Lectures on Stochastic control and applications in finance

Transcription

1 Lecures on Sochasic conrol and applicaions in finance Huyên PHAM Universiy Paris Didero, LPMA Insiu Universiaire de France and CREST-ENSAE hp:// Auumn school on Sochasic conrol problems for FBSDEs and Applicaions Marrakech, December 1-11, 2010

2 Sochasic Conrol and applicaions in finance Absrac. The aim of hese lecures is o presen an inroducion o sochasic conrol, a classsical opic in applied mahemaics, which has known imporan developmens over he las years inspired especially by problems in mahemaical finance. We give an overview of he main mehods and resuls in his area. We firs presen he sandard approach by dynamic programming equaion and verificaion, and poin ou he limis of his mehod. We hen move on o he viscosiy soluions approach: i requires more heory and echnique, bu provides he general mahemaical ool for dealing wih sochasic conrol in a Markovian conex. The las lecure is devoed o an inroducion o he heory of Backward sochasic differenial equaions (BSDEs), which has emerged as a major research opic wih significan conribuions in relaion wih sochasic conrol beyond he Markovian framework. The various mehods presened in hese lecures will be illusraed by several applicaions arising in economics and finance. Lecure 1 : Classical approach o sochasic conrol problem Lecure 2 : Viscosiy soluions and sochasic conrol Lecure 3 : BSDEs and sochasic conrol

3 References for hese lecures: H. Pham (2009): Coninuous-ime sochasic conrol and opimizaion wih financial applicaions, Series SMAP, Springer. I. Kharroubi, J. Ma, H. Pham and J. Zhang (2010): Backward sochasic differenial equaions wih consrained jumps and quasi-variaional inequaliies, Annals of Probabiliy, Vol. 38,

4 Lecure 1 : Classical approach o sochasic conrol problem Inroducion Conrolled diffusion processes Dynamic Programming Principle (DPP) Hamilon-Jacobi-Bellman (HJB) equaion Verificaion heorem Applicaions : Meron porfolio selecion (CRRA uiliy funcions and general uiliy funcions by dualiy approach), Meron porfolio/ consumpion choice Some oher classes of sochasic conrol

5 I. Inroducion Basic srucure of sochasic conrol problem Dynamic sysem in an uncerain environmen: - filered probabiliy space (Ω, F, F = (F ), P): uncerainy and informaion - sae variables X = (X ): F-adaped sochasic process represening he evoluion of he quaniaive variables describing he sysem Conrol: a process α = (α ) whose value is decided a ime in funcion of he available informaion F, and which can influence he dynamics of he sae process X. Performance/crierion: opimize over conrols a funcional J(X, α), e.g. [ T J(X, α) = E f(x, α )d + g(x T ) on a finie horizon or 0 [ J(X, α) = E e β f(x, α )d 0 on an infinie horizon Various and numerous applicaions in economics and finance In parallel, problems in mahemaical finance new developmens in he heory of sochasic conrol

6 Solving a sochasic conrol problem Basic goal: find he opimal conrol (which achieves he opimum of he objecive funcional) if i exiss and he value funcion (he opimal objecive funcional) Tracable characerizaion of he value funcion and opimal conrol - if possible, explici soluions - oherwise: qualiaive descripion and quaniaive resuls via numerical soluions Mahemaical ools Dynamic programming principle and sochasic calculus - PDE characerizaion in a Markovian conex - BSDE in general Sochasic conrol is a opic a he inerface beween probabiliy, sochasic analysis and PDE.

7 II. Conrolled diffusion processes Dynamics of he sae variables in R n : dx s = b(x s, α s )ds + σ(x s, α s )dw s, (1) W d-dimensional Brownian moion on (Ω, F, F = (F ), P). - The conrol α = (α ) is an F-adaped process, valued in A subse of R m, and saisfying some inegrabiliy condiions and/or sae consrains A se of admissible conrols. - Given α A, (, x) [0, T R n, we denoe by X,x = X,x,α he soluion o (1) saring from x a. Performance crierion (on finie horizon) Given a funcion f from R n A ino R, and a funcion g from R n ino R, we define he objecive funcional: [ T J(, x, α) = E and he value funcion: f(xs,x, α s )ds + g(x,x T ), (, x) [0, T R n, α A, v(, x) = sup J(, x, α). α A ˆα A is an opimal conrol if: v(, x) = J(, x, ˆα). A process α in he form α s = a(s, Xs,x ) for some measurable funcion a from [0, T R n ino A is called Markovian or feedback conrol.

8 III. Dynamic programming principle Bellman s principle of opimaliy An opimal policy has he propery ha whaever he iniial sae and iniial decision are, he remaining decisions mus consiue an opimal policy wih regard o he sae resuling from he firs decision (See Bellman, 1957, Ch. III.3) Mahemaical formulaion of he Bellman s principle or Dynamic Programming Principle (DPP) The usual version of he DPP is wrien as [ θ v(, x) = sup E f(xs,x, α s )ds + v(θ, X,x θ ), (2) α A for any sopping ime θ T,T (se of sopping imes valued in [, T ).

9 Sronger version of he DPP In a sronger and useful version of he DPP, θ may acually depend on α in (2). This means: v(, x) = sup sup E α A θ T,T = sup inf E α A θ T,T [ θ [ θ f(xs,x, α s )ds + v(θ, X,x θ ) f(xs,x, α s )ds + v(θ, X,x θ ). (i) For all α A and θ T,T : [ θ v(, x) E f(xs,x, α s )ds + v(θ, X,x θ ). (ii) For all ε > 0, here exiss α A such ha for all θ T,T [ θ v(, x) ε E f(xs,x, α s )ds + v(θ, X,x θ ).

10 Proof of he DPP. (Firs par) 1. Given α A, we have by pahwise uniqueness of he flow of he SDE for X, he Markovian srucure X,x s = X θ,x,x θ s, s θ, for any θ T,T. By he law of ieraed condiional expecaion, we hen ge [ θ J(, x, α) = E f(s, Xs,x, α s )ds + J(θ, X,x θ, α), Since J(.,., α) v, and θ is arbirary in T,T, his implies [ θ J(, x, α) inf E f(s, Xs,x, α s )ds + v(θ, X,x θ ) θ T,T [ θ sup inf f(s, Xs,x, α s )ds + v(θ, X,x θ ). = E α A θ T,T v(, x) sup inf E α A θ T,T [ θ f(s, Xs,x, α s )ds + v(θ, X,x θ ). (3)

11 Proof of he DPP. (Second par) 2. Fix some arbirary conrol α A and θ T,T. By definiion of he value funcions, for any ε > 0 and ω Ω, here exiss α ε,ω A, which is an ε-opimal conrol for v(θ(ω), X,x θ(ω) (ω)), i.e. Le us now define he process v(θ(ω), X,x θ(ω) (ω)) ε J(θ(ω), X,x θ(ω) (ω), αε,ω ). (4) ˆα s (ω) = { αs (ω), s [0, θ(ω) α ε,ω s (ω), s [θ(ω), T. Delicae measurabiliy quesions! By measurable selecion resuls, one can show ha ˆα is F-adaped, and so ˆα A. By using again he law of ieraed condiional expecaion, and from (4): [ θ v(, x) J(, x, ˆα) = E f(s, Xs,x, α s )ds + J(θ, X,x θ, αε ) [ θ E f(s, Xs,x, α s )ds + v(θ, X,x θ ) ε. Since α, θ and ε are arbirary, his implies [ θ v(, x) sup sup f(s, Xs,x, α s )ds + v(θ, X,x θ ). E α A θ T,T

12 IV. Hamilon-Jacobi-Bellman (HJB) equaion The HJB equaion is he infiniesimal version of he dynamic programming principle: i describes he local behavior of he value funcion when we send he sopping ime θ in he DPP (2) o. The HJB equaion is also called dynamic programming equaion. Formal derivaion of HJB We assume ha he value funcion is smooh enough o apply Iô s formula, and we pospone inegrabiliy quesions. For any α A, and a conrolled process X,x, apply Iô s formula o v(s, X,x s ) beween s = and s = + h: v( + h, X,x +h ) = v(, x) + +h ( ) v + Lα s v (s, Xs,x )ds + (local) maringale, where for a A, L a is he second-order operaor associaed o he diffusion X wih consan conrol a: L a v = b(x, a).d x v r ( σ(x, a)σ (x, a)d 2 xv ).

