Does a strong solution to a SDE imply lipschitz condition?Growth condition for Ito diffusionsIs the solution to a driftless SDE with Lipschitz variation a martingale?existence/uniqueness of solution and Ito's formulaExample of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?Strong solution SDE - independence of initial conditiionExistence and uniqueness of solution of a non linear SDEHow to prove a solution of SDE with Lipschitz condition is unbounded?Is the solution to this (simple) Stochastic Differential Equation unique?SDE existence and uniqueness
What does it mean to express a gate in Dirac notation?
Fizzy, soft, pop and still drinks
Why does processed meat contain preservatives, while canned fish needs not?
Is the 5 MB static resource size limit 5,242,880 bytes or 5,000,000 bytes?
How can the Zone of Truth spell be defeated without the caster knowing?
How to reduce LED flash rate (frequency)
How to verbalise code in Mathematica?
Why was the Spitfire's elliptical wing almost uncopied by other aircraft of World War 2?
What route did the Hindenburg take when traveling from Germany to the U.S.?
French for 'It must be my imagination'?
Is the claim "Employers won't employ people with no 'social media presence'" realistic?
Was there a shared-world project before "Thieves World"?
What is the strongest case that can be made in favour of the UK regaining some control over fishing policy after Brexit?
The Defining Moment
Reducing vertical space in stackrel
How do I reattach a shelf to the wall when it ripped out of the wall?
What happened to Captain America in Endgame?
How to have a sharp product image?
Unexpected email from Yorkshire Bank
How can I practically buy stocks?
Apply MapThread to all but one variable
Is there an official tutorial for installing Ubuntu 18.04+ on a device with an SSD and an additional internal hard drive?
Noun clause (singular all the time?)
How come there are so many candidates for the 2020 Democratic party presidential nomination?
Does a strong solution to a SDE imply lipschitz condition?
Growth condition for Ito diffusionsIs the solution to a driftless SDE with Lipschitz variation a martingale?existence/uniqueness of solution and Ito's formulaExample of a regular strong solution of an SDE, which doesn't satisfy a Lyapunov condition?Is the distribution of an Ito diffusion at time t absolutely continuous wrt Lebesgue measure?Strong solution SDE - independence of initial conditiionExistence and uniqueness of solution of a non linear SDEHow to prove a solution of SDE with Lipschitz condition is unbounded?Is the solution to this (simple) Stochastic Differential Equation unique?SDE existence and uniqueness
$begingroup$
Consider $dX_t=b(X_t,t)dt+sigma(X_t,t)dB_t$. I know that,
$|b(x,t)-b(y,t)|+|sigma(x,t)-sigma(y,t)|leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. However in Oksendal, to show that $dX_t=sign(X_t)dB_t$ doesn't have a strong solution another proof is presented, even though $sign(x)$ fails the lipschitz condition. I have seen it at a couple of more places for this equation.
My question: Is Lipschitz condition an iff condition for a strong solution?
stochastic-calculus stochastic-integrals stochastic-analysis
asked 5 hours ago
Who caresWho cares
16713
$endgroup$
add a comment |
$begingroup$
Consider $dX_t=b(X_t,t)dt+sigma(X_t,t)dB_t$. I know that,
$|b(x,t)-b(y,t)|+|sigma(x,t)-sigma(y,t)|leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. However in Oksendal, to show that $dX_t=sign(X_t)dB_t$ doesn't have a strong solution another proof is presented, even though $sign(x)$ fails the lipschitz condition. I have seen it at a couple of more places for this equation.
My question: Is Lipschitz condition an iff condition for a strong solution?
stochastic-calculus stochastic-integrals stochastic-analysis
asked 5 hours ago
Who caresWho cares
16713
$endgroup$
add a comment |
$begingroup$
Consider $dX_t=b(X_t,t)dt+sigma(X_t,t)dB_t$. I know that,
$|b(x,t)-b(y,t)|+|sigma(x,t)-sigma(y,t)|leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. However in Oksendal, to show that $dX_t=sign(X_t)dB_t$ doesn't have a strong solution another proof is presented, even though $sign(x)$ fails the lipschitz condition. I have seen it at a couple of more places for this equation.
My question: Is Lipschitz condition an iff condition for a strong solution?
stochastic-calculus stochastic-integrals stochastic-analysis
asked 5 hours ago
Who caresWho cares
16713
$endgroup$
Consider $dX_t=b(X_t,t)dt+sigma(X_t,t)dB_t$. I know that,
$|b(x,t)-b(y,t)|+|sigma(x,t)-sigma(y,t)|leq D|x-y|$ for some constant D implies the existence and uniqueness of a strong solution. However in Oksendal, to show that $dX_t=sign(X_t)dB_t$ doesn't have a strong solution another proof is presented, even though $sign(x)$ fails the lipschitz condition. I have seen it at a couple of more places for this equation.
