Mixing
Discrete Time “Girsanov” with sub-Gaussian noise
$begingroup$
Context
I seem to recall from a course in stochastic calculus a few years back that for a random vector $X_t$ with dynamics
$$dX_t = f(x_t)dt + g_t dW_t$$
where $W$ is $mathbb{P}$-wiener there exists another measure $mathbb{Q}$ such that $X_t$ is a martingale under this measure (under suitable conditions on $f,g$).
Question
Currently, I find myself in a position where (super)-"martingalizing" a discrete time process seems useful. The setting is as follows: I have given myself a space $(Omega, (mathcal{F_t}), mathbb{P})$ and I consider the process
$$
X_{t+1}=f(X_t)+eta_t Leftrightarrow X_{t+1}-X_t=f(X_t)-X_t+eta_t
$$
where $eta_t$ is independent $sigma^2$-sub-gaussian noise, $mathbb{E} [e^{lambda
p eta_t}] leq e^{lambda ^2 sigma^2/2}$ for all the one-dimensional projections $p eta_t$. The function $f$ should be reasonable for the setting (I guess be measurable and satisfy some discrete analogue of Novikov).
Now, my hunch is that I might not be able to find $mathbb{Q}$ such that $Y_t = CX_t$ becomes a martingale (with a deterministic matrix $C$) - but maybe at least a supermartingale (the end user will be large deviation so I don't really care about it being a martingale, super will do)? I saw the answer to
Discrete and continuous Girsanov
however, the setting to that answer seems considerably simpler (using predictability).
Progress
I was considering that if one defines, for some $phi_t$, analogously to the continuous case
$$
L_t = expleft(sum_{k=1}^tlangle phi_k, W_k rangle-frac{1}{2}sum_{k=1}^tlangle phi_k, phi_k rangle right)
$$
the following computation could go through...
begin{align*}mathbb{E}[CX_{t+1}L_{t+1}|mathcal{F_t}]&=CL_tmathbb{E}[X_{t+1}e^{langlephi_{t+1},phi_{t+1}rangle-langle phi_{t+1},eta_{t+1}rangle/2}|mathcal{F_t}] \
&= dots \
&leq CX_t L_t,
end{align*}
for a clever choice of $phi_t$ by using the sub-gaussian property - but as for the "$dots$" i am stuck on:
- Can I actually deal with the nonlinearity at all? We can in the continuous case so should be possible I feel?
- What is the correct choice of kernel $phi_t$?
I would very much appreciate any help for proving/disproving the above inequality. NB, I know I have been sloppy in defining $f$. If one absolutely requires another property such as convexity to answer the question I am willing to admit the property (even though I obviously would prefer this not to be necessary...).
probability stochastic-processes stochastic-calculus
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
$endgroup$
add a comment |
$begingroup$
Context
I seem to recall from a course in stochastic calculus a few years back that for a random vector $X_t$ with dynamics
$$dX_t = f(x_t)dt + g_t dW_t$$
where $W$ is $mathbb{P}$-wiener there exists another measure $mathbb{Q}$ such that $X_t$ is a martingale under this measure (under suitable conditions on $f,g$).
Question
Currently, I find myself in a position where (super)-"martingalizing" a discrete time process seems useful. The setting is as follows: I have given myself a space $(Omega, (mathcal{F_t}), mathbb{P})$ and I consider the process
$$
X_{t+1}=f(X_t)+eta_t Leftrightarrow X_{t+1}-X_t=f(X_t)-X_t+eta_t
$$
where $eta_t$ is independent $sigma^2$-sub-gaussian noise, $mathbb{E} [e^{lambda
p eta_t}] leq e^{lambda ^2 sigma^2/2}$ for all the one-dimensional projections $p eta_t$. The function $f$ should be reasonable for the setting (I guess be measurable and satisfy some discrete analogue of Novikov).
Now, my hunch is that I might not be able to find $mathbb{Q}$ such that $Y_t = CX_t$ becomes a martingale (with a deterministic matrix $C$) - but maybe at least a supermartingale (the end user will be large deviation so I don't really care about it being a martingale, super will do)? I saw the answer to
Discrete and continuous Girsanov
however, the setting to that answer seems considerably simpler (using predictability).
Progress
I was considering that if one defines, for some $phi_t$, analogously to the continuous case
$$
L_t = expleft(sum_{k=1}^tlangle phi_k, W_k rangle-frac{1}{2}sum_{k=1}^tlangle phi_k, phi_k rangle right)
$$
the following computation could go through...
begin{align*}mathbb{E}[CX_{t+1}L_{t+1}|mathcal{F_t}]&=CL_tmathbb{E}[X_{t+1}e^{langlephi_{t+1},phi_{t+1}rangle-langle phi_{t+1},eta_{t+1}rangle/2}|mathcal{F_t}] \
&= dots \
&leq CX_t L_t,
end{align*}
for a clever choice of $phi_t$ by using the sub-gaussian property - but as for the "$dots$" i am stuck on:
- Can I actually deal with the nonlinearity at all? We can in the continuous case so should be possible I feel?
- What is the correct choice of kernel $phi_t$?