13 Plug ino he DPP: [ +h ( v sup E α A + Lα s v ) (s, Xs,x ) + f(x,x s, α s )ds = 0. Divide by h, send h o zero, and obain by he mean-value heorem, he so-called HJB equaion: v + sup [L a v + f(x, a) = 0, (, x) [0, T ) R n. (5) a A Moreover, if v is coninuous a T, we have he erminal condiion v(t, x) = v(t, x) = g(x), x R n. Classical approach o sochasic conrol: Show if possible he exisence of a smooh soluion o HJB, or even beer obain an explici soluion Verificaion sep: prove ha his smooh soluion o HJB is he value funcion of he sochasic conrol problem, and obain as a byproduc he opimal conrol. Remark. In he classical verificaion approach, we don need o prove he DPP, bu only o ge he exisence of a smooh soluion o he HJB equaion.

14 V. Verificaion approach Theorem Le w be a funcion in C 1,2 ([0, T R n ), soluion o he HJB equaion: w [ (, x) + sup L a w(, x) + f(x, a) = 0, (, x) [0, T ) R n, a A w(t, x) = g(x), x R n. (and saisfying evenually addiional growh condiions relaed o f and g). Suppose here exiss a measurable funcion â(, x), (, x) [0, T ) R n, valued in A, aaining he supremum in HJB i.e. such ha he SDE [ sup L a w(, x) + f(x, a) = Lâ(,x) w(, x) + f(x, â(, x)), a A dx s = b(x s, â(s, X s ))ds + σ(x s, â(s, X s ))dw s,x admis a unique soluion, denoed by ˆX s, given an iniial condiion X = x, and he,x process ˆα = {â(s, ˆX s ) s T } lies in A. Then, and ˆα is an opimal feedback conrol. w = v,

15 Proof of he verificaion heorem. (Firs par) 1. Suppose ha w is a smooh supersoluion o he HJB equaion: w [ (, x) sup L a w(, x) + f(x, a) 0, (, x) [0, T ) R n, (6) a A w(t, x) g(x), x R n. (7) For any α A, and a conrolled process X,x, apply Iô s formula o w(s, X,x s ) beween s = and s = T τ n, and ake expecaion: E [ w(t τ n, X,x T τ n ) = w(, x) + E [ T τn ( w + Lα s w ) (s, Xs,x )ds where (τ n ) is a localizing sequence of sopping imes for he local maringale appearing in Iô s formula. Since w is a supersoluion o HJB (6), his implies: E [ w(t τ n, X,x T τ n ) + E [ T τn f(xs,x, α s )ds w(, x). By sending n o infiniy, and under suiable inegrabiliy condiions, we ge: E [ [ T w(t, X,x T ) + E f(xs,x, α s )ds w(, x). Since w(t,.) g, and α is arbirary, we obain [ T v(, x) = sup E f(xs,x, α s )ds + g(x,x T ) α A w(, x).

16 Proof of he verificaion heorem. (Second par) 2. Suppose ha he supremum in HJB equaion is aained: w (, x) Lâ(,x) w(, x) + f(x, â(, x)) = 0, (, x) [0, T ) R n, (8) w(t, x) = g(x), x R n. (9) Apply Iô s formula o w(s,,x ˆX s ) for he feedback conrol ˆα. By same argumens as in he firs par, we have now he equaliy (afer an evenual localizaion): [ T ( ) w w(, x) = E + Lˆα s,x,x w (s, ˆX s )ds + w(t, ˆX T ) [ T,x,x = E f( ˆX s, ˆα s )ds + g( ˆX T ) ( v(, x)). Togeher wih he firs par, his proves ha w = v and ˆα is an opimal feedback conrol.

17 Probabilisic formulaion of he verificaion approach The analyic saemen of he verificaion heorem has a probabilisic formulaion: Suppose ha he measurable funcion w on [0, T R n saisfies he wo properies: for any conrol α A wih associaed conrolled process X, he process w(, X ) + 0 f(x s, α s )ds is a supermaringale (10) here exiss a conrol ˆα A wih associaed conrolled process X, such ha he process w(, ˆX ) + 0 f( ˆX s, ˆα s )ds is a maringale. (11) Then, w = v, and ˆα is an opimal conrol. Remark. Noice ha in he probabilisic verificaion approach, we do no need smoohness of w, bu we require a supermaringale propery. In he analyic verificaion approach, he smoohness of w is used for applying Iô s formula o w(, X ). This allows us o derive he supermaringale propery as in (10), which is in fac he key feaure for proving ha w v, and hen w = v wih he maringale propery (11).

18 VI. Applicaions 1. Meron porfolio selecion in finie horizon An agen invess a any ime a proporion α of his wealh X in a sock of price S and 1 α in a bond of price S 0 wih ineres rae r. The invesor faces he porfolio consrain ha a any ime, α is valued in A closed convex subse of R. Assuming a Black-Scholes model for S (wih consan rae of reurn µ and volailiy σ > 0), he dynamics of he conrolled wealh process is: dx = X α ds + X (1 α ) S S 0 ds 0 = X (r + α (µ r)) d + X α σdw. The preferences of he agen is described by a uiliy funcion U: increasing and concave funcion. The performance of a porfolio sraegy is measured by he expeced uiliy from erminal wealh Uiliy maximizaion problem a a finie horizon T : v(, x) = sup α A E[U(X,x T ), (, x) [0, T (0, ). Sandard sochasic conrol problem

19 HJB equaion for Meron s problem v + rxv x + sup [a(µ r)xv x + 1 a A 2 x2 a 2 σ 2 v xx = 0, (, x) [0, T ) (0, ) v(t, x) = U(x), x > 0. The case of CRRA uiliy funcions: U(x) = xp p, p < 1, p 0 Relaive Risk Aversion: xu (x)/u (x) = 1 p. We look for a candidae soluion o HJB in he form w(, x) = ϕ()u(x). Plugging ino HJB, we see ha ϕ should saisfy he ODE: ϕ () + ρϕ() = 0, φ(t ) = 1, where [ 1 ρ = rp + p sup a(µ r) a A 2 a2 (1 p)σ 2, ϕ() = e ρ(t ).

20 The value funcion is equal o v(, x) = e ρ(t ) U(x), and he opimal conrol is consan (in proporion of wealh invesed) [ 1 â = arg max a(µ r) a A 2 a2 (1 p)σ 2. When A = R (no porfolio consrain), he values of ρ and â are explicily given by and ρ = (µ r)2 2σ 2 â = p 1 p + rp. µ r σ 2 (1 p),

21 General uiliy funcions: U is C 1, sricly increasing and concave on (0, ), and saisfies he Inada condiions: U (0) =, U ( ) = 0. Convex conjugae of U: where I := (U ) 1 = Ũ. Ũ(y) := sup[u(x) xy = U(I(y)) yi(y), y > 0, x>0 Assume ha A = R (no porfolio consrain and complee marke) and for simpliciy r = 0 so ha HJB is also wrien as v 1 µ 2 vx 2 = 0, 2 σ 2 v xx wih a candidae for he opimal feedback conrol: Recall he erminal condiion: â(, x) = µ σ 2 v x x 2 v xx. v(t, x) = U(x). Fully nonlinear second order PDE Bu remarkably, i can be solved explicily by convex dualiy!

