and Applications Alexander Steinicke University of Graz Vienna Seminar in Mathematical Finance and Probability,

Transcription

1 Backward Sochasic Differenial Equaions and Applicaions Alexander Seinicke Universiy of Graz Vienna Seminar in Mahemaical Finance and Probabiliy, / 31

2 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 2 / 31

3 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Sochasic (ordinary, forward) differenial equaions (driven by a Brownian moion W ): or dx = b(, X )d + σ(, X )dw, X 0 = x 0 X = x 0 + b(s, X s )ds + σ(s, X s )dw s, 0 T / 31

4 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Sochasic (ordinary, forward) differenial equaions (driven by a Brownian moion W ): or dx = b(, X )d + σ(, X )dw, X 0 = x 0 X = x 0 + b(s, X s )ds + σ(s, X s )dw s, 0 T. 0 0 MANY applicaions (disurbed ODEs, fundamenal for finance in con. ime,...) 3 / 31

5 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Sochasic (ordinary, forward) differenial equaions (driven by a Brownian moion W ): or dx = b(, X )d + σ(, X )dw, X 0 = x 0 X = x 0 + b(s, X s )ds + σ(s, X s )dw s, 0 T. 0 0 MANY applicaions (disurbed ODEs, fundamenal for finance in con. ime,...) b, σ progressively measurable,... 3 / 31

6 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Sochasic (ordinary, forward) differenial equaions (driven by a Brownian moion W ): or dx = b(, X )d + σ(, X )dw, X 0 = x 0 X = x 0 + b(s, X s )ds + σ(s, X s )dw s, 0 T. 0 0 MANY applicaions (disurbed ODEs, fundamenal for finance in con. ime,...) b, σ progressively measurable,... x 0 = X 0 is deerminisic 3 / 31

7 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Sochasic (ordinary, forward) differenial equaions (driven by a Brownian moion W ): or dx = b(, X )d + σ(, X )dw, X 0 = x 0 X = x 0 + b(s, X s )ds + σ(s, X s )dw s, 0 T. 0 0 MANY applicaions (disurbed ODEs, fundamenal for finance in con. ime,...) b, σ progressively measurable,... x 0 = X 0 is deerminisic X is F -measurable 3 / 31

8 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Consider he same siuaion backward in ime: X = ξ+ b(s, X s )ds+ σ(s, X s )dw s, X T = ξ, 0 T. 4 / 31

9 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Consider he same siuaion backward in ime: X = ξ+ b(s, X s )ds+ X T = ξ L 2 (F T ) σ(s, X s )dw s, X T = ξ, 0 T. 4 / 31

10 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Consider he same siuaion backward in ime: X = ξ+ b(s, X s )ds+ X T = ξ L 2 (F T ) Is X 0 deerminisic? σ(s, X s )dw s, X T = ξ, 0 T. 4 / 31

11 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Consider he same siuaion backward in ime: X = ξ+ b(s, X s )ds+ X T = ξ L 2 (F T ) Is X 0 deerminisic? Is X F -measurable? σ(s, X s )dw s, X T = ξ, 0 T. 4 / 31

12 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Consider he same siuaion backward in ime: X = ξ+ b(s, X s )ds+ X T = ξ L 2 (F T ) Is X 0 deerminisic? Is X F -measurable? σ(s, X s )dw s, X T = ξ, 0 T. In general: NO! Everyhing is F T -measurable. Problem is no well-posed. 4 / 31

13 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs When is i possible o find an adaped backward soluion ha has he dynamics of his SDE? How o find a seing such ha he problem is well-posed? X = ξ + b(s, X s )ds + σ(s, X s )dw s, 0 T. 5 / 31

14 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs When is i possible o find an adaped backward soluion ha has he dynamics of his SDE? How o find a seing such ha he problem is well-posed? X = ξ + b(s, X s, Z s )ds Z s dw s, 0 T. Sochasic erm has o be conrolled by a process ha subracs he righ amoun of randomness of ξ. No arbirary σ! 6 / 31

