Homogenization and other multi-scale problems in Bellman-Isaacs equations Martino Bardi Dipartimento di Matemetica Pura ed Applicata Università di Padova New Trends in Analysis and Control of Nonlinear PDEs Roma, June 13–15, 2011 Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 1 / 29 Plan 1 2 Homogenization of non-coercive Hamilton–Jacobi equations (joint work with G. Terrone) I Some previous results I Examples of non-homogenization I Homogenization on subspaces I Convex-concave eikonal equations and differential games Optimal control with stochastic volatility (joint work with A. Cesaroni and L. Manca) I Motivation: financial models I Singular perturbations of H-J-B equations I Merton portfolio optimization with stochastic volatility Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 2 / 29 Plan 1 2 Homogenization of non-coercive Hamilton–Jacobi equations (joint work with G. Terrone) I Some previous results I Examples of non-homogenization I Homogenization on subspaces I Convex-concave eikonal equations and differential games Optimal control with stochastic volatility (joint work with A. Cesaroni and L. Manca) I Motivation: financial models I Singular perturbations of H-J-B equations I Merton portfolio optimization with stochastic volatility Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 2 / 29 1. Homogenization of Hamilton–Jacobi Equations Problem: let ε → 0+ in utε + H x, xε , Du ε = 0 in (0, T ) × RN u ε (0, x) = h x, xε . ZN − periodic case: H(x, ξ + k , p) = H(x, ξ, p), h(x, ξ + k ) = h(x, ξ) ∀k ∈ ZN . Goal: find continuous effective Hamiltonian and initial data H, h s.t. uε → u locally uniformly, and u solves ut + H (x, Du) = 0 in (0, T ) × RN u(0, x) = h (x) . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 3 / 29 Classical result for coercive H Coercivity in p is lim H(x, ξ, p) = +∞ |p|→∞ uniformly in x, ξ. P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92: The effective Hamiltonian H = H(x, p) is the unique constant such that the cell problem, with x, p frozen parameters, H(x, ξ, Dξ χ + p) = H, in RN , has a ZN −periodic (viscosity) solution χ(ξ), called the corrector; if h = h(x) is independent of xε , u ε → u locally uniformly. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 4 / 29 Classical result for coercive H Coercivity in p is lim H(x, ξ, p) = +∞ |p|→∞ uniformly in x, ξ. P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92: The effective Hamiltonian H = H(x, p) is the unique constant such that the cell problem, with x, p frozen parameters, H(x, ξ, Dξ χ + p) = H, in RN , has a ZN −periodic (viscosity) solution χ(ξ), called the corrector; if h = h(x) is independent of xε , u ε → u locally uniformly. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 4 / 29 Classical result for coercive H Coercivity in p is lim H(x, ξ, p) = +∞ |p|→∞ uniformly in x, ξ. P.-L. Lions - Papanicolaou - Varadhan ’86, L.C. Evans ’92: The effective Hamiltonian H = H(x, p) is the unique constant such that the cell problem, with x, p frozen parameters, H(x, ξ, Dξ χ + p) = H, in RN , has a ZN −periodic (viscosity) solution χ(ξ), called the corrector; if h = h(x) is independent of xε , u ε → u locally uniformly. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 4 / 29 The effective initial data If the initial data h = h(x, xε ) must find the initial condition h(x). O. Alvarez - M. B. (ARMA ’03): Assume there exists the recession function of H H(x, ξ, λp) . λ→+∞ λ H 0 (x, ξ, p) := lim Consider, for frozen x, If then wt + H 0 (x, ξ, Dξ w) = 0, w(0, ξ) = h(x, ξ). limt→+∞ w(t, ξ) = constant =: h(x), limt→0 limε→0 u ε = h(x). In the coercive case Martino Bardi (Università di Padova) h (x) = minξ h(x, ξ). Multi-scale Bellman-Isaacs Roma, June 15th, 2011 5 / 29 The effective initial data If the initial data h = h(x, xε ) must find the initial condition h(x). O. Alvarez - M. B. (ARMA ’03): Assume there exists the recession function of H H(x, ξ, λp) . λ→+∞ λ H 0 (x, ξ, p) := lim Consider, for frozen x, If then wt + H 0 (x, ξ, Dξ w) = 0, w(0, ξ) = h(x, ξ). limt→+∞ w(t, ξ) = constant =: h(x), limt→0 limε→0 u ε = h(x). In the coercive case Martino Bardi (Università di Padova) h (x) = minξ h(x, ξ). Multi-scale Bellman-Isaacs Roma, June 15th, 2011 5 / 29 The effective initial data If the initial data h = h(x, xε ) must find the initial condition h(x). O. Alvarez - M. B. (ARMA ’03): Assume there exists the recession function of H H(x, ξ, λp) . λ→+∞ λ H 0 (x, ξ, p) := lim Consider, for frozen x, If then wt + H 0 (x, ξ, Dξ w) = 0, w(0, ξ) = h(x, ξ). limt→+∞ w(t, ξ) = constant =: h(x), limt→0 limε→0 u ε = h(x). In the coercive case Martino Bardi (Università di Padova) h (x) = minξ h(x, ξ). Multi-scale Bellman-Isaacs Roma, June 15th, 2011 5 / 29 Bellman-Isaacs equations In control theory the Hamiltonian is, for A compact, H(x, ξ, p) = max {−f (x, ξ, a) · p − l (x, ξ, a)} , a∈A ξ= x ε and in 0-sum differential games it is, for B compact, H(x, ξ, p) = min max {−f (x, ξ, a, b) · p − l (x, ξ, a, b)} . b∈B a∈A The associated control system in an oscillating periodic medium is x ẋ = f x, , a, b ε and cost - payoff Z t x(t) x(s) , a(s), b(s) ds + h x(t), J := l x(s), ε ε 0 that player a wants to minimize and player b to maximize. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 6 / 29 Bellman-Isaacs equations In control theory the Hamiltonian is, for A compact, H(x, ξ, p) = max {−f (x, ξ, a) · p − l (x, ξ, a)} , a∈A ξ= x ε and in 0-sum differential games it is, for B compact, H(x, ξ, p) = min max {−f (x, ξ, a, b) · p − l (x, ξ, a, b)} . b∈B a∈A The associated control system in an oscillating periodic medium is x ẋ = f x, , a, b ε and cost - payoff Z t x(t) x(s) , a(s), b(s) ds + h x(t), J := l x(s), ε ε 0 that player a wants to minimize and player b to maximize. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 6 / 29 Alternative conditions to coercivity coercivity of H is equivalent to Small Time Local Controllability in a time proportional to the distance (by one of the players for any control of the other), it is a STRONG assumption, can be replaced by Controllability in a uniformly Bounded Time (by one of the players for any control of the other) (M.B. - Alvarez Mem. AMS 2010) Example: hypoelliptic eikonal equation utε + k X i=1 |fi x x · Du ε | = l x, ε ε in (0, T ) × RN with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating condition. Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10 for control problems, Cardaliaguet ’10 for games). Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 7 / 29 Alternative conditions to coercivity coercivity of H is equivalent to Small Time Local Controllability in a time proportional to the distance (by one of the players for any control of the other), it is a STRONG assumption, can be replaced by Controllability in a uniformly Bounded Time (by one of the players for any control of the other) (M.B. - Alvarez Mem. AMS 2010) Example: hypoelliptic eikonal equation utε + k X i=1 |fi x x · Du ε | = l x, ε ε in (0, T ) × RN with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating condition. Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10 for control problems, Cardaliaguet ’10 for games). Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 7 / 29 Alternative conditions to coercivity coercivity of H is equivalent to Small Time Local Controllability in a time proportional to the distance (by one of the players for any control of the other), it is a STRONG assumption, can be replaced by Controllability in a uniformly Bounded Time (by one of the players for any control of the other) (M.B. - Alvarez Mem. AMS 2010) Example: hypoelliptic eikonal equation utε + k X i=1 |fi x x · Du ε | = l x, ε ε in (0, T ) × RN with f1 , ..., fk ∈ C ∞ satisfying Hörmander bracket-generating condition. Non-resonance conditions (Arisawa - Lions ’99, M.B. - Alvarez ’10 for control problems, Cardaliaguet ’10 for games). Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 7 / 29 Examples of non-homogenization Example 1 utε + |uxε + uyε |β = cos 2π x−y ε u ε (0, x, y ) = 0 is solved by x, y ∈ R (β ≥ 1) u ε (t, x, y ) = t cos 2π x−y . ε Example 2 utε + |uxε + uyε |β = 0 x, y ∈ R u ε (0, x, y ) = cos 2π x−y ε is solved by u ε (t, x, y ) = cos 2π x−y . ε In both cases u ε has no limit as ε → 0 if x 6= y . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 8 / 29 Example 3 utε + |uxε | − |uyε | = cos 2π x−y ε u ε (0, x, y ) = 0 is solved by x, y ∈ R u ε (t, x, y ) = t cos 2π x−y . ε Example 4 utε + |uxε | − |uyε | = 0 x, y ∈ R u ε (0, x, y ) = 2π x−y ε is solved by Again u ε (t, x, y ) = cos 2π x−y . ε u ε has no limit as ε → 0 if x 6= y . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 9 / 29 H vanishing in a direction The previous examples are special cases of Proposition Assume ∃ p0 ∈ ZN \ {0} such that z , utε + H(Dz u ε ) = l ε H(λp0 ) = 0 ∀ λ ∈ R in z u ε (0, z) = h ε If h = 0 ∃ smooth ZN −periodic l such that limε→0 u ε @ if l = 0 ∃ smooth ZN −periodic h such that limε→0 u ε @ The 1st statement is SHARP if H is 1-homogeneous and convex: ∀ p0 ∈ ZN \ {0} H(p0 ) 6= 0 is nonresonance condition ⇒ homogenization! Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 10 / 29 H vanishing in a direction The previous examples are special cases of Proposition Assume ∃ p0 ∈ ZN \ {0} such that z , utε + H(Dz u ε ) = l ε H(λp0 ) = 0 ∀ λ ∈ R in z u ε (0, z) = h ε If h = 0 ∃ smooth ZN −periodic l such that limε→0 u ε @ if l = 0 ∃ smooth ZN −periodic h such that limε→0 u ε @ The 1st statement is SHARP if H is 1-homogeneous and convex: ∀ p0 ∈ ZN \ {0} H(p0 ) 6= 0 is nonresonance condition ⇒ homogenization! Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 10 / 29 Homogenization on subspaces If H satisfies the assumption of the negative Proposition, must restrict the data l, h to oscillate only in directions where H does not vanish. Let V M–dimensional subspace of RN , zV := ΠV (z) projection z z z V V V utε + H , Dz u ε = l z, , u ε (0, z) = h z, ε ε ε N Assume coercivity in V : ∀ p̄ ∈ R lim |p|→+∞, p∈V H(θ, p + p̄) = +∞, uniformly in θ ∈ V . Theorem ∃ H̄(z, p) continuous such that, ∀h(z, θ) periodic in θ, u ε converges uniformly on compacta of (0, +∞) × RN to u, unique solution of ut + H̄(z, Du) = 0, Martino Bardi (Università di Padova) u(0, z) = min h(z, θ) Multi-scale Bellman-Isaacs θ∈V Roma, June 15th, 2011 11 / 29 Homogenization on subspaces If H satisfies the assumption of the negative Proposition, must restrict the data l, h to oscillate only in directions where H does not vanish. Let V M–dimensional subspace of RN , zV := ΠV (z) projection z z z V V V utε + H , Dz u ε = l z, , u ε (0, z) = h z, ε ε ε N Assume coercivity in V : ∀ p̄ ∈ R lim |p|→+∞, p∈V H(θ, p + p̄) = +∞, uniformly in θ ∈ V . Theorem ∃ H̄(z, p) continuous such that, ∀h(z, θ) periodic in θ, u ε converges uniformly on compacta of (0, +∞) × RN to u, unique solution of ut + H̄(z, Du) = 0, Martino Bardi (Università di Padova) u(0, z) = min h(z, θ) Multi-scale Bellman-Isaacs θ∈V Roma, June 15th, 2011 11 / 29 Related results: Barles 2007, Viterbo 2008 (by symplectic methods). Example 1 utε + |uxε + γuyε | = l x, y , x−y ε u ε (0, x, y ) = h x, y , x−y ε x, y ∈ RN/2 there is homogenization ∀ l, h periodic in θ = x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 1 and 2 of non-homogenization; γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 12 / 29 Related results: Barles 2007, Viterbo 2008 (by symplectic methods). Example 1 utε + |uxε + γuyε | = l x, y , x−y ε u ε (0, x, y ) = h x, y , x−y ε x, y ∈ RN/2 there is homogenization ∀ l, h periodic in θ = x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 1 and 2 of non-homogenization; γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 12 / 29 Related results: Barles 2007, Viterbo 2008 (by symplectic methods). Example 1 utε + |uxε + γuyε | = l x, y , x−y ε u ε (0, x, y ) = h x, y , x−y ε x, y ∈ RN/2 there is homogenization ∀ l, h periodic in θ = x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 1 and 2 of non-homogenization; γ 6= 1 fits in the Thm. with V = {(q, −q) : q ∈ RN/2 }. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 12 / 29 Convex-concave eikonal equation |Dx u ε | − g2 x, y , x−y |Dy u ε | = l(x, y , x−y utε + g1 x, y , x−y ε ε ε ) u ε (0, x, y ) = h(x, y , x−y ε ) there is homogenization if (g1 − g2 )(x, y , θ) ≥ γo > 0 ∀ x, y , θ, or, symmetrically, g2 − g1 ≥ γo . Proof: by the Thm. with V = {(q, −q) : q ∈ RN/2 }. Motivation: Pursuit-Evasion type differential games, with pursuer controlling the x variables, evader the y variables, and cost depending on the distance between them. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 13 / 29 The result is sharp in the next Example 2 utε + |uxε | − γ|uyε | = l x, y , x−y ε there is homogenization ∀ l, h periodic in θ = x, y ∈ RN/2 x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 3 and 4 of non-homogenization; γ 6= 1 fits in the preceding slide. Other results by control-game methods we can treat problems with different assumptions: instead of strongly interacting fast variables, decoupling of xε and yε , e.g., h or l has a saddle point in this entries; many problems on homogenization of convex-concave H-J equations remain open! Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 14 / 29 The result is sharp in the next Example 2 utε + |uxε | − γ|uyε | = l x, y , x−y ε there is homogenization ∀ l, h periodic in θ = x, y ∈ RN/2 x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 3 and 4 of non-homogenization; γ 6= 1 fits in the preceding slide. Other results by control-game methods we can treat problems with different assumptions: instead of strongly interacting fast variables, decoupling of xε and yε , e.g., h or l has a saddle point in this entries; many problems on homogenization of convex-concave H-J equations remain open! Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 14 / 29 The result is sharp in the next Example 2 utε + |uxε | − γ|uyε | = l x, y , x−y ε there is homogenization ∀ l, h periodic in θ = x, y ∈ RN/2 x−y ε ⇐⇒ γ 6= 1. γ = 1 by the Examples 3 and 4 of non-homogenization; γ 6= 1 fits in the preceding slide. Other results by control-game methods we can treat problems with different assumptions: instead of strongly interacting fast variables, decoupling of xε and yε , e.g., h or l has a saddle point in this entries; many problems on homogenization of convex-concave H-J equations remain open! Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 14 / 29 2. Optimal control with stochastic volatility In Financial Mathematics the evolution of a stock S is described by d log Ss = γ ds + σ dWs and the classical Black-Scholes formula for the option pricing problem is derived assuming the parameters are constants. However the volatility σ is not really a constant, it rather looks like an ergodic mean-reverting stochastic process, so it is often modeled as σ(ys ) with ys an Ornstein-Uhlenbeck diffusion process. It is argued in the book Fouque, Papanicolaou, Sircar: Derivatives in financial markets with stochastic volatility, 2000, that the process ys also evolves on a faster time scale than the stock prices. Next picture shows the typical bursty behavior. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 15 / 29 Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 16 / 29 The model with fast stochastic volatility σ proposed in [FPS] is d log Ss = γ ds + σ(ys ) dWs dys = 1ε (m − ys ) + √ν d W̃s ε For the option pricing problem the limit is given by the Black-Scholes formula of the model with (constant) mean historical volatility d log Ss = γ ds + σ dWs , 2 2 e−(y −m) /2ν σ = σ (y ) √ dy , 2πν 2 R 2 Z 2 the mean being w.r.t. the (Gaussian) invariant measure of the Ornstein-Uhlenbeck process driving ys . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 17 / 29 Merton portfolio optimization problem Invest βs in the stock Ss , 1 − βs in a bond with interest rate r . Then the wealth xs evolves as d xs = (r + (γ − r )βs )xs ds + xs βs σ dWs and want to maximize the expected utility at time t, for some h increasing and concave. E[h(xt )], If h(x) = ax δ /δ with a > 0, 0 < δ < 1, a HARA function, and the parameters are constants the problem has an explicit solution. The version with fast stochastic volatility is d xs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs dys = 1ε (m − ys ) + √ν d W̃s ε QUESTIONS: 1. Is the limit as ε → 0 a Merton problem with constant volatility σ? 2. If so, is σ an average of σ(·) ? Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 18 / 29 Merton portfolio optimization problem Invest βs in the stock Ss , 1 − βs in a bond with interest rate r . Then the wealth xs evolves as d xs = (r + (γ − r )βs )xs ds + xs βs σ dWs and want to maximize the expected utility at time t, for some h increasing and concave. E[h(xt )], If h(x) = ax δ /δ with a > 0, 0 < δ < 1, a HARA function, and the parameters are constants the problem has an explicit solution. The version with fast stochastic volatility is d xs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs dys = 1ε (m − ys ) + √ν d W̃s ε QUESTIONS: 1. Is the limit as ε → 0 a Merton problem with constant volatility σ? 2. If so, is σ an average of σ(·) ? Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 18 / 29 By formal asymptotic expansions [FPS] argue that the answers are 1. Yes, the limit is a Merton problem with constant volatility σ 2. σ is a harmonic average of σ Z −1/2 1 , σ := dµ(y ) σ 2 (y ) NOT the linear average as in Black-Scholes! Practical consequence: if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes formula for option pricing I estimate the volatility by the arithmetic mean σa2 = N 1X 2 σi , N i=1 whereas in the Merton problem I use the harmonic mean !