22 Inroduce he convex conjugae of v, also called dual value funcion: ṽ(, y) = sup[v(, x) xy, y > 0. x>0 change of variables: y = v x and x = ṽ y. ṽ saisfies he linear parabolic Cauchy problem: ṽ + 1 µ 2 ṽ 2 σ 2y2 yy = 0 ṽ(t, y) = Ũ(y). From Feynman-Kac formula, ṽ is represened as ṽ(, y) = E [ Ũ(yY T ), where Y is he soluion o dy s = Ys µ σ dw s, Y = 1. Remark. YT [dq/dp = E F is he densiy of he risk-neural probabiliy measure Q, under which S is a maringale: ds = S σdw Q,

23 The primal value funcion is obained by dualiy relaion: [ṽ(, v(, x) = inf y) + xy, x > 0. y>0 From he represenaion of ṽ, we ge: v(, x) = inf y>0 { E [ Ũ(yY T ) + xy } (12) Recalling ha Ũ = (U ) 1 =: I, he infimum in (12) is aained a ŷ = ŷ(, x) s.. E [ Y T I(ŷY T ) = x, (sauraion budge consrain) (13) and we have v(, x) = E [Ũ(ŷYT ) + ŷyt I(ŷYT ). Recalling ha he supremum in Ũ is aained a x = I(y), i.e. Ũ(y) = U(I(y)) yi(y), we obain: [,x v(, x) = E U( ˆX T ), wih ˆX,x T = I(ŷY T ). (14)

24 Consider now he sricly posiive Q-maringale process: [ ˆX s,x := E Q I(ŷYT ) F s, s T. - From he sauraion budge consrain (13), we have ˆX,x = x. - From he maringale represenaion heorem (or since he marke is complee), here exiss ˆα A s.. d ˆX,x s = ˆX,x s σ ˆα s dw Q s =,x ˆX s ˆα s ds s, S s which means ha ˆX,x is a wealh process conrolled by he proporion ˆα, and saring from iniial capial x a ime. From he represenaion (14) of he value funcion, his proves ha ˆX,x is he opimal wealh process: v(, x) = E [ U( ˆX,x T ).

25 2. Meron porfolio/consumpion choice on infinie horizon In addiion o he invesmen α in he sock, he agen can also consume from his wealh: (c ) 0 consumpion per uni of wealh The wealh process, conrolled by (α, c) is governed by: dx = X (r + α (µ r) c ) d + X α σdw. The preferences of he agen is described by a uiliy U from consumpion, and he goal is o maximize over porfolio/consumpion he expeced uiliy from ineremporal consumpion up o a random ime horizon: [ τ v(x) = sup E (α,c) e β U(c X x )d, x > 0. 0 We assume ha τ is independen of F (marke informaion), and follows E(λ). Infinie horizon sochasic conrol problem: [ v(x) = sup E (α,c) e (β+λ) U(c X x )d, x > 0. 0

26 HJB equaion (β + λ)v rxv sup[a(µ r)v + 1 a A 2 a2 x 2 σ 2 v sup[u(cx) cxv = 0, x > 0. c 0 Explici soluion for CRRA uiliy funcion: U(x) = x p /p. Under he condiion ha β + λ > ρ, we have v(x) = K U(x), wih K = ( ) 1 p 1 p. β + λ ρ The opimal porfolio/consumpion sraegies are: â = arg max a A [a(µ r) 1 2 a2 (1 p)σ 2 ĉ = 1 x (v (x)) 1 p 1 = K 1 p 1.

27 VII. Some oher classes of sochasic conrol problems Ergodic and risk-sensiive conrol problems - Risk-sensiive conrol problem: lim sup T 1 θt ln E [ ( T ) exp θ f(x, α )d 0 Applicaions in finance: Bielecki, Pliska, Fleming, Sheu, Nagai, Davis, ec... - Large deviaions conrol problem: lim sup T [ 1 T P XT T x Dual of risk-sensiive conrol problem Applicaions in finance: Pham, Sekine, Nagai, Haa, Sheu

28 Opimal sopping problems: The conrol decision is a sopping ime τ where we decide o sop he process Value funcion of opimal sopping problem (over a finie horizon): [ τ v(, x) = sup E f(xs,x )ds + g(xτ,x ). τ T The HJB equaion is a free boundary or variaional inequaliy: min [ v Lv f, v g = 0, where L is he infiniesimal generaor of he Markov process X. Typical applicaions in finance in American opion pricing

29 Impulse and opimal swiching problems: The conrol is a sequence of increasing sopping imes (τ n ) n associaed o a sequence of acions (ζ n ) n : τ n represens he ime decision when we decide o inervene on he sae sysem X by using an acion ζ n F τn -measurable: X τ n Γ(X τ n, ζ n ) Value funcion: [ T v(, x) = sup E (τ n ζ n ) f(x,x s )ds + g(x,x T ) + n c(x τ n, ζ n ). The HJB equaion is a quasi-variaional inequaliy: min [ v Lv f, v Hv = 0, where L is he infiniesimal generaor of he Markov process X, and H is a nonlocal operaor associaed o he jump and cos induced by an acion: [ Hv(, x) = sup v(, Γ(x, e)) + c(x, e). e E Various applicaions in finance: Transacion coss and liquidiy risk models, where rading imes ake place discreely Real opions and firm invesmen problems, where decisions represen change of regimes or producion echnologies

30 Lecure 2 : Viscosiy soluions and sochasic conrol Non smoohness of value funcions: a moivaing financial example Inroducion o viscosiy soluions Viscosiy properies of he dynamic programming equaion Comparison principles Applicaion: Super-replicaion in uncerain volailiy models

31 I. Non smoohness of value funcions: a moivaing financial example Consider he conrolled diffusion process dx s = α s X s dw s, wih an unbounded conrol α valued in A = R + : Uncerain volailiy model. Consider he sochasic conrol problem v(, x) = sup α A E[g(X,x T ), (, x) [0, T (0, ), Superreplicaion cos of an opion payoff g(x T ). If v were smooh, i would be a classical soluion o he HJB equaion: [ 1 v + sup a R + 2 a2 x 2 v xx = 0, (, x) [0, T ) (0, ). (1) Bu, for he supremum in a R o be finie and HJB equaion (1) o be well-posed, we mus have v xx 0, i.e. v(,.) is concave in x, for any [0, T ).

32 Now, by aking he zero conrol in he definiion of v, we ge v(, x) g(x), which combined wih he concaviy of v(,.), implies: v(, x) ĝ(x), < T, where ĝ is he concave envelope of g: he smalles concave funcion above g. Moreover, since g ĝ, and by Jensen s inequaliy and maringale propery of X, we have Therefore, v(, x) sup E[ĝ(X,x T ) α A sup α A ĝ ( E[X,x T ) = ĝ(x). v(, x) = ĝ(x), (, x) [0, T ) (0, ). There is a conradicion wih he smoohness of v, whenever ĝ is no smooh!, for example when g is concave (hence equal o ĝ) bu no smooh. Need o consider he case where he supremum in HJB can explode (singular case) and o define weak soluions for HJB equaion Noion of viscosiy soluions (Crandall, Ishii, P.L. Lions)

33 II. Inroducion o viscosiy soluions Consider nonlinear parabolic second-order parial differenial equaions: F (, x, w, w, D xw, D 2 xxw) = 0, (, x) [0, T ) O, (2) where O is an open subse of R n and F is a coninuous funcion of is argumens, saisfying he ellipiciy condiion: for all (, x) [0, T ) O, r R, (q, p) R R n, M, M S n, M M = F (, x, r, q, p, M) F (, x, r, q, p, M), (3) and he paraboliciy condiion: for all [0, T ), x O, r R, q, ˆq R, p R n, M S n, q ˆq = F (, x, r, q, p, M) F (, x, r, ˆq, p, M). (4) Typical example: HJB equaion F (, x, r, q, p, M) = q H(x, p, M), where H is he Hamilonian funcion of he sochasic conrol problem: [ 1 H(x, p, M) = sup b(x, a).p + a A 2 r (σσ (x, a)m) + f(x, a)

34 Inuiion for he noion of viscosiy soluions Assume ha w is a smooh supersoluion o (2). Le ϕ be a smooh es funcion on [0, T ) O, and (, x) [0, T ) O be a minimum poin of w ϕ: 0 = (w ϕ)(, x) = min(w ϕ). In his case, he firs and second-order opimaliy condiions imply (w ϕ) (, x) 0 (= 0 if > 0) D x w(, x) = D x ϕ(, x) and Dxw(, 2 x) Dxϕ(, 2 x). From he ellipiciy and paraboliciy condiions (3) and (4), we deduce ha F (, x, ϕ(, x), ϕ (, x), D x ϕ(, x), D 2 xϕ(, x)) F (, x, w(, x), w (, x), D x w(, x), D 2 xw(, x)) 0, Similarly, if w is a classical subsoluion o (2), hen for all es funcions ϕ, and (, x) [0, T ) O such ha (, x) is a maximum poin of w ϕ, we have F (, x, ϕ(, x), ϕ (, x), D x ϕ(, x), D 2 xϕ(, x)) 0.