15 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs When is i possible o find an adaped backward soluion ha has he dynamics of his SDE? How o find a seing such ha he problem is well-posed? X = ξ + b(s, X s, Z s )ds Z s dw s, 0 T. Sochasic erm has o be conrolled by a process ha subracs he righ amoun of randomness of ξ. No arbirary σ! Z is par of he soluion. 6 / 31

16 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Rename b =: f, X =: Y o wrie X = ξ + b(s, X s, Z s )ds Z s dw s, as Y = ξ + f (s, Y s, Z s )ds Z s dw s or, in differenial noaion, dy = f (Y, Z )d + Z dw 0 T 7 / 31

17 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Rename b =: f, X =: Y o wrie X = ξ + b(s, X s, Z s )ds Z s dw s, as Y = ξ + f (s, Y s, Z s )ds Z s dw s or, in differenial noaion, dy = f (Y, Z )d + Z dw 0 T This is called a (sandard) BSDE (backward sochasic differenial equaion) wih generaor f and erminal condiion ξ. 7 / 31

18 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs Rename b =: f, X =: Y o wrie X = ξ + b(s, X s, Z s )ds Z s dw s, as Y = ξ + f (s, Y s, Z s )ds Z s dw s or, in differenial noaion, dy = f (Y, Z )d + Z dw 0 T This is called a (sandard) BSDE (backward sochasic differenial equaion) wih generaor f and erminal condiion ξ. A soluion o a BSDE is a pair of processes (Y, Z) such ha he equaion is saisfied. 7 / 31

19 Applicaions - Why do we need BSDEs? Pricing of coningen claims 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 8 / 31

20 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw 8 / 31

21 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. 8 / 31

22 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. π is he amoun of money invesed in S a ime 8 / 31

23 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. π is he amoun of money invesed in S a ime Y is he wealh of he rader 8 / 31

24 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. π is he amoun of money invesed in S a ime Y is he wealh of he rader he money lend/borrowed is Y π 8 / 31

25 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. π is he amoun of money invesed in S a ime Y is he wealh of he rader he money lend/borrowed is Y π 8 / 31

26 Applicaions - Why do we need BSDEs? Pricing of coningen claims Le S be a risky asse wih evoluion dynamics ds = µ S d + σ S dw A rader may inves in S or borrow/lend money (wihou risk) a an ineres rae r. π is he amoun of money invesed in S a ime Y is he wealh of he rader he money lend/borrowed is Y π dy = π S ds + r (Y π )d = (π (µ r ) + r Y )d + π σ dw 8 / 31

27 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw 9 / 31

28 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw Suppose we have a ime T a payoff ξ L 2 (European call opion) 9 / 31

29 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw Suppose we have a ime T a payoff ξ L 2 (European call opion) Wha is he minimal amoun Y 0 such ha a ime T we can cover ξ by a sraegy π such ha Y T = ξ? 9 / 31

30 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw Suppose we have a ime T a payoff ξ L 2 (European call opion) Wha is he minimal amoun Y 0 such ha a ime T we can cover ξ by a sraegy π such ha Y T = ξ? Pu differenly, we look for a soluion (Y, π) ha solves 9 / 31

31 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw Suppose we have a ime T a payoff ξ L 2 (European call opion) Wha is he minimal amoun Y 0 such ha a ime T we can cover ξ by a sraegy π such ha Y T = ξ? Pu differenly, we look for a soluion (Y, π) ha solves Y = ξ (π s (µ s r s ) + r s Y s ) ds π s σ s dw s 9 / 31

32 Applicaions - Why do we need BSDEs? Pricing of coningen claims dy = (π (µ r ) + r Y )d + π σ dw Suppose we have a ime T a payoff ξ L 2 (European call opion) Wha is he minimal amoun Y 0 such ha a ime T we can cover ξ by a sraegy π such ha Y T = ξ? Pu differenly, we look for a soluion (Y, π) ha solves Y = ξ (π s (µ s r s ) + r s Y s ) ds If λ exiss, such ha µ r = σλ, we ge π s σ s dw s which is a BSDE. Y = ξ (Z s λ s + r s Y s ) ds Z s dw s, 9 / 31