−1 N 1X 1 2 σh = ≤ σa2 . 2 N σ i=1 i Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 19 / 29 By formal asymptotic expansions [FPS] argue that the answers are 1. Yes, the limit is a Merton problem with constant volatility σ 2. σ is a harmonic average of σ Z −1/2 1 , σ := dµ(y ) σ 2 (y ) NOT the linear average as in Black-Scholes! Practical consequence: if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes formula for option pricing I estimate the volatility by the arithmetic mean σa2 = N 1X 2 σi , N i=1 whereas in the Merton problem I use the harmonic mean !−1 N 1X 1 2 σh = ≤ σa2 . 2 N σ i=1 i Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 19 / 29 By formal asymptotic expansions [FPS] argue that the answers are 1. Yes, the limit is a Merton problem with constant volatility σ 2. σ is a harmonic average of σ Z −1/2 1 , σ := dµ(y ) σ 2 (y ) NOT the linear average as in Black-Scholes! Practical consequence: if I have N empirical data σ1 , ..., σN of the volatility, in the Black-Scholes formula for option pricing I estimate the volatility by the arithmetic mean σa2 = N 1X 2 σi , N i=1 whereas in the Merton problem I use the harmonic mean !−1 N 1X 1 2 σh = ≤ σa2 . 2 N σ i=1 i Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 19 / 29 Conclusion: the correct average seems to depend on the problem! QUESTIONS: Why? Can the formula for the Merton problem be rigorously justified and what is the convergence as ε → 0? Is there a unified "formula" for the two problems and for other similar problems? Note: the Black-Scholes model has no control, the associated PDE is a heat-type linear equation; Merton is a stochastic control problem whose Hamilton-Jacobi-Bellman equation is fully nonlinear and degenerate parabolic. I’ll present a result on two-scale Hamilton-Jacobi-Bellman equations that will answer these (and other) questions. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 20 / 29 The H-J approach to Singular Perturbations Control system dxs = f (xs , ys , αs ) ds + σ(xs , ys , αs ) dWs , dys = 1ε g(xs , ys ) ds + x0 = x, √1 ν(xs , ys ) dWs , ε xs ∈ Rn , αs ∈ A, ys ∈ Rm , y0 = y Cost functional J ε (t, x, y , α) := Rt u ε (t, x, y ) := inf E [J ε (t, x, y , α)] 0 l(xs , ys , αs ) ds + h(xt , yt ) Value function α∈A(t) A(t) admisible controls on [0, t]. GOAL: let ε → 0 and find a simplified model. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 21 / 29 The H-J approach to Singular Perturbations Control system dxs = f (xs , ys , αs ) ds + σ(xs , ys , αs ) dWs , dys = 1ε g(xs , ys ) ds + x0 = x, √1 ν(xs , ys ) dWs , ε xs ∈ Rn , αs ∈ A, ys ∈ Rm , y0 = y Cost functional J ε (t, x, y , α) := Rt u ε (t, x, y ) := inf E [J ε (t, x, y , α)] 0 l(xs , ys , αs ) ds + h(xt , yt ) Value function α∈A(t) A(t) admisible controls on [0, t]. GOAL: let ε → 0 and find a simplified model. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 21 / 29 Assumptions on data: h continuous, supy h(x, y ) ≤ K (1 + |x|2 ) f , σ, g, τ, l Lipschitz in (x, y ) (unif. in α) with linear growth Then u ε solves the HJB equation ∂u ε 1 ε 2 ε 1 2 ε + H x, y , Dx u , Dxx u , √ Dxy u − Lu ε = 0 in R+ × Rn × Rm , ∂t ε ε n o H (x, y , p, M, Z ) := max −tr(σσ T M) − f · p − l − tr(σνZ T ) a∈A 2 L := tr(νν T Dyy ) + g · Dy . Theorem (Da Lio - Ley). The value function u ε is the unique viscosity solution with growth u ε (t, x, y ) ≤ C(1 + |x|2 + |y |2 ) of the Cauchy problem with initial condition u ε (0, x, y ) = h(x, y ). Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 22 / 29 Next assume on the linear operator L Ellipticity: Lyapunov: ∃ Λ(y ) > 0 s.t. ∀ x ν(x, y )ν T (x, y ) ≥ Λ(y )I ∃w ∈ C(Rm ), k > 0, R0 > 0 s.t. −Lw ≥ k for |y | > R0 , ∀x, w(y ) → +∞ as |y | → +∞. Then, the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen (FS) dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ , is uniquely ergodic, i.e., has a unique invariant measure µx . Liouville property: any bounded subsolution of −Lv = 0 is constant. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 23 / 29 Next assume on the linear operator L Ellipticity: Lyapunov: ∃ Λ(y ) > 0 s.t. ∀ x ν(x, y )ν T (x, y ) ≥ Λ(y )I ∃w ∈ C(Rm ), k > 0, R0 > 0 s.t. −Lw ≥ k for |y | > R0 , ∀x, w(y ) → +∞ as |y | → +∞. Then, the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen (FS) dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ , is uniquely ergodic, i.e., has a unique invariant measure µx . Liouville property: any bounded subsolution of −Lv = 0 is constant. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 23 / 29 Next assume on the linear operator L Ellipticity: Lyapunov: ∃ Λ(y ) > 0 s.t. ∀ x ν(x, y )ν T (x, y ) ≥ Λ(y )I ∃w ∈ C(Rm ), k > 0, R0 > 0 s.t. −Lw ≥ k for |y | > R0 , ∀x, w(y ) → +∞ as |y | → +∞. Then, the fast subsystem rescaled by τ = s/ε and with x ∈ Rn frozen (FS) dyτ = g(x, yτ ) dτ + ν(x, yτ ) dWτ , is uniquely ergodic, i.e., has a unique invariant measure µx . Liouville property: any bounded subsolution of −Lv = 0 is constant. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 23 / 29 A typical sufficient condition: ellipticity + recurrence condition i h lim sup sup g(x, y ) · y + tr(νν T (x, y )) < 0 |y |→+∞ x Proof by choosing as Lyapunov function w(y ) = |y |2 . Example: Ornstein-Uhlenbeck process dyτ = (m(x) − yτ )dτ + ν(x)dWτ , where m, ν are bounded. GOAL: let ε → 0 in the Cauchy problem for the HJB equation. Similarity with homogenization problems: the "fast" variable y plays the role of xε . Main difference with periodic homogenization: the state variables ys are unbounded and uncontrolled. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 24 / 29 A typical sufficient condition: ellipticity + recurrence condition i h lim sup sup g(x, y ) · y + tr(νν T (x, y )) < 0 |y |→+∞ x Proof by choosing as Lyapunov function w(y ) = |y |2 . Example: Ornstein-Uhlenbeck process dyτ = (m(x) − yτ )dτ + ν(x)dWτ , where m, ν are bounded. GOAL: let ε → 0 in the Cauchy problem for the HJB equation. Similarity with homogenization problems: the "fast" variable y plays the role of xε . Main difference with periodic homogenization: the state variables ys are unbounded and uncontrolled. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 24 / 29 Convergence Theorem Under the previous assumptions, the (weak) limit u(t, x) as ε → 0 of the value function u ε (t, x, y ) solves R ∂u 2 n ∂t + H x, y , Dx u, Dxx u, 0 dµx (y ) = 0 in R+ × R (CP) R u(0, x) = h(x, y ) dµx (y ) If, moreover, either g, ν do not depend on x or g(x, ·), ν(x, ·) ∈ C 1 and g(·, y ), ν(·, y ) ∈ Cb1 Dx g, Dy g, Dx ν, Dy ν are Hölder in y uniformly w.r.t. x, then u is the unique viscosity solution (with quadratic growth) of (CP) and u ε (t, x, y ) → u(t, x) locally uniformly on (0, +∞) × Rn . