Stopping rule for Itô integral
If $Y$ is an Itô-integrable process (w.r.t. Brownian motion $B_t$) and T is a stopping time, then it holds that $int_0^{twedge T} Y_s dB_s = int_0^t textbf{1}_{T geq s} Y_s dB_s$. I have two questions about this:
$1.$ Why does the right side Itô integral exist? If $Y_s$ is simple, bounded then $textbf{1}_{T geq s} Y_s$ is itself a simple predictable process for which the integral exists (assuming $T$ took finitely many values). But in the general case we would need to approximate the right side integral by integrals of some simple processes?
$2.$ Can we replace $dB_s$ with $dM_s$ for a continuous local martingale? Does the right-side integral still make sense?
probability-theory stochastic-processes stochastic-calculus
asked Nov 20 at 11:14
user390884
224
add a comment |
If $Y$ is an Itô-integrable process (w.r.t. Brownian motion $B_t$) and T is a stopping time, then it holds that $int_0^{twedge T} Y_s dB_s = int_0^t textbf{1}_{T geq s} Y_s dB_s$. I have two questions about this:
$1.$ Why does the right side Itô integral exist? If $Y_s$ is simple, bounded then $textbf{1}_{T geq s} Y_s$ is itself a simple predictable process for which the integral exists (assuming $T$ took finitely many values). But in the general case we would need to approximate the right side integral by integrals of some simple processes?
$2.$ Can we replace $dB_s$ with $dM_s$ for a continuous local martingale? Does the right-side integral still make sense?
probability-theory stochastic-processes stochastic-calculus
asked Nov 20 at 11:14
user390884
224
For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03
add a comment |
If $Y$ is an Itô-integrable process (w.r.t. Brownian motion $B_t$) and T is a stopping time, then it holds that $int_0^{twedge T} Y_s dB_s = int_0^t textbf{1}_{T geq s} Y_s dB_s$. I have two questions about this:
$1.$ Why does the right side Itô integral exist? If $Y_s$ is simple, bounded then $textbf{1}_{T geq s} Y_s$ is itself a simple predictable process for which the integral exists (assuming $T$ took finitely many values). But in the general case we would need to approximate the right side integral by integrals of some simple processes?
$2.$ Can we replace $dB_s$ with $dM_s$ for a continuous local martingale? Does the right-side integral still make sense?
probability-theory stochastic-processes stochastic-calculus
asked Nov 20 at 11:14
user390884
224
If $Y$ is an Itô-integrable process (w.r.t. Brownian motion $B_t$) and T is a stopping time, then it holds that $int_0^{twedge T} Y_s dB_s = int_0^t textbf{1}_{T geq s} Y_s dB_s$. I have two questions about this:
$1.$ Why does the right side Itô integral exist? If $Y_s$ is simple, bounded then $textbf{1}_{T geq s} Y_s$ is itself a simple predictable process for which the integral exists (assuming $T$ took finitely many values). But in the general case we would need to approximate the right side integral by integrals of some simple processes?
$2.$ Can we replace $dB_s$ with $dM_s$ for a continuous local martingale? Does the right-side integral still make sense?
probability-theory stochastic-processes stochastic-calculus
probability-theory stochastic-processes stochastic-calculus
asked Nov 20 at 11:14
user390884
224
asked Nov 20 at 11:14
user390884
224
asked Nov 20 at 11:14
user390884
224
asked Nov 20 at 11:14
user390884
224
asked Nov 20 at 11:14
user390884
224
224
For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03
add a comment |
For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03
For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03
For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03
add a comment |
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For Brownian motion you can find a detailed proof in the book Brownian motion - An introduction to stochastic processes by Schilling & Partzsch.
– saz
Nov 20 at 12:03