33 Applicaions - Why do we need BSDEs? Pricing of coningen claims The above problem has an explici soluion: [ E Q e ] T r sds ξ F, where Q is he risk-neural measure. 10 / 31

34 Applicaions - Why do we need BSDEs? Pricing of coningen claims The above problem has an explici soluion: [ E Q e ] T r sds ξ F, where Q is he risk-neural measure. Bu: If he raes for borrowing and lending are differen, he wealh process saisfies Y = ξ ( π s µ s + r l s(y s π s ) + r b s(y s π s ) ) ds σ s π s dw s. 10 / 31

35 Applicaions - Why do we need BSDEs? Pricing of coningen claims The above problem has an explici soluion: [ E Q e ] T r sds ξ F, where Q is he risk-neural measure. Bu: If he raes for borrowing and lending are differen, he wealh process saisfies Y = ξ corresponding BSDE: ( Y =ξ ( π s µ s + r l s(y s π s ) + r b s(y s π s ) ) ds Z s µ s σ s + r l s σ s (σ s Y s Z s ) + r b s σ s (σ s Y s Z s ) Z s dw s ) ds σ s π s dw s. 10 / 31

36 Applicaions - Why do we need BSDEs? Pricing of coningen claims The above problem has an explici soluion: [ E Q e ] T r sds ξ F, where Q is he risk-neural measure. Bu: If he raes for borrowing and lending are differen, he wealh process saisfies Y = ξ corresponding BSDE: ( Y =ξ ( π s µ s + r l s(y s π s ) + r b s(y s π s ) ) ds Z s µ s σ s + r l s σ s (σ s Y s Z s ) + r b s σ s (σ s Y s Z s ) Z s dw s No more explici soluions in he above manner. ) ds σ s π s dw s. 10 / 31

37 Applicaions - Why do we need BSDEs? Pricing of coningen claims Similar approaches for: 11 / 31

38 Applicaions - Why do we need BSDEs? Pricing of coningen claims Similar approaches for: Hedging wih consrains (Sraegy beween given bounds [ m, M]) leads o Y = ξ + f (s, Y s, Z s )ds Z s dw s + A T A for an adaped, nondecreasing process A. 11 / 31

39 Applicaions - Why do we need BSDEs? Pricing of coningen claims Similar approaches for: Hedging wih consrains (Sraegy beween given bounds [ m, M]) leads o Y = ξ + f (s, Y s, Z s )ds Z s dw s + A T A for an adaped, nondecreasing process A. Same equaion is used o hedge American opions: Y ζ, Y T = ζ T. Refleced BSDEs, RBSDEs. 11 / 31

40 Applicaions - Why do we need BSDEs? Pricing of coningen claims Similar approaches for: Hedging wih consrains (Sraegy beween given bounds [ m, M]) leads o Y = ξ + f (s, Y s, Z s )ds Z s dw s + A T A for an adaped, nondecreasing process A. Same equaion is used o hedge American opions: Y ζ, Y T = ζ T. Refleced BSDEs, RBSDEs. Soluion: (Y, Z, A) 11 / 31

41 Applicaions - Why do we need BSDEs? Represenaion of risk measures 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 12 / 31

42 Applicaions - Why do we need BSDEs? Represenaion of risk measures Risk measures nonlinear F-expecaions 12 / 31

43 Applicaions - Why do we need BSDEs? Represenaion of risk measures Risk measures nonlinear F-expecaions A nonlinear expecaion is an operaor E : L 2 R such ha X X E(X ) E(X ), equaliy only if X = X. E(c) = c for consans for each X, here is η X such ha for all A F : E(X 1I A ) = E(η X 1I A ). In his case: η X =: E (X ). 12 / 31

