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Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References

American-style options, stochastic volatility, and degenerate parabolic variational inequalities

Paul Feehan1

1Department of Mathematics

Rutgers University

September 8, 2010 – Modena, Italy Kolmogorov Equations in Physics and Finance Based on joint work with P. Daskalopoulos and C. Pop

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Collaborators

Joint work with

◮ Panagiota Daskalopoulos, Columbia University, and

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Collaborators

Joint work with

◮ Panagiota Daskalopoulos, Columbia University, and ◮ Camelia Pop, Ph.D. student, Rutgers University.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Introduction and motivation from mathematical finance

◮ We consider continuous processes defined by stochastic

differential equations with degenerate/non-Lipschitz coefficients and and motivated by option pricing.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Introduction and motivation from mathematical finance

◮ We consider continuous processes defined by stochastic

differential equations with degenerate/non-Lipschitz coefficients and and motivated by option pricing.

◮ In particular, we stochastic volatility processes, such as the

Heston process, and their generalizations.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Introduction and motivation from mathematical finance

◮ We consider continuous processes defined by stochastic

differential equations with degenerate/non-Lipschitz coefficients and and motivated by option pricing.

◮ In particular, we stochastic volatility processes, such as the

Heston process, and their generalizations.

◮ We consider their Kolmogorov PDEs and initial/boundary

value and obstacle problems arising in option pricing.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Introduction and motivation from mathematical finance

◮ We consider continuous processes defined by stochastic

differential equations with degenerate/non-Lipschitz coefficients and and motivated by option pricing.

◮ In particular, we stochastic volatility processes, such as the

Heston process, and their generalizations.

◮ We consider their Kolmogorov PDEs and initial/boundary

value and obstacle problems arising in option pricing.

◮ We explore questions of existence, uniqueness, and regularity

• f solutions to variational inequalities, as well as the regularity

and geometric properties of the free boundary.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Some difficulties characteristic of option-pricing problems

◮ Processes defined by stochastic differential equations with

◮ Degenerate diffusion coefficients. Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Some difficulties characteristic of option-pricing problems

◮ Processes defined by stochastic differential equations with

◮ Degenerate diffusion coefficients. ◮ Non-Lipschitz coefficients. Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Some difficulties characteristic of option-pricing problems

◮ Processes defined by stochastic differential equations with

◮ Degenerate diffusion coefficients. ◮ Non-Lipschitz coefficients.

◮ Payoff or obstacle functions which are at most Lipschitz.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Some difficulties characteristic of option-pricing problems

◮ Processes defined by stochastic differential equations with

◮ Degenerate diffusion coefficients. ◮ Non-Lipschitz coefficients.

◮ Payoff or obstacle functions which are at most Lipschitz. ◮ Discontinuous data for initial/boundary value problems.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Some difficulties characteristic of option-pricing problems

◮ Processes defined by stochastic differential equations with

◮ Degenerate diffusion coefficients. ◮ Non-Lipschitz coefficients.

◮ Payoff or obstacle functions which are at most Lipschitz. ◮ Discontinuous data for initial/boundary value problems. ◮ Other complications include:

◮ Unbounded domains. ◮ Unbounded coefficients. ◮ Ubounded boundary data and obstacle functions. ◮ Non-local obstacle constraints. Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

A motivating example

These complications arise in simple, low-dimensional examples:

◮ Find the American-style down-and-out put option value

function when the underlying asset price is a Heston stochastic volatility process. We often restrict to this example in our presentation, though many

• f our results admit natural generalizations.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate processes

◮ Feller (1951), McKean (1956), Pinsky (1969)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate processes

◮ Feller (1951), McKean (1956), Pinsky (1969) ◮ Bass & Lavrentiev (2007), Bass & Perkins (2003, 2007)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate processes

◮ Feller (1951), McKean (1956), Pinsky (1969) ◮ Bass & Lavrentiev (2007), Bass & Perkins (2003, 2007) ◮ Bayraktar & Xing (2009), Bayraktar, Kardaras & Xing (2010),

Ekstr¨

• m & Tysk (2010), Nystr¨
• m (2008)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Processes defined by stochastic differential equations with non-Lipschitz coefficients

