[PPT] - Semilinear perturbations of Kolmogorov operators, obstacle problems, PowerPoint Presentation

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Semilinear perturbations of Kolmogorov

perators, obstacle problems, and optimal

stopping

Carlo Marinelli

Hausdorff Institute for Mathematics Universit¨ at Bonn and Facolt` a di Economia Universit` a di Bolzano http://www.uni-bonn.de/∼cm788

Based on joint work with Viorel Barbu and Zeev Sobol

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Outline

1. Motivation and history of the problem

◮ Classical: the price of an American option is (often) the solution of

an obstacle problem

◮ Not so classical (Kholodnyi NA ’97): the price of an American

ption is expected to “solve” a semilinear PDE with a discontinuous

reaction term

◮ Q: Is there a general connection among optimal stopping, obstacle

problems, variational inequalities, semilinear PDEs?

2. Solving obstacle problems via semilinear PDEs

◮ Solution of a semilinear PDE is the solution of the obstacle problem ◮ (Nonlinear) monotone operator techniques give a natural concept of

solution

◮ Equation is globally well-posed

3. Back to optimal stopping

◮ Solution of the semilinear PDE is also (often) the value function of

the original optimal stopping problem

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Starting point: 1D Black-Scholes heuristics

Assume Black-Scholes dynamics dSt = (r − d)St dt + σStdWt and an American option with payoff g : R+ → R+, with price v(t, x) = sup

τ∈[t,T]

Et,xe−r(τ−t)g(Sτ)

(Bensoussan AAM ’84, Karatzas AMO ’88)

By heuristic (clever!) arguments, Kholodnyi “showed” that v solves the PDE vt + 1 2σ2x2vxx + (r − d)xvx − rv = q(x, v), v(T, x) = g(x), where q(x, v) =

−d(x),

v ≤ g(x), 0, v > g(x), d(x) =

− 1

2σ2x2gxx − (r − d)xgx + rg +.

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Issues with Kholodnyi’s equation

★ What do we mean by “solution”? The reaction term q is discontinuous (if at all defined) ★ The space domain is unbounded, coefficients are unbounded ★ d is in general defined only as a distribution in D′(R) As far as we know, no known technique applies in the general case: ✘ PDE with discontinuous terms: does not handle unbounded coefficients ✘ PDE with growing coefficients: does not handle discontinuous terms ✘ PDE with measure data: does not handle “rough” equations ✔ Viscosity solutions approach for call/put options on a single BS asset works! (Benth et al. F&S ’03)

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Semilinear PDEs vs. Variational Inequalities

Classical analytic approach: solve the obstacle problem

vt + L0v − cv ≤ f ,

vt + L0v − cv = f

n {v(t, x) > g(x)}

v(T, x) = g(x) e.g. turning it into a VI dv dt + L0v − cv − Ng(v) ∋ f , v(T) = g(T). Why another approach?

◮ Much easier to do numerical analysis on a PDE rather than on a free

boundary problem/VI (Benth et al. IFB ’04)

◮ Nonlinear discontinuous PDEs have an intrinsic mathematical

interest

◮ New way to solve obstacle problems without a variational setting ◮ Can one do better than call/put options on one BS asset?

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An abstract general framework

Let X be a right Markov process on a Hilbert space H, with semigroup Pt, and consider the optimal stopping problem v(t, x) = sup

τ∈[t,T]

Et,xe−

R τ

t c(Xs) dsg(Xτ)

Goal: characterization of the value function v in terms of the solution of a suitable semilinear equation. Plan:

◮ Construct a state space E ◮ Formulate abstract semilinear (evolution) eq. on E ◮ Specify the concept of solution ◮ Prove well-posedness ◮ Prove that the solution coincides with the solution of the obstacle

problem

◮ Prove that the solution coincides with the value function

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State space: Lp(H, ν), p ≥ 1

Let ν be an excessive probability measure for Pt with full topological support, i.e. such that

H

Ptϕ dν ≤ eωt

H

ϕ dν ∀ϕ ∈ Cb(H)+, and ν(U) > 0 for any nonempty open set U ⊆ H. Such a measure ν always exists! (R¨

ckner-Trutnau IDAQP ’07)

Set A = −Np + cI, where −Np is the generator of Pt on Lp(H, ν). Then −Np (hence also A) is ω-m-accretive, i.e. (let ω = 0 for simplicity) (i) x, J(y) ≥ 0 ∀[x, y] ∈ A; (ii) R(λI + A) = Lp(E, ν) ∀λ > 0.

