On a stochastic differential equation arising in a price impact model

It is the purpose of this paper to derive conditions for existence and uniqueness of solutions to this SDE. A special feature of our study is that the SDE’s coefficients are defined only implicitly and, hence, standard Lipschitz and growth conditions are not easily applicable. We aim to provide readily verifiable criteria in terms of the model primitives: the market makers’ utility functions and initial endowments and stocks’ dividends.

Our main results, stated in Section 3, yield such conditions for locally bounded order flows. Theorem 3.1 shows that if the market makers’ risk aversions are bounded along with sufficiently many of their derivatives then there exist unique maximal local solutions. Its proof relies on Sobolev’s embedding results for stochastic integrals due to Sznitman [7]. For the special case of exponential utilities Theorem 3.2 establishes the existence of a unique global solution. Theorem 3.3 proves this under the alternative assumption that, in a Brownian framework, the market makers’ initial endowments and stocks’ dividends are Malliavin differentiable and risk aversions are bounded along with their first derivatives. The main tool here is the Clark-Ocone formula for D 1,1 from Karatzas, Ocone, and Li [3].

We use the conventions and notations of the parent paper [2, Section 2]. In particular, for a metric space X we denote by C([0, 1], X) the space of continuous maps from [0, 1] to X. For nonnegative integers m and n and an open set V ⊂ R d we denote by C m = C m (V ) = C m (V, R n ) the Fréchet space of m-times continuously differentiable functions f : V → R n with the topology generated by the semi-norms

where C is a compact subset of V , β = (β 1 , . . . , β d ) is a multi-index of non-negative integers, |β| d i=1 β i , and

x∈V is a family of stochastic processes K(x) = (K t (x)) t∈[0,1] , then we say that K has values in C m (V ) if for every t ∈ [0, 1] the stochastic field K t on V has sample paths in C m (V ).

Let u m = u m (x), m = 1, . . . , M , be functions on the real line R satisfying Assumption 2.1. Each u m is strictly concave, strictly increasing, twice continuously differentiable,

and for some constant c > 0 the absolute risk aversion

Denote by r = r(v, x) the v-weighted sup-convolution:

(2) r(v, x) max

The main properties of the saddle function r = r(v, x) are collected in [1, Section 4.1]. In particular, for v ∈ (0, ∞) M , the function r(v, •) has the same properties as the functions u m , m = 1, . . . , M , of Assumption 2.1 and, for c > 0 from (1),

From (3) we deduce the exponential growth property (4)

where for real x we denote by x + max(x, 0) and x -(-x) + the positive and negative parts of x. As r(v, x) → 0 when x → ∞, we also obtain the estimates

Let (Ω, F 1 , (F t ) 0≤t≤1 , P) be a complete filtered probability space satisfying Assumption 2.2. There is a d-dimensional Brownian motion B = (B i ) such that every local martingale M admits an integral representation

for some predictable process

Of course, this assumption holds if the filtration is generated by B.

Let Σ 0 and ψ = (ψ j ) j=1,…,J be random variables. We denote

and assume that

From ( 4) and ( 5) we deduce that this integrability condition holds if

Under Assumptions 2.1 and 2.2 and the integrability condition (6) the stochastic fields

respectively, and for a multi-index β = (β 1 , . . . , β M +1+J ) with |β| ≤ 2

see Theorems 4.3 and 4.4 and Corollary 5.4 in [2]. In view of Assumption 2.2 the martingales ∂ β F , |β| ≤ 2, admit integral representations: ∂v m on A are continuous) and, for u ∈ (-∞, 0) M and q ∈ R J , define an M × ddimensional process K(u, q) by ( 9)

The paper is concerned with the existence and uniqueness of a (strong) solution U = (U m ) m=1,…,M with values in (-∞, 0) M to the stochastic differential equation (10)

parameterized by a predictable process Q with values in R J . This equation arises in the price impact model of [2], where it describes the evolution of the expected utilities U = (U m ) of M market makers who collectively acquire Q = (Q j ) stocks from a “large” investor. The functions u m = u m (x), m = 1, . . . , M , specify the market makers’ utilities for terminal wealth and Σ 0 stands for their total initial random endowment. The cumulative dividends paid by the stocks are given by ψ = (ψ j ). According to [2,Theorem 5.8] a predictable process Q = (Q j ) is a (well-defined) investment strategy for the large investor if and only if (10) has a unique (global) solution U . In this case, the total cash amount received by the market makers (and paid by the investor) up to time t is given by G

Of course, it is easy to state standard conditions on the stochastic field K guaranteeing the existence and uniquen