Mean-field SDE driven by a fractional BM. A related stochastic control problem
|
|
- Augustus Burke
- 5 years ago
- Views:
Transcription
1 Mean-field SDE driven by a fractional BM. A related stochastic control problem Rainer Buckdahn, Université de Bretagne Occidentale, Brest Durham Symposium on Stochastic Analysis, July 1th to July 2th, 217
2 Outline Introduction Preliminaries Mean-field SDE driven by a fbm The mean-field control problem
3 Introduction Our objective: Study of the control problem with the dynamics driven by a fbm B H with Hurst parameter H (1/2, 1) X u t = x + t σ(p X u s )db H s + t b(p (X u s,u s), X u s, u s )ds, where x R, u U(, T ) - adapted control process with values in a convex open set U R m, and with the cost functional: J(u) = E [ f ( P (X u t,u t), Xt u ) ( ) ], u t dt + g X u T, P X u ; T Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 3/25
4 characterisation of an optimal control u U(, T ) s.t. J takes its minimum over U(, T ) at u. Short state-of-art: Several recent works on Pontryagin s maximum principle for mean-field control problems driven by a BM: + Carmona, Delarue (215),..., + Buckdahn, Li, Ma (Pontryagin s principle for a mean-field control problem with partial observation, 216); Several recent works on Peng s maximum principle (using spike control) for mean-field control problems driven by a BM: + Buckdahn, Djehiche, Li (211), + Buckdahn, Li, Ma (216); The fbm is not a semimartingale which makes the analysis much more subtle; much less works on maximum principle for control problems driven by a fbm: + Biagini, Hu, Oksendal, Sulem (22), Hu, Zhou (linear control system, Riccati equation is a linear BSDE driven by a BM and a fbm, 25), Han, Hu, Song Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 4/25
5 (213) (Pontryagin s maximum principle with regularity assumptions -as Malliavin differentiability- on the optimal control process; adjoint equation is a linear BSDE driven by a BM and a fbm); + Buckdahn, Jing: Peng s maximum principle for control problem driven by a BM and a fbm (214). With Shuai Jing we try to avoid regularity conditions on the optimal control. Preliminaries (Ω, F, P ) complete probability space; 1.Fractional BM. A minimalist overview: fbm with Hurst parameter H (, 1): centered Gaussian process B H with cov(b H t, B H s ) = 1 2 (t2h + s 2H t s 2H ), s, t. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 5/25
6 For H (1/2, 1), BM W s.t. B H t = t K H (t, s)dw s, t [, T ], with kernel t K H (t, s) = c H s 1/2 H (u s) H 3/2 u H 1/2 du, t > s, s with constant c H = [H(2H 1)/β(2 2H, H 1/2)] 1/2, where β(α, γ) = Γ(α + γ)/(γ(α)γ(γ)) (Beta function), Γ(α) = x α 1 e x dx (Gamma function). Stochastic integral w.r.t. fbm: Let f L 2 ([, T ] Ω); f Dom(δ H ), if δ H (f) L 2 (Ω, F, P ) s.t., for all G D H 1,2, If fi [s,t] Dom(δ H ), Fractional calculus: E [Gδ H (f)] = t s E [ f(t)d H t G ] dt. f(r)db H r := δ H (fi [s,t] ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 6/25
7 Malliavin derivatives. Dt H instead of a BM; D H s G = - classical Malliavin derivative but now w.r.t. B H H(2H 1) s r 2H 2 D H r Gdr = (K H K HD. H G)(s), G S, { where K H is an operator on H := ( t K H(t, s) ˆf(s)ds) t [,T ], ˆf } L 2 ([, T ]) defined by (K H ψ)(s) = c H Γ(H 1/2)s 1/2 H I H 1/2 + (u H 1/2 ψ(u))(s) x with I+(f)(x) α = 1 f(u) Γ(α) (x u) 1 α du, for almost all x [, T ], and K H is its adjoint operator on H. Important: for all ψ H, ψ(t)db H (t)= (K Hψ)(t)dW (t), ψ(t)dw (t)= (K H 1 ψ)(t)db H (t). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 7/25
8 Therefore, F := F W = F BH. L 2 -norm of stochastic integral: For all f Dom(δ H ), E [ f(t)dbt H 2] = E [ KHf(t) 2 dt ] + 2E [ s D H s f(r)d H r f(s)drds ]. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 8/25
9 Mean-field SDE driven by a fbm Given ξ L 2 (Ω, F, P ), Θ L 2 ([, T ] Ω; R m ), we consider the SDE: (1) X t = ξ + t ( ) t γs X s + σ(s, P (Xs,Θ s) db H s + b(s, P (Xs,Θ s), X s )ds, where σ : [, T ] P 2 (R R m ) R and b : Ω [, T ] P 2 (R R m ) R R satisfy the following conditions: (H1) For any s [, T ], x, x R, η, η L 2 (Ω, F, P ) and Θ L 2 (Ω, F, P ; R m ), there exists a constant C > such that σ(s, P (η,θ) ) C, b(s, P (η,θ), x) C(1 + W 2 (P (η,θ), P (,Θ) ) + x ), σ(s, P (η,θ) ) σ(s, P (η,θ)) W 2 (P (η,θ), P (η,θ)), b(s, P (η,θ), x) b(s, P (η,θ), x ) C(W 2 (P (η,θ), P (η,θ)) + x x ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 9/25
10 The SDE will be solved using Girsanov transformation: For γ L ([, T ]) H, T t (ω) = ω + A t (ω) = ω t t K H(γI [,t] )(s)ds, K H(γI [,t] )(s)ds, t [, T ], ω Ω, Clearly, A t T t (ω) = T t A t (ω) = ω. Moreover, for any F S, from the Girsanov theorem E[F ] = E[F (T t )ε 1 t (T t )] = E[F (A t )ε t ], where ε t { t = exp γ s dbs H 1 t ( K 2 H (γi [,t] ) ) } 2 (s)ds { t = exp KH(γI [,t] )(s)dw s 1 t ( K 2 H (γi [,t] ) ) } 2 (s)ds, Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 1/25
