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I am trying to find references (books, papers, etc.) for calculating $\mathbb E f(X_T)$, where $X_T$ is a diffusion and $f$ is a real function that is not continuous, by means of solving a PDE or Feynman-Kac equation.

Edit adressing the comments: Even if the PDE has a solution it can only be shown to equal the expectation under certain conditions. That is why I am asking for a reference for the caclulation of the expectation as solution to a PDE and not about the PDE and its solutions in itself.

Any such "verification theorem" basically uses the Ito formula for the value function and thus requires twice differentiability. This can only be ensured for the PDE solotion if the end data is continuous. So I am not interested in "it should just work" arguments but rather in answers or references to the "when" and "why.

Thank you

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    $\begingroup$ A silly suggestion, but can't you approximate your discontinous function with a better behaved one that satisfies the conditions? $\endgroup$ – Kiwiakos Mar 7 '15 at 20:33
  • $\begingroup$ That might be one solution. The technical mathematical requirements however, are often there for a reason,... $\endgroup$ – Johannes Gerer Mar 7 '15 at 22:45
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    $\begingroup$ the hear kernel is smoothing and so even if the final data is not continuous the expectation is smooth for $t < T.$ $\endgroup$ – Mark Joshi Mar 8 '15 at 23:35
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if we take a digital option and price under BS then you can do the whole thing by direct verification.

i.e. $N(d_2)$ solves the PDE and converges to the final pay-off pointwise.

So if the final pay-off has a finite number of jump discontinuities then subtract a linear combination of digitals to reduce to the continuous case.

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  • $\begingroup$ That's a good starting point. To get to the more general case, the question now is: Is the cumulative distribution function of $X_t$ (SDE solution) also the solution to a PDE? Any ideas/pointers in that direction? $\endgroup$ – Johannes Gerer Mar 13 '15 at 13:02
  • $\begingroup$ if you consider three times s < t < T. The expectation at time s will be the expectation of the value at time t which is the expectation of the value at time T. the value $T$ will be smooth since it's given by integrating a smooth kernel against an L1 function. So in any interval (0,t] it's smooth for t < T. i.e. it's smooth in (0,T). $\endgroup$ – Mark Joshi Mar 14 '15 at 4:24
  • $\begingroup$ It is still hard to see how this works for a general SDE. Explaining this in a comment is too much to ask, so do you have any references? $\endgroup$ – Johannes Gerer Mar 14 '15 at 11:46
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$E\{f(X_T)\}$ can still exist even if $f$ is not continuous.

For example, $$ \begin{equation} f: x \mapsto \begin{cases} 1, \, x >= 3 \\ 0, \text{ otherwise} \end{cases} \end{equation} $$

Then $$ \begin{equation} E\{f(X_T)\} = P(\{X_T >= 3\}) \end{equation} $$

So, if you're example is a binary option which pays 1 if $X_T >= B$ and zero otherwise, and $\{X_t\}$ follows a standard diffusion, $$ \begin{equation} dX_t = X_t(r dt + \sigma dW_t) \end{equation} $$ you'll get $$ \begin{eqnarray} V_t &=& e^{-r\tau}\mathbb{E}\{1_{\{X_T >= B\}}|\mathcal{F_t}\} \\ &=& e^{-r\tau}P(\{X_T >= B\}) \\ &=& e^{-r\tau}\Phi\left(\frac{\log\left(\frac{X_t}{B}\right) + (r - \frac{\sigma^{2}}{2})\tau}{\sqrt{\tau}\sigma}\right) \end{eqnarray} $$, where $\tau := T - t$.

To calculate $\mathbb{E}\{f(X_T) | \mathcal{F_t}\}$ you would find the stochastic differential of $$\begin{equation} e^{-rT}\mathbb{E}\{f(X_T) | \mathcal{F_t}\} = e^{-rt}V_t. \end{equation} $$ Set the "dt-term" to zero (which gives the Black-Scholes PDE) and set appropriate boundary conditions. The temporal end boundary condition is your payoff function. For upper and lower bounds in the spatial dimension you would need to set "suitably large" values. Actual numerical schemes for solving PDEs can be found in standard textbooks on PDEs.

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  • $\begingroup$ This does not answer the question in any way. $\endgroup$ – Johannes Gerer Mar 12 '15 at 12:32
  • $\begingroup$ "Thats why I am asking for a reference for the caclulation of the expectation and not the PDE solution." $\endgroup$ – jensa Mar 12 '15 at 12:34
  • $\begingroup$ "A reference for calculating $E(f_T)$ by means of solving a PDE or Feynman-Kac equation." $\endgroup$ – Johannes Gerer Mar 12 '15 at 12:35
  • $\begingroup$ Yeah, I read that after =) Is he asking for both? $\endgroup$ – jensa Mar 12 '15 at 12:37
  • $\begingroup$ He wants to know how to price Binary options (or more generally Options with discontinuous payoffs) using a PDE. But thank's for your comments, I made the question more clear! $\endgroup$ – Johannes Gerer Mar 12 '15 at 12:38

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