Semi-Lagrangian schemes for linear and fully non-linear - - PowerPoint PPT Presentation

Introduction SL schemes for HJB Conclusion

Semi-Lagrangian schemes for linear and fully non-linear Hamilton-Jacobi-Bellman equations

Kristian Debrabant1,Espen R. Jakobsen2

1Department of Mathematics and Computer Science, University of Southern

Denmark

2Department of Mathematical Sciences, Norwegian University of Science And

Technology, Trondheim, Norway

Padova, June 28, 2012

1

Introduction SL schemes for HJB Conclusion

Outline

1

Introduction

2

Semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations

3

Conclusion

2

Introduction SL schemes for HJB Conclusion

Outline

1

Introduction

2

Semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations

3

Conclusion

3

Introduction SL schemes for HJB Conclusion

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves HJB equation −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

4

Introduction SL schemes for HJB Conclusion

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves HJB equation −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

4

Introduction SL schemes for HJB Conclusion

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves in the viscosity sense −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

4

Introduction SL schemes for HJB Conclusion

Convergence to viscosity solution via Lax type result

Barles & Souganidis 1991: The numerical approximation U obtained by the scheme S(∆t, ∆x, n, j, un

j , u) = 0 converges uniformly to the viscosity

solution if at least for some sequence (∆t, ∆x) converging to zero it is stable: For all (∆t, ∆x) there exists a solution U with a bound independent of (∆t, ∆x). consistent: lim

ξ→0 ∆t,∆x→0 (n∆t,j∆x)→(t,x)

S(∆t, ∆x, n, j, Φn

j + ξ, Φ + ξ)

ρ(∆t, ∆x) → F

• t, x, Φ(t, x), Φt(t, x), DΦ(t, x), D2Φ(t, x)
• for some positive function ρ, any smooth function Φ and any (x, t)

monotone: S(∆t, ∆x, n, j, un

j , u) ≤ S(∆t, ∆x, n, j, v n j , v) if u ≥ v, un j = v n j ,

for any ∆t, ∆x, n, j, u and v Oberman (2006), Pooley, Forsyth, Vetzal (2003): Nonmonotone methods need not converge or even converge to a wrong function

5

Introduction SL schemes for HJB Conclusion

Outline

1

Introduction

2

Semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations

3

Conclusion

6

Introduction SL schemes for HJB Conclusion

Proposed family of SL schemes

PDE: ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in (0, T] × I RN, u(0, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x) Semi-discretization in (0, T) × X∆x: Ut(t, x) − inf

α∈A

• Lk[IU](t, x, α) + c(t, x, α)U(t, x) + f(t, x, α)
• = 0,

Lk[ϕ](t, x, α) :=

P

• i=1

ϕ

• t, x + y +

k,i(t, x, α)

• − 2ϕ(t, x) + ϕ
• t, x + y −

k,i(t, x, α)

• 2k2

such that

P

• i=1

[y +

k,i + y − k,i] = 2k2b + O(k 4), P

• i=1

[y +

k,iy+ ⊤ k,i

+ y −

k,iy − ⊤ k,i ] = 2k 2σσ⊤ + O(k 4) =

⇒ k|σ| ∼ stencil length

P

• i=1

[⊗3

j=1y+ k,i + ⊗3 j=1y − k,i] = O(k 4), P

• i=1

[⊗3

j=1y + k,i + ⊗3 j=1y − k,i] = O(k4) 7

Introduction SL schemes for HJB Conclusion

Proposed family of SL schemes

PDE: ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in (0, T] × I RN, u(0, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x) Semi-discretization in (0, T) × X∆x: Ut(t, x) − inf

α∈A

• Lk[IU](t, x, α) + c(t, x, α)U(t, x) + f(t, x, α)
• = 0,

Lk[ϕ](t, x, α) :=

P

• i=1

ϕ

• t, x + y +

k,i(t, x, α)

• − 2ϕ(t, x) + ϕ
• t, x + y −

k,i(t, x, α)

• 2k2

= ⇒ |Lk[ϕ] − L[ϕ]| ≤ C

• Dϕ∞ + D2ϕ∞ + D3ϕ∞ + D4ϕ∞
• k2

Important for monotonicity: I monotone, i. e. (Iϕ)(x) =

• j

ϕ(xj)wj(x), wi(xj) = δij, and wj(x) ≥ 0 for all i, j ∈ N In general: Not more than second order accurate interpolation (I linear)

7

Introduction SL schemes for HJB Conclusion

Proposed family of SL schemes

PDE: ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in (0, T] × I RN, u(0, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x) Semi-discretization in (0, T) × X∆x: Ut(t, x) − inf