11 and[ E sup ε p t t [,T ] ] < + and E sup t [,T ] [ sup ε p t (T t ) t [,T ] ] < +, for all p R. Let L 2, ([, T ]; R) be the Banach space of F-adapted processes {ϕ(t), t [, T ]} such that E [ ϕ(t) 2 ε 1 ] t < +. Theorem. The above SDE (1) has a unique solution X L 2, ([, T ]; R). Remark. Proof is based on the following statement with rather technical proof: Proposition. 1) X L 2, ([, T ]; R) is a solution of our SDE iff it solves equ. (2) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 11/25
12 = ξ + X t (T t )ε 1 t (T t ) t σ(s, P (Xs,Θ s))ε 1 s (T s )dbs H + t b(s, T s, P (Xs,Θ s), X s (T s ))ε 1 (T s )ds. 2) For all deterministic Θ L ([, T ]), ( Θ s ε 1 s (T s ) ) s [,T ] Dom(δ H). 3) The above SDE (2) has a unique solution X L 2, ([, T ]; R) Proof. Main tools: + The method of Girsanov transformation; + The formula for E [ f(t)dbh t 2] (involving f and its Malliavin derivative Df); + Picard s iteration with the distance function (E[ X t X t 2 ε 1 t ]) 1/2. s Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 12/25
13 The mean-field control problem Recall: B H fbm with Hurst parameter H (1/2, 1); control state space U R m nonempty, convex, bounded; U([, T ]) := L F ([, T ]; U). Given x R, u U([, T ]) we consider the control problem with dynamics (3) X u t = x + t with the cost functional (4) J(u) = E [ σ(p X u s )db H s + t b(p (X u s,u s), X u s, u s )ds, f ( P (X u t,u t), Xt u ) ( ) ], u t dt + g X u T, P X u. T Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 13/25
14 Assumptions: (H2) σ : P 2 (R) R, b : P 2 (R U) R U R, g : R P 2 (R) R and f : P 2 (R U) R U U are bounded and Lipschitz (in all variables); (H3) σ, b, f and g are C 1 in (x, µ, u) (with µ P 2 (R) and P 2 (R U), respectively), and all first order derivatives are bounded and Lipschitz. This means, e.g., that, for some C R, for any (µ, y), (µ, y ) P 2 (R) R, µ σ(µ, y) µ σ(µ, y ) C(W 2 (µ, µ ) + y y ). Under the above assumptions we have seen the existence and the uniqueness for every u U([, T ]). Assume that there is a u U([, T ]) minimising J : U([, T ]) R; X := X u. Objective: Characterisation of u by convex perturbation: For u U([, T ]), ε, put u ε := u + ε(u u ) U([, T ]), X ε := X uε. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 14/25
15 Lemma The following SDE obtained by formal differentiation of (3) for u = u ε w.r.t. ε at ε =, for t [, T ], Y t = [ ( t Ẽ µ σ P X s, X ) ] s Ỹs dbs H + [ ( t Ẽ µ b P (X s,u s ), Xs, u s, X ) ) ] s, ũ s, (Ỹs, ũ s ũ s ds + t xb ( ) P (X s,u s ), Xs, u s Ys ds + t ub ( P (X s,u s ), Xs, us) (us u s)ds, has a unique solution Y L 2 F ([, [ T ]). Moreover, lim sup E ε Y t Xε t Xt 2] =. ε t [,T ] Above, ( X, Ỹ, ũ, ũ ) is an independent copy of (X, Y, u, u ) defined on a probability space ( Ω, F, P ). The expectation Ẽ[ ] under P only concerns ( X, Ỹ, ũ, ũ ) but not (X, Y, u, u ). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 15/25
16 As (X, u ) is optimal, J(u ε ) J(u ), ε [, 1]. Thus, d dε J(uε ) 1 := lim ε= <ε ε (J(uε ) J(u )), and d dε J(uε ) = E [ x g(xt, P X )Y ] [ T T + E [Ẽ µ g(xt, P X, X ]] T T )ỸT ε= +E [ x f(p (X t,u t ), Xt, u t )Y t dt ] +E [ [ Ẽ ( µ f) 1 (P (X t,u t ), Xt, u t, X ] t, ũ t )Ỹt dt ] +E [ [ Ẽ ( µ f) 2 (P (X t,u t ), Xt, u t, X ] t, ũ t )(ũ t ũ t ) dt ] +E [ u f(p (X t,u t ), Xt, u t )(u t u t )dt ]. (5) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 16/25
17 We introduce the adjoint BSDE with its F-adpated solution (P, β): P t = P T α s ds β s dw s, t [, T ], t t P T = x g(xt, P X [ ) + Ẽ T µ g( X ] T, P X T, X T ), where α L 2 F ([, T ]) will be specified later. Applying Itô s formula (with BH ) taking the expectation and rearranging terms, we get E [ ] Y T P T = E [ {Ẽ[ ( Y s Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs )] + Ps x b ( P (X s,u s ), Xs, u ) s +α s + µ σ ( P X s, Xs ) [ ]}] E D H s P s ds + E [( Ẽ [ ( Ps ( µ b) 2 P(X s,u s ), X s, ũ s, u )] s +P s u b ( P (X s,u s ), X s, u s) ) (u s u s) ] ds. (6) Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 17/25
18 Combining (5) with (6) we obtain E [ {Ẽ[ ( Y s Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs )] +Ps x b ( P (X s,u s ), Xs, u ) s + + µ σ ( P X s, Xs ) [ ] E D H s P s + x f(p (X s,u s ), Xs, u s) + α s +Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, Xs, u s) ]}] ds E [ (u s u s) { Ẽ [ ( µ f) 2 (P (X s,u s ), X s, ũ s, X s, u s) ] (7) + u f(p (X s,u s ), X s, u s) +Ẽ[ Ps ( µ b) 2 ( P(X s,u s ), X s, ũ s, X s, u s)] +Ps u b ( P (X s,u s ), X s, u s)}] ds. Putting the first integral equal to zero, we choose α s = Ẽ[ ( Ps ( µ b) 1 P(X s,u s ), X )] s, ũ s, Xs Ps x b ( ) P (X s,u s ), Xs, u s µ σ ( ) [ ] P X s, Xs E D H s P s x f(p (X s,u s ), Xs, u s) Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, Xs, u s) ]. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 18/25