44 Applicaions - Why do we need BSDEs? Represenaion of risk measures Risk measures nonlinear F-expecaions A nonlinear expecaion is an operaor E : L 2 R such ha X X E(X ) E(X ), equaliy only if X = X. E(c) = c for consans for each X, here is η X such ha for all A F : E(X 1I A ) = E(η X 1I A ). In his case: η X =: E (X ). Theorem: If Y = ξ + f (s, Y s, Z s )ds Z s dw s, (and f is uniformly Lipschiz in (y, z)) hen E f defined by E f (ξ) = Y consiues a nonlinear expecaion f -expecaion. 12 / 31

45 Applicaions - Why do we need BSDEs? Represenaion of risk measures Risk measures nonlinear F-expecaions A nonlinear expecaion is an operaor E : L 2 R such ha X X E(X ) E(X ), equaliy only if X = X. E(c) = c for consans for each X, here is η X such ha for all A F : E(X 1I A ) = E(η X 1I A ). In his case: η X =: E (X ). Theorem: If Y = ξ + f (s, Y s, Z s )ds Z s dw s, (and f is uniformly Lipschiz in (y, z)) hen E f defined by E f (ξ) = Y consiues a nonlinear expecaion f -expecaion. In he case f = 0 we ge back he ordinary condiional expecaion E (X ) = E[X F ] (This will serve as an easy example laer). 12 / 31

46 Applicaions - Why do we need BSDEs? Represenaion of risk measures Converse heorem (parly): 13 / 31

47 Applicaions - Why do we need BSDEs? Represenaion of risk measures Converse heorem (parly): If E is a nonlinear expecaion such ha wih f µ (y, z) = µ z, and if E(X + X ) E(X ) + E fµ (X ), E (X + X ) = E (X ) + X for X L 2 (F ), hen here exiss a generaor f, no depending on y such ha E = E f. 13 / 31

48 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 14 / 31

49 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: 14 / 31

50 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: Le X be he forward soluion o dx = b(, X )d + σ(, X )dw, X 0 = x. 14 / 31

51 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: Le X be he forward soluion o dx = b(, X )d + σ(, X )dw, X 0 = x. Moreover, le L be he operaor Lφ = φ + b x φ + σ2 2 2 xxφ. 14 / 31

52 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: Le X be he forward soluion o dx = b(, X )d + σ(, X )dw, X 0 = x. Moreover, le L be he operaor Lφ = φ + b x φ + σ2 2 2 xxφ. Assume, here is a (nice) soluion v o he PDE 0 = Lφ + f (,, φ, σ x φ), φ(t, x) = G(x). 14 / 31

53 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: Le X be he forward soluion o dx = b(, X )d + σ(, X )dw, X 0 = x. Moreover, le L be he operaor Lφ = φ + b x φ + σ2 2 2 xxφ. Assume, here is a (nice) soluion v o he PDE 0 = Lφ + f (,, φ, σ x φ), φ(t, x) = G(x). Then, he couple Y := v(, X ), Z := x v(, X ) solves he backward equaion Y = G(X T ) + f (s, X s, Y s, Z s )ds Z s dw s. forward-backward SDE (decoupled) Proof: Iô formula. 14 / 31

54 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs There is a connecion beween semilinear parabolic equaions and BSDEs: Le X be he forward soluion o dx = b(, X )d + σ(, X )dw, X 0 = x. Moreover, le L be he operaor Lφ = φ + b x φ + σ2 2 2 xxφ. Assume, here is a (nice) soluion v o he PDE 0 = Lφ + f (,, φ, σ x φ), φ(t, x) = G(x). Then, he couple Y := v(, X ), Z := x v(, X ) solves he backward equaion Y = G(X T ) + f (s, X s, Y s, Z s )ds Z s dw s. forward-backward SDE (decoupled) Proof: Iô formula. If he BSDE has a mos one soluion, hen solving he BSDE and he PDE are equivalen. 14 / 31

55 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs Example: L = + σ2 2 xx + µ x Lu(, x) + u(, x) + σ x u(, x) = 0 u(t, x) = sin(x) ranslaes ino dx = µd + σdw, X 0 = 1 dy = Y + Z + Z dw Y T = sin(x T ) 15 / 31

56 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs The Feynman-Kac approach allows: Solving he BSDE gives rise (in general) o a viscosiy soluion. 16 / 31