◮ Yamada (1978), Yamada & Watanabe (1971)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Processes defined by stochastic differential equations with non-Lipschitz coefficients

◮ Yamada (1978), Yamada & Watanabe (1971) ◮ Bahlali, Mezerdi, & Ouknine (1998)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Processes defined by stochastic differential equations with non-Lipschitz coefficients

◮ Yamada (1978), Yamada & Watanabe (1971) ◮ Bahlali, Mezerdi, & Ouknine (1998) ◮ Berkaoui, Bossy, & Diop (2008), H¨

• pfner (2010)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate elliptic/parabolic partial differential equations

◮ Fabes, Jerison, Kenig & Serapioni (1982), Freidlin (1985),

Kohn & Nirenberg (1967), Levendorskii (1993), Murthy & Stampacchia (1968), Stredulinsky (2009), as well as earlier work of Feller, McKean, and many others . . .

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate elliptic/parabolic partial differential equations

◮ Fabes, Jerison, Kenig & Serapioni (1982), Freidlin (1985),

Kohn & Nirenberg (1967), Levendorskii (1993), Murthy & Stampacchia (1968), Stredulinsky (2009), as well as earlier work of Feller, McKean, and many others . . .

◮ Daskalopoulos & Hamilton (1998), Koch (1999)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate elliptic/parabolic partial differential equations

◮ Fabes, Jerison, Kenig & Serapioni (1982), Freidlin (1985),

Kohn & Nirenberg (1967), Levendorskii (1993), Murthy & Stampacchia (1968), Stredulinsky (2009), as well as earlier work of Feller, McKean, and many others . . .

◮ Daskalopoulos & Hamilton (1998), Koch (1999) ◮ Bayraktar, Xing & Kardaras (2009, 2010), Ekstr¨

• m & Tysk

(2010), Frentz & Nystr¨

• m (2010)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Stationary/evolutionary variational inequalities

◮ Bensoussan & Lions (1982), Friedman (1982), Kinderlehrer &

Stampacchia (1980)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Stationary/evolutionary variational inequalities

◮ Bensoussan & Lions (1982), Friedman (1982), Kinderlehrer &

Stampacchia (1980)

◮ Jaillet, Lamberton, & Lapeyre (1989, 1990), Touzi (1999),

Villeneuve (1999), Zhang (1993)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Stationary/evolutionary variational inequalities

◮ Bensoussan & Lions (1982), Friedman (1982), Kinderlehrer &

Stampacchia (1980)

◮ Jaillet, Lamberton, & Lapeyre (1989, 1990), Touzi (1999),

Villeneuve (1999), Zhang (1993)

◮ Chadam & Chen (2006, 2008, 2010), Ekstr¨

• m (2004),

Laurence & Salsa (2009), Nystr¨

• m (2008)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate stationary/evolutionary variational inequalities

◮ Mastroeni & Matzeu (1996, 1998)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate stationary/evolutionary variational inequalities

◮ Mastroeni & Matzeu (1996, 1998) ◮ Barbu & Marinelli (2008)

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate stationary/evolutionary variational inequalities

◮ Mastroeni & Matzeu (1996, 1998) ◮ Barbu & Marinelli (2008) ◮ . . . and undoubtedly others.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Variational problems with discontinuous boundary data

Motivations from finance are well-known. For example:

◮ A European-style up-and-out call option, with strike K = ek

and upper barrier B = eb, defines an initial/boundary value problem with discontinuous data at (T, b) ∈ ∂pQ if b > k:

◮ u(T, x) = (ex − ek)+, when −∞ < x < b; ◮ u(t, b) = 0, when 0 ≤ t < T. Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Variational problems with discontinuous boundary data

Motivations from finance are well-known. For example:

◮ A European-style up-and-out call option, with strike K = ek

and upper barrier B = eb, defines an initial/boundary value problem with discontinuous data at (T, b) ∈ ∂pQ if b > k:

◮ u(T, x) = (ex − ek)+, when −∞ < x < b; ◮ u(t, b) = 0, when 0 ≤ t < T.