I would very much appreciate any help for proving/disproving the above inequality. NB, I know I have been sloppy in defining $f$. If one absolutely requires another property such as convexity to answer the question I am willing to admit the property (even though I obviously would prefer this not to be necessary...).
probability stochastic-processes stochastic-calculus
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
$endgroup$
add a comment |
$begingroup$
Context
I seem to recall from a course in stochastic calculus a few years back that for a random vector $X_t$ with dynamics
$$dX_t = f(x_t)dt + g_t dW_t$$
where $W$ is $mathbb{P}$-wiener there exists another measure $mathbb{Q}$ such that $X_t$ is a martingale under this measure (under suitable conditions on $f,g$).
Question
Currently, I find myself in a position where (super)-"martingalizing" a discrete time process seems useful. The setting is as follows: I have given myself a space $(Omega, (mathcal{F_t}), mathbb{P})$ and I consider the process
$$
X_{t+1}=f(X_t)+eta_t Leftrightarrow X_{t+1}-X_t=f(X_t)-X_t+eta_t
$$
where $eta_t$ is independent $sigma^2$-sub-gaussian noise, $mathbb{E} [e^{lambda
p eta_t}] leq e^{lambda ^2 sigma^2/2}$ for all the one-dimensional projections $p eta_t$. The function $f$ should be reasonable for the setting (I guess be measurable and satisfy some discrete analogue of Novikov).
Now, my hunch is that I might not be able to find $mathbb{Q}$ such that $Y_t = CX_t$ becomes a martingale (with a deterministic matrix $C$) - but maybe at least a supermartingale (the end user will be large deviation so I don't really care about it being a martingale, super will do)? I saw the answer to
Discrete and continuous Girsanov
however, the setting to that answer seems considerably simpler (using predictability).
Progress
I was considering that if one defines, for some $phi_t$, analogously to the continuous case
$$
L_t = expleft(sum_{k=1}^tlangle phi_k, W_k rangle-frac{1}{2}sum_{k=1}^tlangle phi_k, phi_k rangle right)
$$
the following computation could go through...
begin{align*}mathbb{E}[CX_{t+1}L_{t+1}|mathcal{F_t}]&=CL_tmathbb{E}[X_{t+1}e^{langlephi_{t+1},phi_{t+1}rangle-langle phi_{t+1},eta_{t+1}rangle/2}|mathcal{F_t}] \
&= dots \
&leq CX_t L_t,
end{align*}
for a clever choice of $phi_t$ by using the sub-gaussian property - but as for the "$dots$" i am stuck on:
- Can I actually deal with the nonlinearity at all? We can in the continuous case so should be possible I feel?
- What is the correct choice of kernel $phi_t$?
I would very much appreciate any help for proving/disproving the above inequality. NB, I know I have been sloppy in defining $f$. If one absolutely requires another property such as convexity to answer the question I am willing to admit the property (even though I obviously would prefer this not to be necessary...).
probability stochastic-processes stochastic-calculus
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
$endgroup$
Context
I seem to recall from a course in stochastic calculus a few years back that for a random vector $X_t$ with dynamics
$$dX_t = f(x_t)dt + g_t dW_t$$
where $W$ is $mathbb{P}$-wiener there exists another measure $mathbb{Q}$ such that $X_t$ is a martingale under this measure (under suitable conditions on $f,g$).
Question
Currently, I find myself in a position where (super)-"martingalizing" a discrete time process seems useful. The setting is as follows: I have given myself a space $(Omega, (mathcal{F_t}), mathbb{P})$ and I consider the process
$$
X_{t+1}=f(X_t)+eta_t Leftrightarrow X_{t+1}-X_t=f(X_t)-X_t+eta_t
$$
where $eta_t$ is independent $sigma^2$-sub-gaussian noise, $mathbb{E} [e^{lambda
p eta_t}] leq e^{lambda ^2 sigma^2/2}$ for all the one-dimensional projections $p eta_t$. The function $f$ should be reasonable for the setting (I guess be measurable and satisfy some discrete analogue of Novikov).
Now, my hunch is that I might not be able to find $mathbb{Q}$ such that $Y_t = CX_t$ becomes a martingale (with a deterministic matrix $C$) - but maybe at least a supermartingale (the end user will be large deviation so I don't really care about it being a martingale, super will do)? I saw the answer to
Discrete and continuous Girsanov
however, the setting to that answer seems considerably simpler (using predictability).
Progress
I was considering that if one defines, for some $phi_t$, analogously to the continuous case
$$
L_t = expleft(sum_{k=1}^tlangle phi_k, W_k rangle-frac{1}{2}sum_{k=1}^tlangle phi_k, phi_k rangle right)
$$
the following computation could go through...
begin{align*}mathbb{E}[CX_{t+1}L_{t+1}|mathcal{F_t}]&=CL_tmathbb{E}[X_{t+1}e^{langlephi_{t+1},phi_{t+1}rangle-langle phi_{t+1},eta_{t+1}rangle/2}|mathcal{F_t}] \
&= dots \
&leq CX_t L_t,
end{align*}
for a clever choice of $phi_t$ by using the sub-gaussian property - but as for the "$dots$" i am stuck on:
- Can I actually deal with the nonlinearity at all? We can in the continuous case so should be possible I feel?
- What is the correct choice of kernel $phi_t$?
I would very much appreciate any help for proving/disproving the above inequality. NB, I know I have been sloppy in defining $f$. If one absolutely requires another property such as convexity to answer the question I am willing to admit the property (even though I obviously would prefer this not to be necessary...).
probability stochastic-processes stochastic-calculus
probability stochastic-processes stochastic-calculus
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
asked Jan 13 at 19:46
sortofamathematiciansortofamathematician
516
516
add a comment |
add a comment |
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