35 General definiion of (disconinuous) viscosiy soluions Given a locally bounded funcion w on [0, T O, we define is upper-semiconinuous (usc) envelope w and lower-semiconinuous (lsc) envelope w by w (, x) = lim sup <T,x x w(, x ), w (, x) = lim inf w(, x ). <T,x x Remark. w w w, and w is usc (resp. lsc) on [0, T ) O iff w = w (resp. = w ), and w is coninuous [0, T ) O iff w = w = w. Definiion.1 Le w : [0, T O R be locally bounded. (i) w is a viscosiy supersoluion (resp. subsoluion) of (2) on [0, T ) O if F (, x, ϕ(, x), ϕ (, x), D x ϕ(, x), D 2 xxϕ(, x)) (resp ) 0, for all (, x) [0, T ) O, and es funcions ϕ such ha (, x) is a minimum (resp. maximum) poin of w ϕ (resp. w ϕ). (ii) w is a viscosiy soluion of (2) on [0, T ) O if i is boh a subsoluion and supersoluion of (2).

36 III. Viscosiy properies of he DPE We urn back o he sochasic conrol problem: [ T v(, x) = sup E f(xs,x, α s )ds + g(x,x T ), (, x) [0, T R n, α A wih Hamilonian funcion H on R n R n S n : [ 1 H(x, p, M) = sup b(x, a).p + a A 2 r (σσ (x, a)m) + f(x, a). We inroduce he domain of H as dom(h) = {(x, p, M) R n R n S n : H(x, p, M) < }, and make he following hypohesis (DH): H is coninuous on in(dom(h)) and here exiss a coninuous funcion G on R n R n S n such ha (x, p, M) dom(h) G(x, p, M) 0. Example. In he example considered a he beginning of his lecure: [ 1 H(x, p, M) = sup a R 2 a2 x M 2, and so G(x, p, M) = M.

37 Viscosiy propery inside he domain Theorem.1 The value funcion v is a viscosiy soluion o he HJB variaional inequaliy min [ v H(x, D xv, Dxv) 2, G(x, D x v, Dxv) 2 = 0, on [0, T ) R n. Remark. In he regular case when he Hamilonian H is finie on he whole domain R n R n S n (his occurs ypically when he conrol space is compac), he condiion (DH) is saisfied wih any choice of sricly posiive coninuous funcion G. In his case, he HJB variaional inequaliy is reduced o he regular HJB equaion: v (, x) H(x, D xv, D 2 xv) = 0, (, x) [0, T ) R n, which he value funcion saisfies in he viscosiy sense. Hence, he above Theorem saes a general viscosiy propery including boh he regular and singular case.

38 Proof of viscosiy supersoluion propery Le (, x) [0, T ) R n and le ϕ C 2 ([0, T ) R n ) be a es funcion such ha 0 = (v ϕ)(, x) = min [0,T ) R n (v ϕ). (5) By definiion of v (, x), here exiss a sequence ( m, x m ) m in [0, T ) R n such ha ( m, x m ) (, x) and v( m, x m ) v (, x), when m goes o infiniy. By he coninuiy of ϕ and by (5) we also have ha γ m := v( m, x m ) ϕ( m, x m ) 0. Le α A, a consan process equal o a A, and X m,x m s process. Le τ m = inf{s m : X m,x m s (h m ) be a sricly posiive sequence such ha he associaed conrolled x m η}, wih η > 0 a fixed consan. Le h m 0 and γ m h m 0. We apply he firs par of he DPP for v( m, x m ) o θ m := τ m ( m + h m ) and ge [ θm v( m, x m ) E f(s, X m,x m s, a)ds + v(θ m, X m,x m θ m ). m Equaion (5) implies ha v v ϕ, hus [ θm ϕ( m, x m ) + γ m E f(s, X m,x m s, a)ds + ϕ(θ m, X m,x m θ m ). m

39 Apply Iô s formula o ϕ(s, X m,x m s ) beween m and θ m : [ γ m 1 θm ( + E ϕ ) h m h m La ϕ f (s, X m,x m s, a)ds m 0. (6) Now, send m o infiniy: by he mean value heorem, and he dominaed convergence heorem, we ge ϕ (, x) L a ϕ(, x) f(, x, a) 0. Since a is arbirary in A, and by definiion of H, his means: ϕ (, x) H( x, D x ϕ(, x), D 2 xϕ(, x)) 0. In paricular, ( x, D x ϕ(, x), D 2 xϕ(, x)) dom(h), and so G( x, D x ϕ(, x), D 2 xϕ(, x)) 0. Therefore, [ min ϕ (, x) H( x, D x ϕ(, x), Dxϕ(, 2 x)), G( x, D x ϕ(, x), Dxϕ(, 2 x)) which is he required supersoluion propery. 0,

40 Proof of viscosiy subsoluion propery Le (, x) [0, T ) R n and le ϕ C 2 ([0, T ) R n ) be a es funcion such ha 0 = (v ϕ)(, x) = max [0,T ) R n (v ϕ). (7) As before, here exiss a sequence ( m, x m ) m in [0, T ) R n s.. γ m := v( m, x m ) ϕ( m, x m ) 0. ( m, x m ) (, x) and v( m, x m ) v (, x), We will show he resul by conradicion, and assume on he conrary ha ϕ (, x) H( x, D x ϕ(, x), D 2 xϕ(, x)) > 0, and G( x, D x ϕ(, x), D 2 xϕ(, x)) > 0. Under (DH), here exiss η > 0 such ha ϕ (, y) H(y, D xϕ(, y), D 2 xϕ(, y)) > 0, for (, x) B(, η) B( x, η). (8) Observe ha we can assume w.l.o.g. in (7) ha (, x) achieves a sric maximum so ha max p B((, x),η) (v ϕ) =: δ < 0, (9) where p B((, x), η) = [, + η B( x, η) { + η} B( x, η).

41 We apply he second par of DP: here exiss ˆα m A s.. v( m, x m ) δ [ θm 2 E f( ˆX m,x m s, ˆα s m )ds + v(θ m, ˆX m,x m θ m ) m, (10) where θ m = inf{s m : (s, ˆX m,x m s ) / B(, η) B( x, η)}. Observe by coninuiy of he sae process ha (θ m, ˆX m,x m θ m ) p B((, x), η) so ha from (9)-(10): ϕ( m, x m ) + γ m δ [ θm 2 E f( ˆX m,x m s, ˆα s m )ds + ϕ(θ m, ˆX m,x m θ m ) δ. m Apply Iô s formula o ϕ(s, ˆX m,x m s ) beween m and θ m, we hen ge afer noing ha he sochasic inegral vanishes in expecaion: γ m δ [ θm ( 2 + E ϕ ) Lˆαm s ϕ f Now, from (8) and definiion of H, we have m ϕ (s, ˆX m,x m s ϕ (s, ˆX m,x m s > 0, for m s θ m. Plugging ino (11), his implies (s, ˆX m,x m s, ˆα s m )ds ) Lˆαm s ϕ(s, ˆX m,x m s ) f( ˆX m,x m s, ˆα s m ) ) H(s, D x ϕ(s, ˆX m,x m s ), Dxϕ(s, 2 ˆX m,x m s )) δ. (11) γ m δ 2 δ, (12) and we ge he conradicion by sending m o infiniy: δ/2 δ.

