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We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young, the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young (Link to Young's classical paper), the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young, the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect the condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young, the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...

We consider a stochastic process $\left(X_{t}\right)_{t\geq 0}$, defined as an integral process, s.t. $$X_{t}=\int_{0}^{t}u_{s}\,dB_{s}^{H}.$$ With a fractional Brownian motion $B^H_{t}$. If $H\neq\frac{1}{2}$, the stochastic integral can not be defined in the classical Itô sense, due to Bichteler-Dellacherie theorem.

Using the classical Young theory, $X_{t}$ is well defined, if the trajectories of $u_{t}$ has finite $q$ variation, if $q<\frac{1}{1-H}$.

Question 1: Is it possible to define $X_{t}$ in such a way, that the trajectories of $u_{t}$ don't have to be restricted w.r.t. there regularity? As far as I know, it is possible to use Rough Path Theory, to extend Young's classical result. Up to which extent, is it possible to extend Young's theory by rough path theory?

Edit: More precisely, given an integral $\int_{}{}fdg$, with $f$ having finite $q$-variation and $g$ having finite $p$ variation. According Young, the following holds:

The integral $\int_{}{}fdg$ is well defined if

(Y1) there are no common discontinuities and

(Y2) if $\frac{1}{p}+\frac{1}{q}>1$.

So, how does the transition from Young to RPT affect the condition (Y2)?

Question 2: Which classical stochastic analysis tools are available using the rough path approach? More precisely are there substitutes of the following classical tools?

• Itô formula
• Burkholder inequality (Upper bounds for moments of $X^{*}_{t}=\underset{s\leq t}{\text{sup}}\,X_{s}$ )

Question 3: Is it possible to extend Young's approach using other tools?

• Regularity structures
• Malliavin Calculus (Skorohod integral)
• White Noise Analysis
• ...