◮ An American-style down-and-out put option, strike K = ek

and lower barrier B = eb, defines an initial/boundary value problem with discontinuous data at (T, b) ∈ ∂pQ if b < k:

◮ u(T, x) = (ek − ex)+, when b < x < ∞; ◮ u(t, b) = 0, when 0 ≤ t < T. Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Heston’s stochastic volatility process

The process proposed by Heston (1993) is defined by S(u) = exp(X(u)), where dX(u) = (r − Y (u)/2)) du +

• Y (u) dW1(u),

X(t) = x, dY (u) = κ(θ − Y (u)) du + σ

• Y (u) dW2(u),

Y (t) = y, where (W1(u), dW3(u)) is two-dimensional Brownian motion, W2(u) := ρW1(u) +

• 1 − ρ2 W3(u), κ, θ, σ are positive constants,

ρ ∈ (−1, 1), and r ≥ 0.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Heston’s parabolic PDE

Denote R2

+ := R × (0, ∞), let O ⊂ R2 + be a domain, and let

Q := [0, T) × O. Given h : O → R, define u(t, x, y) := e−r(T−t)Et,x,y

Q

[h(X(T), Y (T))] , and note that −u′ + Au = 0

• n Q,

u(T, ·) = h

• n O,

where −Au := y 2

• uxx + 2ρσuxy + σ2uyy
• +(r −y/2)ux +κ(θ−y)uy −ru.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Degenerate elliptic/parabolic PDEs

Suppose (t, x) ∈ Q = [0, T) × O and O ⊂ Rn, and −Au(t, x) := 1 2

• i,j

aij(t, x) ∂2u ∂xi∂xj (t, x) +

• i

bi(t, x) ∂u ∂xi (t, x) − c(t, x)u(t, x). If ξTA(t, x)ξ ≥ µ(t, x)|ξ|2, ξ ∈ Rn, where µ(x) > 0, then A is elliptic (parabolic) on Q if µ > 0 on Q, and A is uniformly elliptic (parabolic) on Q if µ ≥ δ on Q, for some constant δ > 0. This condition fails for the Heston operator, as µ = 0 along {y = 0} component of ¯ O and the operator is “degenerate”.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Weighted Sobolev spaces

Definition

We need a weight function when defining our Sobolev spaces, w(x, y) := 2 σ2 yβ−1e−γ|x|−µy, β = 2κθ σ2 , µ = 2κ σ2 , for (x, y) ∈ O and a suitable positive constant, γ. Then H1(O, w) := {u ∈ L2(O, w) : (1 + y)1/2u ∈ L2(O, w), and y1/2Du ∈ L2(O, w)}, where u2

H1(O,w) :=

• O

y

• u2

x + u2 y

• w dxdy +
• O

(1 + y)u2 w dxdy.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Weighted Sobolev spaces and the boundary at y = 0

Let H1

0(O, w) be the closure in H1(O, w) of C ∞ 0 (O).

Definition

Denote Γ0 := ¯ O ∩ (R × {0}) and let H1

0(O ∪ Γ0, w) be the closure

in H1(O, w) of C ∞

0 (O ∪ Γ0).

One has the following useful result:

Lemma

If β > 1, then H1

0(O ∪ Γ0, w) = H1 0(O, w).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

A bilinear form

Definition

The Heston generator, −A, may be written in divergence form and defines a bilinear map, a : V × V → R, via a(u, v) := (Au, v)H, u, v ∈ C ∞

0 (O).

where V := H1

0(O, w) and H := L2(O, w).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

G˚ arding inequality

Proposition

Let r, σ, κ, θ ∈ R be constants such that β := 2κθ σ2 > 0, σ = 0, and − 1 < ρ < 1. Then, there are positive constants, C1, C2, depending at most on the coefficients r, κ, θ, ρ, σ, such that for all u ∈ V , a(u, u) ≥ 1 2C2u2

V − C3(1 + y)1/2u2 H.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Difficulties in option pricing problems and a key example Difficulties and survey of related research Heston process and degenerate elliptic/parabolic PDEs Analytical tools: Sobolev spaces and energy estimates

Continuity estimate

Proposition

There is a positive constant, C1, depending at most on the coefficients r, κ, θ, ρ, σ such that |a(u, v)| ≤ C1uV vV , ∀(u, v) ∈ V × V .