α∈A

• Lk[IU](t, x, α) + c(t, x, α)U(t, x) + f(t, x, α)
• = 0,

Lk[ϕ](t, x, α) :=

P

• i=1

ϕ

• t, x + y +

k,i(t, x, α)

• − 2ϕ(t, x) + ϕ
• t, x + y −

k,i(t, x, α)

• 2k2

= ⇒ |Lk[ϕ] − L[ϕ]| ≤ C

• Dϕ∞ + D2ϕ∞ + D3ϕ∞ + D4ϕ∞
• k2

Important for monotonicity: I monotone, i. e. (Iϕ)(x) =

• j

ϕ(xj)wϕ,j(x), wϕ,i(xj) = δij, wϕ,j(x) ≥ 0,

• j

wϕ,j(x) ≡ 1 In general: Not more than second order accurate interpolation (I linear) But: Higher order monotonicity preserving interpolation is possible if ϕ is known to be monotone → nonlinear in ϕ

7

Introduction SL schemes for HJB Conclusion

A unifying framework - examples

1

The approximation of Falcone (1987): k = √ ∆x, y±

k = k 2b

bDϕ ≈ Iϕ(x + ∆xb) − Iϕ(x) ∆x

2

Camilli-Falcone (1995): k = √ ∆x, y ±

k,j = ±kσj + k2 M b

1 2 Tr[σσ⊤D2ϕ] + bDϕ ≈

M

• j=1

Iϕ(x + √ ∆xσj + ∆x

M b) − 2Iϕ(x) + Iϕ(x −

√ ∆xσj + ∆x

M b)

2∆x

3

New version (efficient for σ independent of α): 1 2 Tr[σσ⊤D2ϕ] + bDϕ ≈ Iϕ(x + k 2b) − Iϕ(x) k 2 +

M

• j=1

Iϕ(x + kσj) − 2Iϕ(x) + Iϕ(x − kσj) 2k 2

8

Introduction SL schemes for HJB Conclusion

Monotonicity preserving cubic Hermite interpolation

(Fritsch & Carlson 1980, Eisenstat, Jackson & Lewis 1985) On [xi, xi+1]: (Iϕ)(x) = c0 + c1(x − xi) + c2(x − xi)2 + c3(x − xi)3 with (Iϕ)(xi) = ϕi, (Iϕ)(xi+1) = ϕi+1, (Iϕ)′(xi) ≈ ϕ′

i , (Iϕ)′(xi+1) ≈ ϕ′ i+1

di =

ϕi−2−8ϕi−1+8ϕi+1−ϕi+2 12∆x

, ∆i =

ϕi+1−ϕi ∆x

, βi = di

∆i , γi = di+1 ∆i .

Use (Iϕ)′(xi) = βi∆i, (Iϕ)′(xi+1) = γi∆i = ⇒ (Iϕ)(x) = ϕi + (ϕi+1 − ϕi)Pi(x) where Pi(x) = βi x − xi ∆x + (3 − γi − 2βi) x − xi ∆x 2 − (2 − βi − γi) x − xi ∆x 3 . = ⇒ wϕ,i(x) = (1 − Pi(x))1[xi ,xi+1)(x) + Pi−1(x)1[xi−1,xi )(x) Modify βi and γi such that (βi, γi) ∈ M: 1 2 3 4 1 2 3 4 M βi γi = ⇒ fourth order accurate monotonicity preserving C0 interpolant = ⇒ New second order compact stencil schemes

9

Introduction SL schemes for HJB Conclusion

Monotonicity preserving cubic Hermite interpolation

(Fritsch & Carlson 1980, Eisenstat, Jackson & Lewis 1985) On [xi, xi+1]: (Iϕ)(x) = c0 + c1(x − xi) + c2(x − xi)2 + c3(x − xi)3 with (Iϕ)(xi) = ϕi, (Iϕ)(xi+1) = ϕi+1, (Iϕ)′(xi) ≈ ϕ′

i , (Iϕ)′(xi+1) ≈ ϕ′ i+1

di =

ϕi−2−8ϕi−1+8ϕi+1−ϕi+2 12∆x

, ∆i =

ϕi+1−ϕi ∆x

, βi = di

∆i , γi = di+1 ∆i .

