AMartingaleApproachforFractionalBrownian Motions and Related Path Dependent PDEs

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1 AMaringaleApproachforFracionalBrownian Moions and Relaed Pah Dependen PDEs Jianfeng ZHANG Universiy of Souhern California Join work wih Frederi VIENS Mahemaical Finance, Probabiliy, and PDE Conference Rugers Universiy, 5/17-5/19, 2017

2 Ouline Inroducion 1 Inroducion 2 3 4

3 The sandard risk neural pricing Le S be an underlying asse price, lp ariskneuralmeasure: ds = (, S )db Le = g(s T ) be a payoff a T, hen he price a is : Y = IE [ ] In he above Markovian seing : Y = u(, S u (, x)@ 2 xxu = 0, u(t, x) =g(x). In pah dependen seing : = (, S ), = g(s ), hen Y = u(, S u (,!)@ 2!!u = 0, u(t,!) =g(!).

4 Rough volailiy model Rough volailiy : ds = S db and is rough See e.g. Gaheral-Jaisson-Rosenbaum (2014) Anauralmodel: Goal : characerize Y := IE (hence B H )canbeobserved driven by a fracional Brownian moion B H F B,BH To focus on he main idea we will assume is FT BH -measurable and consider Y = IE F BH Sone relaed recen works : El Euch-Rosenbaum (2017), Fouque-Hu (2017)

5 Ouline Inroducion 1 Inroducion 2 3 4

6 Fracional Brownian Moion Le B H be a fbm wih 0 < H < 1: B H B H s Normal(0, ( s) 2H ) B H = B when H = 1 2 Two main feaures : B H is no Markovian (H 6= 1 2 ) B H is no a semimaringale (H < 1 2 ) Our goal : characerize Y := IE g(b H ) F BH

7 in BM case Le := g(b T ) and Y := IE [g(b T )]. Denoe v(, x) :=IE g(x + B T B ) = R IR g(y)p(t, y x)dy where p(, x) := 1 p 2 e x2 2. p(, x) 1 xxp(, x) v(, x)+ 1 xxv(, x) =0, v(t, x) =g(x). Y = v(, B ),0apple apple T

8 A for fbm Le := g(b H T ) and Y := IE [g(b H T )]. Denoe v(, x) :=IE g(x + B H T B H ) = R IR g(y)p H(T, y x)dy : where p H (, x) := p 1 e x H v(, x)+h 2H xx v(, x) =0, v(t, x) =g(x). Y 0 = v(0, 0)

9 AheaequaionforfBM Le := g(bt H) and Y := IE [g(bt H)]. Denoe v(, x) :=IE g(x + BT H B H ) v(, x)+h 2H xx v(, x) =0, v(t, x) =g(x). Y 0 = v(0, B0 H), Y T = v(t, BT H) However, v(, B H ) is no a maringale : Y 6= v(, B H ) for 0 < < T.

10 AcrucialrepresenaionoffBM Represenaion : B H lf := lf BH = lf W = R 0 K(, r)dw r K(, r) ( r) 2H 1, which blows up a = r when H < 1 2 Decomposiion : B H T = Z T 0 K(T, r)dw r = Z 0 K(T, r)dw r + Z T K(T, r)dw r R 0 K(T, r)dw r is F -measurable R T K(T, r)dw r is independen of F The previous decomposiion B H T = BH +[B H T B H ] does no saisfy his propery

11 An alernaive hea equaion Le := g(b H T ) and Y = IE hg Z 0 K(T, r)dw r + Z T K(T, r)dw r i Denoe v(, x) :=IE h g x + R T K(T, r)dw r i Then Y = v, R 0 K(T, r)dw r, 0 apple apple T Noe : v, R 0 K(T, r)dw r is a maringale v(, x)+ 1 2 K 2 (T, )@ xx v(, x) =0, v(t, x) =g(x).

12 Acloserlook Inroducion T := R 0 K(T, r)dw r = IE [B H T ] is F -measurable T is he forward variance and is observable in marke Three ways o express Y : Y = v 1 (, B H ^ ) =v 2 (, W ^ ) =v(, T ) B H is no a semimaringale W is a maringale (of course) bu v 2 is no coninuous v has desired regulariy and 7! T is a maringale

13 An exension Denoe Y := IE h g(b H T )+R T By previous compuaion : i f (s, Bs H )ds. Y = IE [g(b H T )] + R T = v(t, g;, IE [B H T ]) + R T = u(, {IE [B H s ]} applesapplet ) IE [f (s, B H s )]ds v(s, f (s, );, IE [B H s ])ds Noe : u is pah dependen If H = 1 2, IE [B s ]=B, so V = u(, B ) is sae dependen In more general cases, Y = u, {Bs H } 0applesapple {IE [Bs H ]} applesapplet.

