ファインマン・カッツ公式
最近よく金融工学に現れるファインマンカッツ公式を考えています。これは放物経偏微分方程式の解を確立過程と関係する。The starting point of Feynman Kac seems to be a partial differential differential equation like Black-Scholes, and the point is to relate this PDE to the conditional expectation of an integral and an expo ential, which are functions derived from a Brownian motion.
Given the PDE:
du(x,t)/dt + mu(x,t)*du(x,t)/dx + (1/2)s^2(x,t)d^2u(x,t)/dx^2-V(x,t)+f(x,t)=0
u(x,t) = EQ[INTEGRAL*1f(xr,r)dr + exp(-INTEGRAL(t to T)(V(xtau, tau))dtau)*psi(xT)|xt=x]
where EQ is the expectation under measure Q.
Where this formula is used practically is difficult to say, although it would enable easier discretization of expected future derivatives payoffs dependent on a random underlying X, some event horizon time r, and some maturity time T.
この定理な証明がわかりにくいです。 Incredibly, the proof involves beginning with applying Ito's Lemma to the interior of the expectation above, and ending with substituting u(x,t) for Y(t) (Y(0) in some texts). The initial choice of Y, as the interior of the integral above with psi replaced by u and T replaced by s, is unclear. Y(s) = INTEGRAL*2f(xr,r)dr + exp(-INTEGRAL(t to s)(V(xtau, tau))dtau)*u(Xs,s)
The proof steps involve obtaining the differential dY via Ito's lemma, integrating to obtain a stochastic process, taking the expectation, and substituting u(x,t) for Y(t) (or E(Y(t)|Xt=x)). These steps result in the desired outcome, but it is unclear what the intuition leading to the initial Y formula is. It seems that this technique of creating a formula in terms of the solution to second-order differential equations and an integral arose as a heuristic in quantum physics or thermodynamics.
今度伊藤の補題の証明を復習したいです。