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

Variational inequality problem

Problem

Let f ∈ L2(O, w) and g, ψ ∈ H1(O, w) such that ψ ≤ g on O, and suppose β = 1. Find u ∈ K such that a(u, v − u) ≥ (f , v − u)L2(O,w), with u = g on ∂βO, ∀v ∈ K with v = g on ∂Oβ, where K := {v ∈ H1(O, w) : v ≥ ψ}.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

Remarks

Remark

◮ Here ∂βO := ∂O − Γ0, β > 1, and ∂βO := ∂O, β < 1, where

we recall that Γ0 = ¯ O ∩ (R × {0}).

◮ Recall that H1 0(O ∪ Γ0, w) = H1 0(O, w) when β > 1. Hence,

for β = 1, the boundary conditions for u, v are given by u − g, v − g ∈ H1

0(O, w).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

Existence and uniqueness of solutions

Theorem

There exists a unique solution to the stationary variational inequality for the Heston generator. The result is proved by the penalization method and adapting the arguments of Bensoussan and Lions (1982).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

Another weighted Sobolev space

Definition

Let H2(O, w) := {u ∈ L2(O, w) : (1+y)1/2u, y1/2Du, yD2u ∈ L2(O, w)}, where u2

H2(O,w) :=

• O
• y2

u2

xx + 2u2 xy + u2 yy

• +y
• u2

x + u2 y

• + (1 + y)u2

w dxdy. Let H2

loc(O, w) denote the space of functions u ∈ H2(O′, w) for all

O′ ⋐ O.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

H2 regularity and solution to the strong problem

Suppose ψ(x, y) = (K − ex)+ or (ex − K)+. Then,

Theorem

If u is the solution to the stationary variational inequality for the Heston generator, then u ∈ H2(O, w) and Au − f ≥ 0, u − ψ ≥ 0, (Au − f )(u − ψ) = 0 on O.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Existence and uniqueness Regularity

Remarks

Remark

◮ The preceding result is proved by adapting the arguments of

Bensoussan and Lions (1982) and Jaillet, Lamberton, and Lapeyre (1990).

◮ We expect, by work in progress, that u ∈ W 2,p(O, w), p > 2,

and u ∈ C 1,1(O).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Analytical tools Existence and uniqueness for the strong problem Regularity

A failure of coercivity

◮ Simple attempts to adapt the argument Bensoussan and Lions

(1982) in their proof existence and uniqueness of solutions to the “strong” variational inequality to the Heston generator, −A, fail because the bilinear map defined by A is non-coercive.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Analytical tools Existence and uniqueness for the strong problem Regularity

A change of dependent variable

◮ To circumvent the lack of coerciveness, we employ the change

• f dependent variable

˜ u(t, x, y) = e−λ(1+y)(T−t)u(t, x, y), u ∈ V , (t, x, y) ∈ Q, by analogy with the familiar exponential shift change of dependent variable ˜ u = e−λ(T−t)u.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Analytical tools Existence and uniqueness for the strong problem Regularity

Transformation of the equations

◮ One finds that the non-coercive parabolic problem,

−u′ + Au = f on Q, u(T) = h on O, u = g on Σ, is transformed, for t ∈ [T − δ, T] and sufficiently small δ, into an equivalent coercive parabolic problem, −˜ u′ + ˜ A˜ u = ˜ f on Q, ˜ u(T) = h on O, ˜ u = ˜ g on Σ,

◮ An obstacle condition u ≥ ψ is transformed into an equivalent

• bstacle condition ˜

u ≥ ˜ ψ.

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Analytical tools Existence and uniqueness for the strong problem Regularity

Transformation of the bilinear form

The bilinear form on V × V (defined by the weight w) associated to the operator ˜ A(t) (with suitable boundary conditions) is ˜ a(t; ˜ u(t), v) := (˜ A(t)˜ u(t), v)L2(O,w). (1) We then obtain the key continuity estimate and G˚ arding inequality for ˜ a(t).

