Dear George Lowther, I don’t have comments on the contents of this article. I am a PhD student in mathematic(Wits University South-Africa) and desire to get article on quasi-martingales and stochastic integrals.

Regards

Francis

It is dominated by which has finite conditional expectation as local martingales are locally integrable https://almostsuremath.com/2009/12/23/localization/#scn_local_lem11

]]>Dear George,

Well yes, but who is the dominator of the sequence ?

]]>If we have convergence of random variables and dominated as where (a.s.) then (a.s.).

]]>In the proof of lemma 6, when you treat the case of local martingales, on the second to last string of equalities you say that in order to prove it you use dominated convergence for conditional expectations, can you please explain what exactly mean by that ?

]]>I checked a few sources. Rogers & Williams (Diffusions, Markov Processes, and Martingales), Metivier (Semimartingales), Jacod & Shiryaev (Limit Theorems for stochastic processes), Kallenberg (Foundations of Modern Probability) all have quadratic variation starting from 0. Protter is the outlier here, but either convention is fine IMO as long as you are consistent.

]]>Hi Jakob,

Yes, this is just a matter of convention about what you do at time 0. Obviously any statement using one convention can be converted to an equivalent statement using the other. I’ll have to check for explicit definitions in the literature, but I am sure both are used. Similar issues occur in many places, such as the integral (as you mentioned), the definition of predictable processes and stopping times, the Doob-Meyer decomposition, dual projections, etc. In a way, you could consider extending your processes to negative times, by either being constant equal to their time 0 values, or equal to 0 there. I.e, do you consider your processes to have a jump at time zero or not? I made a choice (no jump at time 0) and stuck to it throughout these notes.

Thanks

George

first: thank you a lot for this blog, it’s an amazing source for stochastic calculus!

I am seriously confused about one point where your definition of the quadratic variation seems to be different from the one in Protter’s book. Apparently Protter defines the quadratic variation (and quadratic covariation) such that there is an additional term X_0 Y_0. Do you know what I mean? Otherwise I can provide more details… But there are a lot of instances there where it really makes a difference if one integrates over [0, t] or (0, t], for example… Is it possible that there are different conventions in the literature? ]]>

The whole proof could be tidied. We are really only constructing a single process X, but doing this by inductively extending it over a sequence of time steps. Due to notation issues, I denoted each of these newly extended processes as a sequence of processes, rather than just one process. Also, extending across each time step involves evaluating F(X), but it is well defined as the value of F(X) only depends on the values of X already constructed.

]]>I don’t think it is a problem, just the explanation should be tidied up. The point is that we can define F(X) over the interval [0,tau), even though we do not yet know that X has a left limit at tau. This is because F is backwards looking, F(X)_t only depends on X at times before t. More precisely, we actually define a process which is defined on each interval [0,tau_r] as equalling F(X^r), which I denoted as F(X).

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