57 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs The Feynman-Kac approach allows: Solving he BSDE gives rise (in general) o a viscosiy soluion. Numerical schemes for BSDEs as an alernaive o solve PDEs by MC mehods (especially in higher dimensions). 16 / 31

58 Applicaions - Why do we need BSDEs? Feynman-Kac represenaion of PDEs The Feynman-Kac approach allows: Solving he BSDE gives rise (in general) o a viscosiy soluion. Numerical schemes for BSDEs as an alernaive o solve PDEs by MC mehods (especially in higher dimensions). Similar approaches exis for SPDEs. They lead o DSBSDEs (doubly sochasic backward SDEs). 16 / 31

59 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 17 / 31

60 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion BSDEs emerged in he 1970s (Bismu) from his field. 17 / 31

61 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion BSDEs emerged in he 1970s (Bismu) from his field. Goal: Maximize an expeced gain of he form ] J(ν) = E [g(x νt ) + f (X ν, ν())dw, 0 wih respec o ν. Here, X ν is he soluion of dx ν = b (X, ν )d + σ (X, ν )dw 17 / 31

62 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion ] J(ν) = E [g(x νt ) + f (X ν, ν())dw 0 18 / 31

63 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion ] J(ν) = E [g(x νt ) + f (X ν, ν())dw 0 Sufficien/necessary condiions for opimaliy given by BSDEs: Le us define he Hamilonian H (x, u, p, q) = b (x, u)p + σ (x, u)q + f (x, u). 18 / 31

64 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion ] J(ν) = E [g(x νt ) + f (X ν, ν())dw 0 Sufficien/necessary condiions for opimaliy given by BSDEs: Le us define he Hamilonian H (x, u, p, q) = b (x, u)p + σ (x, u)q + f (x, u). Find argmax u (H (x, u, p, q)) = ˆν (x, p, q). 18 / 31

65 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion ] J(ν) = E [g(x νt ) + f (X ν, ν())dw 0 Sufficien/necessary condiions for opimaliy given by BSDEs: Le us define he Hamilonian H (x, u, p, q) = b (x, u)p + σ (x, u)q + f (x, u). Find argmax u (H (x, u, p, q)) = ˆν (x, p, q). Solve an associaed BSDE P = x g(x ˆν ) + x Ĥ (Xs ˆν, P s, Q s )ds Q s dw s. 18 / 31

66 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion ] J(ν) = E [g(x νt ) + f (X ν, ν())dw 0 Sufficien/necessary condiions for opimaliy given by BSDEs: Le us define he Hamilonian H (x, u, p, q) = b (x, u)p + σ (x, u)q + f (x, u). Find argmax u (H (x, u, p, q)) = ˆν (x, p, q). Solve an associaed BSDE P = x g(x ˆν ) + x Ĥ (Xs ˆν, P s, Q s )ds Q s dw s. ge opimal conrol ˆν by argmax u (H (X, u, P s, Q s )) 18 / 31

67 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion Uiliy maximizaion: Given: Sock: S = S µ r d r + 0 σ r dw r Wealh up o now: X π = x + 0 π sds s, x > 0. Uiliy funcion: U : R R. Liabiliy: F L 2 19 / 31

68 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion Uiliy maximizaion: Given: Sock: S = S µ r d r + 0 σ r dw r Wealh up o now: X π = x + 0 π sds s, x > 0. Uiliy funcion: U : R R. Liabiliy: F L 2 Maximize he expeced uiliy [ ( )] sup E U x + π s ds s F 0 19 / 31

69 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion Uiliy funcions for example: logarihmic: U(x) = log(x) power: U(x) = xp p, p ]0, 1[. exponenial: U(x) = exp( γx), γ > / 31

70 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion Uiliy funcions for example: logarihmic: U(x) = log(x) power: U(x) = xp p, p ]0, 1[. exponenial: U(x) = exp( γx), γ > 0. Analyical approach: Hamilon-Jacobi-Bellman: Resriced o Markov seing Convex dualiy: mosly non consrucive 20 / 31