19 This gives the following form of the BSDE for P t = P T t α s ds t β s dw s : { P t = P T + Ẽ [ ( Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs, u )] s t +P s x b ( P (X s,u s ), Xs, u ) s + µ σ ( P X s, Xs ) [ ] E D H s P s + x f(p (X s,u s ), Xs, u s) +Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, X s, u s) ]} ds t (8) β s dw s, which is a mean-field BSDE driven by the standard Brownian motion W. Let us assume that the above BSDE has an F-adapted solution (P, β) (Discussion on existence and unique for the BSDE later). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 19/25
20 Then (7) becomes E [ (u s u s) { Ẽ [ ( µ f) 2 (P (X s,u s ), X s, ũ s, X s, u s) ] (9) + u f(p (X s,u s ), X s, u s) +Ẽ[ Ps ( µ b) 2 ( P(X s,u s ), X s, ũ s, X s, u s)] +Ps u b ( P (X s,u s ), X s, u s)}] ds. As U is open and u U([, T ]) arbitrary, we have = Ẽ[ ( µ f) 2 (P (X t,u t ), X t, ũ t, Xt, u t ) ] + u f(p (X t,u t ), Xt, u t ) +Ẽ[ ( Pt ( µ b) 2 P(X t,u t ), X )] t, ũ t, Xt, u t + Pt u b ( ) P (X t,u t ), Xt, u t, dp -a.s., dt-a.e. Hence, we have following necessary conditions of Pontryagin-type. Theorem. If (X, u ) is an optimal pair of our control problem, then (X, u ) satisfies the following system: Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 2/25
21 X t = x + P T = P t = t σ(p X s )db H s + t b(p (X s,u s ), X s, u s)ds, (1) x g(xt, P X ) + Ẽ[ T µ g( X T, P X, X T T ) ], { P T β s dw s + Ẽ [ ( Ps ( µ b) 1 P(X s,u s ), X s, ũ s, Xs, u )] s t t +P s x b ( P (X s,u s ), Xs, u s) + µ σ ( P X, ) [ ] X s s E D H s P s + x f(p (X s,u s ), X s, u s) + Ẽ[ ( µ f) 1 (P (X s,u s ), X s, ũ s, X s, u s) ]} ds, = Ẽ[ ( Pt ( µ b) 2 P(X t,u t ), X t, ũ t, Xt, u )] t + Pt u b ( P (X t,u t ), Xt, u ) t +Ẽ[ ( µ f) 2 (P (X t,u t ), X t, ũ t, Xt, u t ) ] + u f(p (X t,u t ), Xt, u t ), dp -a.s., dt-a.e. After this necessary condition we can give also a sufficient one: Theorem. In addition to our standard assumptions, assume that g = g(x, µ) is Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 21/25
22 convex in (x, µ) and the Hamiltonian H(µ, x, u, y, z) := f(µ, x, u) + b(µ, x, u)y + σ(µ)z be jointly convex in (µ, x, u). Let (u, X ) satisfy the above system. Then (u, X ) is optimal: J(u ) = inf u U([,T ]) J(u). Remark. The convexity assumption on the Hamiltonian implies practically the linearity of b and σ. Another assumption in which b and σ don t need to be linear: + g : R P 2 (R) R jointly convex in (x, µ), with x g, µ g ; + b(µ, x, u) : P 2 (R U) R U R is jointly convex in (µ, x, u) with ( µ b) 1 (µ, x, u, y), strictly convex in (µ, u); + f(µ, x, u) : P 2 (R U) R U R is jointly convex in (µ, x, u), and strictly convex in (µ, u), with ( µ f) 1 (η, x, u, y), x f(η, x, u, y) ; + σ(µ) σ R. Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 22/25
23 Recall (Carmona, Delarue, 215): g convex (strictly convex), if there exists exists λ (resp., > ) such that for all x, x R, ξ, ξ L 2 (Ω, F, P ), g(x, P ξ ) g(x, P ξ ) x g(x, P ξ )(x x) + E [ µ g(x, P ξ, ξ)(ξ ξ) ] + λ ( x x 2 + E[ ξ ξ 2 ] ). Theorem. If in addition to our standard assumptions the above assumption is satisfied and (u, X ) solves the above system, then (u, X ) is optimal: J(u ) = inf u U([,T ]) J(u). Concerning the solvability of the coupled forward-backward system (1) under the conditions of the preceding Theorem, we proceed as follows: For any given (P, ξ) L 2 (F t ) L 2 (F t ), we suppose that there is some η L 2 (F t ; U) such that: Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 23/25
24 = Ẽ[ ( µ f) 2 ( P(ξ,η), ξ, η, ξ, η )] + u f(p (ξ,η), ξ, η) +Ẽ[ P ( µ b) 2 ( P(ξ,η), ξ, η, ξ, η )] + P ( u b)(p (ξ,η), ξ, η). Lemma. The mapping (P, ξ) η is unique and η = η(p, ξ) is Lipschitz in (P, ξ) under L 2 -norm. Theorem. Under the assumptions of the preceding theorem (i.e., in particular σ(µ) = σ R) and the preceding assumption, then, if the time horizon T > is sufficiently small, there exists a unique solution (X, (P, β), u = η(x, P )) of the coupled FBSDE (1). Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 24/25
25 THANK YOU VERY MUCH! Rainer Buckdahn, Université de Bretagne Occidentale, Brest France 25/25
Weak solutions of mean-field stochastic differential equations
Weak solutions of mean-field stochastic differential equations Juan Li School of Mathematics and Statistics, Shandong University (Weihai), Weihai 26429, China. Email: juanli@sdu.edu.cn Based on joint works
More informationNonlinear representation, backward SDEs, and application to the Principal-Agent problem
Nonlinear representation, backward SDEs, and application to the Principal-Agent problem Ecole Polytechnique, France April 4, 218 Outline The Principal-Agent problem Formulation 1 The Principal-Agent problem
More informationOn continuous time contract theory
Ecole Polytechnique, France Journée de rentrée du CMAP, 3 octobre, 218 Outline 1 2 Semimartingale measures on the canonical space Random horizon 2nd order backward SDEs (Static) Principal-Agent Problem
More informationLAN property for sde s with additive fractional noise and continuous time observation
