Are the Azéma-Yor processes truly remarkable?

Are the Azéma-Yor processes truly remarkable? Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Swiss Probability Seminar, 5 Dec 2007

Are the Azéma-Yor processes truly remarkable? A 40-minutes long affirmative answer Jan Obłój j.obloj@imperial.ac.uk based on joint works with L. Carraro, N. El Karoui, A. Meziou and M. Yor Swiss Probability Seminar, 5 Dec 2007

Table of Contents 1 Introduction Definition and group properties Examples 2 Applications to solving the Skorokhod embedding problem solving SDEs 3 Intrinsic characterizations via optimal properties via structural properties 4 Further martingales functions of" General project MM-martingales conjecture Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 3 / 22

1 Introduction Definition and group properties Examples 2 Applications to solving the Skorokhod embedding problem solving SDEs 3 Intrinsic characterizations via optimal properties via structural properties 4 Further martingales functions of" General project MM-martingales conjecture Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 4 / 22

Azéma-Yor processes Lemma Let (X t ) be a càdlàg semimartingale with continuous running supremum X t = sup s t X s and u a locally bounded function. Then M U (X) t := U(X t ) u(x t )(X t X t ) = where U(y) = y 0 u(x)dx. Furthermore if u 0 then MU (X) t = U(X t ). Remarks: (M U (X) t ) is a semimartingale, (M U (X) t ) is a local martingale when (X t ) is a local martingale t 0 u(x s )dx s, (1) when (X t ) is a continuous local martingale (1) holds for any locally integrable u. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 5 / 22

Azéma-Yor processes Lemma Let (X t ) be a càdlàg semimartingale with continuous running supremum X t = sup s t X s and u a locally bounded function. Then M U (X) t := U(X t ) u(x t )(X t X t ) = where U(y) = y 0 u(x)dx. Furthermore if u 0 then MU (X) t = U(X t ). t 0 u(x s )dx s, (1) Proof: Assume u C 1, general u via monotone class them. Integrating by parts M U (X) t =U(X t ) = t 0 t 0 u(x s )dx s (X s X s ) du(x s ) }{{} 0 on dx s t 0 u(x s )dx s + t 0 t 0 u(x s )d(x s X s ) [u(x), X X] t }{{} u(x s )dx s = t 0 u(x s )dx s. u 0 implies M U (X) t U(X t ) and equality in inf{t : X t = y}, for any y > 0. 0 Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 5 / 22

The set of Azéma-Yor processes Recall: M U (X) t = U(X t ) u(x t )(X t X t ), X t = sup s t X s. We have natural operations M U (X) t + M F (X) t = M U+F (X) t, αm U (X) t = M αu (X) t, α R. More importantly, we the following group structure: Lemma For (X t ) as previously consider u, v > 0. Then M V (M U (X)) t = M V U (X) t, in particular when V = U 1, i.e. v(y) = 1/u(V(y)), M V (M U (X)) t = X t. Proof: Note that M U (X) has a continuous running supremum. We have ( ) dm V (M U (X)) t = v M U (X) t ) dm U (X) t = v ( U(X t ) ) u(x t )dx t. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 6 / 22

Examples and easy applications Let (X t ) be a continuous martingale and T a stopping time Let u(x) = 1 x k, so that U(x) = (x k) +. M U (X) t = U(X t ) u(x t )(X t X t ) = (X t k)1 1 Xt k X (X t k t X t ) = X t 1 k1 Xt k X, t k and stopping in T we get kp(x T k) = EX T 1 Xt k, Doob s max equality. Let u(x) = px p 1, p > 1. M U (X) t = X p t px p 1 t (X t X t ) = px p 1 t X t (p 1)X p t. This gives Doob s L p inequality. It is also useful for (p = 2): EX T = E(X T X T ) E(X T X T ) 2 = E(X 2 T 2X T X T ) +EXT 2 }{{} = E X T. =0 Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 7 / 22

