Regarding notation of functions

I have little a problem in understanding some notation, when a function is defined.

It sometimes says

Assume furthermore that we are given a $C^{1,2}$ function $$f:R_+\times R\to R$$

What does $C^{1,2}$ mean? This notation appear in Arbitrage Theory in Continuous Time by Thomas Bjork. Another example is below.

Theorem 4.10 (Itˆo’s formula) Assume that the process X has a stochastic differential given by

dX(t) = μ(t)dt + σ(t)dW(t), (4.27)

where μ and σ are adapted processes, and let f be a $C^{1,2}$-function.

3 Answers

I've never seen this notation but I'd guess that it means functions that are $C^1$ as a function of the first variable and $C^2$ as a function of the second variable, in both cases the other variable being held fixed.

• Makes sense: Itô's formula involves $\frac{\partial}{\partial t} f$ and $\frac{\partial^2}{\partial x^2} f$. – dafinguzman Nov 17 '15 at 15:01
• Just to clear my doubt. One silly question. When we say "functions that are C1" and "functions that are C2" what exactly do we mean? please explain, – Javed Nov 23 '15 at 21:55
• @Javed, $C^k$ means functions differentiable to order $k$ with continuous derivatives of all orders $0,\dots,k$. – lhf Nov 23 '15 at 23:10

Other suggestions:

1) Perhaps this notation means that $f$ is $C^1$ and has a derivative that has finite norm in $L^2$.

2) It's also possible that this notation means that $f$ is $C^1$ and has a derivative that is Lipschitz continuous with Lipschitz constant 2.

For any constant $k \in R$, the function $:R_+ \to R$ defined by $x \mapsto f(x,k)$ is $C^1$ and for any constant $k \in R_+$, the function $:R \to R$ defined by $y \mapsto f(k,y)$ is $C^2$