From c11da650bf6deac6bd0ea2ec13e8e252ca75c2fa Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?V=C3=ADctor?= Date: Sat, 28 Oct 2023 15:47:37 +0200 Subject: [PATCH] corrected typo stochastic control --- Mathematics/5th/Stochastic_control/Stochastic_control.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/Mathematics/5th/Stochastic_control/Stochastic_control.tex b/Mathematics/5th/Stochastic_control/Stochastic_control.tex index 306d978..8933876 100644 --- a/Mathematics/5th/Stochastic_control/Stochastic_control.tex +++ b/Mathematics/5th/Stochastic_control/Stochastic_control.tex @@ -113,7 +113,7 @@ \vf{X}_{t_0}=\vf{x}_0 \end{cases} \end{equation} - where $\vf{b}:\RR_{\geq 0}\times\RR^d\times A\to\RR^d$, $\vf{\sigma}:\RR_{\geq 0}\times\RR^d\times A\to\RR^{d\times m}$ are continuous, $A$ is a compact metric space and $\alpha_t\in \text{ct}_t:=\{\rho:[0,t]\times\Omega\to A:\rho\text{ is progressively measurable}\}$ is a \emph{control parameter}. + where $\vf{b}:\RR_{\geq 0}\times\RR^d\times A\to\RR^d$, $\vf{\sigma}:\RR_{\geq 0}\times\RR^d\times A\to\RR^{d\times m}$ are continuous, $A$ is a compact metric space and $\alpha_t\in \text{ct}_t:=\{\rho:[t,T]\times\Omega\to A:\rho\text{ is progressively measurable}\}$ is a \emph{control parameter}. \end{definition} From here on we will assume that both $\vf{b}$ and $\vf{\sigma}$ are uniformly Lipschitz-continuous in the second variable. \begin{theorem} @@ -128,14 +128,14 @@ \begin{definition}[Finite horizon problem] Let $T>0$, $g:\RR^d\to\RR$ be continuous and bounded and $\ell:[0,T]\times\RR^d\times A\to\RR$ be continuous and bounded. We define the following problem: $$ - \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^T \ell(r,\vf{X}_r^{t_0,\vf{x}_0,\alpha},\alpha_r)\dd{r}+g(\vf{X}_T^{t_0,\vf{x}_0,\alpha})\right) + \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^T \ell(s,\vf{X}_s^{t_0,\vf{x}_0,\alpha},\alpha_s)\dd{s}+g(\vf{X}_T^{t_0,\vf{x}_0,\alpha})\right) $$ The first term in the expectation is called \emph{running cost} and the second one \emph{terminal cost}. \end{definition} \begin{definition}[Infinte horizon problem] - Let $g:\RR^d\to\RR$ be continuous and bounded and $\ell:[0,\infty)\times\RR^d\times A\to\RR$ be continuous and bounded. We define the following problem: + Let $\ell:[0,\infty)\times\RR^d\times A\to\RR$ be continuous and bounded and $r>0$. We define the following problem: $$ - \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^\infty \exp{-\tau r}\ell(r,\vf{X}_r^{t_0,\vf{x}_0,\alpha},\alpha_r)\dd{r}\right) + \inf_{\alpha\in \text{ct}_{t_0}}\Exp\left(\int_{t_0}^\infty \exp{-r s}\ell(s,\vf{X}_s^{t_0,\vf{x}_0,\alpha},\alpha_s)\dd{s}\right) $$ \end{definition} \end{multicols}