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The semilinear PDE

Given g ∈ Lp(H, ν) and d ∈ Lp(H, ν)+ define the nonlinear multivalued

perator Bd on Lp(H, ν) by

[Bdy](x) =      −d(x), y(x) < g(x), [−d(x), 0], y(x) = g(x), 0, y(x) > g(x). We are going to establish well-posedness in Lp(E, ν) of the evolution equation du dt (t) + Au(t) + Bdu(t) ∋ 0, u(0) = g. (1) We are to “use” several concepts of solutions:

◮ Strong: (1) satisfied a.e. on (0, T) ◮ Generalized: SOLA ◮ Mild in the sense of Crandall-Liggett: limit of a discrete scheme ◮ Mild in the sense of Duhamel’s principle

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Well-posedness

Theorem. The equation

du dt (t) + Au(t) + Bdu(t) ∋ 0, u(0) = g admits a unique CL-mild solution in Lp(E, ν), p ≥ 1 (which is SOLA for p > 1). Moreover, if g ∈ D(A) and p > 1, then it admits a unique strong solution u ∈ W 1,∞([0, T] → Lp(H, ν)) ∩ L∞([0, T] → D(A)) which is also right-differentiable.

Proof. Have to show that A + Bd is ω-m-accretive...

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A + Bd is ω-m-accretive in Lp(H, ν)

Lemma. Bd is m-accretive in Lp(H, ν).
Proof. Bd is accretive because

y → d(x)(H(y − g(x)) − 1) is a maximal monotone graph in R × R for each x ∈ H. Maximality: equation y + By = f , f ∈ Lp(H, ν) easily admits a solution.

Theorem. A + Bd is ω-m-accretive in Lp(H, ν).
Proof. Three different cases:

◮ p = 2: follows by Rockafellar’s criterion: D(A) ∩ int D(Bd) = ∅. ◮ p > 1: solve uλ + Aλuλ + Bduλ ∋ f , get a priori estimates on uλ, let

λ → 0 (reflexivity of Lp is crucial)

◮ p = 1: solve uε + Auε + Bd,εuε = f , get monotonicity for uε, let

ε → 0.

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Solution of the PDE is a solution of the obstacle problem

Have to choose d first! Two cases: Theorem.

◮ If g ∈ D(A), let d := (Ag)+. ◮ If g ∈ Lp(H, ν) = D(A), assume that (Aλg)+ is weakly compact in

Lp(H, ν), and let d be such that (Aλg)+ ⇀ d. Then u(t) ≥ g ν-a.s..

Proof. Prove that S(t) := e−t(A+Bd) leaves invariant

Kg :=

ϕ ∈ Lp(H, ν) : ϕ ≥ g

ν-a.e.

.

Enough to prove that (I + λA + λBd)−1Kg ⊆ Kg for all λ ∈]0, ω−1[. Use sub-Markovianity of A and definition of Bd. Key observation: By definition of Bd, u(t) ≥ g ν-a.e. implies that u is the solution of the obstacle problem!

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Solution as value function of the optimal stopping problem

Theorem 1. Assume that Pt is strong Feller (or L(Xt) ≪ ν). Then u(T − t) = v(t) ν-a.e. ∀t ≤ T

Proof. Two steps:
1. Establish a Duhamel representation for CL-mild solutions.
2. Refine the proof in Barbu-M (AMO ’08)
Remark. Still true for Markov processes that are limits of strong Feller

processes (hence always true for solutions of SDEs on Rd) Q:

◮ Can one approximate any right Markov process by a strong Feller

Markov process?

◮ Counterexamples?

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Further properties

◮ Under very mild assumptions v is continuous, hence u has a

continuous ν-modification. Moreover, recall that ν has full topological support!

◮ Convexity assumptions on g are needed also for the 1D viscosity

solutions approach.

◮ Finite-difference schemes converge to the mild solution

(Crandall-Liggett theorem)

◮ If dim H < ∞ one expects further regularity for u ◮ Infinite-horizon optimal stopping problems ( elliptic PDEs) are

automatically included

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An example

Let X be the solution of an SDE on H Xs = x + s

t

b(Xr) dr + s

t

σ(Xr) dW (r) with Kolmogorov operator −N0φ = 1 2 Tr[(σQ1/2)(σQ1/2)∗D2φ] + b(x), DφH , φ ∈ C 2

b (H).

Consider an optimal stopping problem as before. Let Ptφ(x) := Exφ(X(t)), φ ∈ Cb(H). Let P∗

t ν ≤ eωtν, extend Pt to

L2(H, ν), and set −N2φ := limh↓0 h−1(Phφ − φ) in L2(H, ν).

Lemma. Let b ∈ C 2(H) ∩ L2(H, ν), σ ∈ C 2(H, L(H, H)), and

|Db(x)|H + |Dσ(x)|L(H,H) ≤ C ∀x ∈ H. Then −N0 is ω-accretive and −N2 is the closure in L2(H, ν) of −N0 defined on D(N0) = C 2

b (H).