LAN property for sde s with additive fractional noise and continuous time observation Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Samy Tindel (Purdue University) Vlad s 6th birthday,
More informationLarge Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm
Large Deviations Principles for McKean-Vlasov SDEs, Skeletons, Supports and the law of iterated logarithm Gonçalo dos Reis University of Edinburgh (UK) & CMA/FCT/UNL (PT) jointly with: W. Salkeld, U. of
More informationThe stochastic heat equation with a fractional-colored noise: existence of solution
The stochastic heat equation with a fractional-colored noise: existence of solution Raluca Balan (Ottawa) Ciprian Tudor (Paris 1) June 11-12, 27 aluca Balan (Ottawa), Ciprian Tudor (Paris 1) Stochastic
More informationAn adaptive numerical scheme for fractional differential equations with explosions
An adaptive numerical scheme for fractional differential equations with explosions Johanna Garzón Departamento de Matemáticas, Universidad Nacional de Colombia Seminario de procesos estocásticos Jointly
More informationLECTURES ON MEAN FIELD GAMES: II. CALCULUS OVER WASSERSTEIN SPACE, CONTROL OF MCKEAN-VLASOV DYNAMICS, AND THE MASTER EQUATION
LECTURES ON MEAN FIELD GAMES: II. CALCULUS OVER WASSERSTEIN SPACE, CONTROL OF MCKEAN-VLASOV DYNAMICS, AND THE MASTER EQUATION René Carmona Department of Operations Research & Financial Engineering PACM
More informationStrong Solutions and a Bismut-Elworthy-Li Formula for Mean-F. Formula for Mean-Field SDE s with Irregular Drift
Strong Solutions and a Bismut-Elworthy-Li Formula for Mean-Field SDE s with Irregular Drift Thilo Meyer-Brandis University of Munich joint with M. Bauer, University of Munich Conference for the 1th Anniversary
More informationInformation and Credit Risk
Information and Credit Risk M. L. Bedini Université de Bretagne Occidentale, Brest - Friedrich Schiller Universität, Jena Jena, March 2011 M. L. Bedini (Université de Bretagne Occidentale, Brest Information
More informationA Concise Course on Stochastic Partial Differential Equations
A Concise Course on Stochastic Partial Differential Equations Michael Röckner Reference: C. Prevot, M. Röckner: Springer LN in Math. 1905, Berlin (2007) And see the references therein for the original
More informationDifferential Games II. Marc Quincampoix Université de Bretagne Occidentale ( Brest-France) SADCO, London, September 2011
Differential Games II Marc Quincampoix Université de Bretagne Occidentale ( Brest-France) SADCO, London, September 2011 Contents 1. I Introduction: A Pursuit Game and Isaacs Theory 2. II Strategies 3.
More informationIntegral representations in models with long memory
Integral representations in models with long memory Georgiy Shevchenko, Yuliya Mishura, Esko Valkeila, Lauri Viitasaari, Taras Shalaiko Taras Shevchenko National University of Kyiv 29 September 215, Ulm
More informationBackward Stochastic Differential Equations with Infinite Time Horizon
Backward Stochastic Differential Equations with Infinite Time Horizon Holger Metzler PhD advisor: Prof. G. Tessitore Università di Milano-Bicocca Spring School Stochastic Control in Finance Roscoff, March
More informationRough paths methods 4: Application to fbm
Rough paths methods 4: Application to fbm Samy Tindel Purdue University University of Aarhus 2016 Samy T. (Purdue) Rough Paths 4 Aarhus 2016 1 / 67 Outline 1 Main result 2 Construction of the Levy area:
More informationSolvability of G-SDE with Integral-Lipschitz Coefficients
Solvability of G-SDE with Integral-Lipschitz Coefficients Yiqing LIN joint work with Xuepeng Bai IRMAR, Université de Rennes 1, FRANCE http://perso.univ-rennes1.fr/yiqing.lin/ ITN Marie Curie Workshop
More informationAn Introduction to Malliavin calculus and its applications
An Introduction to Malliavin calculus and its applications Lecture 3: Clark-Ocone formula David Nualart Department of Mathematics Kansas University University of Wyoming Summer School 214 David Nualart
More informationp 1 ( Y p dp) 1/p ( X p dp) 1 1 p
Doob s inequality Let X(t) be a right continuous submartingale with respect to F(t), t 1 P(sup s t X(s) λ) 1 λ {sup s t X(s) λ} X + (t)dp 2 For 1 < p
More informationThe Wiener Itô Chaos Expansion
1 The Wiener Itô Chaos Expansion The celebrated Wiener Itô chaos expansion is fundamental in stochastic analysis. In particular, it plays a crucial role in the Malliavin calculus as it is presented in
More informationA Class of Fractional Stochastic Differential Equations
Vietnam Journal of Mathematics 36:38) 71 79 Vietnam Journal of MATHEMATICS VAST 8 A Class of Fractional Stochastic Differential Equations Nguyen Tien Dung Department of Mathematics, Vietnam National University,
More informationBernardo D Auria Stochastic Processes /12. Notes. March 29 th, 2012
1 Stochastic Calculus Notes March 9 th, 1 In 19, Bachelier proposed for the Paris stock exchange a model for the fluctuations affecting the price X(t) of an asset that was given by the Brownian motion.