14 Ouline Inroducion 1 Inroducion 2 3 4

15 The canonical seup Inroducion Recall Y = u, {Bs H } 0applesapple {IE [Bs H ]} applesapplet. For 2 [0, T ],! 2 ld 0 ([0, )), and 2 C 0 ([, T ]), define : (! ) s :=! s 1 [0,) (s)+ s 1 [,T ] (s), 0 apple s apple T. The canonical space : n o := (,! ): 2 [0, T ],! 2 ld 0 ([0, )), 2 C 0 ([, T ]) ; o 0 := n(,! ) 2 :! 2 C 0 ([0, ]),! 0 = 0, =!.

16 Coninuous mapping Inroducion Recall n o := (,! ): 2 [0, T ],! 2 ld 0 ([0, )), 2 C 0 ([, T ]). The meric : d((,! ), ( 0,! )) := p 0 + sup 0applesappleT (! ) s (! ) s. C 0 ( ) : coninuous mapping u :! IR C 0 b ( ) : bounded u 2 C 0 ( )

17 Pah derivaives Inroducion Time derivaive u(,! ):=lim #0 u( +,! ) u(,! u is he righ ime derivaive! Firs order spaial derivaive : Fréche derivaive wih respec o 1 h i h@ u(,! ), i := lim u(,! ( + " )) u(,! ), "!0 " for all (,! ) 2, 2 C 0 ([, T ]).

18 Pah derivaives (con) Second order spaial derivaive : bilinear operaor on C 0 ([, T ]) : h@ 2 u(,! ), ( 1, 2 )i h 1 := lim "!0 " h@ u(,! ( + " 1 )), 2 i i h@ u(,! ), 2 i. for all (,! ) 2, 1, 2 2 C 0 ([, T ]). Define he spaces C 1,2 ( ) and C 1,2 b ( ) in obvious sense

19 Funcional Io formula : H 1 2 Regular case : K(, ) is finie and hus s 2 [, T ] 7! K s := K(s, ) is in C 0 ([, T ]). Denoe : X s := B H s,0apple s apple ; s := IE [B H s ], apple s apple T Funcional Io formula : du(, X ) u( )d + h@ u( ), K idw h@2 u( ), (K, K )id. If H = 1 2, K = 1, his is exacly Dupire s funcional Io formula

20 Funcional Io formula : H < 1 2 K(s, ) (s ) H 1 s K(s, ) (s ) H 3 2,0apple < s apple T For some > 1 2 H, for any (,! ) 2 0, any < 1 < 2 apple T, any 2 C 0 ([, T ] wih suppor in [ 1, 2 ], h@ u(,! ), i applec[ 2 1 ] k k 1, h@ 2 u(,! ), (, )i applec[ 2 1 ] 2 k k 2 1. Roughly speaking, we u(,! )=0. Denoe Ks, := K(+ )_s. Then he following limis exis : h@ u(,! ), K i := lim!0 h@ u(,! ), K, i; h@ 2 u(,! ), (K, K )i := lim h@ 2 u(,! ), (K,, K, )i.!0 Funcional Io formula sill holds

21 Linear pah dependen PDE Y := IE h g(b H T )+R T i f (s, Bs H )ds = u(, X ) Y + R 0 f (s, BH s )ds is a maringale Linear PPDE u(,! )+ 1 2 h@2 u(,! ), (K, K )i + f (,! )=0, u(t,!) =g(! T ). Theorem. Assume f and g are smooh, hen he above PPDE has a unique classical soluion u.

22 Ouline Inroducion 1 Inroducion 2 3 4

23 Nonlinear dynamics Inroducion Forward dynamics : Volerra SDE X = x + Z 0 b(; r, X )dr + Backward dynamics : BSDE Z 0 (; r, X )dw r Y = g(x )+ Z T f (s, X, Y s, Z s )ds Z T Z s dw s. The backward one iself is ime consisen. If we consider Volerra ype of BSDEs, see a series of works by Jiongmin Yong. Y = u(, X ), where s := x + Z 0 b(s; r, X )dr + Z 0 (s; r, X )dw r, apple s apple T.

24 Nonlinear PPDE Represenaion : u(,! ):=Y,!, where X,! s Y,! s = s + R s b(s; r,! X,! )dr + R s = g(! X,! ) + R T s (s; r,! X,! )dw r R T s Z,! r dw r f (r,! X,!, Y,! r, Z,! r )dr. Semilinear PPDE : ',! s := '(s;,!), apple s apple T, for ' = u h@2 u, (,!,,! )i + h@ u, b,! i + f,!, u, h@ u,,! i)=0, u(t,!) =g(!).

25 Furher research Inroducion Conrolled problems (fully nonlinear PPDE) Viscosiy soluion Efficien numerical algorihms

26 Thank you very much for your aenion!