Mean Field Games. Math 581 Project


 Rosaline McKenzie
 2 years ago
 Views:
Transcription
1 Mean Field Games Tiago Miguel Saldanha Salvador Mah 58 Projec April 23
2 Conens Inroducion 2 2 Analysis of second order MFG 3 2. On he FokkerPlank equaion Exisence of soluions o a 2 nd MFG Uniqueness of soluions of a 2 nd order MFG Analysis of firs order MFG 3. On he HamilonJacobi equaion On he coninuiy equaion Exisence of soluions o a s order MFG A Sochasic Calculus 25 A. Brownian Moion and filraion A.2 Sochasic inegral and Iô s formula A.3 Sochasic differenial equaions B Auxiliary resuls 27 Inroducion Mean Field Games (MFG) is a class of sysems of parial differenial equaions ha are used o undersand he behaviour of muliples agens each individually rying o opimize heir posiion in space and ime, bu wih heir preferences being parly deermined by he choices of all oher agens, in he asympoic limi when he number of agens goes o infiniy. This heory has been recenly developed by J. M. Lasry and P. L. Lions in a series of papers [6, 7, 8, 9] and presened hrough several lecures of P. L. Lions a he Collège de France. The ypical model for MFG is he following: u ν u + H(x, m, D x u) = F (x, m) in R d [, T ], m ν m div(d p H(x, m, D x u)m) = in R d [, T ], (MFG) m() = m, u(x, T ) = G(x, m(t )) in R d. where ν is a nonnegaive parameer. The firs equaion is an HamilonJacobi equaion evolving backward in ime whose soluion is he value funcion of each agen. Indeed, he inerpreaion is he following: an average agen moves accordingly o he sochasic differenial equaion dx = α d + 2νdW where W = {W : R + } is a sandard Brownian moion and α is he conrol o be chosen by he agen. He hen wishes o minimize [ ] T E L(X s, m(s), α(s)) + F (X s, m(s))ds + G(X T, m(t ))
3 where L is he Legendre ransform of H wih respec o he las variable. The second equaion is a FokkerPlanck ype equaion evolving forward in ime ha governs he evoluion of he densiy funcion m of he agens. In his repor we will focus on sudying he exisence and uniqueness of soluions of MFG. In Secion 2 we consider (MFG) wih ν = and he Hamilonian H(p) = 2 p 2, proving he exisence and uniqueness of classical soluions. In Secion 3 we consider he same Hamilonian bu wih ν = and prove exisence and uniqueness of (weak) soluions. For boh secions we follow closely [3], rying o provide more deail in he proofs where i fel needed. Finally in he Appendix we review some basic definiions and resuls of sochasic calculus, as well as some resuls from measure heory ha are used. 2 Analysis of second order MFG Our goal in his Secion is o prove he exisence of classical soluions for he following MFG: u u + 2 D xu 2 = F (x, m) in R d (, T ) m m div(md x u) = in R d (, T ) () m() = m, u(x, T ) = G(x, m(t )) in R d Here D x u denoes he parial gradien wih respec o x. We need o inroduce some definiions. Definiion 2.. A pair (u, m) is a classical soluion o () if u, m C 2, (R d (, T )) C(R d [, T ]) and (u, m) saisfies () in he classical sense. Definiion 2.2. P is he se of Borel probabiliy measures m on R d wih finie firs order momen, i.e., R d x dm(x) <. We endow P wih he following (KanorovichRubisein) disance d(µ, ν) = inf x y dγ(x, y) γ Π(µ,ν) R 2d where Π(µ, ν) is he se of Borel probabiliy measures on R 2d such ha for any Borel se A R d. γ(a R d ) = µ(a) and γ(r d A) = ν(a) We can now sae he main heorem of his Secion: Theorem 2.3. Suppose here is some consan C such ha (Bounds on F and G) F and G are uniformly bounded by C over R d P, (Lipschiz coninuiy of F and G) For all (x, m ), (x 2, m 2 ) R d P, we have F (x, m ) F (x 2, m 2 ) C ( x x 2 + d(m, m 2 )) and G(x, m ) G(x 2, m 2 ) C ( x x 2 + d(m, m 2 )), 3