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 25 / 29 Examples Denote hφi := R φ(y )dµx (y ). Got simple formulas for effective H and h H(x, p, M) = hH(x, ·, p, M, 0)i, h(x) = hh(x, ·)i. Corollary [see also Kushner, book 1990] For split systems, i.e., σ = σ(x, y ), f = f0 (x, y ) + f1 (x, a), l = l0 (x, y ) + l1 (x, a), the linear averaging of the data is the correct limit, i.e., Z t ε lim u (t, x, y ) = u(t, x) := inf E hli(xs , αs ) ds + hhi(xt ) , α. ε→0 0 dxs = hf i(xs , αs ) ds + hσσ T i1/2 (xs ) dWs Proof: H(x, p, M) = −trace(Mhσσ T i)/2 + maxA {−hf i · p − hli} . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 26 / 29 Examples Denote hφi := R φ(y )dµx (y ). Got simple formulas for effective H and h H(x, p, M) = hH(x, ·, p, M, 0)i, h(x) = hh(x, ·)i. Corollary [see also Kushner, book 1990] For split systems, i.e., σ = σ(x, y ), f = f0 (x, y ) + f1 (x, a), l = l0 (x, y ) + l1 (x, a), the linear averaging of the data is the correct limit, i.e., Z t ε lim u (t, x, y ) = u(t, x) := inf E hli(xs , αs ) ds + hhi(xt ) , α. ε→0 0 dxs = hf i(xs , αs ) ds + hσσ T i1/2 (xs ) dWs Proof: H(x, p, M) = −trace(Mhσσ T i)/2 + maxA {−hf i · p − hli} . Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 26 / 29 Example: we recover the Black-Scholes formula with stochastic volatility. In general, for system or cost NOT split, H(x, p, M) = h max{...}i > max h{...}i A A and the limit control problem is not obvious. We try to write H as a Bellman Hamiltonian in some other way in order to find an explicit effective control problem approximating the singularly perturbed one as ε → 0. We did it for Merton problem with stochastic volatility (see next slides), Ramsey model of optimal economic growth with (fast) random parameters, Vidale - Wolfe advertising model with random parameters, advertising game in a duopoly with Lanchester dynamics and random parameters. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 27 / 29 Merton portfolio optimization with stochastic volatility Given a control βs the wealth xs evolves as dxs = (r + (γ − r )βs )xs ds + xs βs σ(ys ) dWs dys = 1ε (m − ys ) ds + and the value functions is √ν d W̃s ε x0 = x y0 = y V ε (t, x, y ) := supβ. E[h(xt )]. Let ρ = correlation of Ws and W̃s . The HJB equation is ∂V ε b2 x 2 σ2 ε bxρσν ε − rxVxε − max (γ − r )bxVxε + Vxx + √ Vxy ∂t 2 ε b ε (m − y )Vyε + ν 2 Vyy = ε Assume the utility h has h0 > 0 and h00 < 0. Then expect a value ε < 0. function strictly increasing and concave in x, i.e., Vxε > 0, Vxx Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 28 / 29 The HJB equation becomes ε √ V ε ]2 [(γ − r )Vxε + xρσν (m − y )Vyε + ν 2 Vyy xy ∂V ε ε − rxVxε + = ε ∂t ε σ 2 (y )2Vxx By the Theorem, V ε (t, x, y ) → V (t, x) as ε → 0 and V solves (γ − r )2 Vx2 ∂V − rxVx + ∂t 2Vxx Z 1 σ 2 (y ) dµ(y ) = 0 in R+ × R+ This is the HJB equation of a Merton problem with the harmonic average of σ as constant volatility Z σ := −1/2 1 dµ(y ) . σ 2 (y ) Therefore this is the limit control problem. The limit of the optimal control βsε,∗ as ε → 0 can also be studied. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 29 / 29 The HJB equation becomes ε √ V ε ]2 [(γ − r )Vxε + xρσν (m − y )Vyε + ν 2 Vyy xy ∂V ε ε − rxVxε + = ε ∂t ε σ 2 (y )2Vxx By the Theorem, V ε (t, x, y ) → V (t, x) as ε → 0 and V solves (γ − r )2 Vx2 ∂V − rxVx + ∂t 2Vxx Z 1 σ 2 (y ) dµ(y ) = 0 in R+ × R+ This is the HJB equation of a Merton problem with the harmonic average of σ as constant volatility Z σ := −1/2 1 dµ(y ) . σ 2 (y ) Therefore this is the limit control problem. The limit of the optimal control βsε,∗ as ε → 0 can also be studied. Martino Bardi (Università di Padova) Multi-scale Bellman-Isaacs Roma, June 15th, 2011 29 / 29

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# Homogenization and other multi-scale problems in Bellman