Feehan Stochastic volatility and degenerate variational inequalities

Introduction Stationary variational inequalities for Heston generator Evolutionary variational inequalities for Heston generator Stochastic representation of solutions to variational problems References Analytical tools Existence and uniqueness for the strong problem Regularity

Continuity estimate and G˚ arding inequality for the transformed Heston operator

Proposition

For a sufficiently large positive constant λ, depending only the coefficients of A, and a sufficiently small positive constant δ < T, depending only on λ and the coefficients of A, the bilinear map ˜ a(t) : V × V → R obeys |˜ a(t; u, v)| ≤ CuV vV , ˜ a(t; v, v) ≥ α 2 v2

V ,

for all u, v ∈ V and t ∈ [T − δ, T].

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Change of Sobolev weight and transformation back to

• riginal problem

The weight in our previous definition of weighted Sobolev spaces, w(x, y) := 2 σ2 yβ−1e−γ|x|−µy, (x, y) ∈ O, is replaced, when transforming back from a solution ˜ u to a solution u to the original problem, by ˜ w(x, y) := e−2λM(1+y)w(x, y) = 2 σ2 yβ−1e−γ|x|−µy−2δλ(1+y), (x, y) ∈ O, where M > T is a constant.

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Setup for the evolutionary problem

Definition

◮ Recall that V = H1 0(O, w) and H = L2(O, w). Denote

V := L2(0, T; V ), V ′ := L2(0, T; V ′), H := L2(0, T; H), and K := {v ∈ V : v ≥ ψ}, given ψ ∈ V .

◮ The transformed Heston generator, −A(t), defines a linear

map A (t) ∈ L (V , V ′) and a bilinear map a(t) : V × V → R by a(t; u, v) := A (t)u(v), u, v ∈ V , and A (t)u(v) := (A(t)u, v)H, u, v ∈ C(0, T; C ∞

0 (O)).

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Boundary conditions

Definition

◮ Given u, g ∈ L2(0, T; H1(O, w)), then u = g on

∂βO × [0, T), β = 1, means that u − g ∈ L2(0, T; H1

0(O, w)). ◮ As usual, we may assume, without loss, that g = 0, and the

condition ψ ≤ g on [0, T) × ∂O is replaced by ψ ≤ 0 on [0, T) × ∂O.

◮ Recall that ∂βO := ∂O − Γ0, β > 1, and ∂βO := ∂O, β < 1,

where Γ0 = ¯ O ∩ (R × {0}).

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Evolutionary variational inequality problem

Problem

Suppose f , ψ ∈ H and h ∈ V with h ≥ ψ(T, ·) on O and ψ ≤ 0

• n [0, T) × ∂O. Find u ∈ K , with u′ ∈ H , such that

−(u′(t), v − u(t))H + a(t; u(t), v − u(t)) ≥ (f (t), v − u(t))H, ∀v ∈ V with v ≤ ψ(t, ·), t ∈ [0, T].

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Existence and uniqueness of solutions

Theorem

There exists a unique solution to the evolutionary variational inequality for the Heston generator. As with the stationary variational inequality, the result is proved by the penalization method and adapting the arguments of Bensoussan and Lions (1982).

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Regularity for solutions to the strong problem for the parabolic Heston variational inequality

Suppose ψ(t, x, y) = (ex − K)+ or (K − ex)+, (t, x, y ∈ Q. Using

• ur weighted Sobolev spaces and estimates, we adapt the

Bensoussan-Lions regularity theory to establish

Theorem

If u is the solution to the evolutionary variational inequality for the Heston generator, then u ∈ L2(0, T; H2(O, w)). Given this regularity, a solution to the strong problem for the parabolic Heston variational inequality is a solution to the more familiar “complementarity” or strong form of the obstacle problem.

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Solution to the strong form of the obstacle problem

Theorem

Given f ∈ L2(0, T; L2(O, w)), g ∈ L2(0, T; H1(O, w)), and h ∈ H1(O, w) obeying g ≥ ψ on Q, h ≥ ψ on O, there is a unique u ∈ L2(0, T; H2(O, w)) solving −u′ + Au ≥ f on Q, u ≥ ψ on Q, (−u′ + Au − f )(u − ψ) = 0 on Q, u = g on ∂βO × [0, T), u(T) = h on O.