4 The probabiliy measure m is absoluely coninuous wih respec o he Lebesgue measure, denoed by L d and has a Hölder coninuous densiy, sill denoe by m, which saisfies R d x 2 m (x)dx C. Then here is a leas one classical soluion o (). We will firs rea wo PDE s in () separaely: we obain some esimaes on he FokkerPlanck equaion and recall some known facs of he hea equaion. 2. On he FokkerPlank equaion In his Secion we will derive some resuls on he following FokkerPlanck equaion m m div(mb) = in R d (, T ) (2) m() = m where b : R d [, T ] R is a given vecor field. We can look a i as an evoluion equaion on he space of probabiliy measures. We will assume ha he vecor field b is coninuous, uniformly Lipschiz in space and bounded. The reason for his is ha in he proof of Theorem 2.3 we will ake b = D x u. Definiion 2.4. We say ha m is a weak soluion o (2) if m L ([, T ], P) is such ha for any es funcion ϕ D(R d [, T )) we have R d ϕ(x, )dm (x) + T R d ( ϕ(x, ) + ϕ(x, ) D x ϕ(x, ) b(x, )) dm()(x). Consider he following sochasic differenial equaion (SDE) dx = b(x, )d + 2dW [, T ] (3) X = Z where W is a sandard ddimensional Brownian moion and he iniial condiion Z L is random variable independen of W. Under he assumpion on b by Theorem A. here is a unique soluion o (3). The nex Lemma shows ha he soluion of (3) is closely relaed o he soluion of (2). Lemma 2.5. If L(Z ) = m, hen m() := L(X ) is a weak soluion of (2), where X is he soluion of (3). Here L(X) denoes he law (densiy funcion) of he random variable X. Proof. This is a sraighforward consequence of Iô s formula. Indeed, le ϕ C 2, (R d [, T ]). Then by Iô s formula (Theorem A.9) T T ϕ(x, ) = ϕ(z, ) + ( ϕ(x s, s) D x ϕ(x s, s) b(x s, s) + ϕ(x s, s)) ds + D x ϕ(x s, s) dw s. We know ha [ ] T E D x ϕ(x s, s) dw s =. 4
5 Hence aking he expecaion on he above equaliy leads o [ E [ϕ(x, )] = E ϕ(z, ) + So by he definiion of m we have ϕ(x, )dm()(x) = ϕ(x, )dm (x)+ R d R d Therefore for ϕ D ( R d [, T ) ) and aking = T we have R d ϕ(x, )dm (x) + T i.e., m is a weak soluion of (2). ] ( ϕ(x s, s) D x ϕ(x s, s) b(x s, s) + ϕ(x s, s)) ds. R d ( ϕ(x, s) Dϕ(x, s) b(x, s) + ϕ(x, s)) dm(s)(x)ds. R d ( ϕ(x, ) Dϕ(x, ) b(x, ) + ϕ(x, )) dm()(x)ds =, The above inerpreaion of m as he probabiliy densiy of he soluion of (3) allows us o show ha he map m() is Hölder coninuous. Lemma 2.6. Le m() := L(X ) where X is he soluion of (3). Then here is a consan c = c (T ) (i.e., depending only on T ), such ha for all s, [, T ] d(m(), m(s)) c ( + b ) s /2. Proof. We sar by observing ha he probabiliy measure γ of he pair (X, X s ) belongs o Π(m(), m(s)). Therefore d(m(), m(s)) x y dγ(x, y) = E [ X X s ]. R 2d Wihou loss of generaliy suppose s <. Then [ E [ X X s ] = E b(x τ, τ)dτ + ] 2(W W s ) s [ E b(x τ, τ) dτ + ] 2 W W s s b ( s) + 2 s π ( ) 2 s b T + π { } T 2 s ( b + ) max, π So by aking c = max { T, 2 π } we are done. We can also obain easily an esimae on he second order momen of m Lemma 2.7. Le m() := L(X ) where X is he soluion of (3). Then here is a consan c = c (T ) such ha for all [, T ] ( ) x 2 dm()(x) c R d x 2 dm (x) + + b 2 R d. 5
6 Proof. By definiion of m we have Hence R d x 2 dm()(x) = E [ X 2] [ ] 2 x 2 dm()(x) 3E X 2 + b(x s, s)ds + 2 W 2 R d ( ) 3 x 2 dm (x) + b R d ) c x (R 2 dm (x) + b 2 + d where c = max{3, 3T 2, 6T }. 2.2 Exisence of soluions o a 2 nd MFG In his Secion we prove Theorem 2.3. In order o do ha we need firs o recall some exisence and uniqueness resuls for he following hea equaion w w + a(x, ) Dw + b(x, )w = f(x, ) in R d [, T ] (4) w(x, ) = w (x) in R d where a, b, f : R d [, T ] R and w : R d R. For his we inroduce some noaion. Definiion 2.8. Le s be an ineger and α (, ). We denoe by C s,α (R d [, T ]) he se of funcions f : R d [, T ] R such ha for any pair (k, l) wih 2k + l s, k D l xf exiss and such ha hese derivaives are bounded, αhölder coninous in space and α/2hölder coninous in ime. We hen have he following heorem whose proof can be found in [5]: Theorem 2.9. Suppose ha a, b, f C,α (R d [, T ]) and ha w C,α (R d ) (he classical Hölder space). Then (4) has a unique weak soluion u C 2,α (R d [, T ]). We also have he following inerior esimae: Theorem 2.. Suppose a b and ha f C(R d [, T ]) is bounded. Then any classical bounded soluion w of (4) saisfies, for any compac se K R d (, T ) D x w(x, ) D x w(y, s) sup C f (x,),(y,s) K x y β + s β/2 where β (, ) depends only on he dimension d and C = C(K, w, d). The idea of he proof is o consruc a map Ψ such ha a fixed poin of Ψ is a soluion of he sysem (). Then we use he Schauder fixed poin heorem o prove he exisence of he fixed poin. Theorem 2. (Schauder fixed poin). Le K be a convex, closed and compac subspace of a opological vecor space V and Ψ : K K a coninuous map. Then Ψ has a fixed poin. 6
