Backward martingale representation and endogenous completeness in finance

Transcription

1 Backward martingale representation and endogenous completeness in finance Dmitry Kramkov (with Silviu Predoiu) Carnegie Mellon University 1 / 19

2 Bibliography Robert M. Anderson and Roberto C. Raimondo. Equilibrium in continuous-time financial markets: endogenously dynamically complete markets. Econometrica, 76(4):841 97, 28. J. Hugonnier, S. Malamud, and E. Trubowitz. Endogenous completeness of diffusion driven equilibrium markets. Econometrica, 8(3): , 212. Frank Riedel and Frederik Herzberg. Existence of financial equilibria in continuous time with potentially complete markets. arxiv:127.21v1, July 212. Dmitry Kramkov and Silviu Predoiu. Integral representation of martingales motivated by the problem of endogenous completeness in financial economics. arxiv: v3, October / 19

3 Martingale Integral Representation (Ω, F 1, F = (F t ) t [,1], P): a complete filtered probability space. Q: an equivalent probability measure. S = (S j t ): J-dimensional martingale under Q. We want to know whether any local martingale M = (M t ) under Q admits an integral representation with respect to S, that is, M t = M + H u ds u, t [, 1], for some predictable S-integrable process H = (H j t). Completeness in Mathematical Finance. Jacod s Theorem (2nd FTAP): the integral representation holds iff Q is the only martingale measure for S. Easy to verify if S is given in terms of local characteristics ( forward description). 3 / 19

4 Backward Martingale Representation Inputs: random variables ζ > and ψ = (ψ j ) The density of the martingale measure Q is defined by dq dp = const ζ. ψ is the terminal value for S: S t E Q [ψ F t ], t [, 1]. Problem Determine (easily verifiable) conditions on ζ and ψ so that the martingale representation property holds under Q and S. 4 / 19

5 Radner equilibrium Inputs: M agents with utility functions U m and initial random endowments Λ m, interest rate r =, J stocks with terminal values (dividends) ψ = (ψ j ). Output: stocks price process S = (St j ) such that 1. S 1 = ψ. 2. Given the S-market, the agents optimal strategies (stock s quantities) H m = (Ht m,j ) satisfy the clearing condition: M m=1 H m,j t =, t [, 1], j = 1,..., J. 5 / 19

6 Construction of Radner equilibrium Two steps: 1. Find static (Arrow-Debreu) equilibrium, that is, find a pricing measure Q such that if economic agents can trade any payoff ξ at the price p = E Q [ξ], then the clearing condition holds: the total wealth does not change. Main tool: Brower s fixed point theorem. 2. Define S t = E Q [ψ F t ], t [, 1], (ψ is the terminal dividend) and verify endogenous completeness of the S-market = Radner equilibrium. Other applications: equilibrium based price impact models (a project with Peter Bank; David German) and model s completion with options (Davis and Obloj). 6 / 19

7 Diffusion framework The random variables ψ = S 1 and ζ = const dq dp 1 ζ G(X 1 )e β(t,xt)dt, ψ j F j 1 (X 1 )e αj (t,x t)dt where Y t E[ζ F t ] and 1 are given by f j (t, X t )e αj (s,x s)ds dt g j (t, X t ) e (αj (s,x s)+β(s,x s))ds dt, j = 1,..., J, Y t F j, G : R d R and f j, g j, α j, β : [, 1] R d R are deterministic functions; X = (X i t ) is a d-dimensional diffusion: X t = X + b(s, X s )ds + σ(s, X s )dw s, t [, 1], with drift and volatility functions b i, σ ij : [, 1] R d R. 7 / 19

8 Assumptions on functions 1. The functions F j = F j (x) and G = G(x) are weakly differentiable and have exponential growth: 2. The Jacobian matrix F j + G Ne N x. ( F j x i ) the Lebesgue measure on R d. has rank d almost surely under 3. The maps t e N f j (t, ) ( e N x f j (t, x) ) x R d, t e N g j (t, ) and t α j (t, ), t β(t, ) of [, 1] to L are analytic on (, 1) and Hölder continuous on [, 1]. Careful with item 3: stronger than pointwise analyticity! Seems to be overlooked in the cited literature. 8 / 19

9 Assumptions on the diffusion X 1. The map t b i (t, ) of [, 1] to L is analytic on (, 1) and Hölder continuous on [, 1]. 2. The map t σ ij (t, ) of [, 1] to C is analytic on (, 1) and Hölder continuous on [, 1]. Moreover, σ = σ(t, x) is uniformly continuous with respect to x: σ(t, x) σ(t, y) ω( x y ). for some strictly increasing function ω = (ω(ɛ)) ɛ> such that ω(ɛ) as ɛ, and has a bounded inverse: σ 1 (t, x) N (uniform ellipticity for σσ ). Counter-example on t-analyticity condition in σ = σ(t, x). 9 / 19

10 Main result Theorem Assume that F = F X. Under the conditions above the martingale representation property holds for the probability measure Q with the density and the Q-martingale dq dp ζ E[ζ], S t E Q [ψ F t ], t [, 1]. 1 / 19

