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 Welsh Probability Seminar, 17 Jan 28

Are the Azéma-Yor processes truly remarkable? An hour 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 Welsh Probability Seminar, 17 Jan 28

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 Swansea 17.1.8 3 / 23

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 Swansea 17.1.8 4 / 23

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 u(x)dx. Furthermore if u 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 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 Swansea 17.1.8 5 / 23

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 u(x)dx. Furthermore if u then MU (X) t = U(X t ). t 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 t u(x s )dx s (X s X s ) du(x s ) }{{} on dx s t u(x s )dx s + t t u(x s )d(x s X s ) [u(x), X X] t }{{} u(x s )dx s = t u(x s )dx s. u implies M U (X) t U(X t ) and equality in inf{t : X t = y}, for any y >. Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 5 / 23

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 >. 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 Swansea 17.1.8 6 / 23

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 ) and stopping in T we get Let u(x) = px p 1, p > 1. = (X t k)1 Xt k 1 X t k (X t X t ) = X t 1 Xt k k1 X t k, kp(x T k) = EX T 1 Xt k, Doob s max equality. 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. Optional stopping yields (p > 1): p 1 p EXp T = EX T X p 1 T ( E X T p) 1 ( p EX p ) p 1 p T, which gives Doob s L p inequality. Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 7 / 23

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. 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. = Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 7 / 23

AY solution to the Skorokhod embedding problem Let (X t ) be a continuous local martingale and µ a centred probability measure on R. Let T Ψ = inf{t : X t Ψ(X t )}. We look for an increasing Ψ such that X TΨ µ. Applying stopping theorem to M U at T Ψ and writing X TΨ ν we have U(x)ν(dx) u(x)(x Ψ 1 (x))ν(dx) =, u C 2, where we used X TΨ = Ψ 1 (X TΨ ). Putting ν = µ Ψ 1 and solving for Ψ we get 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. Another way of proving the result is to observe that actually T µ = inf{t : M U (X) t = } for a suitably chosen U. Then 1 1 = Uniform[, 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 Swansea 17.1.8 9 / 23

AY solution (optimal properties) We just saw that T Ψµ = inf{t : X t Ψ µ (X t )} satisfies X TΨµ µ. Suppose that T is another stopping time with X T µ and (X t T ) is UI. (Using AY martingales) we saw Doob s max equality: [ ] kp(x T k) = E X T 1 XT k. Writing µ(x) = µ([x, )) and p := P(X T k) we have [ ] ( ) kp E X T 1 XT µ 1 (p) = pψ µ µ 1 (p), hence ( ) µ Ψ 1 µ (k) p = P(X T k), since µ is decreasing. This bound is attained by the Azéma-Yor stopping time: P(X TΨµ k) = µ(ψ 1 µ (k)) P(X T k). Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 1 / 23

The Bachelier equation Theorem Let (X t ) be a semimartingale with X t. Consider ϕ : R R +, locally bounded away from zero, and put V(y) = y dx ϕ(x). The Bachelier equation dy t = ϕ(y t )dx t, Y =, (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 > essential. Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 11 / 23

The Bachelier equation Theorem Let (X t ) be a semimartingale with X t. Consider ϕ : R R +, locally bounded away from zero, and put V(y) = y dx ϕ(x). The Bachelier equation dy t = ϕ(y t )dx t, Y =, (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, 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 Swansea 17.1.8 11 / 23

The Drawdown equation (1) Drawdown condition arrises as a natural restriction for fund managers on their portfolios. Assume: X t > is a stock pice processes (a martingale), M t = t A udx u is an investment strategy, (A u ) predictable, portfolio requirement: t T, M t αm t, for some α (, 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 Swansea 17.1.8 12 / 23

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 >. 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, where w(x) = x V(x) V (x), Proof: Immediate, as u(x t )X t, 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 Swansea 17.1.8 13 / 23

The Drawdown equation (3) More to the point we can reverse the procedure. Theorem Let X t > be as above, with X = 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 = 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 Swansea 17.1.8 14 / 23

Characterization via optimality Let U : R + R + increasing concave, sublinear at. Let (N t ) be a non-negative local martingale, N t t a.s., EN <. 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 = U(N ). Then [ P t M U (N) t = U(N t ), and E h ( M U ) ] (N) E [h ( ) ] P for all h concave increasing. Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 16 / 23

Characterization via structure Theorem Let ( (X t ) be a continuous local martingale, X = a.s. The process H(Xt, X t ) : t ) 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 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 u(x s )dx s, Large classes of martingales can be described, e.g. u 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 Swansea 17.1.8 17 / 23

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 h sdb s. Note that H, B t = t 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 Swansea 17.1.8 18 / 23

"Martingales functions of..." (X t : t ) is a continuous local martingale with X =, X = a.s. X t = sup s t X s, X t = inf s t N s, L X t - local time at. 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 Swansea 17.1.8 2 / 23

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 allowed us to prove earlier 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 Swansea 17.1.8 21 / 23

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 Swansea 17.1.8 22 / 23

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 9 115. Springer, Berlin, 1979. L. Bachelier. Théorie des probabilités continues. Journal des Mathématiques Pures et Appliquées, pages 259 327, 196. 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, 26. 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, 26. Jan Obłój () On the Azéma-Yor processes Swansea 17.1.8 23 / 23