More informationApproximation of BSDEs using least-squares regression and Malliavin weights
Approximation of BSDEs using least-squares regression and Malliavin weights Plamen Turkedjiev (turkedji@math.hu-berlin.de) 3rd July, 2012 Joint work with Prof. Emmanuel Gobet (E cole Polytechnique) Plamen
More informationMalliavin Calculus in Finance
Malliavin Calculus in Finance Peter K. Friz 1 Greeks and the logarithmic derivative trick Model an underlying assent by a Markov process with values in R m with dynamics described by the SDE dx t = b(x
More informationTopics in fractional Brownian motion
Topics in fractional Brownian motion Esko Valkeila Spring School, Jena 25.3. 2011 We plan to discuss the following items during these lectures: Fractional Brownian motion and its properties. Topics in
More informationContinuous Time Finance
Continuous Time Finance Lisbon 2013 Tomas Björk Stockholm School of Economics Tomas Björk, 2013 Contents Stochastic Calculus (Ch 4-5). Black-Scholes (Ch 6-7. Completeness and hedging (Ch 8-9. The martingale
More informationStochastic Differential Equations.
Chapter 3 Stochastic Differential Equations. 3.1 Existence and Uniqueness. One of the ways of constructing a Diffusion process is to solve the stochastic differential equation dx(t) = σ(t, x(t)) dβ(t)
More informationWeak Convergence of Numerical Methods for Dynamical Systems and Optimal Control, and a relation with Large Deviations for Stochastic Equations
Weak Convergence of Numerical Methods for Dynamical Systems and, and a relation with Large Deviations for Stochastic Equations Mattias Sandberg KTH CSC 2010-10-21 Outline The error representation for weak
More informationBSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, March 2009
BSDEs and PDEs with discontinuous coecients Applications to homogenization K. Bahlali, A. Elouain, E. Pardoux. Jena, 16-20 March 2009 1 1) L p viscosity solution to 2nd order semilinear parabolic PDEs
More informationOn the Multi-Dimensional Controller and Stopper Games
On the Multi-Dimensional Controller and Stopper Games Joint work with Yu-Jui Huang University of Michigan, Ann Arbor June 7, 2012 Outline Introduction 1 Introduction 2 3 4 5 Consider a zero-sum controller-and-stopper
More informationSome Properties of NSFDEs
Chenggui Yuan (Swansea University) Some Properties of NSFDEs 1 / 41 Some Properties of NSFDEs Chenggui Yuan Swansea University Chenggui Yuan (Swansea University) Some Properties of NSFDEs 2 / 41 Outline
More informationIntroduction to numerical simulations for Stochastic ODEs
Introduction to numerical simulations for Stochastic ODEs Xingye Kan Illinois Institute of Technology Department of Applied Mathematics Chicago, IL 60616 August 9, 2010 Outline 1 Preliminaries 2 Numerical
More informationMalliavin calculus and central limit theorems
Malliavin calculus and central limit theorems David Nualart Department of Mathematics Kansas University Seminar on Stochastic Processes 2017 University of Virginia March 8-11 2017 David Nualart (Kansas
More informationBernardo D Auria Stochastic Processes /10. Notes. Abril 13 th, 2010
1 Stochastic Calculus Notes Abril 13 th, 1 As we have seen in previous lessons, the stochastic integral with respect to the Brownian motion shows a behavior different from the classical Riemann-Stieltjes
More informationELEMENTS OF STOCHASTIC CALCULUS VIA REGULARISATION. A la mémoire de Paul-André Meyer
ELEMENTS OF STOCHASTIC CALCULUS VIA REGULARISATION A la mémoire de Paul-André Meyer Francesco Russo (1 and Pierre Vallois (2 (1 Université Paris 13 Institut Galilée, Mathématiques 99 avenue J.B. Clément
More informationHJB equations. Seminar in Stochastic Modelling in Economics and Finance January 10, 2011
Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics, Charles University in Prague petrasek@karlin.mff.cuni.cz Seminar in Stochastic Modelling in Economics and Finance
More informationA MODEL FOR THE LONG-TERM OPTIMAL CAPACITY LEVEL OF AN INVESTMENT PROJECT
A MODEL FOR HE LONG-ERM OPIMAL CAPACIY LEVEL OF AN INVESMEN PROJEC ARNE LØKKA AND MIHAIL ZERVOS Abstract. We consider an investment project that produces a single commodity. he project s operation yields
More informationON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WITH MONOTONE DRIFT
Elect. Comm. in Probab. 8 23122 134 ELECTRONIC COMMUNICATIONS in PROBABILITY ON A SDE DRIVEN BY A FRACTIONAL BROWNIAN MOTION AND WIT MONOTONE DRIFT BRAIM BOUFOUSSI 1 Cadi Ayyad University FSSM, Department
More informationStochastic optimal control with rough paths
Stochastic optimal control with rough paths Paul Gassiat TU Berlin Stochastic processes and their statistics in Finance, Okinawa, October 28, 2013 Joint work with Joscha Diehl and Peter Friz Introduction
More informationTheoretical Tutorial Session 2
1 / 36 Theoretical Tutorial Session 2 Xiaoming Song Department of Mathematics Drexel University July 27, 216 Outline 2 / 36 Itô s formula Martingale representation theorem Stochastic differential equations
