Russo–Vallois integral

Definitions

Definition: A sequence H_n of stochastic processes converges uniformly on compact sets in probability to a process H,

:H=\text{ucp-}\lim_{n\rightarrow\infty}H_n,

if, for every \varepsilon0 and T0,

:\lim_{n\rightarrow\infty}\mathbb{P}(\sup_{0\leq t\leq T}|H_n(t)-H(t)|\varepsilon)=0.

One sets:

:I^-(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^tf(s)(g(s+\varepsilon)-g(s)),ds :I^+(\varepsilon,t,f,dg)={1\over\varepsilon}\int_0^t f(s)(g(s)-g(s-\varepsilon)) , ds

and

:[f,g]_\varepsilon (t)={1\over \varepsilon}\int_0^t(f(s+\varepsilon)-f(s))(g(s+\varepsilon)-g(s)),ds.

Definition: The forward integral is defined as the ucp-limit of

:I^-: \int_0^t fd^-g=\text{ucp-}\lim_{\varepsilon\rightarrow\infty (0?)}I^-(\varepsilon,t,f,dg).

Definition: The backward integral is defined as the ucp-limit of

:I^+: \int_0^t f , d^+g = \text{ucp-}\lim_{\varepsilon\rightarrow\infty (0?)}I^+(\varepsilon,t,f,dg).

Definition: The generalized bracket is defined as the ucp-limit of

:[f,g]\varepsilon: [f,g]\varepsilon=\text{ucp-}\lim_{\varepsilon\rightarrow\infty}[f,g]_\varepsilon (t).

For continuous semimartingales X,Y and a càdlàg function H, the Russo–Vallois integral coincidences with the usual Itô integral:

:\int_0^t H_s , dX_s=\int_0^t H , d^-X.

In this case the generalised bracket is equal to the classical covariation. In the special case, this means that the process

:[X]:=[X,X] ,

is equal to the quadratic variation process.

Also for the Russo-Vallois Integral an Ito formula holds: If X is a continuous semimartingale and

:f\in C_2(\mathbb{R}),

then

:f(X_t)=f(X_0)+\int_0^t f'(X_s) , dX_s + {1\over 2}\int_0^t f''(X_s) , d[X]_s.

By a duality result of Triebel one can provide optimal classes of Besov spaces, where the Russo–Vallois integral can be defined. The norm in the Besov space

:B_{p,q}^\lambda(\mathbb{R}^N)

is given by

:||f||{p,q}^\lambda=||f||{L_p} + \left(\int_0^\infty {1\over |h|^{1+\lambda q}}(||f(x+h)-f(x)||_{L_p})^q , dh\right)^{1/q}

with the well known modification for q=\infty. Then the following theorem holds:

Theorem: Suppose

:f\in B_{p,q}^\lambda, :g\in B_{p',q'}^{1-\lambda}, :1/p+1/p'=1\text{ and }1/q+1/q'=1.

Then the Russo–Vallois integral

:\int f , dg

exists and for some constant c one has

:\left| \int f , dg \right| \leq c ||f||{p,q}^\alpha ||g||{p',q'}^{1-\alpha}.

Notice that in this case the Russo–Vallois integral coincides with the Riemann–Stieltjes integral and with the Young integral for functions with finite p-variation.

References