7 Proof of Theorem 2.3. Le C be a large consan o be fixed laer and le M be he se of maps µ C([, T ], P) such ha and o d(µ(), µ(s)) sup C s, [,T ] s /2 s sup x 2 dµ()(x) C. [,T ] R d To any µ M we associae an m = Ψ(µ) M in he following way: le u be he unique soluion u u + 2 D xu 2 = F (x, µ()) in R d [, T ] (5) u(x, T ) = G(x, µ(t ) in R d Then we define m = ψ(µ) M as he unique soluion of he FokkerPlank equaion m m div(md x u) = in R d [, T ] (6) m() = m (x) in R d In order o apply he Schauder fixed poin heorem, we need o show ha: M is a convex closed and compac subse of C([, T ], P), Ψ is well defined and Ψ is coninuous. ) M is a convex closed and compac subse of C([, T ], P). Le λ [, ], µ, µ 2 M, γ Π(µ (), µ (s)) and γ 2 Π(µ 2 (), µ 2 (s)). We have ha and herefore λγ + ( λ)γ 2 Π(λµ () + ( λ)µ 2 (), λµ (s) + ( λ)µ 2 (s)) d(λµ () + ( λ)µ 2 (), λµ (s) + ( λ)µ 2 (s)) x y d(λγ (x, y) + ( λ)γ 2 (x, y)) R 2d = λ x y dγ (x, y) + ( λ) x y dγ 2 (x, y)). R 2d R 2d Then aking he infimum over γ Π(µ (), µ (s)) and γ 2 Π(µ 2 (), µ 2 (s)) shows ha d(λµ () + ( λ)µ 2 (), λµ (s) + ( λ)µ 2 (s)) λd(µ (), µ (s)) + ( λ)d(µ 2 (), µ 2 (s)). We also have x 2 d(λµ + ( λ)µ 2 )()(x) = λ R d x 2 dµ ()(x) + ( λ) R d x 2 dµ 2 ()(x). R d From he las wo equaliies i s now easy o see ha, indeed, λµ + ( λ)µ 2 M and so M is convex. Now le µ n M such ha µ n µ in C([, T ], P). To prove ha M is closed we need o show ha µ M. 7
8 I s easy o show ha and from his i follows easily ha d(µ(), µ(s)) d(µ() µ n (), µ(s) µ n (s)) + d(µ n (), µ n (s)) d(µ(), µ(s)) sup C. s, [,T ] s /2 s As for he second order momen esimae we noe ha x 2 dµ()(x) = R d x 2 d(µ() µ n ())(x) + R d x 2 dµ n ()(x) R d Taking he supremum for [, T ] we ge sup x 2 dµ()(x) sup x 2 d(µ() µ n ())(x) + sup [,T ] R d [,T ] R d sup x 2 d(µ() µ n ())(x) + C [,T ] R d Now since µ n µ in C([, T ], P), by aking n we ge as desired. sup [,T ] R d x 2 dµ()(x) C [,T ] For he proof ha M is compac we refer he reader o Lemma 5.7 of [3]. 2) ψ is welldefined. Firs we need o see ha a soluion of (5) exiss and is unique. R d x 2 dµ n ()(x) Consider hen he HopfCole ransformaion given by w = e u/2. Then i is easy o check ha u is a soluion of (5) if and only if w is a soluion of The map (x, ) F (x, m()) belongs o C,/2 bounded over R d P and w w = wf (x, µ()) in R d [, T ] (7) w(x, T ) = e G(x,µ(T ))/2 in R d since F is Lipschiz in boh variables, uniformly d(µ(), µ(s)) sup C. s, [,T ] s /2 s because µ M. The map x e G(x,µ(T ))/2 is in C,/2 (R d ) since G is Lipschiz in x and uniformly bounded over R d P. Then appealing o Theorem 2.9 here is a unique soluion in C 2,/2 o (7) which implies ha here is a unique soluion in C 2,/2 o (5). Recall ha, by assumpion, he maps (x, ) F (x, m()) and x G(x, µ(t ) are bounded by C. Hence a sraighforward applicaion of he comparison principle implies ha u is bounded by ( + T )C. Similarly he maps x F (x, m()) and x G(x, µ(t ) are C Lipschiz coninuous (again by our assumpions on F and G) and so u is also C Lipschiz coninuous. Hence D x u is bounded by C. 8
9 Now we look a he FokkerPlanck equaion (6). By expanding he divergence erm, we can wrie i ino he form m m D x m D x u(x, ) m u(x, ) = in R d (, T ) m() = m Since u C 2,/2, he maps (x, ) D x u(x, ) and (x, ) u(x, ) belong o C,/2. Also by assumpion m C,α (R d ). Hence by Theorem (2.9) here is a unique soluion m C 2,/2 o (6). Moreover, by Lemma 2.6, for all s, [, T ] and by Lemma 2.7 for all [, T ] d(m(), m(s)) c ( + C ) s 2 R d x 2 dm()(x) c (C + + C 2 ) where c depends only on T. So if we choose C = max{c ( + C ), c (C + + C 2 )}, m M and Ψ is hen welldefined. 