42 Terminal condiion Due o he singulariy of he Hamilonian H, he value funcion may be disconinuous a T, i.e. v(t, x) may be differen from g(x). The righ erminal condiion is given by he relaxed erminal condiion: Theorem.2 The value funcion v is a viscosiy soluion o min [ v g, G(x, D x v, Dxv) 2 = 0, on {T } R n. (13) This means ha v (T,.) is a viscosiy supersoluion o min [ v (T, x) g(x), G(x, D x v (T, x), Dxv 2 (T, x)) 0, on R n. (14) and v (T,.) is a viscosiy subsoluion o min [ v (T, x) g(x), G(x, D x v (T, x), Dxv 2 (T, x)) 0, on R n. (15) Remark. Denoe by ĝ he upper G-envelope of g, defined as he smalles funcion above g, and viscosiy supersoluion o G(x, Dĝ, D 2 ĝ) 0. Then v (T, x) ĝ(x). On he oher hand, since ĝ is a viscosiy supersoluion o (14), and if a comparison principle holds for (13), hen v (T, x) ĝ(x). This implies v(t, x) = v (T, x) = v (T, x) = ĝ(x). In he regular case, we have ĝ = g, and v is coninuous a T.

43 IV. Srong comparison principles and uniqueness Consider he DPE saisfied by he value funcion [ min v H(x, D xv, Dxv) 2, G(x, D x v, Dxv) 2 = 0, on [0, T ) R n. (16) min [ v(t, x) g(x), G(x, D x v, D 2 xv) = 0, on {T } R n. (17) We say ha a srong comparison principle holds for (16)-(17) when he following saemen is rue: If u is an usc viscosiy subsoluion o (16)-(17) and w is a lsc viscosiy supersoluion o (16)-(17), saisfying some growh condiion, hen u w. Remark. The argumens for proving comparison principles are: - dedoubling variables echnique - Ishii s Lemma Sandard reference: user s guide of Crandall, Ishii s Lions (92).

44 Consequence of srong comparison principles Uniqueness and coninuiy Suppose ha v and w are wo viscosiy soluions o (16)-(17). This means ha v is a viscosiy subsoluion o (16)-(17), and w is a viscosiy supersoluion o (16)-(17), and vice-versa. By he srong comparison principle, we ge: v w and w v. Since w w, v v, his implies: v = v = w = w. Therefore, v = w, i.e. uniqueness v = v, i.e. coninuiy of v on [0, T ) R n. Conclusion The value funcion of he sochasic conrol problem is he unique coninuous viscosiy soluion o (16)-(17) (saisfying some growh condiion).

45 V. Applicaion: superreplicaion in uncerain volailiy model Consider he conrolled diffusion dx s = α s X s dw s, s T, wih he conrol process α valued in A = [a, ā, where 0 a ā. Given a coninuous funcion g on R +, we consider he sochasic conrol problem: v(, x) = sup E [ g(x,x T ), (, x) [0, T (0, ). α A Financial inerpreaion α represens he uncerain volailiy process of he sock price X, and he funcion g represens he payoff of an European opion of mauriy T. The value funcion v is he superreplicaion cos for his opion, ha is he minimum capial required o superhedge (by means of rading sraegies on he sock) he opion payoff a mauriy T whaever he realizaion of he uncerain volailiy.

46 The Hamilonian of his sochasic conrol problem is H(x, M) = [1 sup a [a,ā 2 a2 x 2 M, (x, M) (0, ) R. We shall hen disinguish wo cases: ā finie or no. Bounded volailiy: ā <. In his regular case, H is finie on he whole domain (0, ) R, and is given by H(x, M) = 1 2â2 (M)x 2 M, wih { ā if M 0 â(m) = a if M < 0. v is coninuous on [0, T (0, ), and is he unique viscosiy soluion wih linear growh condiion o he so-called Black-Scholes-Barenbla equaion v + 2â2 1 (v xx ) x 2 v xx = 0, (, x) [0, T ) (0, ), saisfying he erminal condiion v(t, x) = g(x), x (0, ). Remark. If g is convex, hen v is equal o he Black-Scholes price wih volailiy ā, which is convex in x, so ha â(v xx ) = ā.

47 Unbounded volailiy: ā =. In his singular case, he Hamilonian is given by H(x, M) = { 1 2 a2 x 2 M if G(M) := M 0 if M < 0. v is he unique viscosiy soluion o he HJB variaional inequaliy min [ v 1 2 a2 x 2 v xx, v xx = 0, on [0, T ) (0, ), (18) min [ v g, v xx = 0, on {T } (0, ). (19) Explici soluion o (18)-(19) Denoe by ĝ he concave envelope of g, i.e. he soluion o min[ĝ g, ĝ xx = 0. Le us consider he Black-Scholes price wih volailiy a of he opion payoff ĝ, i.e. [ w(, x) = E ĝ ( ) ˆX,x T, where Then, d ˆX s = a ˆX s dw s, s T, ˆX = x. v = w, on [0, T ) (0, ).

48 Proof. Indeed, he funcion w is soluion o he Black-Scholes equaion: w a2 x 2 w xx = 0, on [0, T ) (0, ) w(t, x) = ĝ(x), x (0, ). Moreover, w inheris from ĝ he concaviy propery, and so (This holds rue in he viscosiy sense) w xx 0, (, x) [0, T ) (0, ). Therefore, w saisfies he same HJB variaional inequaliy as v: min [ w 1 2 a2 x 2 w xx, w xx = 0, min [ w g, w xx = 0, on [0, T ) (0, ), on {T } (0, ). We conclude by uniqueness resul. Remark. When a = 0, we have w = ĝ, and so v(, x) = ĝ(x) on [0, T ) (0, ).

49 Lecure 3 : Backward Sochasic Differenial Equaions and sochasic conrol Inroducion General properies of BSDE The Markov case : nonlinear Feynman-Kac formula. Simulaion of BSDE Applicaion: CRRA uiliy maximizaion Refleced BSDE and opimal sopping problem BSDE wih consrained jumps and quasi-variaional inequaliies

50 I. Inroducion BSDEs firs inroduced by Bismu (73): adjoin equaion in Ponryagin maximum principle (linear BSDEs) Emergence of he heory since he seminal paper by Pardoux and Peng (90): general BSDEs BSDEs widely used in sochasic conrol and mahemaical finance Replicaion problem linear BSDE Porfolio opimizaion, risk measure nonlinear BSDE, refleced and consrained BSDEs Improve exisence and uniqueness of BSDEs, especially quadraic BSDEs BSDE provide a probabilisic represenaion of nonlinear PDEs: nonlinear Feynman- Kac formulae Numerical mehods for nonlinear PDEs

51 II. General resuls on BSDEs Le W = (W ) 0 T be a sandard d-dimensional Brownian moion on (Ω, F, F, P ) where F = (F ) 0 T is he naural filraion of W, and T is a fixed finie horizon. Noaions P: se of progressively measurable processes on Ω [0, T S 2 (0, T ): se of elemens Y P such ha E [ sup Y 2 <, 0 T H 2 (0, T ) d : se of elemens Z P, R d -valued, such ha [ T E Z 2 d <. Definiion of BSDE 0 A (one-dimensional) Backward Sochasic Differenial Equaion (BSDE in shor) is wrien in differenial form as dy = f(, Y, Z )d Z.dW, Y T = ξ, (1) where he daa is a pair (ξ, f), called erminal condiion and generaor (or driver): ξ L 2 (Ω, F T, P), f(, ω, y, z) is P B(R R d )-measurable. A soluion o (1) is a pair (Y, Z) S 2 (0, T ) H 2 (0, T ) d such ha Y = ξ + T f(s, Y s, Z s )ds T Z s.dw s, 0 T.