71 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion Uiliy funcions for example: logarihmic: U(x) = log(x) power: U(x) = xp p, p ]0, 1[. exponenial: U(x) = exp( γx), γ > 0. Analyical approach: Hamilon-Jacobi-Bellman: Resriced o Markov seing Convex dualiy: mosly non consrucive BSDE approach: some numerics available (Lipschiz, quadraic generaors) 20 / 31

72 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion There are......many more applicaions (principal-agen problem,...) 21 / 31

73 Applicaions - Why do we need BSDEs? Sochasic conrol / Uiliy maximizaion There are......many more applicaions (principal-agen problem,...) Mea-heorem : Any problem in mahemaical finance can be reduced (in some sense) o a (cerain ype of) BSDE. 21 / 31

74 Mahemaical reamen An easy example 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 22 / 31

75 Mahemaical reamen An easy example Suppose ha ξ = Eξ + 0 Z sdw s (predicable represenaion propery.) 22 / 31

76 Mahemaical reamen An easy example Suppose ha ξ = Eξ + 0 Z sdw s (predicable represenaion propery.) Then, wih Y = Eξ + 0 Z sdw s, we have Y = ξ which is a BSDE wih f = 0. Noe also ha Y = E [ξ F ]. Z s dw s, 22 / 31

77 Mahemaical reamen Ieraing schemes 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 23 / 31

78 Mahemaical reamen Ieraing schemes Theoreical (ypical) mehod o prove exisence and uniqueness (usually Lipschiz generaors): 23 / 31

79 Mahemaical reamen Ieraing schemes Theoreical (ypical) mehod o prove exisence and uniqueness (usually Lipschiz generaors): Sar wih (Y 0, Z 0 ) = (0, 0) 23 / 31

80 Mahemaical reamen Ieraing schemes Theoreical (ypical) mehod o prove exisence and uniqueness (usually Lipschiz generaors): Sar wih (Y 0, Z 0 ) = (0, 0) Ge Y n+1 by equivalen o Y n+1 = ξ + f (s, Ys n, Zs n )ds Y n+1 [ = E ξ + f (s, Ys n, Zs n )ds Zs n+1 dw s F Z n+1 by Maringale represenaion of ξ + 0 f (s, Y n s, Z n s )ds. ], 23 / 31

81 Mahemaical reamen Ieraing schemes Theoreical (ypical) mehod o prove exisence and uniqueness (usually Lipschiz generaors): Sar wih (Y 0, Z 0 ) = (0, 0) Ge Y n+1 by equivalen o Y n+1 = ξ + f (s, Ys n, Zs n )ds Y n+1 [ = E ξ + f (s, Ys n, Zs n )ds Zs n+1 dw s F Z n+1 by Maringale represenaion of ξ + 0 f (s, Y n s, Z n s )ds. Show convergence of (Y n, Z n ) n 0 by Banach s fixed-poin heorem ], 23 / 31

82 Mahemaical reamen Numerics 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 24 / 31

83 Mahemaical reamen Numerics To obain a numerical scheme for Y = ξ + f (s, Y s, Z s )ds we rewrie he equaion for one sep in a ime-ne: Y i 1 = Y i + i i Z s dw s, f (s, Y s, Z s )ds Z s dw s, i 1 i 1 24 / 31

84 Mahemaical reamen Numerics To obain a numerical scheme for Y = ξ + f (s, Y s, Z s )ds we rewrie he equaion for one sep in a ime-ne: Y i 1 = Y i + i Then, discreize he equaion: i Z s dw s, f (s, Y s, Z s )ds Z s dw s, i 1 i 1 Ŷ i 1 = Ŷ i + ( i )f ( i, Ŷ i, Ẑ i ) ( W i )Ẑ i 1, Ŷ T = ξ, 24 / 31

