Dynamically Consistent Nonlinear Evaluations and Expectations

How an agent (or a firm, an investor, a financial market) evaluates a contingent claim, say a European type of derivatives X, with maturity t? In this paper we study a dynamic evaluation of this problem. We denote by {Ft}t≥0, the information acquired by this agent. The value X is known at the maturity t means that X is an Ft–measurable random variable. We denote by Es,t[X] the evaluated value of X at the time s ≤ t. Es,t[X] is Fs–measurable since his evaluation is based on his information at the time s. Thus Es,t[·] is an operator that maps an Ft–measurable random variable to an Fs–measurable one. A system of operators {Es,t[·]}0≤s≤t<∞ is called Ft–consistent evaluations if it satisfies the following conditions: (A1) Es,t[X] ≥ Es,t[Y ], if X ≥ Y ; (A2) Et,t[X] = X; (A3) Er,sEs,t[X] = Er,t[X], for r ≤ s ≤ t; (A4) 1AEs,t[X1A] = 1AEs,t[X], if A ∈ Fs. In the situation where Ft is generated by a Brownian motion, we propose the so-called g–evaluation defined by E s,t[X] := ys, where y is the solution of the backward stochastic differential equation with generator g and with the terminal condition yt = X. This g–evaluation satisfies (A1)–(A4). We also provide examples to determine the function g = g(y, z) by testing. The main result of this paper is as follows: if a given Ft–consistent evaluation is Eμ–dominated, i.e., (A5) Es,t[X]−Es,t[X ] ≤ Eμ [X−X ], for a large enough μ > 0, where gμ = μ(|y| + |z|), then Es.t[·] is a g– evaluation