Does a strong solution to a SDE imply lipschitz condition?












2












$begingroup$


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







share|cite|improve this question











$endgroup$

















    2












    $begingroup$


    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







    share|cite|improve this question











    $endgroup$















      2












      2








      2


      3



      $begingroup$


      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







      share|cite|improve this question











      $endgroup$




      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






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 4 hours ago







      Who cares

















      asked 4 hours ago









      Who caresWho cares

      16713




      16713






















          1 Answer
          1






          active

          oldest

          votes


















          4












          $begingroup$

          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 $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag{1}$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt{|x-y|} quad text{and} quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbb{R}$, 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.






          share|cite|improve this answer









          $endgroup$














            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
            });


            }
            });














            draft saved

            draft discarded


















            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









            4












            $begingroup$

            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 $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag{1}$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt{|x-y|} quad text{and} quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbb{R}$, 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.






            share|cite|improve this answer









            $endgroup$


















              4












              $begingroup$

              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 $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag{1}$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt{|x-y|} quad text{and} quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbb{R}$, 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.






              share|cite|improve this answer









              $endgroup$
















                4












                4








                4





                $begingroup$

                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 $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag{1}$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt{|x-y|} quad text{and} quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbb{R}$, 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.






                share|cite|improve this answer









                $endgroup$



                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 $$dX_t = b(t,X_t) , dt + sigma(t,X_t) , dB_t, qquad X_0 = x_0, tag{1}$$ are bounded, $b$ is measurable, $sigma$ is continuous and there exist constants $C>0$ and $epsilon>0$ such that $$|sigma(t,x)-sigma(t,y)| leq C sqrt{|x-y|} quad text{and} quad |sigma(t,x)| geq epsilon$$ for all $t geq 0$, $x,y in mathbb{R}$, 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.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                sazsaz

                83.1k863132




                83.1k863132






























                    draft saved

                    draft discarded




















































                    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.




                    draft saved


                    draft discarded














                    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





















































                    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







                    -

                    Popular posts from this blog

                    Callistus I

                    Image
                    Callistus I Res apud Vicidata repertae: Nativitas : unknown value ; Roma Obitus : 223 ; Roma Pontificatus 217–222 Praecessor : Zephyrinus Successor : Urbanus I Sanctus Callistus qui ieiunium instituit; e libro manu scripto Francogallico saeculo XIV Callistus (natus die ignoto; obiit anno 222), fuit episcopus Romae et Papa Ecclesiae Catholicae Romanae ab anno 217. Vita | Secundum adversarium suum, presbyterem, fuit Hippolytus Callistus, servus argentarii Christiani. Papae Zephyrino, Callistus Romae arcessivit. Callistus fuit primus diaconus ac dux catacumbae apud Viam Appiam, quae hodie Sancti Callisti appellatur. Pontificatus | Nexus externi | Vicimedia Communia plura habent quae ad Callistus I spectant. Fontes Latinae de papis usque ad annum 530 (Papa Felix IV) Liber pontificalis Opera Omnia .mw-parser-output .stipula{padding:3px;background:#F7F8FF;border:1px solid grey;margin:auto}.mw-parser-output .stipula td.cell1{bac...
                    Read more

                    Tabula Rosettana

                    Image
                    Tabula multilinguis Rosettana in Museo Britannico ostenditur. Tabula Rosettana, [1] etiam titulo OGIS 90 agnita, est stela decreto de rebus sacris in Aegypto anno 196 a.C.n. lato inscripta. Tabula iuxta Rosettam Aegypti, urbem in delta Nili et ad oram maris Mediterranei iacentem, anno 1799 a milite Francico reperta est. Inventio stelae, linguis duabus scripturisque tribus inscriptae, eruditis Instituti Aegypti statim nuntiata est; ibi enim iussu imperatoris Napoleonis eruditi omnium scientiarum (sub aegide Commissionis Scientiarum et Artium) properaverant cum expeditione Francica. Qua a Britannis mox debellata, tabula Rosettana Londinium missa hodie apud Museum Britannicum iacet. Textus Graecus cito lectus interpretationi textuum Aegyptiorum (in formis hieroglyphica et demotica expressorum) gradatim adiuvit. Denique textum plene interpretatus est Ioannes Franciscus Champollion. Ab opere eruditorum cumulativo coepit hodiernus scripturae hieroglyphicae linguaeque Aegyptiae a...
                    Read more

                    How to label and detect the document text images

                    Image
                    1 $begingroup$ This is what I mean as document text image: I want to label the texts in image as separate blocks and my model should detect these labels as classes. NOTE: This is how the end result should be like: The labels like Block 1, Block 2, Block 3,.. should be Logo, Title, Date,.. Others, etc. Work done: First approach : I tried to implement this method via Object Detection, it didn't work. It didn't even detect any text. Second approach : Then I tried it using PixelLink. As this model is build for scene text detection, it detected each and every text in the image. But this method can detect multiple lines of text if the threshold values are increased. But I have no idea how do I add labels to the text blocks. PIXEL_CLS_WEIGHT_all_ones = 'PIXEL_CLS_WEIGHT_all_ones' PIXEL_C...
                    Read more