Skorokhod embedding problem Recall the Azéma-Yor solution to the Skorokhod embedding problem. Let (X t ) be a continuous local martingale. For a centred probability measure µ on R define 1 Ψ µ (x) = yµ(dy). µ([x, )) [x, ) Then the Azéma-Yor stopping time T µ = inf{t : X t Ψ µ (X t )} satisfies X Tµ µ and (X t Tµ ) is a UI martingale. One way of proving the result is to observe that actually T µ = inf{t : M U (X) t = 0} for a suitably chosen U. Then 1 1 = Uniform[0, 1]. M U (X) Tµ U(X Tµ ) One then recovers the law of X Tµ and in consequence the law of X Tµ = Ψ 1 µ (X Tµ ). Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 9 / 22

The Bachelier equation Theorem Let (X t ) be a semimartingale with X t 0. Consider ϕ : R R +, locally bounded away from zero, and put V(y) = y dx 0 ϕ(x). The Bachelier equation dy t = ϕ(y t )dx t, Y 0 = 0, (2) has a strong, pathwise unique solution given by Y t = M U (X) t, where U = V 1. Remarks: Solution to (2) is defined up to its explosion time inf{t : X t = V( )}. We could also allow X to explode. As everywhere, we can easily take arbitrary fixed, or random, starting points. Assumption ϕ > 0 essential. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 10 / 22

The Bachelier equation Theorem Let (X t ) be a semimartingale with X t 0. Consider ϕ : R R +, locally bounded away from zero, and put V(y) = y dx 0 ϕ(x). The Bachelier equation dy t = ϕ(y t )dx t, Y 0 = 0, (2) has a strong, pathwise unique solution given by Y t = M U (X) t, where U = V 1. Proof: We have dm U (X) t = u(x t )dx t = ϕ(u(x t ))dx t = ϕ(m U (X) t )dx t. Now let (Y t ) be any solution of (2). Since Y t 0, we can write dm V (Y) t = 1 ϕ(y t ) dy t = dx t, so that M V (Y) t = X t and thus Y t = M U (X) t. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 10 / 22

The Drawdown equation (1) Drawdown condition arrises as a natural restriction for fund managers on their portfolios. Assume: X t > 0 be a stock pice processes (a martingale), M t = t 0 A udx u is an investment strategy, (A u ) predictable, portfolio requirement: t T, M t αm t, for some α (0, 1). We call this the α-dd (drawdown) condition. Aim: among strategies which satisfy the α-dd condition find the maximizer of the final utility (cf. Cvitanić and Karatzas). Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 11 / 22

The Drawdown equation (2) Let w : R R be increasing with w(y) < y. We say that (M t ) satisfies w drawdown (w-dd) condition if M(t) w(m t ). Lemma Let u > 0. The Azéma-Yor process M U (X) satisfies w-dd constraint where V = U 1. ) M U (X) t w (M U (X) t, t 0, where w(x) = x V(x) V (x), Proof: Immediate, as u(x t )X t 0, we have: M U (X) t U(X t ) u(x t )X t = w(u(x t )) = w ( M U (X) t ). Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 12 / 22

The Drawdown equation (3) More to the point we can reverse the procedure. Theorem Let X t > 0 be as above, with X 0 = 1. Let w : R R be increasing with w(y) < y and put V(y) = exp ( y ds 1 s w(s)) assuming it is well defined for y [1, ) with V( ) =. The drawdown equation dy t = ( Y t w(y t ) ) dx t X t, Y 0 = 1, (3) admits a strong, pathwise unique, solution which satisfies the w-dd constraint: Y t > w(y t ) a.s. The solution is given explicitly by Y t = M U (X) t, where U = V 1. Remark: Equation (3) gives a natural interpretation to the investment strategy associated to M U (X). Proof: Note that w(x) = x V(x) V (x) and it follows that MU (X) satisfies the w-dd constraint. It solves the equation and uniqueness follows taking the inverse among Azéma-Yor processes and from the uniqueness of the Doléans-Dade exponential. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 13 / 22

Characterization via optimality Let U : R + R + increasing concave, sublinear at. Let (N t ) be a non-negative local martingale, N t t 0 a.s., EN 0 <. U(N t ) is the benchmark process. M U (N) is a UI martingale and concavity of U implies M U (N) t U(N t ). Furthermore, it is optimal: Theorem Let (P t ) be a UI martingale with P t U(N t ), P 0 = U(N 0 ). Then [ P t M U (N) t = U(N t ), and E h ( M U ) ] (N) E [h ( ) ] P for all h 0 concave increasing. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 15 / 22