More informationHarnack Inequalities and Applications for Stochastic Equations
p. 1/32 Harnack Inequalities and Applications for Stochastic Equations PhD Thesis Defense Shun-Xiang Ouyang Under the Supervision of Prof. Michael Röckner & Prof. Feng-Yu Wang March 6, 29 p. 2/32 Outline
More informationThe multidimensional Ito Integral and the multidimensional Ito Formula. Eric Mu ller June 1, 2015 Seminar on Stochastic Geometry and its applications
The multidimensional Ito Integral and the multidimensional Ito Formula Eric Mu ller June 1, 215 Seminar on Stochastic Geometry and its applications page 2 Seminar on Stochastic Geometry and its applications
More informationRough Burgers-like equations with multiplicative noise
Rough Burgers-like equations with multiplicative noise Martin Hairer Hendrik Weber Mathematics Institute University of Warwick Bielefeld, 3.11.21 Burgers-like equation Aim: Existence/Uniqueness for du
More informationTools of stochastic calculus
slides for the course Interest rate theory, University of Ljubljana, 212-13/I, part III József Gáll University of Debrecen Nov. 212 Jan. 213, Ljubljana Itô integral, summary of main facts Notations, basic
More information1. Stochastic Processes and filtrations
1. Stochastic Processes and 1. Stoch. pr., A stochastic process (X t ) t T is a collection of random variables on (Ω, F) with values in a measurable space (S, S), i.e., for all t, In our case X t : Ω S
More informationRepresentations of Gaussian measures that are equivalent to Wiener measure
Representations of Gaussian measures that are equivalent to Wiener measure Patrick Cheridito Departement für Mathematik, ETHZ, 89 Zürich, Switzerland. E-mail: dito@math.ethz.ch Summary. We summarize results
More informationJoint Parameter Estimation of the Ornstein-Uhlenbeck SDE driven by Fractional Brownian Motion
Joint Parameter Estimation of the Ornstein-Uhlenbeck SDE driven by Fractional Brownian Motion Luis Barboza October 23, 2012 Department of Statistics, Purdue University () Probability Seminar 1 / 59 Introduction
More informationPathwise volatility in a long-memory pricing model: estimation and asymptotic behavior
Pathwise volatility in a long-memory pricing model: estimation and asymptotic behavior Ehsan Azmoodeh University of Vaasa Finland 7th General AMaMeF and Swissquote Conference September 7 1, 215 Outline
More informationON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME
ON THE POLICY IMPROVEMENT ALGORITHM IN CONTINUOUS TIME SAUL D. JACKA AND ALEKSANDAR MIJATOVIĆ Abstract. We develop a general approach to the Policy Improvement Algorithm (PIA) for stochastic control problems
More informationON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS
PORTUGALIAE MATHEMATICA Vol. 55 Fasc. 4 1998 ON THE PATHWISE UNIQUENESS OF SOLUTIONS OF STOCHASTIC DIFFERENTIAL EQUATIONS C. Sonoc Abstract: A sufficient condition for uniqueness of solutions of ordinary
More informationMartingale Problems. Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi
s Abhay G. Bhatt Theoretical Statistics and Mathematics Unit Indian Statistical Institute, Delhi Lectures on Probability and Stochastic Processes III Indian Statistical Institute, Kolkata 20 24 November
More informationBrownian Motion. 1 Definition Brownian Motion Wiener measure... 3
Brownian Motion Contents 1 Definition 2 1.1 Brownian Motion................................. 2 1.2 Wiener measure.................................. 3 2 Construction 4 2.1 Gaussian process.................................
More informationRough paths methods 1: Introduction
Rough paths methods 1: Introduction Samy Tindel Purdue University University of Aarhus 216 Samy T. (Purdue) Rough Paths 1 Aarhus 216 1 / 16 Outline 1 Motivations for rough paths techniques 2 Summary of
More informationNEUTRAL STOCHASTIC FUNCTIONAL DIFFERENTIAL EQUATION DRIVEN BY FRACTIONAL BROWNIAN MOTION AND POISSON POINT PROCESSES
Gulf Journal of Mathematics Vol 4, Issue 3 (216) 1-14 NETRAL STOCHASTIC FNCTIONAL DIFFERENTIAL EQATION DRIVEN BY FRACTIONAL BROWNIAN MOTION AND POISSON POINT PROCESSES EL HASSAN LAKHEL 1 AND SALAH HAJJI
More informationMA8109 Stochastic Processes in Systems Theory Autumn 2013
Norwegian University of Science and Technology Department of Mathematical Sciences MA819 Stochastic Processes in Systems Theory Autumn 213 1 MA819 Exam 23, problem 3b This is a linear equation of the form
More informationUnexpected Default in Information based Model
Rainer Buckdahn, Université de Bretange Occidentale, Brest, France/ Shandong University, Jinan, P.R. China University of Loughborough, December, 5-9, 2016 Outline Information process β for default time
More informationMathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio ( )
Mathematical Methods for Neurosciences. ENS - Master MVA Paris 6 - Master Maths-Bio (2014-2015) Etienne Tanré - Olivier Faugeras INRIA - Team Tosca November 26th, 2014 E. Tanré (INRIA - Team Tosca) Mathematical
More informationConvergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics
