Analyzing model robustness via a distortion of the stochastic root: A Dirichlet prior approach

It is standard in quantitative risk management to model a random vector X:={X_tk} k=1,..,d X:=X_tkk=1,d of consecutive log-returns to ultimately analyze the probability law of the accumulated return X_t1 +X_tdX_t1+X_td. By the Markov regression representation (see [25]), any stochastic model for X can be represented as X_tk =f_k (X_t1,..,X_tk-1,U_k)X_tk=f_k(X_t1,X_tk-1,U_k), k=1,..,d k=1,d, yielding a decomposition into a vector:=Ukk=1,..,d U:=U_kk=1,d of i.i.d. random variables accounting for the randomness in the model, and a function f:={f k } k=1,..,d f:=f_kk=1,d representing the economic reasoning behind. For most models, f is known explicitly and U_k may be interpreted as an exogenous risk factor affecting the return X_tk in time step k. While existing literature addresses model uncertainty by manipulating the function f, we introduce a new philosophy by distorting the source of randomness U and interpret this as an analysis of the model's robustness. We impose consistency conditions for a reasonable distortion and present a suitable probability law and a stochastic representation for U based on a Dirichlet prior. The resulting framework has one parameter c[0]c[0]tuning the severity of the imposed distortion. The universal nature of the methodology is illustrated by means of a case study comparing the effect of the distortion to different models for X. As a mathematical byproduct, the consistency conditions of the suggested distortion function reveal interesting insights into the dependence structure between samples from a Dirichlet prior.