Use (Iϕ)′(xi) = βi∆i, (Iϕ)′(xi+1) = γi∆i = ⇒ (Iϕ)(x) = ϕi + (ϕi+1 − ϕi)Pi(x) where Pi(x) = βi x − xi ∆x + (3 − γi − 2βi) x − xi ∆x 2 − (2 − βi − γi) x − xi ∆x 3 . = ⇒ wϕ,i(x) = (1 − Pi(x))1[xi ,xi+1)(x) + Pi−1(x)1[xi−1,xi )(x) Modify βi and γi such that (βi, γi) ∈ M: 1 2 3 4 1 2 3 4 M βi γi = ⇒ fourth order accurate monotonicity preserving C0 interpolant = ⇒ New second order compact stencil schemes

9

Introduction SL schemes for HJB Conclusion

Discretization in time

Semi-discretization in (0, T) × X∆x: Ut(t, x) − inf

α∈A

• Lk[IU](t, x, α) + c(t, x, α)U(t, x) + f(t, x, α)
• = 0

Lk[ϕ](t, x, α) :=

P

• i=1

ϕ

• t, x + y +

k,i(t, x, α)

• − 2ϕ(t, x) + ϕ
• t, x + y −

k,i(t, x, α)

• 2k 2

, Fully discrete scheme: θ ∈ [0, 1], Un

i − Un−1 i

∆t = inf

α∈A

• Lα

k [I ¯

Uθ,n

·

]n−1+θ

i

+ cα,n−1+θ

i

¯ Uθ,n

i

+ f α,n−1+θ

i

• where ¯

Uθ,n

·

= (1 − θ)Un−1

·

+ θUn

· , U0 i = g(xi)

in X∆x.

10

Introduction SL schemes for HJB Conclusion

Proposed schemes - general properties

Theorem (D./Jakobsen) The considered scheme has a truncation error bounded by O

• (1 − 2θ)∆t + ∆t2 + ∆xr

k 2 + k2 . r = 2 : k ∼ √ ∆x r = 4 : k ∼ ∆x Under the CFL condition (1 − θ)∆t P k 2 − cα,n−1+θ

i

• ≤ 1 and θ∆t cα,n−1+θ

i

≤ 1 for all α, n, i it has a unique bounded solution U, which is stable when 2θ∆t supα |cα,+|0 ≤ 1: |Un|0 ≤ e2 supα |cα,+|0tn |g|0 + tn sup

α

|f α|0

• .

For monotone interpolation it is monotone under the above CFL condition. Barles-Souganidis result = ⇒ for linear interpolation, U converges uniformly to the viscosity solution u as ∆t, k, ∆xr

k2 → 0. 11

Introduction SL schemes for HJB Conclusion

Test problem with non-smooth solution

ut − Tr

• sin2 x1

sin x1 sin x2 sin x1 sin x2 sin2 x2

• D2u
• = f(t, x)

u(t, x) = (1 + t) sin x2 2

• sin x1

2

for − π < x1 < 0, sin x1

4

for 0 < x1 < π.

−2.5 −2 −1.5 −1 −3.5 −3 −2.5 −2 log10 ∆x log10 |u(T, ·) − U·(T)|0 linear: pnum ≈ 0.50 cubic: pnum ≈ 0.95

12

Introduction SL schemes for HJB Conclusion

An application from finance

Superreplication problem of European options: Stochastic volatility model with gamma constraints (Bruder, Bokanowski, Maroso, Zidani, SIAM J.

• Numer. Anal. 2009)

inf

α2

1+α2 2=1

• α2

1ut(t, x) − 1

2 Tr

• σα(t, x)σα ⊤(t, x)D2u(t, x)
• = 0,

0 < x1, x2, t u(0, x1, x2) = max(0, 1 − x1), 0 ≤ x1, x2 ≤ 3

1 2 3 0 1 2 3 0.5 1 x1 x2 U t = 1

σα(t, x) =

• α1x1

√x2 α2η(t, x1, x2)

• for arbitrary η(t, x1, x2) > 0.

Solution monotone in x1 and x2 direction

13

Introduction SL schemes for HJB Conclusion

An application from finance

Superreplication problem of European options: Stochastic volatility model with gamma constraints (Bruder, Bokanowski, Maroso, Zidani, SIAM J.

• Numer. Anal. 2009)

inf

α2

1+α2 2=1

• α2

1ut(t, x) − 1

2 Tr

• σα(t, x)σα ⊤(t, x)D2u(t, x)
• = f(t, x),

0 < x1, x2 < 3 u(0, x1, x2) = max(0, 1 − x1), 0 ≤ x1, x2 ≤ 3

−1.5 −1 −0.5 −3 −2 −1 log10 ∆x log10 |u(T, ·) − U·(T)|0 linear: pnum ≈ 1.13 cubic: pnum ≈ 1.98

σα(t, x) =

• α1x1

√x2 α2η(t, x1, x2)

