On the control of the difference between two Brownian motions: a dynamic\n copula approach

2016/01/18 by Thomas Deschatre, Deschatre, Thomas
Decision Sciences · Economics, Econometrics and Finance · #FOS: Mathematics #Financial Risk and Volatility Modeling #Probability (math.PR) #Probability and Risk Models #Stochastic processes and financial applications

paper · pdf · doi:10.48550/arxiv.1601.04546

openalex publication_date 2016/01/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We propose new copulae to model the dependence between two Brownian motions\nand to control the distribution of their difference. Our approach is based on\nthe copula between the Brownian motion and its reflection. We show that the\nclass of admissible copulae for the Brownian motions are not limited to the\nclass of Gaussian copulae and that it also contains asymmetric copulae. These\ncopulae allow for the survival function of the difference between two Brownian\nmotions to have higher value in the right tail than in the Gaussian copula\ncase. Considering two Brownian motions B_t¹ and B_t², the main result is\nthat the range of possible values for \ℙ\(B_t¹-B²_t \≥\n\η\) with \η > 0 is the same for Markovian pairs and all pairs of\nBrownian motions, that is\n\[0,2\Φ\(\(-\η)/(2\√(t))\)\] with \Φ being\nthe cumulative distribution function of a standard Gaussian random variable.\n

Citations

• MODELLING ASYMMETRIC EXCHANGE RATE DEPENDENCE*

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