Convergence of Particle Filtering Method for Nonlinear Estimation of Vortex Dynamics Meng Xu Department of Mathematics University of Wyoming February 20, 2010 Outline 1 Nonlinear Filtering Stochastic Vortex
More informationHalf of Final Exam Name: Practice Problems October 28, 2014
Math 54. Treibergs Half of Final Exam Name: Practice Problems October 28, 24 Half of the final will be over material since the last midterm exam, such as the practice problems given here. The other half
More informationRandom G -Expectations
Random G -Expectations Marcel Nutz ETH Zurich New advances in Backward SDEs for nancial engineering applications Tamerza, Tunisia, 28.10.2010 Marcel Nutz (ETH) Random G-Expectations 1 / 17 Outline 1 Random
More informationProf. Erhan Bayraktar (University of Michigan)
September 17, 2012 KAP 414 2:15 PM- 3:15 PM Prof. (University of Michigan) Abstract: We consider a zero-sum stochastic differential controller-and-stopper game in which the state process is a controlled
More informationDiscrete approximation of stochastic integrals with respect to fractional Brownian motion of Hurst index H > 1 2
Discrete approximation of stochastic integrals with respect to fractional Brownian motion of urst index > 1 2 Francesca Biagini 1), Massimo Campanino 2), Serena Fuschini 2) 11th March 28 1) 2) Department
More informationOn the submartingale / supermartingale property of diffusions in natural scale
On the submartingale / supermartingale property of diffusions in natural scale Alexander Gushchin Mikhail Urusov Mihail Zervos November 13, 214 Abstract Kotani 5 has characterised the martingale property
More informationNon-Zero-Sum Stochastic Differential Games of Controls and St
Non-Zero-Sum Stochastic Differential Games of Controls and Stoppings October 1, 2009 Based on two preprints: to a Non-Zero-Sum Stochastic Differential Game of Controls and Stoppings I. Karatzas, Q. Li,
More informationStability of optimization problems with stochastic dominance constraints
Stability of optimization problems with stochastic dominance constraints D. Dentcheva and W. Römisch Stevens Institute of Technology, Hoboken Humboldt-University Berlin www.math.hu-berlin.de/~romisch SIAM
More informationMULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES
MULTIDIMENSIONAL WICK-ITÔ FORMULA FOR GAUSSIAN PROCESSES D. NUALART Department of Mathematics, University of Kansas Lawrence, KS 6645, USA E-mail: nualart@math.ku.edu S. ORTIZ-LATORRE Departament de Probabilitat,
More informationUniformly Uniformly-ergodic Markov chains and BSDEs
Uniformly Uniformly-ergodic Markov chains and BSDEs Samuel N. Cohen Mathematical Institute, University of Oxford (Based on joint work with Ying Hu, Robert Elliott, Lukas Szpruch) Centre Henri Lebesgue,
More informationSolving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review
Solving Stochastic Partial Differential Equations as Stochastic Differential Equations in Infinite Dimensions - a Review L. Gawarecki Kettering University NSF/CBMS Conference Analysis of Stochastic Partial
More informationThe Pedestrian s Guide to Local Time
The Pedestrian s Guide to Local Time Tomas Björk, Department of Finance, Stockholm School of Economics, Box 651, SE-113 83 Stockholm, SWEDEN tomas.bjork@hhs.se November 19, 213 Preliminary version Comments
More informationStochastic Differential Equations
Chapter 5 Stochastic Differential Equations We would like to introduce stochastic ODE s without going first through the machinery of stochastic integrals. 5.1 Itô Integrals and Itô Differential Equations
More informationOn Stochastic Adaptive Control & its Applications. Bozenna Pasik-Duncan University of Kansas, USA
On Stochastic Adaptive Control & its Applications Bozenna Pasik-Duncan University of Kansas, USA ASEAS Workshop, AFOSR, 23-24 March, 2009 1. Motivation: Work in the 1970's 2. Adaptive Control of Continuous
More informationI forgot to mention last time: in the Ito formula for two standard processes, putting
I forgot to mention last time: in the Ito formula for two standard processes, putting dx t = a t dt + b t db t dy t = α t dt + β t db t, and taking f(x, y = xy, one has f x = y, f y = x, and f xx = f yy
More informationNumerical scheme for quadratic BSDEs and Dynamic Risk Measures
Numerical scheme for quadratic BSDEs, Numerics and Finance, Oxford Man Institute Julien Azzaz I.S.F.A., Lyon, FRANCE 4 July 2012 Joint Work In Progress with Anis Matoussi, Le Mans, FRANCE Outline Framework
More informationAn introduction to Birkhoff normal form
An introduction to Birkhoff normal form Dario Bambusi Dipartimento di Matematica, Universitá di Milano via Saldini 50, 0133 Milano (Italy) 19.11.14 1 Introduction The aim of this note is to present an
More informationUNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE
Surveys in Mathematics and its Applications ISSN 1842-6298 (electronic), 1843-7265 (print) Volume 5 (2010), 275 284 UNCERTAINTY FUNCTIONAL DIFFERENTIAL EQUATIONS FOR FINANCE Iuliana Carmen Bărbăcioru Abstract.