• Test problem: u(t, x) = 1 + t2 − e−x2

1 −x2 2

13

Introduction SL schemes for HJB Conclusion

Proposed schemes - convergence result

Theorem (D./Jakobsen) Under the above assumptions, |u − U| ≤ C(|1 − 2θ|∆t1/4 + ∆t1/3 + k1/2 + ∆x k2 ). k ∼ ∆x2/5, ∆t ∼ k2 ∼ ∆x4/5 = ⇒ |u − U| = O(∆x1/5) applies to both linear and monotonicity preserving cubic interpolation no convergence for k ∼ √ ∆x (r = 2) and k ∼ ∆x (r = 4) holds also for unstructured grids Ideas of proof:

1

|U − V| ≤ C ∆x

k2

2

|u − V|: extension of ideas used in the stationary case by Camilli/Falcone, Barles/Jakobsen

14

Introduction SL schemes for HJB Conclusion

Proposed schemes - convergence result

Theorem (D./Jakobsen) Under the above assumptions, |u − U| ≤ C(|1 − 2θ|∆t1/4 + ∆t1/3 + k1/2 + ∆x k2 ). k ∼ ∆x2/5, ∆t ∼ k2 ∼ ∆x4/5 = ⇒ |u − U| = O(∆x1/5) applies to both linear and monotonicity preserving cubic interpolation no convergence for k ∼ √ ∆x (r = 2) and k ∼ ∆x (r = 4) holds also for unstructured grids Ideas of proof:

1

|U − V| ≤ C ∆x

k2

2

|u − V|: extension of ideas used in the stationary case by Camilli/Falcone, Barles/Jakobsen

14

Introduction SL schemes for HJB Conclusion

Outline

1

Introduction

2

Semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations

3

Conclusion

15

Introduction SL schemes for HJB Conclusion

Conclusion

class of semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations unifying framework for several known first order schemes, as well as new versions, including new second order compact stencil schemes for essentially monotone solutions for linear interpolation: stability and convergence in the general case, also for HJB-Isaacs equations both linear and cubic interpolation: convergence with rate ∆x

1 5 16

Introduction SL schemes for HJB Conclusion

Conclusion

class of semi-Lagrangian schemes for Hamilton-Jacobi-Bellman equations unifying framework for several known first order schemes, as well as new versions, including new second order compact stencil schemes for essentially monotone solutions for linear interpolation: stability and convergence in the general case, also for HJB-Isaacs equations both linear and cubic interpolation: convergence with rate ∆x

1 5

Thank you very much for your attention!

16

HJB equations and viscosity solutions

Outline

4

Hamilton-Jacobi-Bellman equations and viscosity solutions

17

HJB equations and viscosity solutions

Stochastic control problem - example (Merton 1969)

Investment and consumption problem: dR(t) = rR(t) dt dS(t) = µS(t) dt + σS(t) dW(t) dX(t) = r

• 1 − π(t)
• X(t) dt + π(t)X(t)
• µ dt + σ dW(t)
• − C(t) dt

=

• r + (µ − r)π(t)
• X(t) − C(t)
• dt + π(t)X(t)σ dW(t)

X(t0) = X0. u(t0, X0) = sup

(π,C)∈A

E T

t0

e−ρ(t−t0)f

• C(t)
• dt+e−ρ(T−t0)g
• X(T)
• 18

HJB equations and viscosity solutions

Stochastic control problem - example (Merton 1969)

Investment and consumption problem: dR(t) = rR(t) dt dS(t) = µS(t) dt + σS(t) dW(t) dX(t) = r

• 1 − π(t)
• X(t) dt + π(t)X(t)
• µ dt + σ dW(t)
• − C(t) dt

=

• r + (µ − r)π(t)
• X(t) − C(t)
• dt + π(t)X(t)σ dW(t)

X(t0) = X0. u(t0, X0) = sup

(π,C)∈A

E T

t0

e−ρ(t−t0)f

• C(t)
• dt+e−ρ(T−t0)g
• X(T)
• 18

HJB equations and viscosity solutions

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves HJB equation −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

19

HJB equations and viscosity solutions

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves HJB equation −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

19

HJB equations and viscosity solutions

Stochastic control problem = ⇒ HJB equation

dX(t) = b

• t, X(t), α(t)
• dt +

M

• i=1

σi

• t, X(t), α(t)
• dWi(t),

t0 ≤ t ≤ T, X(t0) = X0 ∈ I RN. u(t0, X0) = inf

α∈A E

T

t0

β(t0, t)f

• t, X(t), α(t)
• dt+β(t0, T)g
• X(T)
• where

β(t0, t) = e

t

t0 c

• s,X(s),α(s)
• ds.