52 Under some specific assumpions on he generaor f, here is exisence and uniqueness of a soluion o he BSDE (1). Sandard Lipschiz assumpion (H1) f is uniformly Lipschiz in (y, z), i.e. here exiss a posiive consan C s.. for all (y, z, y, z ): f(, y, z) f(, y, z ) C The process {f(, 0, 0), [0, T } H 2 (0, T ) ( ) y y + z z, d dp a.e. Theorem (Pardoux and Peng 90) Under (H1), here exiss a unique soluion (Y, Z) o he BSDE (1). Proof. (a) Assume firs he case where f does no depend on (y, z), and consider he maringale M [ T = E ξ + 0 f(, ω)d F, which is square-inegrable under (H1), i.e. M S 2 (0, T ). By he maringale represenaion heorem, here exiss a unique Z H 2 (0, T ) d s.. Then, he process [ := E ξ + Y M = M 0 + T saisfies (wih Z) he BSDE (1). 0 f(s, ω)ds F Z s.dw s, 0 T. = M 0 f(s, ω)ds, 0 T,

53 Proof. (b) Consider now he general Lipschiz case. As in he deerminisic case, we give a proof based on a fixed poin mehod. Le us consider he funcion Φ on S 2 (0, T ) m H 2 (0, T ) d, mapping (U, V ) S 2 (0, T ) H 2 (0, T ) d o (Y, Z) = Φ(U, V ) defined by Y = ξ + T f(s, U s, V s )ds T Z s.dw s. This pair (Y, Z) exiss from Sep (a). We hen see ha (Y, Z) is a soluion o he BSDE (1) if and only if i is a fixed poin of Φ. Le (U, V ), (U, V ) S 2 (0, T ) H 2 (0, T ) d and (Y, Z) = Φ(U, V ), (Y, Z ) = Φ(U, V ). We se (Ū, V ) = (U U, V V ), (Ȳ, Z) = (Y Y, Z Z ) and f = f(, U, V ) f(, U, V ). Take some β > 0 o be chosen laer, and apply Iô s formula o e βs Ȳs 2 beween s = 0 and s = T : Ȳ0 2 = T 0 T By aking he expecaion, we ge [ E Ȳ0 2 T + E [ T 2C f E 0 [ 4CfE 2 T e βs ( β Ȳs 2 2Ȳs. f s ) ds e βs Z s 2 ds 2 T ( e βs β Ȳs 2 + Z s )ds 2 e βs Ȳs ( Ūs + V s )ds e βs Ȳs 2 ds E [ T 0 0 e βs Ȳ s Z s.dw s. [ T = 2E e βs Ȳ s. f s ds 0 e βs ( Ūs 2 + V s 2 )ds

54 Proof coninued. (b) Now, we choose β = 1 + 4Cf 2, and obain [ T ( E e βs Ȳs 2 + Z s )ds E [ T 0 e βs ( Ūs 2 + V s 2 )ds. This shows ha Φ is a sric conracion on he Banach space S 2 (0, T ) H 2 (0, T ) d endowed wih he norm (Y, Z) β = ( [ T E e βs( Y s 2 + Z s 2) ds 0 We conclude ha Φ admis a unique fixed poin, which is he soluion o he BSDE (1). ) 1 2. Non-Lipschiz condiions on he generaor f is coninuous in (y, z) and saisfies a linear growh condiion on (y, z). Then, here exiss a minimal soluion o he BSDE (1). (Lepelier and San Marin 97) f is coninuous in (y, z), linear in y, and quadraic in z, and ξ is bounded. Then, here exiss a unique bounded soluion o he BSDE (1) (Kobylanski 00).

55 III. The Markov case: non-linear Feynman-Kac formulae Linear Feynman-Kac formula Consider he linear parabolic PDE v (, x) + Lv(, x) + f(, x) = 0, on [0, T ) Rd (2) v(t,.) = g, on R d, (3) where L is he second-order differenial operaor Lv = b(x).d x v r(σσ (x)d 2 xv). Consider he (forward) diffusion process dx = b(x )d + σ(x )dw. Then, by Iô s formula o v(, X ) beween and T, wih v smooh soluion o (2)-(3): v(, X ) = g(x T ) + T f(s, X s )ds T D x v(s, X s ) σ(x s )dw s. I follows ha he pair (Y, Z ) = (v(, X ), σ (X )D x v(, X )) solves he linear BSDE: Remark Y = g(x T ) + T f(s, X s )ds T Z s dw s. We can compue he soluion v(0, X 0 ) = Y 0 by he Mone-Carlo expecaion: [ T = E g(x T ) + f(s, X s )ds. Y 0 0

56 Non linear Feynman-Kac formula Consider he semilinear parabolic PDE The corresponding BSDE is v + Lv + f(, x, v, σ D x v) = 0, on [0, T ) R d (4) v(t,.) = g, on R d, (5) Y = g(x T ) + in he sense ha: T f(s, X s, Y s, Z s )ds T Z s dw s, (6) he pair (Y, Z ) = (v(, X ), σ (X )D x v(, X )) solves (6) Conversely, if (Y, Z) is a soluion o (6), hen Y = v(, X ) for some deerminisic funcion v, which is a viscosiy soluion o (4)-(5). The ime discreizaion and simulaion of he BSDE (6) provides a numerical mehod for solving he semilinear PDE (4)-(5)

57 Simulaion of BSDE: ime discreizaion Time grid π = ( i ) on [0, T : i = i, i = 0,..., N, = T/N Forward Euler scheme X π for X : saring from X π 0 = x, X π i+1 := X π i + b(x π i ) + σ(x π i ) ( W i+1 W i ) Backward Euler scheme (Y π, Z π ) for (Y, Z) : saring from Y π N = g(x π N ), Y π i = Y π i+1 + f(x π i, Y π i+1, Z π i ) Z π i. ( W i+1 W i ) (7) and ake condiional expecaion: [ = E Y π i Y π i+1 + f(x π i, Y π i+1, Z π i ) X π i To ge he Z-componen, muliply (7) by W i+1 W i and ake expecaion: Z π i = 1 [ E Y π i+1 (W i+1 W i ) X π i

58 Simulaion of BSDE: numerical mehods How o compue hese condiional expecaions! several approaches: Regression based algorihms (Longsaff, Schwarz) Choose q deerminisic basis funcions ψ 1,..., ψ q, and approximae Z π i [ = E Y π i+1 (W i+1 W i ) X π i q α k ψ k (X π i ) where α = (α k ) solve he leas-square regression problem: [ q 2 arg inf Ē Y π α R q i+1 (W i+1 W i ) α k ψ k (X π i ) Here Ē is he empirical mean based on Mone-Carlo simulaions of Xπ i, X π i+1, W i+1 W i. k=1 k=1 Efficiency enhanced by using he same se of simulaion pahs o compue all condiional expecaions. Oher mehods: Malliavin Mone-Carlo approach (P.L. Lions, Regnier) Quanizaion mehods (Pagès) Imporan lieraure: Kohasu-Higa, Peersson (01), Ma, Zhang (02), Bally and Pagès (03), Bouchard, Ekeland, Touzi (04), Gobe e al. (05), Soner and Touzi (05), Peng, Xu (06), Delarue, Menozzi (07), Bender and Zhang (08), ec...

59 IV. Applicaion: CRRA uiliy maximizaion Consider a financial marke model wih one riskless asse S 0 = 1, and n socks of price process ( ds = diag(s ) µ d + σ dw ), wher W is a d-dimensional Brownian moion (wih d n), b, σ bounded adaped processes, σ of full rank n. Consider an agen invesing in he socks a fracion α of his wealh X a any ime: dx = X α diag(s ) 1 ds = X (α µ d + α σ dw ) (8) A 0 : se of F-adaped processes α valued in A closed convex se of R n, and saisfying: T 0 α µ d + T 0 α σ 2 d <, (8) is well-defined. Given a uiliy funcion U on (0, ), and saring from iniial capial X 0 > 0, he objecive of he agen is: V 0 := sup E[U(XT α ). (9) α A Here, X α is he soluion o (8) conrolled by α A 0, and saring from X 0 a ime 0, and A is he subse of elemens α A 0 s.. {U(X α τ ), τ T 0,T } is uniformly inegrable. We solve (9) by dynamic programming and BSDE.

