2

Let

  • $(U,\langle\;\cdot\;,\;\cdot\;\rangle)$ and $H$ be separable Hilbert spaces
  • $Q\in\mathfrak L(U)$$^1$ be nonnegative and symmetric with finite trace
  • $f:[0,\infty)\times H\to\mathbb R$ be Fréchet differentiable in time (first argument) and twice Fréchet differentiable in space (second argument)
  • $(\Omega,\mathcal A,\operatorname P)$ be a probability space
  • $X_0$ and $Y$ be $H$-valued random variables on $(\Omega,\mathcal A,\operatorname P)$
  • $Z$ be a $\mathfrak L(U,H)$-valued random variable on $(\Omega,\mathcal A,\operatorname P)$
  • $(W_t)_{t\ge 0}$ be a $Q$-Wiener process on $(\Omega,\mathcal A,\operatorname P)$

I want to derive an Itō formula for $$Y_t:=f(t,X_t)\;\;\;\text{for }t\ge 0\;,$$ where $$X_t:=X_0+tY+ZW_t\;\;\;\text{for }t>0\;.$$

I've tried the following:

  • Let $t\ge 0$ and $t_0,\ldots,t_n\ge 0$ with $$0=t_0<\cdots<t_n=t$$
  • Then, $$Y_t-Y_0=\sum_{i=1}^n\underbrace{\left[Y_{t_i}-f\left(t_{i-1},X_{t_i}\right)\right]}_{=:\;S_i}+\sum_{i=1}^n\underbrace{\left[f\left(t_{i-1},X_{t_i}\right)-Y_{t_{i-1}}\right]}_{=:\;T_i}$$
  • By Taylor's formula$^2$, we obtain $$S_i=\Delta t_i\frac{\partial f}{\partial t}\left(t_{i-1}+\theta_i\Delta t_i,X_{t_i}\right)$$ for some $\theta_i\in [0,1]$, where $\Delta t_i:=t_i-t_{i-1}$, and$^3$ $$T_i=\langle\Delta X_i,\frac{\partial f}{\partial x}\left(t_{i-1},X_{t_{i-1}}\right)\rangle+\frac 12\langle\Delta X_i,\frac{\partial^2f}{\partial x^2}\left(t_{i-1},X_{t_{i-1}}+\vartheta_i\Delta X_i\right)\Delta X_i\rangle$$ for some $[0,1]$-valued random variable $\vartheta_i$ on $(\Omega,\mathcal A,\operatorname P)$, where $\Delta X_i:=X_{t_i}-X_{t_{i-1}}$

It might be a stupid question, but I would like to know how I need to characterize $\vartheta_i$ in the right way (I've made no statement about its relationship to $X$ or how it is distributed).

$^1$ Let $\mathfrak L(A,B)$ be the space of bounded, linear operators from $A$ to $B$.

$^2$ See also this question about Taylor's formula in a normed space.

$^3$ See also this question about the Fréchet derivative of a real-valued function defined on a Hilbert space.

Community
  • 1
0xbadf00d
  • 14,208
  • A quick comment that may help: the only separable Hilbert spaces are either finite-dimensional (i.e. isometric to $\mathbb{R}^d$ for some $d$) or isometric to $\ell^2$ (this fact is noted here). Thus, you can work more concretely with $U$ and $H$ if you'd like. – Marcus M Apr 23 '16 at 17:10
  • @MarcusM Later, I want to work with $U=H$ and $H=\left[L^2(\mathcal V)\right]^3$ or $H=\left[H_0^2(\mathcal V)\right]^3$. However, the statement presented here is provable for any $U,H$ (and of course, the process $X$ can be way more general, but even in that way more general case the corresponding statement is reducible to $X$'s of the form presented here). – 0xbadf00d Apr 23 '16 at 18:14

0 Answers0