Local Martingale on $[0,T]$











up vote
1
down vote

favorite












What is the definition of a local martingale on $[0,T]?$



I guess the definition should be: $M = (M_t)_{t in [0,T]}$ is a local martingale if there exists a localizing sequence $tau_n$ such that for all $n$ the process $M^{tau_n}$ is a martingale.



A sequence of stopping times $(tau_n)_n$ is a localizing sequence if $P$-almost surely $tau_n leq tau_{n+1}$ and $lim_{n to infty} tau_n = T.$



Is this definition correct?






stochastic-calculus







share|cite|improve this question


























    up vote
    1
    down vote

    favorite












    What is the definition of a local martingale on $[0,T]?$



    I guess the definition should be: $M = (M_t)_{t in [0,T]}$ is a local martingale if there exists a localizing sequence $tau_n$ such that for all $n$ the process $M^{tau_n}$ is a martingale.



    A sequence of stopping times $(tau_n)_n$ is a localizing sequence if $P$-almost surely $tau_n leq tau_{n+1}$ and $lim_{n to infty} tau_n = T.$



    Is this definition correct?






    stochastic-calculus







    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      What is the definition of a local martingale on $[0,T]?$



      I guess the definition should be: $M = (M_t)_{t in [0,T]}$ is a local martingale if there exists a localizing sequence $tau_n$ such that for all $n$ the process $M^{tau_n}$ is a martingale.



      A sequence of stopping times $(tau_n)_n$ is a localizing sequence if $P$-almost surely $tau_n leq tau_{n+1}$ and $lim_{n to infty} tau_n = T.$



      Is this definition correct?






      stochastic-calculus







      share|cite|improve this question













      What is the definition of a local martingale on $[0,T]?$



      I guess the definition should be: $M = (M_t)_{t in [0,T]}$ is a local martingale if there exists a localizing sequence $tau_n$ such that for all $n$ the process $M^{tau_n}$ is a martingale.



      A sequence of stopping times $(tau_n)_n$ is a localizing sequence if $P$-almost surely $tau_n leq tau_{n+1}$ and $lim_{n to infty} tau_n = T.$



      Is this definition correct?






      stochastic-calculus




      stochastic-calculus






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked 21 hours ago









      White

      789




      789



























          active

          oldest

          votes











          Your Answer





          StackExchange.ifUsing("editor", function () {
          return StackExchange.using("mathjaxEditing", function () {
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          });
          });
          }, "mathjax-editing");

          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',
          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%2f3004775%2flocal-martingale-on-0-t%23new-answer', 'question_page');
          }
          );

          Post as a guest















          Required, but never shown






























          active

          oldest

          votes













          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes
















           

          draft saved


          draft discarded



















































           


          draft saved


          draft discarded














          StackExchange.ready(
          function () {
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3004775%2flocal-martingale-on-0-t%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

          Can a sorcerer learn a 5th-level spell early by creating spell slots using the Font of Magic feature?

          Image
          28 6 $begingroup$ Per the Font of Magic feature, sorcerer can use Flexible Casting to create 5th-level spell slots at level 7, even though they are not typically available until level 9. You can transform unexpended sorcery points into one spell slot as a bonus action on your turn. The Creating Spell Slots table shows the cost of creating a spell slot of [5th level is 7] If such a sorcerer levels up while still having this 5th-level spell slot, can they choose a 5th-level spell as the spell they gain upon levelling up? The Spellcasting feature states: Additionally, when you gain a level in this class, you can choose one of the sorcerer spells you know and replace it with another spell from the sorcerer spell list, which also must be of a level for which you have spell slots.
          Read more

          Does disintegrating a polymorphed enemy still kill it after the 2018 errata?

          Image
          up vote 13 down vote favorite 1 After the 2018 errata, the disintegrate spell description now reads: A creature targeted by this spell must make a Dexterity saving throw. On a failed save, the target takes 10d6 + 40 force damage. The target is disintegrated if this damage leaves it with 0 hit points. If you were to polymorph an enemy into a rat and then disintegrate it, would the enemy be disintegrated or would it just return to its original form? I know that for Druids, it’s not an instant kill anymore, but is this the case for polymorph as well? dnd-5e spells polymorph errata share | improve this question edited 18 hours ago
          Read more

          A Topological Invariant for $pi_3(U(n))$

          Image
          3 3 $begingroup$ Recently, I saw a construction of topological invariant for $pi_3(U(n))$ with $ngeq 2$ : $$ N=frac{1}{24pi^2}int_{S^3} d^3x epsilon^{ijk} Tr[(U^{-1}partial_{x_i}U)(U^{-1}partial_{x_j}U)(U^{-1}partial_{x_k}U)] , $$ where $Uin U(n)$ depends on $boldsymbol{x}=(x_1,x_2,x_3)in S^3$ , $epsilon^{ijk}$ is the Levi-Civita symbol, $i,j,k=1,2,3$ , and the duplicated indexes are summed over. It is claimed that $N$ is an integer, but why? Update 02/02/2019 I think I got an argument for $n=2$ . In this case, $U=e^{i varphi} q$ with $qin SU(2)$ . Due to the trace and the Levi-Civita symbol in $N$ , $varphi$ does not contribute to $N$ . As $Tr[q^{dagger}partial_i q]=0$ and $(q^{dagger}partial_i q)^{dagger}=-q^{dagger}partial_i q$ , $q^{dagger}partial_i q$ in geneeral has the form $q^{dagger}partia
          Read more