On \(L^p\)-estimates of stochastic integrals (Q2769702)

Let \(\Sigma:= (\Omega,{\mathcal F},({\mathcal F}_t)_{t\in R_+},P)\) be a probability system equipped with a one-parameter filtration \(\{{\mathcal F}_t\), \(t \geq 0\}\) and let \((Z_t,{\mathcal F}_t)\) be a one-dimensional symmetric stable process of order \(\alpha\in (0,2]\). The author investigates the following optimal control problem. Control strategies are modeled as the class \({\mathcal B}\) of predictable processes for \(\Sigma\) whose absolute values are \(\leq 1\), and the controlled process \((\rho^b_t,X^b_t)_{t\in R_+}\) with \(b\in{\mathcal B}\) is defined by the stochastic integrals NEWLINE\[NEWLINE\rho_t^b= \int^t_0 \bigl(1-|b_u |^\alpha \bigr)du, \quad X_t^b=\int^t_0 b_udZ_u.NEWLINE\]NEWLINE The payoff function \(\nu= \nu(s,x)\) on \(\mathbb{R}^2\) defined by \(\nu(s,x)= \sup_{b\in{\mathcal B}}\nu^b(s,x)\), where \(\nu^b(s,x)= E\int^\infty_0 e^{-\lambda t}\varphi^b_t f(s+\rho^b_t, x+ X^b_t)dt\), \(\varphi^b_t= (1-|b_t|^\alpha)^{1/p} \cdot|b_t |^{1 /q}\), \(\lambda>0\), \(1/p+1/q=1\), \(p>1+1/ \alpha\) and \(f:\mathbb{R}^2\to R_+\) is smooth and null for \(t\notin [0,T[\) or \(x\notin] -R,R[\). The following main result is shown by adaptations to the actual case of methods given by Krylov (1987) to prove \(L^p\)-estimates of stochastic integrals w.r.t. diffusion processes: For \(\lambda >0\), \((s,x)\in \mathbb{R}^2\) and \(f:\mathbb{R}^2\to R_+\) Borel measurable, NEWLINE\[NEWLINE\begin{multlined} E\int^\infty_0 e^{-\left(\lambda t\int^t_0|b_u |^\alpha du\right)} |b_t|^{g/2} f(s+t,x+X^b_t) dt\\ \leq N\lambda^{-1/2+1/2 \alpha}\left\{ \int^\infty_s e^{-2\lambda (t-s)}\left[ \int^\infty_{-\infty}f^2(t,x)dx \right]dt \right\}^{1/2}. \end{multlined}NEWLINE\]NEWLINE{}.NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].