2
$\begingroup$

Consider a generic diffusion of the form $$dX_t=f(t,X_t)dt+dB_t,$$ where $f$ is some nice function and $B_t$ is a standard Brownian motion.

The marginal distributions of the integrals $$I:=\int_0^TB_t~dt\qquad J:=\int_0^TX_t~dt$$ can in principle be computed with fairly straightforward methods: $I$ is Gaussian by a classical Riemann sum argument, and $J$ can (again, in principle) be computed by Feynman-Kac.

Question. What about the joint distribution of $(I,J)$?

Are there general techniques that are well adapted to the solution of such problems? I'm also interested in any kind of nontrivial example where such computations have been made.

Improve this question
$\endgroup$
2
  • $\begingroup$ How does Feynman-Kac give the distribution of $J$? In the form I am familiar with, it gives the expected value of J. $\endgroup$
    – Nawaf Bou-Rabee
    Commented Oct 28, 2016 at 21:50
  • 2
    $\begingroup$ For example, in Wikipedia's formulation of F-K (en.wikipedia.org/wiki/Feynman-Kac_formula), I would set $f=0$, $\psi=1$, and $V(x)=\mu x$, and thus potentially obtain the Laplace transform. $\endgroup$
    – user78370
    Commented Oct 28, 2016 at 21:55

1 Answer 1

Reset to default
3
$\begingroup$

Consider the SDE: $$ \begin{cases} d X_t = f(t,X_t) dt + d B_t \\ d Y_t = d B_t \end{cases} $$ The infinitesimal generator of the process $(X_t,Y_t)$ is given by: $$ L_t g(x,y) = f(t,x) \partial_x g(x,y) + \frac{1}{2} \partial_{xx} g(x,y) + \partial_{xy} g(x,y) + \frac{1}{2} \partial_{yy} g(x,y) $$ Let $t_1$ and $t_2$ be real constants. At least formally, the solution to the PDE: $$ \partial_t u_t(x,y) = L_t u_t(x,y)+ (t_1 x + t_2 y) u_t(x,y)\;, \quad \left. u \right|_{t=0} = 1 $$ admits the stochastic representation $$ u_t(x,y)= \mathbb{E}_x \left\{ \exp( t_1 I + t_2 J ) \right\} $$ which we recognize as the joint moment generating function of $I$ and $J$.

Improve this answer
$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .