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Mar
11 |
comment |
analysis of the regularity using Hormander condition
It does for $t\in (0,T)$, not at $t=0$, simply because the domain is not an open set anymore, so you can write the integral but it doesn't have a domain of definition for $t=0$ and therefore doesn't make sense at that time. Am I wrong? |
Mar
5 |
comment |
analysis of the regularity using Hormander condition
@Bazin: there is something important is missing that I do not understand in your answer. Change of variables works great but we should track the change in the domain along with change of variables and as you can see the domain in $z$ is shrinking. It does it to such an extent that at $t=T$ it becomes simply a point. So, you can't define that the solution there as it is not a domain anymore. How would you argue that? |
Feb
14 |
comment |
analysis of the regularity using Hormander condition
yes, I can find the solution however, the regularity is harder to observe. Even though the tranformed function $v$ is perfectly regular due to satisfaction of the Hormander condition, I can't transfer those properties to the original function, can I? I agree I can conclude that I have a solution up to a change of variables, but I don't think regularity follows the same way. I might be wrong, but if you could please add a few lines to your answer with respect to regularity of the original question that would be very helpful as this is the main answer I am looking for! |
Feb
6 |
comment |
Limit of the stochastic process at time 0
@The Bridge: what if $S_u=u^{1/2}$, don't I have a problem then? Is the statement holds for "any" continuous process $S_u$ regardless how wild it is? |
Feb
5 |
comment |
analysis of the regularity using Hormander condition
I would like to have estimates in $L^2$ space. |
Feb
3 |
comment |
analysis of the regularity using Hormander condition
yes, I have gotten to that point too, I must agree the transformed equation has a lot of nice properties, but I need to know if I can utilize those to state the properties of the original equation as this is the one I need to analyze. |
Feb
3 |
asked | analysis of the regularity using Hormander condition |
Feb
3 |
awarded | Commentator |
Feb
3 |
comment |
well-posedness of the transport equation
due to the shape of characteristics. yes, I think I did the computations correct and they asymptotically converge to the level $t=T$ and never cross it. |
Jan
15 |
comment |
well-posedness of the transport equation
I am staring at this pde for a while but I still can't make it clear. It is seems to be straightforward to show that $||w(t,x)||_2\leq ||w(0,x)||_{\inty}$, just by observing the underlying characteristics. However, does it imply that it is $L^2$ stable? If so, how to explain the fact that energy methods tell me the opposite? |
Jan
13 |
awarded | Student |
Jan
13 |
comment |
well-posedness of the transport equation
"continuous dependence on initial data" means to have an estimate $||w(t,x)||^2\leq C ||w(0,x)||^2$ in some norm. I have to choose a norm and I happen to like $L^2$ spaces(because all these integration by parts tricks work), so I would like to have such an estimate there. But at the moment, I can't show it. But, I have a suspicion it is true even in $L^2$ since I know all about solution and the equation just "flattens" any initial data I provided, so it should not be a problem, but energy estimates show there is a big problem at $t=T$ regardless how smooth or integrable the initial data is. |
Jan
13 |
asked | well-posedness of the transport equation |
Nov
23 |
comment |
Limit of the stochastic process at time 0
right, there is something missing: $dY_t=\frac{S_t−Y_t}{t}dt$, this simply comes from writing the SDE for the process defined in the original question. So it does have a solution: it is an average of the path over time interval $[0,t]$. But not clear how that affects the solution of the pde at $t=0$, at least to me. |
Nov
22 |
comment |
Limit of the stochastic process at time 0
related to time $0$ question: If I assume $dS_t = adt+bdB_t$ and from above $dY_t = \frac{1}{t}(S_t-Y_t)$ then I can use Ito formula for $u(t,S_t,Y_t)$. I would like that to be a martingale and thus require $u_t+0.5b^2u_{ss}+au_s+\frac{1}{t}(s-y)u_y=0$. The last term is not defined at $t=0$ but from above answer as $t$ approaches $0$ I have $\frac{1}{t}(s-y)=\frac{0}{0}$. Is there anything can be said about the last term in the pde? |
Nov
21 |
awarded | Scholar |
Nov
21 |
accepted | Limit of the stochastic process at time 0 |
Nov
21 |
comment |
Limit of the stochastic process at time 0
yes, thanks the Bridge and Guillaume, I think the answer now is complete as I was looking for some omega mentioning. Thus, I have the convergence for every single omega and therefore convergence in a.s. sense. |
Nov
20 |
comment |
Limit of the stochastic process at time 0
ok, let's take a particular case, where $S_t$ is according to the following dynamics $dS_t=adt+bdB_t$ where $B_t$ is a Brownian motion, so that $S_t$ has a lognormal distribution. |
Nov
20 |
asked | Limit of the stochastic process at time 0 |