3) Ψ is coninuous. Le µ n M converge o some µ. Le (u n, m n ) and (u, m) be he corresponding soluions. Noe ha (x, ) F (x, µ n ()) and x G(x, µ n (T )) converge locally uniformly o (x, ) F (x, µ()) and x G(x, µ(t ) respecively. Hence we can conclude ha (u n ) converges locally uniformly o u by a sandard argumen wih viscosiy soluions. Since he (D x u n ) are uniformly bounded (by C ), he (u n ) solve an equaion of he form u n u n = f n where f n = 2 D xu n 2 F (x, m n ) is uniformly bounded in x and n. Then by Theorem 2. (D x u n ) is locally uniform Hölder coninuous and herefore converge locally uniform o D x u. This implies ha any converging subsequence of he relaively compac sequence (m n ) is a weak soluion of (6). Bu m is he unique soluion of (6), which proves ha (m n ) converges o m. Hence Ψ is coninuous. Finally, by he Schauder fixed poin heorem, he coninuous map µ m = Ψ(µ) has a fixed poin in M. To his fixed poin m M corresponds a pair (u, m) ha is a classical soluion o () and so we are done. 2.3 Uniqueness of soluions of a 2 nd order MFG In his Secion we prove a uniqueness resul o he sysem (). Theorem 2.2. Besides he assumpions of Theorem 2.3, assume ha For all m, m 2 P wih m m 2 we have R d (F (x, m ) F (x, m 2 ))d(m m 2 )(x) >, 9
10 For all m, m 2 P Then here is a unique soluion o (). R d (G(x, m ) G(x, m 2 ))d(m m 2 )(x). Proof. Le (u, m ) and (u 2, m 2 ) be wo classical soluions of (). We se u = u u 2 and m = m m 2. Then u u + 2 ( D xu 2 D x u 2 2 ) (F (x, m ) F (x, m 2 )) = m m div(m D x u m 2 D x u 2 ) = Since u C 2, (R d (, T )), we can muliply he second equaion by u, inegrae over R d [, T ], followed by par inegraion o ge m(t )u(x, T )dx + R d m (x)u(x, )dx + R d T (8) R d ( u + u)m Du (m D x u m 2 D x u 2 )dxd. Muliplying now he firs equaion by m, inegraing over R d [, T ] and adding o he previous equaliy, leads o m(t )(G(x, m (T )) G(x, m 2 (T ))dx R d T ( + m ) 2 D xu D x u 2 2 m(f (x, m ) F (x, m 2 ))) dxd = R d where we used he fac ha m() = and ha m 2 ( D xu 2 Du 2 2 ) D x u (m D x u m 2 D x u 2 ) = m 2 D xu D x u 2 2. By assumpion m(t )(G(x, m (T )) G(x, m 2 (T ))dx R d and herefore T m(f (x, m ) F (x, m 2 ))dxd. R d Hence, by our assumpion on F, his implies ha m = and herefore u = since u and u 2 (now) solve he same equaion. We finish his Secion by menioning ha he exisence of soluions for second order MFG hold under more general assumpions. Indeed, in [7, 8] he auhors consider equaions of he form u(x, ) u + H(x, Du) = F (x, m)) in Q (, T ) m(x, ) m div(m H p (x, D xu)) = in Q (, T ) m() = m, u(x, T ) = G(x, m(t )) in Q where Q = [, ] d (wih periodic boundary condiions), H : R d R d is Lipschiz coninuous wih respec o x and uniformly bounded in p, convex and of class C wih respec o p. The condiions on F and G are one of he following: F and G are regularizing, i.e., saisfy he same condiions as in Theorem 2.3. F (x, m) = f(x, m(x)) and G(x, m) = g(x, m(x)), where f = f(x, λ) and g = g(x, λ) saisfy suiable growh condiions wih respec o λ and H is sufficienly sricly convex.
11 3 Analysis of firs order MFG In his Secion we will prove he exisence of soluions o he following firs order MFG: u(x, ) + 2 Du(x, ) 2 = F (x, m()) in R d (, T ) m(x, ) div(du(x, )m(x, ) = in R d (, T ) (9) m() = m, u(x, T ) = G(x, m(t )) in R d We consider he following definiion of weak soluions. Definiion 3.. We call he pair (u, m) a weak soluion of (9) if u W, loc (Rd [, T ]), m L (R d (, T )) such ha he firs equaion of (9) is saisfied in he viscosiy sense and he second in saisfied in he sense of disribuions. Noe ha here we don look any more for classical soluions mainly because we no longer have he smoohing erms u and m. Our goal is hen o prove he following. Theorem 3.2. Suppose ha. F and G are coninuous over R d P, 2. There is a consan C such ha for any m P, F (, m), G(, m) C 2 (R d ) and F (, m) C2 (R d ) C G(, m) C2 (R d ) C where for f C 2 (R d ) we denoe C 2 (R d ) by f C 2 (R d ) = sup x R d { f(x) + Df(x) + D 2 f(x) }, 3. m is absoluely coninuous wih respec o he Lebesgue measure and has a densiy, sill denoed by m, which is bounded and has a compac suppor. Then here is a leas one weak soluion of (9). Remark 3.3. Under he assumpions of Theorem 2.2 we can show ha he soluion is unique. The proof is he same wih he only difference being ha now we use he Lipschiz coninuous map u as a es funcion because he densiy m is bounded and has compac suppor. As in Secion 2, we will sudy he wo equaions separaely. 3. On he HamilonJacobi equaion In his Secion we sudy he HamilonJacobi equaion u + 2 D xu 2 = f(x, ) in R d (, T ) () u(x, T ) = g(x) in R d We will sar by recalling some basic facs abou he noion of semiconcaviy which will play a role here. The proofs for hese resuls can be found in [2].