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Improved regularity

Remark

We expect, by work in progress, that u ∈ L2(0, T; W 2,p(O, w)), p > 2, and u ∈ C 1,1(Q).

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Stochastic representations and their consequences

◮ Given a stochastic representation of a solution, certain

regularity and geometric properties follow readily.

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Stochastic representations and their consequences

◮ Given a stochastic representation of a solution, certain

regularity and geometric properties follow readily.

◮ Paradigms for such technques may be found in the work of

Broadie & Detemple (1997), Detemple (2006), Jaillet, Lamberton, & Lapeyre (1990), Villeneuve (1999), and Laurence & Salsa (2009).

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Stochastic representations and their consequences

◮ Given a stochastic representation of a solution, certain

regularity and geometric properties follow readily.

◮ Paradigms for such technques may be found in the work of

Broadie & Detemple (1997), Detemple (2006), Jaillet, Lamberton, & Lapeyre (1990), Villeneuve (1999), and Laurence & Salsa (2009).

◮ The following stochastic representations may be derived by

adapting methods of Bensoussan & Lions (1982), Friedman (1976), and Øksendal (2003).

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Probabilistic solutions to stationary variational equalities I

Problem (Stationary variational equality)

Let O ⊂ R × (0, ∞) be a domain with C 2 boundary, ∂O, let f ∈ C(O) obey |f (x, y)| ≤ C1(1 + y), (x, y) ∈ O, and let g ∈ Cb(∂O). Find u ∈ C 2(O) ∩ C( ¯ O) such that Au = f

• n O,

u = g

• n ∂βO.

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Probabilistic solutions to stationary variational equalities II

Theorem (Uniqueness of solutions to Problem 4.1)

Let u be a solution to Problem 4.1. Then u = u∗ on ¯ O, with u∗(x, y) := EQ

• e−rτg(X(τ), Y (τ))1{τ<∞}
• + EQ

τ e−rsf (X(s), Y (s)) ds

• ,

(x, y) ∈ O, (2) where r > 0 and τ is the exit time from O of the process (X(s), Y (s))s≥0 starting at (x, y) ∈ O.

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Probabilistic solutions to stationary var’l inequalities I

Problem (Stationary variational inequality)

Let O, f , g be as in Problem 4.1 and let ψ ∈ C( ¯ O) obey ψ ≤ g on ∂βO, and |f (x, y)| ≤ C1(1 + y) and |ψ(x, y)| ≤ C2(1 + eC3x), (x, y) ∈ O. Find u ∈ C 2(O) ∩ C( ¯ O) such that Au ≥ f and u ≥ ψ on O, (Au − f )(u − ψ) = 0 on O, u = g on ∂βO.

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Probabilistic solutions to stationary var’l inequalities II

Theorem (Uniqueness of solutions to Problem 4.3)

Let u be a solution to Problem 4.3. Then u = u∗ on ¯ O, with u∗(x, y) := sup

θ∈T

• EQ

τ∧θ e−rsf (X(s), Y (s)) ds

• + EQ
• e−rθψ(X(θ), Y (θ))1{θ<τ}
• + EQ
• e−rτg(X(τ), Y (τ))1{τ≤θ}
• ,

(x, y) ∈ ¯ O, where r > 0, τ is the exit time from O of the process (X(s), Y (s))s≥0 starting at (x, y) ∈ O, and T is the set of F-stopping times with values in [0, ∞).

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Probabilistic solutions to evolutionary var’l equalities I

Problem (Evolutionary variational equality)

Let O be as in Problem 4.1 and let 0 < T < ∞ and Q := [0, T) × O. Let f ∈ C(Q) obey |f (t, x, y)| ≤ C1(1 + y), (t, x, y) ∈ Q, let g ∈ Cb((0, T) × ∂O), and let h ∈ C(O) obey |h(x, y)| ≤ C2(1 + eC3x), (x, y) ∈ O, Find u ∈ C 1,2(Q) ∩ C( ¯ Q) such that u′ + Au = f on Q, u = g on (0, T) × ∂βO, u(T, ·) = h

• n O.

