mathrm{\mid }-1{)}^2\ge 0$, so the drift term has absolute value at most $\sqrt{2}pKV(x)$. For each diffusion column ${\sigma }_{\cdot k}$, positive semidefiniteness and the operator-norm bound give $0\le {\sigma }_{\cdot k}^{\mathsf{T}}{D}^{2}V{\sigma }_{\cdot k}\le \mathrm{\parallel }{D}^{2}V{\mathrm{\parallel }}_{op}\mathrm{\mid }{\sigma }_{\cdot k}{\mathrm{\mid }}^2$. Summing over the finite $m$ columns yields $0\le \frac{1}{2}\mathrm{tr}(\sigma {\sigma }^{\mathsf{T}}{D}^{2}V)\le p(p-1)K^{2}V(x)$. Thus ${\mathcal{L}}_{s}V(x)\le C_{p}V(x)$.

Apply the multidimensional Itô formula [F2] to $V(X)$ and evaluate the common continuous identity at $t\wedge {\tau }_R$. By the stopping identity and mean-zero finite-energy martingale facts in [F3], the stopped stochastic integral has expectation zero. With $g_R(t):=EV(X_{t\wedge {\tau }_R})$, nonnegativity of $V$ and step 2.1 therefore give $g_R(t)=EV(X_0)+E{\int }_0^t{1}_{[0,{\tau }_R]}(s){\mathcal{L}}_{s}V(X_s)ds\le EV(X_0)+C_p{\int }_0^t{g}_R(s)ds\mathrm{.}$ The right side is finite: before stopping $X$ is bounded, and on paths stopped at zero the value is $X_0$ with finite $p$th moment.

Set $H_R(t)=EV(X_0)+C_p{\int }_0^t{g}_R(s)ds$. By step 3.1, $g_R\le H_R$; also $H_R$ is absolutely continuous and $H_R^{\mathrm{\prime }}(t)=C_p{g}_R(t)\le C_p{H}_R(t)$ almost everywhere. Thus $(e^{-C_{p}t}{H}_R(t){)}^{\mathrm{\prime }}\le 0$ almost everywhere, giving $g_R(t)\le EV(X_0)e^{C_{p}t}$.

Take $R=n\in \mathbb{N}$ and let $n\to \mathrm{\infty }$. Every continuous path on $[0,T]$ is bounded, so for each fixed $t$ one has $X_{t\wedge {\tau }_n}\to X_t$ almost surely. Fatou's lemma [F5] and step 4.1 give $EV(X_t)\le EV(X_0)e^{C_{p}T}$. Also $V(X_0)=(1+\mathrm{\mid }{X}_0{\mathrm{\mid }}^2{)}^{p\mathrm{/}2}\le 2^{p\mathrm{/}2}(1+\mathrm{\mid }{X}_0{\mathrm{\mid }}^p)$ and $\mathrm{\mid }{X}_t{\mathrm{\mid }}^p\le V(X_t)$. Taking the supremum over $0\le t\le T$ proves the stated moment bound. AC is only the inherited interface assumption [F6].