85 Mahemaical reamen Numerics To obain a numerical scheme for Y = ξ + f (s, Y s, Z s )ds we rewrie he equaion for one sep in a ime-ne: Y i 1 = Y i + i Then, discreize he equaion: i Z s dw s, f (s, Y s, Z s )ds Z s dw s, i 1 i 1 Ŷ i 1 = Ŷ i + ( i )f ( i, Ŷ i, Ẑ i ) ( W i )Ẑ i 1, Ŷ T = ξ, and find Ŷ i 1 by aking he condiional expecaion ] Ŷ i 1 = E [Ŷi + ( i )f ( i, Ŷ i, Ẑ i ) F i 1 24 / 31

86 Mahemaical reamen Numerics How o find he Z process: 25 / 31

87 Mahemaical reamen Numerics How o find he Z process: Muliply Ŷ i 1 = Ŷ i + ( i )f ( i, Ŷ i, Ẑ i ) ( W i )Ẑ i 1, Ŷ T = ξ, by W i o ge ( W i )Ŷ i 1 = ( W i )Ŷ i +( W i )( i )f ( i, Ŷ i, Ẑ i ) ( W i ) 2 Ẑ i / 31

88 Mahemaical reamen Numerics How o find he Z process: Muliply Ŷ i 1 = Ŷ i + ( i )f ( i, Ŷ i, Ẑ i ) ( W i )Ẑ i 1, Ŷ T = ξ, by W i o ge ( W i )Ŷ i 1 = ( W i )Ŷ i +( W i )( i )f ( i, Ŷ i, Ẑ i ) ( W i ) 2 Ẑ i 1. Taking he condiional expecaion brings us o [ Z i 1 = E ( W i )Ŷ i + ( W i )( i )f ( i, Ŷ i, Ẑ i ) F i 1 ] 1 i 25 / 31

89 Mahemaical reamen Numerics The schemes have order 1 N, N = max i i 26 / 31

90 Mahemaical reamen Numerics The schemes have order 1 N, N = max i i Theoreical rae of convergence since calculaions of condiional expecaions are involved! 26 / 31

91 Mahemaical reamen Numerics The schemes have order 1 N, N = max i i Theoreical rae of convergence since calculaions of condiional expecaions are involved! Oher ype of numerical schemes: Involve Picard ieraions of he equaions and chaos decomposiions of random variables. 26 / 31

92 Mahemaical reamen Numerics The schemes have order 1 N, N = max i i Theoreical rae of convergence since calculaions of condiional expecaions are involved! Oher ype of numerical schemes: Involve Picard ieraions of he equaions and chaos decomposiions of random variables. Applicable codes/schemes do exis! 26 / 31

93 My field wihin BSDE heory 1 Wha is a BSDE? SDEs - he differenial dynamics approach o BSDEs 2 Applicaions - Why do we need BSDEs? Pricing of coningen claims Represenaion of risk measures Feynman-Kac represenaion of PDEs Sochasic conrol / Uiliy maximizaion 3 Mahemaical reamen An easy example Ieraing schemes Numerics 4 My field wihin BSDE heory 27 / 31

94 My field wihin BSDE heory Basically I follow hree main opics (wih Ch. Geiss, Universiy of Jyväskylä): 27 / 31

95 My field wihin BSDE heory Basically I follow hree main opics (wih Ch. Geiss, Universiy of Jyväskylä): BSDEs wih jumps (or Lévy driven BSDEs, BSDEJ, BSDEL): Y =ξ + f (s, Y s, Z s, U s )ds Z s dw s U s (x)ñ(ds, dx) ],T ] R 0 e.g. if Brownian moion in models is replaced by a Lévy process. 27 / 31

96 My field wihin BSDE heory Basically I follow hree main opics (wih Ch. Geiss, Universiy of Jyväskylä): BSDEs wih jumps (or Lévy driven BSDEs, BSDEJ, BSDEL): Y =ξ + f (s, Y s, Z s, U s )ds Z s dw s U s (x)ñ(ds, dx) ],T ] R 0 e.g. if Brownian moion in models is replaced by a Lévy process. Shock phenomena, PDEs (Brown) become PDIEs (Lévy), / 31