More informationNonparametric Drift Estimation for Stochastic Differential Equations
Nonparametric Drift Estimation for Stochastic Differential Equations Gareth Roberts 1 Department of Statistics University of Warwick Brazilian Bayesian meeting, March 2010 Joint work with O. Papaspiliopoulos,
More information-variation of the divergence integral w.r.t. fbm with Hurst parameter H < 1 2
/4 On the -variation of the divergence integral w.r.t. fbm with urst parameter < 2 EL ASSAN ESSAKY joint work with : David Nualart Cadi Ayyad University Poly-disciplinary Faculty, Safi Colloque Franco-Maghrébin
More informationOn the stochastic nonlinear Schrödinger equation
On the stochastic nonlinear Schrödinger equation Annie Millet collaboration with Z. Brzezniak SAMM, Paris 1 and PMA Workshop Women in Applied Mathematics, Heraklion - May 3 211 Outline 1 The NL Shrödinger
More informationStability of Stochastic Differential Equations
Lyapunov stability theory for ODEs s Stability of Stochastic Differential Equations Part 1: Introduction Department of Mathematics and Statistics University of Strathclyde Glasgow, G1 1XH December 2010
More informationSecond order forward-backward dynamical systems for monotone inclusion problems
Second order forward-backward dynamical systems for monotone inclusion problems Radu Ioan Boţ Ernö Robert Csetnek March 6, 25 Abstract. We begin by considering second order dynamical systems of the from
More informationControlled Diffusions and Hamilton-Jacobi Bellman Equations
Controlled Diffusions and Hamilton-Jacobi Bellman Equations Emo Todorov Applied Mathematics and Computer Science & Engineering University of Washington Winter 2014 Emo Todorov (UW) AMATH/CSE 579, Winter
More informationStochastic nonlinear Schrödinger equations and modulation of solitary waves
Stochastic nonlinear Schrödinger equations and modulation of solitary waves A. de Bouard CMAP, Ecole Polytechnique, France joint work with R. Fukuizumi (Sendai, Japan) Deterministic and stochastic front
More informationITO FORMULAS FOR FRACTIONAL BROWNIAN MOTIONS. Robert J. Elliott. Haskayne School of Business, University of Calgary
ITO FORMULAS FOR FRACTIONAL BROWNIAN MOTIONS Robert J. Elliott Haskayne School of Business, University of Calgary 1 ABSTRACT The difficulty of studying financial models with fractional Brownian motions
More informationA maximum principle for optimal control system with endpoint constraints
Wang and Liu Journal of Inequalities and Applications 212, 212: http://www.journalofinequalitiesandapplications.com/content/212/231/ R E S E A R C H Open Access A maimum principle for optimal control system
More informationarxiv: v2 [math.pr] 22 Aug 2009
On the structure of Gaussian random variables arxiv:97.25v2 [math.pr] 22 Aug 29 Ciprian A. Tudor SAMOS/MATISSE, Centre d Economie de La Sorbonne, Université de Panthéon-Sorbonne Paris, 9, rue de Tolbiac,
More informationAn homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted pendulum
9 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June -, 9 FrA.5 An homotopy method for exact tracking of nonlinear nonminimum phase systems: the example of the spherical inverted
More informationLarge Deviations for Perturbed Reflected Diffusion Processes
Large Deviations for Perturbed Reflected Diffusion Processes Lijun Bo & Tusheng Zhang First version: 31 January 28 Research Report No. 4, 28, Probability and Statistics Group School of Mathematics, The
More informationFinite element approximation of the stochastic heat equation with additive noise
p. 1/32 Finite element approximation of the stochastic heat equation with additive noise Stig Larsson p. 2/32 Outline Stochastic heat equation with additive noise du u dt = dw, x D, t > u =, x D, t > u()
More informationWiener integrals, Malliavin calculus and covariance measure structure
Wiener integrals, Malliavin calculus and covariance measure structure Ida Kruk 1 Francesco Russo 1 Ciprian A. Tudor 1 Université de Paris 13, Institut Galilée, Mathématiques, 99, avenue J.B. Clément, F-9343,
More informationReflected Brownian Motion
Chapter 6 Reflected Brownian Motion Often we encounter Diffusions in regions with boundary. If the process can reach the boundary from the interior in finite time with positive probability we need to decide
More informationINITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL DIFFERENTIAL EQUATIONS
Applied Mathematics and Stochastic Analysis, 6:2 23, 9-2. Printed in the USA c 23 by North Atlantic Science Publishing Company INITIAL AND BOUNDARY VALUE PROBLEMS FOR NONCONVEX VALUED MULTIVALUED FUNCTIONAL
More informationSmall ball probabilities and metric entropy
Small ball probabilities and metric entropy Frank Aurzada, TU Berlin Sydney, February 2012 MCQMC Outline 1 Small ball probabilities vs. metric entropy 2 Connection to other questions 3 Recent results for
More informationOPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS
APPLICATIONES MATHEMATICAE 29,4 (22), pp. 387 398 Mariusz Michta (Zielona Góra) OPTIMAL SOLUTIONS TO STOCHASTIC DIFFERENTIAL INCLUSIONS Abstract. A martingale problem approach is used first to analyze
More informationLAN property for ergodic jump-diffusion processes with discrete observations
LAN property for ergodic jump-diffusion processes with discrete observations Eulalia Nualart (Universitat Pompeu Fabra, Barcelona) joint work with Arturo Kohatsu-Higa (Ritsumeikan University, Japan) &
More informationPROBABILITY: LIMIT THEOREMS II, SPRING HOMEWORK PROBLEMS
PROBABILITY: LIMIT THEOREMS II, SPRING 218. HOMEWORK PROBLEMS PROF. YURI BAKHTIN Instructions. You are allowed to work on solutions in groups, but you are required to write up solutions on your own. Please
More informationMaximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments
Austrian Journal of Statistics April 27, Volume 46, 67 78. AJS http://www.ajs.or.at/ doi:.773/ajs.v46i3-4.672 Maximum Likelihood Drift Estimation for Gaussian Process with Stationary Increments Yuliya
More informationRegularity of the density for the stochastic heat equation
Regularity of the density for the stochastic heat equation Carl Mueller 1 Department of Mathematics University of Rochester Rochester, NY 15627 USA email: cmlr@math.rochester.edu David Nualart 2 Department
More informationHomogenization with stochastic differential equations
Homogenization with stochastic differential equations Scott Hottovy shottovy@math.arizona.edu University of Arizona Program in Applied Mathematics October 12, 2011 Modeling with SDE Use SDE to model system
More informationMalliavin Calculus: Analysis on Gaussian spaces
Malliavin Calculus: Analysis on Gaussian spaces Josef Teichmann ETH Zürich Oxford 2011 Isonormal Gaussian process A Gaussian space is a (complete) probability space together with a Hilbert space of centered
More informationZdzislaw Brzeźniak. Department of Mathematics University of York. joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York)
Navier-Stokes equations with constrained L 2 energy of the solution Zdzislaw Brzeźniak Department of Mathematics University of York joint works with Mauro Mariani (Roma 1) and Gaurav Dhariwal (York) LMS
More informationPoisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation
Poisson Jumps in Credit Risk Modeling: a Partial Integro-differential Equation Formulation Jingyi Zhu Department of Mathematics University of Utah zhu@math.utah.edu Collaborator: Marco Avellaneda (Courant
More information