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Let $X_t$ be a stochastic process in $\mathbb{C}^n$ such that $$ dX_t = a(X_t,t)dt + b(X_t,t)dW_t.$$ And let $f:\mathbb{C}^n \to \mathbb{C}$.

Then how to compute $df(X_t)$ in complex?

If $f$ is a real $C^2$ function, one can readily apply Ito formula. However, I am not sure how one can compute $df(X_t)$ in complex case.

For example, let $n=2$ and $f(x) = x_1x_2^*$ where $*$ is conjugate, and $x=(x_1,x_2) \in \mathbb{C}^2$.

In real case, $f(x) = x_1x_2$ which is in $C^2$. Since $\nabla_xf(x) = [x_2,x_1]^T$ and $\nabla_x^2 f(x) = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, we have $$ df(X_t) =\left( [(X_t)_2,(X_t)_1]a(X_t,t) + (b(X_t,t)b(X_t,t)^T)_{1,2}\right)dt + [(X_t)_2,(X_t)_1]b(X_t,t)dW_t$$ where $X_t = [(X_t)_1, (X_t)_2]^T$.

However, in complex, $f(x)$ is not differentiable in $x_2$ becasue $$ \frac{\partial}{\partial x_2}f(x) = x_1\frac{\partial x_2^*}{\partial x_2} = \text{not defined}.$$

Any suggestions/comments/answers will very be appreciated.

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  • $\begingroup$ Maybe you deal with the real and the imaginary part separately? $X_t$ is a complex-valued stochastic process, for which it observes a decomposition $X_t=Y_t+iZ_t$, where both $Y_t$ and $Z_t$ are real-valued processes. Similarly, $f:\mathbb{C}^n\to\mathbb{C}$ observes $f=g+ih$, where $g$ and $h$ are both mappings from $\mathbb{C}^n=\mathbb{R}^{2n}$ to $\mathbb{R}$. In this way, you get back to the real-valued-real-variate-function case. $\endgroup$
    – hypernova
    Commented Apr 10, 2018 at 6:25

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