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(contd.)

Proof. −N0 is ω-accretive, hence closable. Fix f ∈ C 2

b (H) and consider

the eq. (λI + N0)ϕ = f . Candidate solution is ϕ(x) = E ∞ e−λtf (X x

t ) dt,

λ > ω. By second order differentiability of x → X x

t and Itˆ

’s formula, we see that

φ actually is the solution. Then R(I + λN0) ⊂ L2(H, ν) densely, so −N0 is ω-m-accretive by Lumer-Phillips theorem, so it must be N2 = N0 because −N2 is also ω-m-accretive. Remarks.

◮ Similar results for Kolmogorov operators go through under much

weaker assumptions, and also for equations of the type Xs = x + s

t

b(Xr) dr + s

t

σ(Xr) dW (r)+ s

t

Z

g(z, Xr−) ¯ µ(dz, dr) (M-Pr´ evˆ

t-R¨
ckner JFA ’10)

◮ Enough to take N∗ 0 ν ≤ ων.

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American options with general volatility I

Let Q be EMM, n assets with price-per-share Xt, dXt = rXt dt + σ(Xt) dWt, X0 = x ≥ 0, t ∈ [0, T], W is Rm-valued Wiener process, and σ : Rn → L(Rm, Rn) is the volatility function. We do not assume that σ satisfies any nondegeneracy condition! Pricing an American contingent claim with payoff g : Rn → R is equivalent to the optimal stopping problem v(t, x) = sup

τ∈[t,T]

Et,x[e−rτg(Xτ)], Kolmogorov operator: −N0f (x) = 1 2 Tr[σ(x)σ∗(x)D2f (x)] + rx, Df (x)Rn , f ∈ C 2

b (Rn).

Classical (analytic) approach (VI in Sobolev spaces w.r.t. Lebesgue measure) does not apply, essentially because σ can be degenerate.

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American options with general volatility II

Lemma. Assume that

σ ∈ C 2(Rn), |σxi| + |σxixj| ≤ C. Then there exists an excessive probability measure ν for Pt of the form ν(dx) = a 1 + |x|2(n+1) dx.

Proof. Try to solve N∗

0 ρ ≤ ωρ in D′(Rn) ...

Lemma. The infinitesimal generator of Pt in L2(Rn, ν) is −N2 := −N0.

Moreover one has

Rn(N0f )f dν ≤ −1

2

Rn |σ∗Df |2 dν + ω
Rn f 2 dν

∀f ∈ C 2

b (Rn).

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American options with general volatility III

We can now apply the abstract existence results: v is uniquely identified by the mild solution in L2(Rn, ν) of the nonlinear parabolic equation du dt + N2u + ru + Bu ∋ 0, u(0) = g, provided g satisfies Dg ∈ L∞(Rn), Tr[σσ∗D2g] ∈ M(Rn), Tr[σσ∗D2g] ≥ 0 in M(Rn).

◮ Exchange options and basket put options can be covered as

examples.

◮ American options on assets with stochastic volatility can be treated

in a similar way (augmenting the state space).

◮ Asian options with American feature also covered (adding an

auxiliary process) v(x) = sup

τ∈[0,T]

Ex

K −

1 τ + δ τ Xs ds + .

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Path-dependent American options I

Price of n assets under an EMM Q:

dX(t) = rX(t) + σ(X(t), Xs(t)) dW (t),

0 ≤ t ≤ T X(0) = x0, Xs(0) = x1(s), −T ≤ s ≤ 0, where Xs(t) := X(t + s), s ∈ (−T, 0). Consider an American option with payoff g : Rn × L2([−T, 0]) → R: v(s, x0, x1) = sup

τ∈[0,T]

Es,(x0,x1)[e−rτg(X(τ), Xs(τ))], e.g. g(x0, x1) = α0g0(x0) + α2g1(x1), with α1, α2 ≥ 0 and g0(x0) = (K0 − x0)+, g1(x1) =

K1 −

−T

x1(s) ds +

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Path-dependent American options II

◮ Formulate as an infinite dimensional problem in

H = Rn × L2([−T, 0], Rn)

◮ Look for an excessive measure ν of the form ν = ν1 ⊗ ν2, with ν1, ν2

probability measures on Rn and L2([−T, 0], Rn). Choose ν1(dx0) = ρ(x0) dx0, ρ(x0) = a 1 + |x0|2n , a =

Rn

1 1 + |x0|2n dx0 −1 , and ν2 a Gaussian measure on L2([−T, 0], Rn).

◮ Impose assumptions on σ so that N0 = N2 in L2(H, ν) (but still no

nondegeneracy!)

◮ Impose assumptions on g such that (Aλg)+ w.c. in L2(H, ν).

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