Characterization via structure Theorem Let ( (X t ) be a continuous local martingale, X = a.s. The process H(Xt, X t ) : t 0 ) is a local martingale if and only if there exists u Lloc 1 such that H(X t, X t ) = M U (X) t = U(X t ) u(x t )(X t X t ) = where U(y) = y 0 u(x)dx. Put differently: Azéma-Yor martingales are the only M-martingales. Naturally we have the analogue for H(X t, X t ). t 0 u(x s )dx s, Large classes of martingales can be described, e.g. u 0 and U( ) <. Note that given H t = H(X t, X t ), knowing X t we can recover u. As a consequence: H(X + t, X t ) is a local martingale in the filtration of (X t ) iff H const. It suffices to prove the thm for X = B a Brownian motion. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 16 / 22

The key steps of the proof For H C 2 the thm follows from Itô s lemma. The "if" part follows from the monotone class theorem. Write H t = H(B t, B t ) = t 0 h sdb s. Note that H, B t = t 0 h sds is an additive functional of the Markov process (B t, B t ). Apply Mooto s theorem and deduce that h s = u(b s, B s ). Use the fact that for large (stochastic) intervals, H(B t, B t ) is a function of B t only, and a local martingale. In consequence it is affine in B t (with B t fixed). Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 17 / 22

"Martingales functions of..." (X t : t 0) is a continuous local martingale with X 0 = 0, X = a.s. X t = sup s t X s, X t = inf s t N s, L X t - local time at 0. B = (B t ) is a BM. The Problem: For what functions H : R 4 R is the process H X t = H(X t, X t, X t, L X t ) a local martingale? A simplification via the DDS Theorem: { } H X t is a local martingale in natural filtration of X { H B t is a local martingale in natural filtration of B } More generally, we are interested in a complete characterization of A-martingales, that is local martingales H(A t, B t ), where (A t, B t ) is Markov and (A t ) has singular support w.r.t. the Lebesgue measure. Previous theorem solved the problem for M-martingales. We also have complete description of L-martingales (O.; Fitzsimmons and Wroblewski ). Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 19 / 22

Motivation - martingale inequalities Say (X t ) is a UI martingale and X µ. Is is interesting to know what is the possible law of X. AY martingale X t 1 y1 Xt y X t y allows to prove easily that P(X y) µ ( [Ψ 1 µ (y), ) ) = P(B Tµ y). In order to obtain a similar bound for P(X y and X z) we would need martingales which are function of (X t, X t, X t ), that is MM-martingales. In particular, we want a martingale featuring 1 Xt y 1 X t z. In a recent work with A. Cox we obtained sharp bounds for P(X y and X z). However the structural dependence of the bound on (µ, y, z) suggests that there are no non-trivial MM-martingales. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 20 / 22

MM-martingales conjecture Conjecture H(B t, B t, B t ) is a local martingle if and only if H(b, y, z) = H 1 (b, y) + H 2 (y, z), where H 1 (B t, B t ) and H 2 (B t, B t ) are local martingales. Put differently: an MM-martingale is a sum of an M-martingale and an M-martingale. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 21 / 22

Bibliography [] J. Azéma and M. Yor. Une solution simple au problème de Skorokhod. In Séminaire de Probabilités, XIII, volume 721 of Lecture Notes in Math., pages 90 115. Springer, Berlin, 1979. L. Bachelier. Théorie des probabilités continues. Journal des Mathématiques Pures et Appliquées, pages 259 327, 1906. L. Carraro, N. El Karoui, A. Meziou, and J. Obłój. On Azéma-Yor martingales: further properties and applications. in preparation. J. Cvitanić and I. Karatzas. On portfolio optimization under "drawdown" constraints. IMA Lecture Notes in Mathematics & Applications, 65:77 88, 1995. J. Obłój. A complete characterization of local martingales which are functions of Brownian motion and its supremum. Bernoulli, 12(6):955 969, 2006. J. Obłój and M. Yor. On local martingale and its supremum: harmonic functions and beyond. In Y. Kabanov, R. Lipster, and J. Stoyanov, editors, From Stochastic Calculus to Mathematical Finance, pages 517 534. Springer-Verlag, 2006. Jan Obłój () On the Azéma-Yor processes Bern 05.12.07 22 / 22