Stochastic differential games with inside information

Abstract

MSC(2010): 60H40, 60H07, 60H05, 60J75, 60J75, 60Gxx, 91G80, 93E20, 93E10Keywords: Optimal control, inside information, white noise, Hida-Malliavin calculus, Donsker delta functional, anticipative stochastic calculus, maximum principle, BSDE, optimal insider consumption and optimal insider portfolio under model uncertainty. AbstractWe study stochastic differential games of jump diffusions, where the players have access to inside information. Our approach is based on anticipative stochastic calculus, white noise, Hida-Malliavin calculus, forward integrals and the Donsker delta functional. We obtain a characterization of Nash equilibria of such games in terms of the corresponding Hamiltonians. This is used to study applications to insider games in finance, specifically optimal insider consumption and optimal insider portfolio under model uncertainty.We now explain this in more detail:The system we consider, is described by a stochastic differential equation driven by a Brownian motion B(t) and an independent compensated Poisson random measureÑ (dt, dζ), jointly defined on a filtered probability space (Ω, F = {F t } t≥0 , P) satisfying the usual conditions. We assume that the inside information is of initial enlargement type. Specifically, we assume that the two inside filtrations H 1 , H 2 representing the information flows available to player 1 and player 2, respectively, have the formfor all t, where Y i is a given F T 0 -measurable random variable, for some fixed T 0 > T > t.Here the insider control process u(t) = (u 1 (t), u 2 (t)), where u i (t) is the control of player i; i=1,2. Thus we assume that the value at time t of our insider control process u i (t) is allowed to depend on both Y i and F t ; i = 1, 2. In other words, u i is assumed to be H i -adapted for i = 1, 2. Therefore they have the formFor simplicity (albeit with some abuse of notation) we will in the following write u i in stead of u i ; i = 1, 2. Consider a controlled stochastic processwhere u i (t) = u i (t, y i ) y i =Y i is the control process of insider i; i = 1, 2, and the (anticipating) stochastic integrals are interpreted as forward integrals, as introduced in [RV] (Brownian motion case) and in [DMØP1] (Poisson random measure case). A motivation for using forward integrals in the modelling of insider control is given in [BØ]. Let A i denote a given set of admissible H i −adapted controls u i of player i, with values inThe performance functional J i (u); u = (u 1 , u 2 ) of player i is defined by, writing y = (y 1 , y 2 ) and dy = dy 1 dy 2 ,x(t, y 1 , y 2 ), u 1 (t, y 1 ), u 2 (t, y 2 ), y)E[δ Y 1 ,Y 2 (y 1 , y 2 )|F t ]dt + g i (x(T, y 1 , y 2 ), y 1 , y 2 )E[δ Y 1 ,Y 2 (y 1 , y 2 )|F T ]}dy]; i = 1, 2.(1.4)A Nash equilibrium for the game (3.3)-(3.8) is a pairû = (û 1 ,û 2 ) ∈ A 1 × A 2 such that sup u 1 ∈A 1 J 1 (u 1 ,û 2 ) ≤ J 1 (û 1 ,û 2 ) (1.5) {eq2.10}

Optimal portfolios for different anticipating integrals under insider information

“…We anticipate that the forward integral is the only one among these that provides an optimal investment strategy that takes advantage of the anticipating condition in the financial sense. Therefore our analysis further supports the use of this integral in a financial context as employed, for instance, in [4,8,9,10,11,23,25].…”

Optimal Portfolios for Different Anticipating Integrals under Insider Information

“…We anticipate that the forward integral is the only one among these that provides an optimal investment strategy that takes advantage of the anticipating condition in the financial sense. Therefore our analysis further supports the use of this integral in a financial context as employed, for instance, in [13][14][15][16][17][18][19]. It is also worth mentioning that the problem of insider trading has been studied by means of different approaches, notably enlargement of filtrations [20,21], but semimartingale treatments are not as general as the use of anticipating stochastic calculus [14].…”

“…The second is that we want to maximize the clarity of our derivations. Once the role of the different anticipating integrals in the context of finance is clear, one is able to approach much more general questions as done for instance in [13][14][15][16][17][18][19] with the forward integral. Herein we aim to highlight some of the contrasts between the uses of these integrals in a financial context, so the nature of this work is fundamentally methodological.…”

“…Now taking σ s = σ and µ s = µ as two constant processes, and η = M f (B(T)), the different factors in (16) become:…”

A simple comparison between Skorokhod & Russo-Vallois integration for insider trading

“…The problem of discerning the strategies of a dishonest trader who possesses privileged information in a financial market, "the insider", is a venerable one in the field of stochastic analysis applied to finance [2,4,9,11,14,16] and continues to be of current interest [5,6,7]. Within this work, a much simplified version of this problem is considered, as our goal is to favor the accessibility to the comparison between the two anticipating stochastic integrals in the context of finance.…”