60 Value funcion processes: For [0, T, and α A, we denoe by: A (α) = {β A : β. = α. }, and define he family of F-adaped processes V (α) := ess sup E [U(X βt ) F, 0 T. β A (α) Dynamic programming (DP) For any α A, he process {V (α), 0 T } is a supermaringale There exiss an opimal conrol ˆα A o V 0 if and only if he maringale propery holds, i.e. he process {V (ˆα), 0 T } is a maringale. In he sequel, we exploi he DP in he case of CRRA uiliy funcions: U(x) = x p /p, p < 1. The key observaion is he propery ha he F-adaped process Y := V (α) U(X α ) > 0 does no depend on α A, and Y T = 1. We adop a BSDE verificaion approach: we are looking for (Y, Z) soluion o Y = 1 + T f(s, ω, Y s, Z s )ds for some generaor f o be deermined such ha T Z s dw s, (10) For any α A, he process {U(X α )Y, 0 T } is a supermaringale There exiss ˆα A for which {U(X ˆα )Y, 0 T } is a maringale.

61 By applying Iô s formula o U(X α )Y, he supermaringale propery for all α A, and he maringale propery for some ˆα imply ha f should be equal o [ f(, Y, Z ) = p sup (µ Y + σ Z ).a 1 p a A 2 Y σ a 2, (11) wih a candidae for he opimal conrol given by ˆα arg max a A [ (µ Y + σ Z ).a 1 p 2 Y σ a 2, 0 T. (12) Exisence and uniqueness of a soluion o he BSDE (10)-(11): Change of variables Ỹ = ln Y, Z = Z/Y (Ỹ, Z) saisfy a quadraic BSDE. Then, we rely on resuls by Kobylanski (00) Exisence and uniqueness of (Y, Z) S (0, T ) H 2 (0, T ) d Verificaion argumen: le (Y, Z) be he soluion o (10)-(11) By consrucion U(X α )Y is a (local)-supermaringale + inegrabiliy condiions on α A: i is a supermaringale sup α A E[U(XT α) U(X 0)Y 0. By BMO echniques, we show ha ˆα defined in (12) lies in A U(X ˆα )Y is a maringale E[U(XT ˆα ) = U(X 0)Y 0 We conclude ha V 0 := sup α A E[U(XT α) = U(X 0)Y 0, and ˆα is an opimal conrol.

62 Markov cases Meron model: he coefficiens of he sock price µ() and σ() are deerminisic In his deerminisic case, he BSDE (10)-(11) is reduced o an ODE: Y () = 1 + T f(s, Y (s))ds, [ f(, y) = y p sup µ().a 1 p a A 2 σ()a 2 =: yρ() and he soluion is given by: Y () = e T Meron problem: V 0 ρ(s)ds we find again he soluion o he = U(X 0 ) exp ( T ρ(s)ds ). 0

63 Facor model: he coefficiens of he sock price µ(, L ) and σ(, L ) depend on a facor process dl = η(l )d + dw. In his case, he BSDE for (Ỹ, Z) = (ln Y, Z/Y ) is wrien as: Y = T wih a quadraic (in z) generaor f(, l, y, z) = p sup a A f(s, L s, Ỹs, Z s )ds T Z s dw s [ (µ(, l) + σ(, l)z).a 1 p 2 σ(, l)a z2. Ỹ = ϕ(, L ), wih a corresponding semilinear PDE for ϕ: ϕ + η(l).d lϕ r(d2 lϕ) + f(, l, ϕ, D l ϕ) = 0, ϕ(t, l) = 0. Value funcion: V 0 = U(X 0 ) exp ( ϕ(0, L 0 ) ). Remark The BSDE approach and dynamic programming is also well-suiable for exponenial uiliy maximizaion: Many papers: El Karoui, Rouge (00), Hu, Imkeller, Muller (04), Sekine (06), Becherer (06), ec...

64 V. Refleced BSDE and opimal sopping problem We consider a class of BSDEs where he soluion Y is consrained o say above a given process L, called obsacle. An increasing process K is inroduced for pushing he soluion upwards, above he obsacle Noion of refleced BSDE: Given pair of erminal condiion/generaor (ξ, f) and a coninuous obsacle process (L ) s.. ξ L T, find a riple of adaped processes (Y, Z, K) wih K nondecreasing s.. Y = ξ + T f(s, Y s, Z s )ds T Z s dw s + K T K, 0 T, (13) Y L, 0 T, (14) T 0 (Y L )dk = 0. (15) Connecion wih opimal sopping problem: in he case where f(, ω) does no depend on (y, z), here exiss a unique soluion o (13)-(14)-(15) given by [ τ F Y = ess sup E f(s)ds + L τ 1 τ<t + ξ1 τ=t, 0 T. (16) τ T,T

65 Argumens of proof. Snell envelope of H := 0 f(s)ds + L 1 <T + ξ1 =T, i.e. [ τ F S := ess sup E f(s)ds + L τ 1 τ<t + ξ1 τ=t, 0 T. τ T,T 0 Doob-Meyer decomposiion for he coninuous supermaringale S, and maringale represenaion heorem here exiss (Z, K) s.. ds = Z dw dk Denoe by Y = S 0 f(s)ds. Then (Y, Z) saisfies: dy = f()d + Z dw dk, Y T = ξ Y L. Consider he opimal sopping ime for he Snell envelope, i.e. τ = inf{s : S = H } T = inf{s : Y = L } T, which means ha he sopped process S τ K τ = K, i.e. is a maringale. This implies ha T 0 (Y L )dk = 0. (Y, Z, K) is soluion o he refleced BSDE (13)-(14)-(15).

66 General case: (ξ, f) saisfying sandard assumpion (H1) wih Lipschiz generaor, and L is a coninuous obsacle in S 2 (0, T ). Exisence and approximaion by penalizaion For each n N, we consider he (unconsrained) BSDE where K n Y n = ξ + T f(s, Y n s, Z n s )ds + K n T K n T = n T (Y s n L s ) ds exisence and uniqueness of (Y n, Z n ). Z n s.dw s, (17) Sae a priori uniform esimaes on he sequence (Y n, Z n, K n ): here exiss ome posiive consan C s.. [ E sup 0 T Y n 2 + T 0 Z n 2 d + K n T 2 C, n N. By comparison principle for BSDE, (Y n ) n is an increasing sequence, and i converges o some Y S 2 (0, T ), and he convergence also holds in H 2 (0, T ), i.e. [ T lim n E Y 2 d = 0. Moreover, Y L. 0 Y n Prove ha (Z n, K n ) n is a Cauchy sequence in H 2 (0, T ) d S 2 (0, T ): use Iô s formula o Y n Y m 2, and inequaliy 2ab εa ε b2 for suiable ε. Consequenly, (Z n, K n ) n converges o some (Z, K) in H 2 (0, T ) d S 2 (0, T ). Pass o he limi in (17) in order o obain he exisence of (Y, Z, K) soluion o he refleced BSDE.

67 Remark: Alernaive formulaion of he Skorokhod condiion. The definiion of a soluion o he refleced BSDE (13)-(14) wih he Skorokhod condiion (15) T 0 (Y L )dk = 0 can be formulaed equivalenly in erms of minimal soluion: We say ha (Y, Z, K) is a minimal soluion o (13)-(14) if for any oher soluion (Ỹ, Z, K) soluion o (13)-(14), we have Y Ỹ, 0 T. Any soluion o he refleced BSDE (13)-(14)-(15) is a minimal soluion o (13)- (14), and he converse is also rue.

