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I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance.

To be specific, suppose we want to compute the expected value of $f(S_T)$, where $f(S)$ is a scalar function with a uniform Lipschitz bound, i.e., there exists a constant $c$ such that $$ \mid f(U) - f(V) \mid \leq c \, \mid\mid U-V\mid\mid $$ for every $U,V$.

The question is, why do we need to guarantee that an instrument's payoff must satisfy this condition? If this condition is not satisfied, does it mean we can't construct an efficient pricing model?

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    $\begingroup$ it's not unusual to have discontinuous pay-offs so I don't agree with the premise of the question e,g, digital options, barrier options, TARNs, trigger swaps, credit derivatives $\endgroup$ – Mark Joshi Jun 16 '15 at 21:36
  • $\begingroup$ Questions about assumptions are relevant but if you want "rigorous" answers you should state it more carefully. What means "major" part and what is an "efficient" pricing model? We probably all agree that one does not need Lipschitz to calculate expected values. As Mark Joshi points out you might be able (or even need) to go a long way with somewhat weaker assumptions. $\endgroup$ – g g Jun 17 '15 at 11:42
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In this case it is just the notion that your payoff function should not explode at some point - made mathematically rigorous.

Have a look at the following picture from wikipedia:

enter image description here

Intuitively the Lipschitz condition (or Lipschitz continuity) ensures that your payoff function always remains entirely outside the white cone, so it cannot e.g. become infinitely steep at one point. Depending on the kinds of payoff functions you want to price and the implementation of the system this is something that could enhance the efficiency of the pricing model.

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