My question: Is Lipschitz condition an iff condition for a strong solution?
stochastic-calculus stochastic-integrals stochastic-analysis
stochastic-calculus stochastic-integrals stochastic-analysis
asked 5 hours ago
Who caresWho cares
16713
asked 5 hours ago
Who caresWho cares
16713
edited 4 hours ago
Who cares
asked 5 hours ago
Who caresWho cares
16713
asked 5 hours ago
Who caresWho cares
16713
asked 5 hours ago
Who caresWho cares
16713
16713
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin:
Let $(B_t)_t geq 0$ be a one-dimensional Bownian motion. If the coefficients of the SDE $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag1$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt quad textand quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbbR$, then the SDE $(1)$ has a (unique) strong solution.
Note that the result does not require any regularity assumptions on the drift $b$ (except from measurability) and only Hölder continuity of order $1/2$ for the diffusion coefficient $sigma$.
A nice overview on known existence and uniqueness results for SDEs can be found in the book Singular Stochastic Differential Equations by Cherny & Engelbert.
answered 3 hours ago
sazsaz
83.1k863132
$endgroup$
add a comment |
Your Answer
StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);
else
createEditor();
);
function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader:
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
,
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);
);
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3204672%2fdoes-a-strong-solution-to-a-sde-imply-lipschitz-condition%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin:
Let $(B_t)_t geq 0$ be a one-dimensional Bownian motion. If the coefficients of the SDE $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag1$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt quad textand quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbbR$, then the SDE $(1)$ has a (unique) strong solution.
Note that the result does not require any regularity assumptions on the drift $b$ (except from measurability) and only Hölder continuity of order $1/2$ for the diffusion coefficient $sigma$.
A nice overview on known existence and uniqueness results for SDEs can be found in the book Singular Stochastic Differential Equations by Cherny & Engelbert.
answered 3 hours ago
sazsaz
83.1k863132
$endgroup$
add a comment |
$begingroup$
No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin:
Let $(B_t)_t geq 0$ be a one-dimensional Bownian motion. If the coefficients of the SDE $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag1$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt quad textand quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbbR$, then the SDE $(1)$ has a (unique) strong solution.
Note that the result does not require any regularity assumptions on the drift $b$ (except from measurability) and only Hölder continuity of order $1/2$ for the diffusion coefficient $sigma$.
A nice overview on known existence and uniqueness results for SDEs can be found in the book Singular Stochastic Differential Equations by Cherny & Engelbert.
answered 3 hours ago
sazsaz
83.1k863132
$endgroup$
add a comment |
$begingroup$
No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin:
Let $(B_t)_t geq 0$ be a one-dimensional Bownian motion. If the coefficients of the SDE $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag1$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt quad textand quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbbR$, then the SDE $(1)$ has a (unique) strong solution.
Note that the result does not require any regularity assumptions on the drift $b$ (except from measurability) and only Hölder continuity of order $1/2$ for the diffusion coefficient $sigma$.
A nice overview on known existence and uniqueness results for SDEs can be found in the book Singular Stochastic Differential Equations by Cherny & Engelbert.
answered 3 hours ago
sazsaz
83.1k863132
$endgroup$
No, Lipschitz continuity is not a necessary condition for the existence of a strong solution. There is, for instance, the following general result which goes back to Zvonkin:
Let $(B_t)_t geq 0$ be a one-dimensional Bownian motion. If the coefficients of the SDE $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag1$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt quad textand quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbbR$, then the SDE $(1)$ has a (unique) strong solution.
Note that the result does not require any regularity assumptions on the drift $b$ (except from measurability) and only Hölder continuity of order $1/2$ for the diffusion coefficient $sigma$.
A nice overview on known existence and uniqueness results for SDEs can be found in the book Singular Stochastic Differential Equations by Cherny & Engelbert.
answered 3 hours ago
sazsaz
83.1k863132
answered 3 hours ago
sazsaz
83.1k863132
answered 3 hours ago
sazsaz
83.1k863132
answered 3 hours ago
sazsaz
83.1k863132
83.1k863132
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3204672%2fdoes-a-strong-solution-to-a-sde-imply-lipschitz-condition%23new-answer', 'question_page');
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