12 Definiion 3.4. A map w : R d R is semiconcave if here is some consan C > such ha one of he following equivalen condiions is saisfied:. he map x w(x) C 2 x 2 is concave in R d. 2. w(λx + ( λ)y) λw(x) + ( λ)w(y) Cλ( λ) x y 2 for any x, y R d and λ [, ]. 3. D 2 w CI d in he sense of disribuions. 4. (p q) (x y) C x y 2 for any x, y R d, [, T ], p D + x w(x) and q D + x w(y), where D + x w denoes he superdifferenial of w wih respec o he x variable, namely D + x w(x) = {p R d : lim sup y x w(y) w(x) p (y x) y x }. Lemma 3.5. Le w : R d R be semiconcave. Then w is locally Lipschiz coninuous in R d. Moreover D + x w(x) is he closed convex hull of he se D xw(x) of reachable gradiens defined by D xw(x) = {p R d : (x n ) wih x n x such ha D x w(x n ) exiss and converges o p} In paricular, D + x w(x) is compac, convex and non empy subse of R d for any x R d. Finally w is differeniable a x if and only if D + w(x) is a singleon. Lemma 3.6. Le (w n ) be a sequence of uniformly semiconcave maps on R d which converge poinwiseo a map w : R d R. Then he convergence is locally uniform and w is semiconcave. Moreover, for any x n x and any p n D + w n (x n ), he se of accumulaion poins of (p n ) is conained in D + w(x). Finally, Dw n (x) converges o Dw(x) for a.e. x R d. Definiion 3.7. Le (x, ) R d [, T ]. We denoe by A(x, ) he nonempy se of opimal conrols o u(x, ), i.e., α L 2 ([, T ], R d ) such ha T u(x, ) = 2 α(s) 2 + f(x(s), s) ds + g(x(t )) where x(s) = x + s α(τ)dτ. We call x( ) he associaed rajecory o he conrol α. Lemma 3.8. If (x n, n ) (x, ) wih α n A(x n, n ), hen, up o a subsequence, (α n ) weakly converges in L 2 o some α A(x, ). We can now sudy equaion (). Lemma 3.9. Le f : R d [, T ] R and g : R d R be coninuous funcions. For any [, T ], f(, ), g C 2 (R d ) wih f(, ) C 2 C, g C 2 C () for some consan C. Then equaion () has a unique bounded uniformly coninuous viscosiy soluion which is given by he represenaion formula u(x, ) = T inf α L 2 ([,T ],R d ) 2 α(s) 2 + f(x(s), s) ds + g(x(t )), 2
13 where x(s) = x + α(τ)dτ. Moreover u is Lipschiz coninuous and saisfies s D x, u C, D 2 xxu C I d where he las inequaliy holds in he sense of disribuions. Proof. From he heory of HamilonJacobi equaions we already know ha () has a unique bounded uniformly coninuous viscosiy soluion given by u(x, ) = α L 2 ([,T ],R d ) T 2 α(s) 2 + f(x(s), s) ds + g(x(t )). Hence we only need o check ha u is Lipschiz coninuous wih D x, u C and D 2 xxu C I d in he sense of disribuions for some consan C = C (T ). ) u is Lipschiz coninuous wih respec o x. Le x, x 2 R d, [, T ] and α A(x, ). We hen have u(x 2, ) T T 2 α(s) 2 + f(x(s) + x 2 x, s) ds + g(x(t ) + x 2 x ) 2 α(s) 2 + f(x(s), s) + C x 2 x ds + g(x(t )) + C x 2 x u(x, ) + C(T + ) x 2 x Thus u is Lipschiz coninuous wih respec o x wih Lipschiz consan C(T + ). 2) u is Lipschiz coninuous wih respec o. Fix x R d and [, T ]. From he dynamic programming principle we have for any < s T u(x, ) = s 2 α(τ) 2 + f(x(τ), τ)dτ + u(x(s), s) where α A(x, ) and x( ) is is associaed rajecory. We have u(x, ) u(x, s) u(x, ) u(x(s), s) + u(x(s), s) u(x, s) s 2 α(τ) 2 + f(x(τ), τ) dτ + C(T + ) x(s) x (s ) 2 α 2 + f + C(T + ) where in he second inequaliy we used he fac u is C(T + )Lipschiz coninuous wih respec o x. In Lemma 3., we show ha α is bounded by a consan C 2 = C 2 (T ). Hence he inequaliy above proves ha u is Lipschiz coninuous wih respec o. 3) D x, u C. I follows easily from ) and 2). 3
14 4) D 2 xxu C I d in he sense of disribuions. Le x, y R d, [, T ], λ [, ] and se x λ = λx + ( λ)y. By Definiion 3.4 i s enough o show ha λu(x, ) + ( λ)u(y, ) u(x λ, ) + Cλ( λ) x y 2 where C = C(T ) is a consan. Le α A(x λ, ) and x λ ( ) is associaed rajecory. Then [ T ] λu(x, ) + ( λ)u(y, ) λ 2 α(s) 2 + f(x λ (s) + x x λ, s) ds + g(x λ (T ) + x x λ ) [ T ] + ( λ) 2 α(s) 2 + f(x λ (s) + y x λ, s) ds + g(x λ (T ) + y x λ ) T 2 α(s) 2 + f(x λ (s), s) ds + g(x λ (T ) + C(T + )λ( λ) x y 2 Hence u is semiconcave. = u(x λ, ) + α (T + )λ( λ) x y 2. Lemma 3. (EulerLagrange opimaliy condiion). If α A(x, ), hen α is of class C ([, T ]) wih α (s) = Df(x(s), s) in [, T ] α(t ) = Dg(x(T )) In paricular, here is a consan C = C (C) such ha for (x, ) R d [, T ) and any α A(x, ) we have α C, where C saisfies (). Lemma 3. (Regulariy of u along opimal soluion). Le (x, ) R d [, T ], α A(x, ) and le us se x(s) = x + α(τ)dτ. Then s. (Uniqueness of he opimal conrol along opimal rajecories) for any s (, T ], he resricion of α o [s, T ] is he unique elemen of A(x(s), s). 