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Probabilistic solutions to evolutionary var’l equalities II

Theorem (Uniqueness of solutions to Problem 4.5)

Let u be a solution to Problem 4.5. Then u = u∗ on ¯ Q, with u∗(t, x, y) := EQ τ

t

e−rsf (X(s), Y (s)) ds

• + EQ
• e−r(τ−t)h(X(T), Y (T))1{τ=T}
• + EQ
• e−r(τ−t)g(τ, X(τ), Y (τ))1{τ<T}
• ,

where (t, x, y) ∈ Q, r > 0, τ is the exit time from O of (X(s), Y (s))s≥t starting at (t, x, y) ∈ Q, if such a time exists and τ = T otherwise.

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Probabilistic solutions to evolutionary var’l inequalities I

Problem (Evolutionary variational inequality)

Let O, T, Q, f , g, h be as in Problem 4.5, and let ψ obey |ψ(t, x, y)| ≤ C4(1 + eC5x), (t, x, y) ∈ Q, ψ ≤ g on (0, T) × ∂βO and ψ(T, ·) ≤ h on O. Find u ∈ C 1,2(Q) ∩ C( ¯ Q) such that u′ + Au ≥ f

• n Q,

u ≥ ψ

• n Q,

(u′ + Au − f )(u − ψ) = 0

• n Q,

u = g

• n (0, T) × ∂βO,

u(T, ·) = h

• n O.

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Probabilistic solutions to evolutionary var’l inequalities II

Theorem (Uniqueness of solutions to Problem 4.7)

Let u be a solution to Problem 4.7. Then u = u∗ on ¯ Q, with u∗(t, x, y) := sup

θ∈Tt,T

• EQ

τ∧θ

t

e−rsf (s, X(s), Y (s)) ds

• + EQ
• e−r(θ−t)ψ(θ, X(θ), Y (θ))1{θ<τ∧T}
• + EQ
• e−r(T−t)h(X(T), Y (T))1{T=τ∧θ}
• + EQ
• e−r(τ−t)g(τ, X(τ), Y (τ))1{τ≤θ,τ<T}
• ,

where τ is exit time from O of (X(s), Y (s))s≥t starting at (t, x, y) ∈ Q, and Tt,T is set of F-stopping times valued in [t, T].

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Applications to finance

◮ In applications to option pricing, we need only consider

solutions to the evolutionary variational inequality with f = 0, while ψ(x, y) = (K − ex)+ or (ex − K)+ and h(x, y) = ψ(x, y).

◮ We denote U(t, S, y) = u(t, x, y) and Ψ(t, S, y) = ψ(t, x, y),

where S = ex. The results on the next few slides provide a small sample of what may be proved by adapting arguments of Broadie & Detemple (1997), Jaillet, Lamberton, and Lapeyre (1990), Laurence and Salsa (2009), Touzi (1999), and Villeneuve (1999).

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Properties of the solution

Lemma

Let U(t, S, y) be as above. Then

• 1. U(t, S, y) is a non-increasing function of t ∈ [0, T].
• 2. If Ψ(S) is a convex function of S ∈ (0, ∞), then U(t, S, y) is

a convex function of S ∈ (0, ∞), ∀(t, y) ∈ [0, T] × (0, ∞).

• 3. If Ψ(S) is a non-increasing (non-decreasing) function of

S ∈ (0, ∞), then U(t, S, y) is a non-increasing (non-decreasing) function of S ∈ (0, ∞), ∀(t, y) ∈ [0, T] × (0, ∞).

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Properties of the derivative

Lemma

Suppose Ψ(S), S ∈ (0, ∞), obeys m(S2 − S1) ≤ Ψ(S2) − Ψ1(S1) ≤ M(S2 − S1), 0 < S1 < S2 < ∞, for given −∞ < m ≤ M < ∞. Then, for each (t, y) ∈ [0, T) × (0, ∞), U(t, S, y) is a differentiable function of S ∈ (0, ∞) and m ≤ ∂U ∂S ≤ M, ∀(t, S, y) ∈ [0, T) × (0, ∞) × (0, ∞).