68 Connecion wih variaional inequaliies in he Markov case Consider he case where: ξ = g(x T ), f(, ω, Y, Z ) = f(, X, Y, Z ), L = h(x ), 0 T, wih g h, and where X is a diffusion process on R n dx = b(x )d + σ(x )dw. Then, he soluion o he refleced BSDE is given by Y = v(, X ) for some deerminisic funcion v, viscosiy soluion o he variaional inequaliy: min [ v Lv f(, x, v, σ D x v), v h = 0, on [0, T ) R n v(t,.) = g on R n.

69 VI. BSDE wih consrained jumps and quasi-variaional inequaliies Consider he impulse conrol problem: wih [ T v(, x) = sup E g(xt α ) + f(xs α )ds + c(x α τ i α <τ i T conrols: α = (τ i, ξ i ) i where (τ i ) i ime decisions: nondecreasing sequence of sopping imes, ξ i ) (ξ i ) i acion decisions: sequence of r.v. s.. ξ i F τi valued in E, conrolled process X α defined by X α s = x + s b(x α u )du + s σ(x α u )dw u + <τ i s γ(x α τ i, ξ i ) The corresponding dynamic programming equaion is he quasi-variaional inequaliy (QVI): min [ v Lv f, v Hv = 0, v(t,.) = g, (18) where L is he second-order local operaor: and H is he nonlocal operaor Lv(, x) = b(x).d x v(, x) r(σσ (x)d 2 xv(, x)) wih Hv(, x) = sup H e v(, x) e E H e v(, x) = v(, x + γ(x, e)) + c(x, e).

70 The QVI (18) divides he ime-space domain ino: a coninuaion region C in which v(, x) > Hv(, x) and v Lv f = 0 an acion region D in which: v(, x) = Hv(, x) = sup v(, x + γ(x, e)) + c(x, e). e E Various applicaions of impulse conrol problems: Financial modelling wih discree ransacion daes, due e.g. o fixed ransacion coss or liquidiy consrains Opimal muliple sopping: swing opions Projec s invesmen and real opions: managemen of power plans, valuaion of gas sorage and naural resources, fores managemen, Impulse conrol: widespread economical and financial seing wih many pracical applicaions More generally o models wih conrol policies ha do no accumulae in ime.

71 Main heoreical and numerical difficuly in he QVI (18) : The obsacle erm conains he soluion iself I is nonlocal Classical approach : Decouple he QVI (18) by defining by ieraion he sequence of funcions (v n ) n : [ min v n+1 Lv n+1 f, v n+1 Hv n = 0, v n+1 (T,.) = g associaed o a sequence of opimal sopping ime problems (refleced BSDEs) Furhermore, o compue v n+1, we need o know v n on he whole domain heavy compuaions, especially in high dimension (sae space discreizaion): numerically challenging!

72 Idea of our approach Insead of viewing he obsacle erm as a reflecion of v ono Hv (or v n+1 ino Hv n ), consider i as a consrain on he jumps of v(, X ) for some suiable forward jump process X: Le us inroduce he unconrolled jump diffusion X : dx = b(x )d + σ(x )dw + γ(x, e)µ(d, de), E where µ is a Poisson random measure whose inensiy λ is finie and suppors he whole space E. We randomize he sae space! Take some smooh funcion v(, x) and define: Y := v(, X ), Z := σ(x ) D x v(, X ), U (e) := v(, X + γ(x, e)) v(, X ) + c(x, e) = (H e v v)(, X )

73 Apply Iô s formula o Y = v(, X ) beween and T : where Y = Y T + + T K := T E f(x s )ds + K T K T [U s (e) c(x s, e)µ(ds, de), 0 ( v Lv f)(s, X s)ds Z s.dw s Now, suppose ha min[ v Then (Y, Z, U, K) saisfies Lv f, v Hv 0, and v(t,.) = g : T Y = g(x T ) + f(x s )ds + K T K Z s.dw s T + [U s (e) c(x s, e)µ(ds, de), (19) E K is a nondecreasing process, and U saisfies he nonposiiviy consrain : U (e) 0, 0 T, e E. (20) T View (19)-(20) as a Backward Sochasic Equaion wih jump consrains We expec o rerieve he soluion o he QVI (18) by solving he minimal soluion o his consrained BSE.

74 General definiion of BSDEs wih consrained jumps Minimal Soluion : find a soluion (Y, Z, U, K) S 2 H 2 (0, T ) d L 2 ( µ) A 2 o wih Y = g(x T ) + T T E f(x s, Y s, Z s )ds + K T K T Z s.dw s (U s (e) c(x s, Y s, Z s, e))µ(ds, de) (21) h(u (e), e) 0, dp d λ(de) a.e. (22) such ha for any oher soluion (Ỹ, Z, Ũ, K) o (21)-(22) : Y Ỹ, 0 T, a.s. Assumpions on he coefficiens: Forward SDE : b and σ Lipschiz coninuous, γ bounded and Lipschiz coninuous w.r.. x uniformly in e: γ(x, e) γ(x, e) k x x e E Backward SDE : f, g and c have linear growh, f and g Lipschiz coninuous, c Lipschiz coninuous w.r.. y and z uniformly in x and e c(x, y, z, e) c(x, y, z, e) k c ( y y + z z ) Consrain : h Lipschiz coninuous w.r.. u uniformly in e: h(u, e) h(u, e) k h u u and u h(u, e) nonincreasing. (e.g. h(u, e) = u)

75 Exisence and approximaion via penalizaion Consider for each n he BSDE wih jump: Y n = g(x T ) + T wih a penalizaion erm where h = max( h, 0). K n T E f(x s, Y n s, Z n s )ds + K n T K n T Z n s.dw s [U n s (e) c(x s, Y n s, Zn s, e)µ(ds, de) (23) = n h (Us n (e), e)λ(de)ds 0 E For each n, exisence and uniqueness of (Y n, Z n, U n ) soluion o (23) from Tang and Li (94), and Barles e al. (97). Convergence of he penalized soluions Theorem.1 Under (H1), here exiss a unique minimal soluion wih K predicable, o (21)-(22). (Y, Z, U, K) S 2 H 2 (0, T ) d L 2 ( µ) A 2 Y is he increasing limi of (Y n ) and also in S 2 (0, T ), K is he weak limi of (K n ) in S 2 (0, T ), and for any p [1, 2), as n goes o infiniy. Z n Z H p (0,T ) + U n U L p ( µ) 0,

76 Skech of proof. Convergence of (Y n ): by comparison resuls (under he nondecreasing propery of h) Y n Y n+1 Convergence of (Z n, U n, K n ) : more delicae! A priori uniform esimaes on (Y n, Z n, U n, K n ) n in L 2 weak convergence of (Z n, U n, K n ) in L 2 Moreover, in general, we need some srong convergence o pass o he limi in he nonlinear erms f(x, Y n, Z n ), c(x, Y n, Z n ) and h(u n (e), e). Conrol jumps of he predicable process K via a random pariion of he inerval (0,T) and obain a convergence in measure of (Z n, U n, K n ) Convergence of (Z n, U n, K n ) in L p, p [1, 2)

77 Relaed semilinear QVI By Markov propery, he minimal soluion o he consrained BSDE wih jumps is Y = v(, X ) for some deerminisic funcion v. The funcion v is he unique viscosiy soluion o he QVI: [ min v Lv f(., v, σ D x v), inf e E h(he v v, e) = 0 on [0, T ) R n, (24) ogeher wih he relaxed erminal condiion: min [ v g, inf e E h(he v v, e) = 0 on {T } R n. (25) Probabilisic represenaion of semilinear QVIs, and in paricular of impulse conrol problems by means of BSDEs wih consrained jumps. Numerical implicaions for he resoluion of QVIs by means of simulaion of he penalized BSDE: PhD hesis of M. Bernhar, in parnership wih EDF for he valuaion of swing opions and gas sorage conacs.