2. (Uniqueness of he opimal rajecories) D x u(x, ) exiss if and only if A(x, ) is a reduced o singleon. In his case, D x u(x, ) = α() where A(x, ) = {α}. Remark 3.2. In paricular, if we combine he above saemens, we see ha u(, s) is always differeniable a x(s) for s (, T ) wih D x u(x(s), s) = α(s). Proof. Le α A(x(s), s) and le x ( ) be is associaed rajecory. For any h > sufficienly small we define α h L 2 ([, T ], R d ) in he following way α(τ) if τ [, s h) x α h (τ) = (s+h) x(s h) 2h if τ [s h, s + h) α (τ) if τ [s + h, T ] 4
15 Then one easily checks ha x(τ) if τ [, s h) x h (τ) = x(s h) + (τ (s h)) x(s+h) x(s h) 2h if τ [s h, s + h) x (τ) if τ [s + h, T ] Since boh α [s,t ] and α are opimal for u(x(s), s), α, which is nohing bu he concaenaion of α [,s] and α, is also opimal for u(x, ). Also observe ha x (τ) = x + τ α (σ)dσ is given by x(τ) on [, s] and x (τ) on [s, T ]. Hence and u(x, ) = s T 2 α(τ) 2 + f(x(τ), τ)dτ + s 2 α (τ) 2 + f(x (τ), τ) dτ + g(x (T )) u(x, ) T s 2 α h(τ) 2 + f(x h (τ), τ) dτ + g(x h (T )). Using he definiions of α h and x h we can wrie he above inequaliy as s s h 2 α(τ) 2 + f(x(τ), τ) dτ + s+h s s+h s h Now dividing h and aking h + shows ha 2 α (τ) 2 + f(x (τ), τ) dτ ( ) 2 x (s + h) x(s h) 2 2h + f(x h (τ), τ) dτ 2 α(s) α (s) 2 4 α(s) + α (s) 2 since lim h x h (s) = x(s) = x (s). Therefore α(s) α (s) 2, i.e., α(s) = α (s). In paricular x( ) and x ( ) saisfy he same second order differenial equaion y (τ) = D x f(y(τ), τ) y (s) = x (s) = α(s) = α (s) = x (s) y(s) = x(s) = x (s) Hence x( ) = x ( ) and α = α on [s, T ]. This means ha he opimal soluion for u(x(s), s) is unique, hus proving poin. We now show ha if D x u(x, ) exiss, hen A(x, ) is reduced o singleon and D x u(x, ) = α() where A(x, ) = {α}. Indeed, le α A(x, ) and x( ) be he associaed rajecory. Then for any v R d u(x + v, ) T 2 α(s) 2 ds + f(x(s) + v, s)ds + g(x(t ) + v). Since equaliy holds for v = and since boh sides of he inequaliy are differeniable wih respec o v a v = we ge D x u(x, ) = T D x f(x(s), s)ds + D x g(x(t )). 5
16 Then by Lemma 3. we have D x u(x, ) = α(). Therefore x( ) has o be he unique soluion of he second order differenial equaion x (s) = D x f(x(s), s) x () = D x u(x, ) x() = x which in urn implies ha α = x is unique. Conversely, suppose ha A(x, ) is a singleon. We wan o show ha u(, ) is differeniable a x. For his we noe ha, if p belongs o Dxu(x, ) (he se of reachable gradiens of he map u(, )), hen he soluion x (s) = D x f(x(s), s) x () = p x() = x is opimal. Indeed, by definiion of p here is a sequence x n x such ha u(, ) is differeniable a x n and D x u(x n, ) p. Now since u(, ) is differeniable a x n, we know ha he unique soluion x n ( ) of x n(s) = D x f(x n (s), s) x n () = x x n() = Du(x n, ) is opimal. Passing o he limi as n implies by Lemma 3.8 ha x( ), which is he uniform limi of he x n ( ), is also opimal. Bu from our assumpions, here is a unique opimal soluion in A(x, ). Hence Dxu(x, ) has o be reduced o a singleon and since u(, ) is semiconcave by Lemma 3.9, we have ha u(, ) is differeniable a x by Lemma 3.5. Le us consider again (x, ) R d [, T ), α A(x, ) and x( ). Then we have jus proved ha u(, s) is differeniable a x(s) for any s (, T ) wih x (s) = α(s) = D x u(x(s), s). So given α opimal, is associaed rajecory x( ) is a soluion of he differenial equaion x (s) = D x u(x(s), s) on [, T ] x() = x The following Lemma, saes ha he reverse also holds. This is an opimal synhesis resul since i says he opimal conrol can be obained a each posiion y and a each ime s as by he synhesis α (y, s) = D x u(y, s). 6