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Continuation and exericise regions

Definition

Given a solution U(t, S, y) to the evolutionary variational inequality for an obstacle function Ψ(t, S, y), the continuation and exericise regions are defined by C (U) := {(t, S, y) ∈ Q : U(t, S, y) > Ψ(t, S, y)}, E (U) := {(t, S, y) ∈ Q : U(t, S, y) = Ψ(t, S, y)}, and similarly for C (u) and E (u), given u(t, x, y) and ψ(t, x, y).

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Characterization of the free boundary

The results of Touzi (1999) may be adapted to show

Proposition

If Ψ(t, S, y) = (K − S)+, there is a S∗ : [0, T) × (0, ∞) → [0, K] such that C (U) = {(t, S, y) ∈ [0, T) × (0, ∞) × (0, ∞) : S > S∗(t, y)}.

Lemma

If Ψ(t, S, y) = (K − S)+, then S∗ : [0, T) × (0, ∞) → [0, K] is decreasing with respect to y ∈ (0, ∞).

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Properties of the free boundary

We expect, by work in progress, that

◮ S∗(t, y) is a continuous function of t ∈ [0, T), ∀y ∈ (0, ∞). ◮ If Ψ(S) = (K − S)+ (respectively, (S − K)+), then S∗(t, y) is

a non-decreasing (respectively, non-increasing) function of t ∈ [0, T), ∀y ∈ (0, ∞).

◮ If s∗(t, y) = log S∗(t, y), then s∗(t, ·) is Lipschitz, uniformly

with respect to t ∈ [0, T).

◮ S∗ : [0, T) × (0, ∞) → [0, K] is differentiable with respect to

y ∈ (0, ∞).

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Selected references

◮ V. Barbu and C. Marinelli, Variational inequalities in Hilbert

spaces with measures and optimal stopping problems, Appl.

• Math. Optim. 57 (2008), 237–262.

◮ A. Bensoussan and J. L. Lions, Applications of variational

inequalities in stochastic control, 1982.

◮ L. A. Caffarelli, The obstacle problem revisited, J. Fourier

• Anal. Appl. 4 (1998), pp. 383–402.

◮ P. Daskalopoulos and R. Hamilton, Regularity of the boundary

for the porous medium equation, J. American Mathematical Society 11, 1998, pp. 899–965.

◮ A. Friedman, Variational principles and free boundary

problems, Wiley, 1982, New York.

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Selected references (continued)

◮ P. Jaillet, D. Lamberton, and B. Lapeyre, Variational

inequalities and the pricing of American options, Acta Appl.

• Math. 21 (1990), pp. 263–289.

◮ I. Karatzas and S. E. Shreve, Methods of mathematical

finance, Springer, New York, 1998.

◮ D. Kinderlehrer and G. Stampacchia, An Introduction to

Variational Inequalities and Their Applications, 1980.

◮ P. Laurence and S. Salsa, Regularity of the free boundary of

an American option on several assets, Comm. Pure Appl.

• Math. 62, 2009, pp. 969–994.

◮ L. Mastroeni and M. Matzeu, Parabolic variational inequalities

with degenerate elliptic part, Riv. Mat. Univ. Parma 5 (1996), 223–234.

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Selected references (continued)

◮ M. K. V. Murthy and G. Stampacchia, Boundary value

problems for some degenerate elliptic operators, Ann. Mat. Pura Appl. 80 (1968), 1–122.

◮ R. E. Showalter, Monotone operators in Banach space and

nonlinear partial differential equations, 1996.

◮ N. Touzi, American options exercise boundary when the

volatility changes randomly, Applied Mathematics and Optimization 39 (1999), 411–422.

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Mathematical Finance and Partial Differential Equations 2010 Rutgers University, New Brunswick, NJ December 10, 2010 Web: finmath.rutgers.edu Email: mfpde@rci.rutgers.edu 11 invited speakers and up to 6 contributed talks

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THANK YOU!

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