Abstract
We consider a discrete time analog of $G$--expectations and we prove that in the case where the time step goes to 0 the corresponding values converge to the original $G$--expectation. Furthermore we provide error estimates for the convergence rate. This paper is continuation of [4]. Our main tool is a strong approximation theorem which we derive for general discrete time martingales.
Highlights
In this paper we study numerical schemes for G–expectations, which were introduced recently by Peng
The motivation to study G–expectations comes from mathematical finance, and in particular from risk measures and pricing under volatility uncertainty
Our starting point is the dual view on G–expectation via volatility uncertainty, which yields the representation ξ → supP ∈P EP [ξ] where P is the set of probabilities on C([0, T ]; Rd) such that under any P ∈ P, the canonical process B is a martingale with volatility d B /dt taking values in a compact convex subset D ⊂ spaces (Rd)n+1 and (Sd)+ of positive definite matrices
Summary
In this paper we study numerical schemes for G–expectations, which were introduced recently by Peng (see [7] and [8]). Ξ can represent an award of a path dependent European contingent claim In this case the reward is a functional of the stock price, which is equal to the Doolean exponential of the canonical process, and so quadratic variation appears naturally. In [4] the authors introduced a volatility uncertainty in discrete time and an analog of the Peng G–expectation. They proved that the discrete time values converge to. By deriving a strong invariance principle for general discrete time martingales, we are able to provide error estimates for the convergence rate of the current scheme.
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