17 Lemma 3.3 (Opimal synhesis). Le (x, ) R d [, T ) and x( ) be an absoluely coninuous soluion o he differenial equaion x (s) = D x u(x(s), s), a.e. in [, T ] x() = x Then he conrol α := x is opimal for u(x, ), i.e., α A(x, ). In paricular, if u(, ) is differeniable a x, hen equaion (2) has a unique soluion, corresponding o he opimal rajecory. Proof. We sar by observing ha x( ) is Lipschiz coninuous because u is. Le s (, T ) be such ha equaion (2) holds. Hence u is differeniable wih respec o x a (x(s), s) and he Lipschiz coninuous map s u(x(s), s) has a derivaive a s. Since u is Lipschiz coninuous, Lebourg s mean value heorem ([4], Th. (2) 2.3.7), saes ha, for any h > small enough here is some (y h, s h ) [(x(s), s), (x(s + h), s + h)] and some (ξ h x, ξ h ) CoD xu(y h, s h ) wih u(x(s + h), s + h) u(x(s), s) = ξ h x (x(s + h) x(s)) + ξ h h, (3) where CoDx,u(y, s) denoes he closure of he convex hull of he se of reachable gradiens Dx,u(y, s). Now from Carahéodory Theorem, here are (λ h,i, ξx h,i, ξ h,i ) i=,...,d+2 such ha λ h,i, d+2 i= d+2 λ h,i =, (ξx h,i, ξ h,i ) Dxu(y h, s h ) and (ξx, h ξ h ) = i= λ h,i (ξx h,i, ξ h,i ). For each i =,..., d + 2, he ξx h,i converges o D x u(x(s), s) as h because, from Lemma 3.6, any accumulaion poin of (ξx h,i ) h mus belong o D x + u(x(s), s) which is reduced D x u(x(s), s) since u(, s) is differeniable a x(s). Therefore ξ x,h = i λ h,i ξ h,i x D x u(x(s), s) as h. Since u is a viscosiy soluion o () and (ξ h,i x Therefore d+2 ξ h = i= λ h,i ξ h,i = d+2 2 ξ h,i + 2 ξh,i x 2 = f(y h, s h ). as h. Then dividing (3) by h and leing h we ge i=, ξ h,i ) Dx,u(x(s), s) we have λ h,i ξ h,i x 2 f(y h, s h ) 2 D xu(x(s), s) 2 f(x(s), s) d ds u(x(s), s) = D xu(x(s), s) x (s) + 2 D xu(x(s), s) 2 f(x(s), s). and, since x (s) = D x u(x(s), s), we have d ds u(x(s), s) = 2 x (s) 2 f(x(s), s) a.e. in (, T ). Inegraing he above inequaliy over [, T ] we finally obain u(x, ) = T 2 x (s) 2 + f(x(s), s)ds + g(x(t )) where we used he fac ha u(y, T ) = g(y) for y R d. Therefore α := x is opimal. The las saemen of he Lemma is a jus direc consequence of poin 2. of Lemma 3.. 7
18 From he sabiliy of opimal soluions, he graph map (x, ) A(x, ) is closed when he se L 2 ([, T ], R d ) is endowed wih he weak opology. This implies ha he map (x, ) A(x, ) is measurable wih nonempy closed values, so ha i has a Borel measurable selecion ᾱ: namely ᾱ(x, ) A(x, ) for any (x, ) (see []). Fix (x, ) R d (, T ). We define he flow for all s [, T ]. equaion. Φ(x,, s) = x + s ᾱ(x, )(τ)dτ We will use i in he nex Secion o consruc a soluion o he FokkerPlanck Lemma 3.4. The flow Φ has he semigroup propery Φ(x,, s ) = Φ(Φ(x,, s), s, s ) (4) for all s s T. Moreover for any x R d and s, s (, T ) s Φ(x,, s) = D x u(φ(x,, s), s) and Φ(x,, s ) Φ(x,, s) D x u s s. Proof. For any s (, T ) we know from Lemma 3. ha A(Φ(x,, s), s) = {ᾱ(x, ) [s,t ] } and so (4) holds. Moreover, Lemma 3. also saes ha u(, s) is differeniable a Φ(x,, s) wih D x u(φ(x,, s), s) = ᾱ(x, )(s). Bu by definiion s Φ(x,, s) = ᾱ(x, )(s) and so s Φ(x,, s) = D x u(φ(x,, s), s). Finally his las equaliy also implies he D x u Lipschiz coninuiy of Φ(x,, ) on (, T ). We finish his Secion wih he following conracion propery of he flow Φ. Lemma 3.5. If C saisfies (), hen here is some consan C 2 = C 2 (C) such ha, if u is a soluion of (), hen for all s T and x, y R d x y C 2 Φ(x,, s) Φ(y,, s). In paricular, he map x Φ(x,, s) has a Lipshiz coninuous inverse on he se Φ(R d,, s). Proof. Le u be he soluion of (). Then by Lemma 3.9 D 2 xxu C I d on R d (, T ) in he sense of disribuions. Le x(τ) = Φ(x,, s τ) and y(τ) = Φ(y,, s τ) for τ [, s ]. Then from Lemma 3.4, x( ) and y( ) saisfy respecively x (τ) = D x u(x(τ, s τ) τ [, s ) x() = Φ(x,, s) and y (τ) = D x u(y(τ, s τ) τ [, s ) y() = Φ(y,, s) We observe ha for almos all τ [, s ] we have d (x y)(τ) 2 = ((x y )(τ)) ((x y)(τ)) C (x y)(τ) 2 dτ 2 8 (5)
19 where he las inequaliy comes from (5) and he fac ha Dxxu 2 C I d (see Definiion 3.4). Hence by Grownwall s inequaliy (x y)(τ) e C/2τ x() y() for all τ [, s ]. In paricular for τ = s we ge x y e C/2τT Φ(x,, s) Φ(y,, s) hus proving he claim. 3.2 On he coninuiy equaion Our aim is now o show ha, given a soluion () and under assumpion (), he coninuiy equaion µ(x, s) div(d x u(x, s)µ(x, s)) = in R d (, T ) (6) µ(x, ) = m (x) in R d has a unique soluion which is he densiy of he measure µ(s) = Φ(,, s) m for s [, T ], where Φ(,, s) m denoes he pushforward of he measure m by he map Φ(,, s), i.e., he measure defined by Φ(,, s) m (A) = m (Φ(,, s) (A)) for any Borel se A R d. We sar by observing ha he measure Φ(,, s) m is absoluely coninuous wih respec o he Lebesgue measure. Lemma 3.6. Le C be a consan such ha () holds and such ha m is absoluely coninuous, has a compac suppor conained in he ball B(, C) and saisfies m L C. Le us se µ(s) := Φ(,, s) m for s [, T ]. Then here is a