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On our platform, we delve into the analysis of stochastic differential equations (SDEs) with a focus on understanding the regularity of solutions. Allow me to introduce a scenario involving a $q$-dimensional Brownian motion $\mathrm{W}$ on a filtered probability space $(\Omega, \mathcal{F}, \mathbb{F}=(\mathcal{F}_t)_{t \in [0, T]}, \mathbb{P})$, where $x_0$ is a bounded random variable in $\mathbb{R}^d$.

We consider three processes: $(V_t)_{t \in [0, T]}$, $b(t,.,v): [0, T] \times \Omega \rightarrow \mathbb{R}^d$, and $\sigma(t,.,v): [0, T] \times \Omega \rightarrow \mathbb{R}^{d \times q}$. Notably, $b$, $\sigma$, and $V$ are all members of the $\mathbb{L}^{2}$ space. Furthermore, for any $N \in \mathbb{N}^{*}$, if $V \leq N$, then both $b$ and $\sigma$ are bounded.

Our focus lies in the study of solutions to the stochastic differential equation:

$$dX_{t} = b(t,V_{t})X_{t} dt + \sigma(t,V_{t})X_{t} dW_{t}, \quad X_{0} = x_{0}.$$

While the uniqueness of the solution is evident, I am currently seeking references or insights into determining whether $X$ belongs to $\mathbb{L}^{p}$ for some $p \geq 2$.

Your guidance or suggested references on this matter would be immensely appreciated.

Oussama
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1 Answers1

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We follow Application of the Burkholder Davis Gundy inequality and René Schilling, Lothar Partzsch: Brownian Motion - An Introduction to Stochastic Processes, Chapter 19 (2nd edition).

Using the estimate

$$(a+b+c)^p \leq 3^p (a^p+b^p+c^p), \qquad a,b,c \geq 0,$$

we again have

$$|X_t|^p \leq 3^p |X_0|^p + 3^p \left| \int_0^t b(s,V_s)X_{s} \, ds \right|^p + 3^p \left| \int_0^t \sigma(s,V_s)X_{s} \, dB_s \right|^p. \tag{1}$$

By using Jensen's inequality and boundedness we have

$$\begin{align*} \left| \int_0^t b(s,V_s)X_{s} \, ds \right|^p \leq c \int_0^t \left( 1+ \sup_{r \leq s} |X_r|^p \right) \, ds\end{align*},$$

and by the Burkholder-Davis-Gundy and Jensen's inequality we obtain

$$\begin{align*} \mathbb{E} \left( \sup_{r \leq t} \left| \int_0^r \sigma(s,V_s)X_{s} \, dB_s \right|^p \right) &\leq C \mathbb{E} \left( \left| \int_0^t |\sigma(s,V_s)X_{s}|^2 \, ds \right)^{p/2} \right)\\ &\leq C\int_0^t \left(1+ \mathbb{E}\left[ \sup_{r \leq s} |X_r|^p \right] \right) \, ds. \tag{3} \end{align*}$$

So for $u(t) := \mathbb{E} \left( \sup_{r \leq t} |X_r|^p \right)$

we have

$$u(t) \leq c_{1} \mathbb{E}|X_0|^p + c_{2}+ c_{3}\int_0^t u(s) \, ds.$$

and so the claim follows from Gronwall's lemma.

Thomas Kojar
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  • Thank you for your answer, but we don't have the boundness of $b$ and $\sigma$, and this is the heart of the problem, so it's so easy to show that inside the ball(i.e. when the control $V \leq N$), but we can't show that outside of it. – Oussama Dec 22 '23 at 19:31
  • @Oussama in that case, it is unclear what to do without more information on $b,\sigma$. – Thomas Kojar Dec 22 '23 at 19:35
  • So we can't guarantee that $X$ is in $\mathbb{L}^{p}$ for some $p \geq 2$! because this is all what we know about $b$ and $\sigma$ – Oussama Dec 22 '23 at 19:39
  • @Oussama I understand that, but we need more information on the coefficients b(t,x),sigma(t,x) as functions of (t,x) eg. Lipschitz growth. And also decay of the tail $P[V>N]$. – Thomas Kojar Dec 22 '23 at 19:45