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Approximating left continuous process $(L_t)_{0 leq t leq T}$ uniformly on $[0,T]$ by step functions on the...
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The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 leq t leq T}$ with $L_0 = 0$ on the probability space $(Omega, mathcal{F},P)$, is it possible to approximate it uniformly on $[0,T]$ on the dyadics by step functions, i.e. if $D_n=2^{-n} T mathbb{N} cap [0,T]$ is the set of all $n$-th dyadic points, do I have the following convergence
$$
sup_{t in [0,T]}left|L_t(omega) - sum_{t_i in D_n}L_{t_i}(omega) 1_{((t_i,t_i+1]]}(t, omega)right| rightarrow 0, text{ for } n rightarrow infty
$$
for almost every $omega in Omega$ ? The function $1_{((t_i,t_{i+1}]]}$ denotes the left half open stochastic interval.
Thanks a lot in advance!
probability probability-theory stochastic-processes stochastic-calculus stochastic-analysis
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
$endgroup$
add a comment |
$begingroup$
The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 leq t leq T}$ with $L_0 = 0$ on the probability space $(Omega, mathcal{F},P)$, is it possible to approximate it uniformly on $[0,T]$ on the dyadics by step functions, i.e. if $D_n=2^{-n} T mathbb{N} cap [0,T]$ is the set of all $n$-th dyadic points, do I have the following convergence
$$
sup_{t in [0,T]}left|L_t(omega) - sum_{t_i in D_n}L_{t_i}(omega) 1_{((t_i,t_i+1]]}(t, omega)right| rightarrow 0, text{ for } n rightarrow infty
$$
for almost every $omega in Omega$ ? The function $1_{((t_i,t_{i+1}]]}$ denotes the left half open stochastic interval.
Thanks a lot in advance!
probability probability-theory stochastic-processes stochastic-calculus stochastic-analysis
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
$endgroup$
add a comment |
$begingroup$
The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 leq t leq T}$ with $L_0 = 0$ on the probability space $(Omega, mathcal{F},P)$, is it possible to approximate it uniformly on $[0,T]$ on the dyadics by step functions, i.e. if $D_n=2^{-n} T mathbb{N} cap [0,T]$ is the set of all $n$-th dyadic points, do I have the following convergence
$$
sup_{t in [0,T]}left|L_t(omega) - sum_{t_i in D_n}L_{t_i}(omega) 1_{((t_i,t_i+1]]}(t, omega)right| rightarrow 0, text{ for } n rightarrow infty
$$
for almost every $omega in Omega$ ? The function $1_{((t_i,t_{i+1}]]}$ denotes the left half open stochastic interval.
Thanks a lot in advance!
probability probability-theory stochastic-processes stochastic-calculus stochastic-analysis
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
$endgroup$
The question is closely related to optional quadratic variation. If I have a left-continuous adapted stochastic process $(L_t)_{0 leq t leq T}$ with $L_0 = 0$ on the probability space $(Omega, mathcal{F},P)$, is it possible to approximate it uniformly on $[0,T]$ on the dyadics by step functions, i.e. if $D_n=2^{-n} T mathbb{N} cap [0,T]$ is the set of all $n$-th dyadic points, do I have the following convergence
$$
sup_{t in [0,T]}left|L_t(omega) - sum_{t_i in D_n}L_{t_i}(omega) 1_{((t_i,t_i+1]]}(t, omega)right| rightarrow 0, text{ for } n rightarrow infty
$$
for almost every $omega in Omega$ ? The function $1_{((t_i,t_{i+1}]]}$ denotes the left half open stochastic interval.
Thanks a lot in advance!
probability probability-theory stochastic-processes stochastic-calculus stochastic-analysis
probability probability-theory stochastic-processes stochastic-calculus stochastic-analysis
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
edited Jan 2 at 14:43
Davide Giraudo
125k16150261
125k16150261
asked Jan 2 at 12:01
vaoyvaoy
537210
asked Jan 2 at 12:01
vaoyvaoy
537210
asked Jan 2 at 12:01
vaoyvaoy
537210
537210
add a comment |
add a comment |
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