11 Comparison with the literature Assumptions on functions: In Anderson and Raimondo (28), Hugonnier et al. (212), and Riedel and Herzberg (212) The Jacobian matrix ( F j x i ) needs to have full rank only on some open set (counter-example in our setting). The small letter functions, that is, the functions of (t, x) should be (t, x)-analytic. Assumptions on diffusion X : In Anderson and Raimondo (28) X is a Brownian motion. In Hugonnier et al. (212) the diffusion coefficients b = b(t, x) and σ = σ(t, x) are either analytic with respect to (t, x) or the transitional probability is C / 19

12 Idea of the proof when P = Q ψ j F j 1 (X 1 )e αj (t,x t)dt + Ito s formula implies that 1 St j E[ψ j F t ] = e αj (s,x s)ds u j (t, X t )+ e αj (s,x s)ds f j (t, X t )dt, j = 1,..., J. where u j = u j (t, x) solves the parabolic equation: u j t + (L(t) + α j )u j + f j =, u j (T, x) = F j (x). Here L(t) the infinitesimal generator of X : L(t) = 1 2 d (σσ ) ij 2 (t, x) x i x j + i,j=1 r e αj (s,x s)ds f j (r, X r )dr, d i=1 b i (t, x) x i. 12 / 19

13 Idea of the proof when P = Q Our assumptions imply that 1. u j (t, ) F j as t T in Sobolev s spaces W 1 p for all p > t u j (t, ) is an analytic map of (, 1) to W 2 p. It follows that (t, x) det u x (t, x) is 1. continuous on [, 1] R d ; 2. t-analytic on (, 1); 3. det u x (1, x) = det F x (x), x R d a.s. Then det u x (t, x) on [, 1] R d a.s.. As s S t = u(, X ) + e α(r,xr )dr u x (s, X s )σ(s, X s )dw s, we deduce that every martingale is a stochastic integral under S. 13 / 19

14 Elements of the proof Parabolic PDEs: Evolution equations in L p spaces (maximal regularity, analyticity theorem by Kato and Tanabe). Elliptic equations in Sobolev spaces (sectoriality property). Interpolation theory (W 1 p is the midpoint of L p and W 2 p in complex interpolation). Stochastic Analysis: Krylov s variant of Ito s formula (instead of C 2 we can have W 2 p with p d under the uniform ellipticity condition). 14 / 19

15 Counter-example on t-analyticity in σ = σ(t, x) X t = X + b(s, X s )ds + σ(s, X s )dw s, t [, 1]. Recall that b = b(t, x) and σ = σ(t, x) have only minimal classical regularity conditions with respect to x; very strong analyticity assumptions with respect to t. We gave an explicit (!) example which shows that the t-analyticity assumption on the volatility matrix can not be removed. In our construction, dimension d = J = 2 (will not work for d = 1) both σ and its inverse σ 1 are C -matrices on [, 1] R 2 which are bounded with all their derivatives and have analytic restrictions to [, 1 2 ) R2 and ( 1 2, 1] R2. 15 / 19

16 Counter-example on t-analyticity in σ = σ(t, x) g = g(t) is a C -function on [, 1] which equals on [, 1 2 ], while it is analytic and strictly positive on ( 1 2, 1]. h = h(t, y) is an analytic function on [, 1] R such that h 1, h(1, ) const, and For instance, we can take h t h 2 y 2 =. h(t, y) = 1 t 1 (1 + e 2 sin y). 2 2-dimensional diffusion (X, Y ) on [, 1]: X t = 1 + g(s)h(s, Ys )db s, Y t = W t, where B and W are independent Brownian motions. 16 / 19

17 Counter-example on t-analyticity in σ = σ(t, x) Set where Q = P, ψ = (F (X 1, Y 1 ), H(X 1, Y 1 )), F (x, y) = x, H(x, y) = x 2 1 h(1, y) 1 g(t)dt. The determinant of the Jacobian matrix for (F, H): F H x y F H y x = h (1, y) y as h(1, ) is non-constant and analytic. 1 g(t)dt, 17 / 19

18 Counter-example on t-analyticity in σ = σ(t, x) Ito s formula shows that S t E[F (X 1, Y 1 ) F t ] = X t, R t E[H(X 1, Y 1 ) F t ] = X 2 t t h(t, Y t ) As g(t) = for t [, 1 2 ], it follows that on [, 1 2 ] S t = B t, R t = B 2 t t. g(s)ds, Clearly, the Brownian motion Y = W can not be written as a stochastic integral with respect to (S, R). 18 / 19

19 Summary We gave conditions on diffusion X and functions F j, G, α j, β = β(t, x), and f j, g j so that the integral representation holds for the measure Q and the Q-martingale S = (S j ) defined by dq/dp = const ζ and S j 1 = ψj, where 1 ζ G(X 1 )e β(t,xt)dt, ψ j F j 1 (X 1 )e αj (t,x t)dt and Y t E[ζ F t ]. 1 f j (t, X t )e αj (s,x s)ds dt g j (t, X t ) e (αj (s,x s)+β(s,x s))ds dt, j = 1,..., J, Y t The diffusion coefficients of X have only minimal x-regularity. However, they are t-analytical (a counter-example for σ). The study is motivated by the problem of endogenous completeness in financial economics. 19 / 19