Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Peak Value-at-Risk Estimation of Stochastic Processes using Occupation Measures

Jared Miller, Matteo Tacchi, Mario Sznaier, Ashkan Jasour

Abstract

This paper formulates algorithms to upper-bound the maximum Value-at-Risk (VaR) of a state function along trajectories of stochastic processes. The VaR is upper bounded by two methods: minimax tail-bounds (Cantelli/Vysochanskij-Petunin) and Expected Shortfall/Conditional Value-at-Risk (ES). Tail-bounds lead to a infinite-dimensional Second Order Cone Program (SOCP) in occupation measures, while the ES approach creates a Linear Program (LP) in occupation measures. Under compactness and regularity conditions, there is no relaxation gap between the infinite-dimensional convex programs and their nonconvex optimal-stopping stochastic problems. Upper-bounds on the SOCP and LP are obtained by a sequence of semidefinite programs through the moment-Sum-of-Squares hierarchy. The VaR-upper-bounds are demonstrated on example continuous-time and discrete-time polynomial stochastic processes.

Peak Value-at-Risk Estimation of Stochastic Processes using Occupation Measures

Abstract

This paper formulates algorithms to upper-bound the maximum Value-at-Risk (VaR) of a state function along trajectories of stochastic processes. The VaR is upper bounded by two methods: minimax tail-bounds (Cantelli/Vysochanskij-Petunin) and Expected Shortfall/Conditional Value-at-Risk (ES). Tail-bounds lead to a infinite-dimensional Second Order Cone Program (SOCP) in occupation measures, while the ES approach creates a Linear Program (LP) in occupation measures. Under compactness and regularity conditions, there is no relaxation gap between the infinite-dimensional convex programs and their nonconvex optimal-stopping stochastic problems. Upper-bounds on the SOCP and LP are obtained by a sequence of semidefinite programs through the moment-Sum-of-Squares hierarchy. The VaR-upper-bounds are demonstrated on example continuous-time and discrete-time polynomial stochastic processes.
Paper Structure (39 sections, 23 theorems, 76 equations, 6 figures, 17 tables)

This paper contains 39 sections, 23 theorems, 76 equations, 6 figures, 17 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation
  4. Probability Tail Bounds and Value-at-Risk
  5. Concentration Bounds/Minimax
  6. Conditional Value-at-Risk / Expected Shortfall
  7. Stochastic Processes and Occupation Measures
  8. Chance-Peak Problem Statement
  9. Assumptions
  10. VaR Problem
  11. Tail-Bound Program
  12. Tail-Bound Problem
  13. Nonlinear Measure Program
  14. Measure Second-Order Cone Program
  15. Dual Second-Order Cone Program
  16. ...and 24 more sections

Key Result

Lemma 2.1

Defining the positive part $f_+ = f\vee 0$ of a function $f$, the CVAR is the solution to the parametric problem

Figures (6)

  • Figure 1: VAR and CVAR of a unit normal distribution at $\epsilon = 0.1$
  • Figure 2: Trajectories of \ref{['eq:flow_sde']} with mean (dash-dot red), CVAR $\epsilon=0.15$ (dotted black), and VP $\epsilon=0.15$ (solid red) bounds
  • Figure 3: Trajectories of \ref{['eq:twist_sde']} with mean VP (solid red), mean CVAR (translucent black), and $\epsilon=0.15$ (translucent red) bounds
  • Figure 4: Trajectories of \ref{['eq:scatter_discrete']} with $\epsilon = \{0.5, \textrm{ES} \ 0.15\}$ bounds
  • Figure 5: Trajectories of the switched system \ref{['eq:switched_sde']} with $\epsilon=\{0.5, \text{\ac{CVAR}} \ 0.15, \text{\ac{VP}} \ 0.15\}$ bounds
  • ...and 1 more figures

Theorems & Definitions (51)

  • Definition 2.1
  • Remark 1
  • Lemma 2.1: Equations 4 and 5 of rockafellar2002conditional
  • Lemma 2.2: Equation 5.5 of follmer2010convex
  • Remark 2
  • Remark 3
  • Theorem 4.2
  • proof
  • Remark 4
  • Lemma 4.3
  • ...and 41 more