97 My field wihin BSDE heory Basically I follow hree main opics (wih Ch. Geiss, Universiy of Jyväskylä): BSDEs wih jumps (or Lévy driven BSDEs, BSDEJ, BSDEL): Y =ξ + f (s, Y s, Z s, U s )ds Z s dw s U s (x)ñ(ds, dx) ],T ] R 0 e.g. if Brownian moion in models is replaced by a Lévy process. Shock phenomena, PDEs (Brown) become PDIEs (Lévy),... Exisence and uniqueness for non-lipschiz generaors (one-sided Lipschiz, locally Lipschiz, quadraic growh and beyond) 27 / 31

98 My field wihin BSDE heory Applicaion of Malliavin calculus o BSDEs. 28 / 31

99 My field wihin BSDE heory Applicaion of Malliavin calculus o BSDEs. Malliavin derivaive of a RV ξ = sochasic derivaive wih respec o Brownian moion. Denoed as D s ξ, 0 s T. 28 / 31

100 My field wihin BSDE heory Applicaion of Malliavin calculus o BSDEs. Malliavin derivaive of a RV ξ = sochasic derivaive wih respec o Brownian moion. Denoed as D s ξ, 0 s T. Example: D s W 2 = 2W 1I [0,] (s) 28 / 31

101 My field wihin BSDE heory Applicaion of Malliavin calculus o BSDEs. Malliavin derivaive of a RV ξ = sochasic derivaive wih respec o Brownian moion. Denoed as D s ξ, 0 s T. Example: D s W 2 = 2W 1I [0,] (s) If a BSDE is Malliavin differeniable, he differeniaed soluions is again a BSDE. 28 / 31

102 My field wihin BSDE heory Applicaion of Malliavin calculus o BSDEs. Malliavin derivaive of a RV ξ = sochasic derivaive wih respec o Brownian moion. Denoed as D s ξ, 0 s T. Example: D s W 2 = 2W 1I [0,] (s) If a BSDE is Malliavin differeniable, he differeniaed soluions is again a BSDE. The ideniy D Y = Z allows explici access o he Z-process (rading sraegy,...). Numerical improvemens for BSDEs (wih G. Leobacher, KFU Graz) 28 / 31

103 My field wihin BSDE heory References: Overview: Bruno Bouchard: Lecure Noes on BSDEs, Exisence and Main Resuls, Lecure given a he London School of Economics. N. El Karoui, S. Peng, M. Quenez: Backward Sochasic Differenial Equaions in Finance, Mah. Finance, 7 (1), El Karoui, Hamadène, Maoussi: Backward Sochasic Differenial Equaions and Applicaions, Lecures CIMPA school, Nonlinear expecaions: S. Peng: Backward Sochasic differenial equaions, nonlinear expecaion and heir applicaions, Proceedings of he Inernaional congress of Mahemaicians, 1, / 31

104 My field wihin BSDE heory Soch. conrol: S. Peng: A general maximum principle for opimal conrol problems, Siam J. of Conrol and Opimizaion, 28 (4), S. Peng: Backward sochasic differenial equaions and applicaions o opimal conrol, Appl. Mah. Opim., 1993 Connecion o PDEs: É. Pardoux: Backward sochasic differenial equaions and viscosiy soluions of sysems of semiliinear parabolic and ellipic pdes of second order, Socahsic Analysis and Relaed Topics VI, / 31

105 My field wihin BSDE heory Numerics: Bouchard, Elie: Discree-ime Approximaions of Decoupled Forward-Backward SDE wih Jumps, Soch. Proc. Appl. 118, 2008 Bouchard, Elie, Touzi: Discree-ime approximaion of BSDEs and probabilisic schemes for fully nonlinear PDEs, Radon Series Comp. Appl. Mah 8, 2009 Ch. Geiss, C. Labar: Simulaion of BSDEs wih jumps by Wiener Chaos Expansion, Soch. Proc. Appl. 126 (7), Malliavin calculus and BSDEs: C. Geiss, A.S.: Malliavin derivaive of random funcions and applicaions o Lévy driven BSDEs, El. J. Probab., 21 (10), / 31