Setup
Suppose I have a data set with a categorical variable $Y$ (with possible values $j=1,\dots,J$) and another variable $X$. I wish to classify $Y$ based on the information in $X$. I am facing a loss function $l(\hat y,y)$ where $\hat y$ is the predicted class and $y$ is the actual/true class. $l$ can be represented by a matrix $\mathbf{L}$ where rows correspond to actual classes and columns to predicted ones. Each off-diagonal cell $l_{ij}$ ($i\neq j$) of $\mathbf{L}$ contains the loss associated with the specific misclassification.

Steps taken
I fit a (multinomial) logistic regression to the data. Given the fitted coefficients and a new data point with a known $X$ value $x_0$ but an unknown $Y$ value $y_0$, I obtain the fitted class probabilities $\hat p_0$ (a vector). I wish to classify $y_0$, i.e. obtain $\hat y_0$ so as to minimize the expected loss.

Questions
What is the optimal classification rule in this setting?
Could you also recommend a textbook chapter on the topic?

My problem is somewhat similar to Classification optimal decisions considering a loss function but my setting is frequentist and I do not have prior distribution of classes.