Under appropriate assumptions: u solves in the viscosity sense −ut(t, x) − inf

α∈A

• L[u](t, x, α) + c(t, x, α)u(t, x) + f(t, x, α)
• = 0

in [0, T) × I RN, u(T, x) = g(x) in I RN, L[u](t, x, α) = 1 2 Tr[σ(t, x, α)σ⊤(t, x, α)D2u(t, x)] + b(t, x, α)Du(t, x)

19

HJB equations and viscosity solutions

A ⊂ I Rk Control processes A0: set of all progressively measurable processes ν = {ν(t), t ≥ 0} valued in A. Admissible control processes A: subset of all ν ∈ A0 for which E T

t0

• |b(t, x, ν(t))| + |σ(t, x, ν(t))|2

dt < ∞ for x ∈ I Rn (guarantees existence of a controlled process for each given initial condition and control). Cost functional: c∞ < ∞, |f(t, x, α)| + |g(x)| ≤ C(1 + x2) for some constant C independent of (t, α)

20

HJB equations and viscosity solutions

Viscosity solutions (Crandall & Lions 1983)

F

• t, x, u(t, x), ut(t, x), Du(t, x), D2u(t, x)
• = 0 on (0, T)×Ω (⋆)

fulfilling the degenerate ellipticity condition: (A, B ∈ SN) F(t, x, r, s, p, A) ≤ F(t, x, r, s, p, B) whenever A ≥ B ∀(t, x, r, s, p) ∈ [0, T] × Ω × I R × I R × I RN. Example (HJB equation) F(t, x, r, s, p, A) = −s − inf

α∈A

1 2 Tr[σ(t, x, α)σ⊤(t, x, α)A] + b(t, x, α)p + c(t, x, α)r + f(t, x, α)

• 21

HJB equations and viscosity solutions

Viscosity solutions (Crandall & Lions 1983)

F

• t, x, u(t, x), ut(t, x), Du(t, x), D2u(t, x)
• = 0 on (0, T)×Ω (⋆)

fulfilling the degenerate ellipticity condition: (A, B ∈ SN) F(t, x, r, s, p, A) ≤ F(t, x, r, s, p, B) whenever A ≥ B ∀(t, x, r, s, p) ∈ [0, T] × Ω × I R × I R × I RN. Then v ∈ C1,2 (0, T) × Ω

• is a classical supersolution of (⋆) iff

for all (t0, x0, ϕ) ∈ (0, T) × Ω × C1,2 (0, T) × Ω

• such that

(t0, x0) is a minimizer of v − ϕ on (t0, Ω) F

• t0, x0, v(t0, x0), ϕt(t0, x0), Dϕ(t0, x0), D2ϕ(t0, x0)
• ≥ 0.

21

HJB equations and viscosity solutions

Viscosity solutions (Crandall & Lions 1983)

F

• t, x, u(t, x), ut(t, x), Du(t, x), D2u(t, x)
• = 0 on (0, T)×Ω (⋆)

fulfilling the degenerate ellipticity condition: (A, B ∈ SN) F(t, x, r, s, p, A) ≤ F(t, x, r, s, p, B) whenever A ≥ B ∀(t, x, r, s, p) ∈ [0, T] × Ω × I R × I R × I RN. Then v ∈ C0,0 (0, T) × Ω

• is a classical supersolution

subsolution of (⋆) iff for all (t0, x0, ϕ) ∈ (0, T) × Ω × C1,2 (0, T) × Ω

• such that

(t0, x0) is a minimizer maximizer of v − ϕ on (t0, Ω) F

• t0, x0, v(t0, x0), ϕt(t0, x0), Dϕ(t0, x0), D2ϕ(t0, x0)

≥ ≤ 0.

21

HJB equations and viscosity solutions

Viscosity solutions (Crandall & Lions 1983)

F

• t, x, u(t, x), ut(t, x), Du(t, x), D2u(t, x)
• = 0 on (0, T)×Ω (⋆)

fulfilling the degenerate ellipticity condition: (A, B ∈ SN) F(t, x, r, s, p, A) ≤ F(t, x, r, s, p, B) whenever A ≥ B ∀(t, x, r, s, p) ∈ [0, T] × Ω × I R × I R × I RN. Then v ∈ C0,0 (0, T) × Ω

• is a viscosity supersolution

subsolution of (⋆) iff for all (t0, x0, ϕ) ∈ (0, T) × Ω × C1,2 (0, T) × Ω

• such that

(t0, x0) is a minimizer maximizer of v − ϕ on (t0, Ω) F

• t0, x0, v(t0, x0), ϕt(t0, x0), Dϕ(t0, x0), D2ϕ(t0, x0)

≥ ≤ 0.

21