consan C 3 = C 3 (C) such ha, for any s [, T ], µ(s) is absoluely coninuous, has a compac suppor conained in he ball B(, C 3 ) and saisfies µ(s) L C 3. Moreover d(µ(s ), µ(s)) D x u s s for all s s T. Proof. By definiion µ saisfies d(µ(s ), µ(s)) Φ(x,, s ) Φ(x,, s) dm (x) D x u (s s). R d Recall ha Φ is given by Φ(x,, s) = x + s ᾱ(x, )(τ)dτ where ᾱ(x, )(τ) = D x u(φ(x,, τ), τ). Also since u is a soluion of (), D x u C. Addiionally, m has compac suppor conained in B(, C). Hence he (µ(s)) have a compac suppor conained in B(, R) where R = C + T C. Le us now fix [, T ]. From Lemma 3.5, we know ha here is some C 2 = C 2 (T ) such ha he map x Φ(x,, ) has a C 2 Lipschiz coninuous inverse on he se Φ(R d,, ). Le us denoe his inverse by Ψ. Then, if A is a Borel subse of R d we have µ(s)(a) = m (Φ (,, )(A)) = m (Ψ(A)) m L d (Ψ(A)) m C 2 L d (A). 9
The Transport Equation
The Transpor Equaion Consider a fluid, flowing wih velociy, V, in a hin sraigh ube whose cross secion will be denoed by A. Suppose he fluid conains a conaminan whose concenraion a posiion a ime will be
More informationNotes on Mean Field Games
Notes on Mean Field Games (from P.L. Lions lectures at Collège de France) Pierre Cardaliaguet January 5, 202 Contents Introduction 2 2 Nash equilibria in games with a large number of players 4 2. Symmetric
More informationStochastic Optimal Control Problem for Life Insurance
Sochasic Opimal Conrol Problem for Life Insurance s. Basukh 1, D. Nyamsuren 2 1 Deparmen of Economics and Economerics, Insiue of Finance and Economics, Ulaanbaaar, Mongolia 2 School of Mahemaics, Mongolian
More informationOptimal Stock Selling/Buying Strategy with reference to the Ultimate Average
Opimal Sock Selling/Buying Sraegy wih reference o he Ulimae Average Min Dai Dep of Mah, Naional Universiy of Singapore, Singapore Yifei Zhong Dep of Mah, Naional Universiy of Singapore, Singapore July
More informationnonlocal conditions.
ISSN 17493889 prin, 17493897 online Inernaional Journal of Nonlinear Science Vol.11211 No.1,pp.39 Boundary Value Problem for Some Fracional Inegrodifferenial Equaions wih Nonlocal Condiions Mohammed
More informationMTH6121 Introduction to Mathematical Finance Lesson 5
26 MTH6121 Inroducion o Mahemaical Finance Lesson 5 Conens 2.3 Brownian moion wih drif........................... 27 2.4 Geomeric Brownian moion........................... 28 2.5 Convergence of random
More informationMathematics in Pharmacokinetics What and Why (A second attempt to make it clearer)
Mahemaics in Pharmacokineics Wha and Why (A second aemp o make i clearer) We have used equaions for concenraion () as a funcion of ime (). We will coninue o use hese equaions since he plasma concenraions
More informationRandom Walk in 1D. 3 possible paths x vs n. 5 For our random walk, we assume the probabilities p,q do not depend on time (n)  stationary
Random Walk in D Random walks appear in many cones: diffusion is a random walk process undersanding buffering, waiing imes, queuing more generally he heory of sochasic processes gambling choosing he bes
More informationA UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS
A UNIFIED APPROACH TO MATHEMATICAL OPTIMIZATION AND LAGRANGE MULTIPLIER THEORY FOR SCIENTISTS AND ENGINEERS RICHARD A. TAPIA Appendix E: Differeniaion in Absrac Spaces I should be no surprise ha he differeniaion
More informationA Generalized Bivariate OrnsteinUhlenbeck Model for Financial Assets
A Generalized Bivariae OrnseinUhlenbeck Model for Financial Asses Romy Krämer, Mahias Richer Technische Universiä Chemniz, Fakulä für Mahemaik, 917 Chemniz, Germany Absrac In his paper, we sudy mahemaical
More informationOptimal Investment and Consumption Decision of Family with Life Insurance
Opimal Invesmen and Consumpion Decision of Family wih Life Insurance Minsuk Kwak 1 2 Yong Hyun Shin 3 U Jin Choi 4 6h World Congress of he Bachelier Finance Sociey Torono, Canada June 25, 2010 1 Speaker
More informationAn Optimal Selling Strategy for Stock Trading Based on Predicting the Maximum Price
An Opimal Selling Sraegy for Sock Trading Based on Predicing he Maximum Price Jesper Lund Pedersen Universiy of Copenhagen An opimal selling sraegy for sock rading is presened in his paper. An invesor
More informationANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS
ANALYSIS AND COMPARISONS OF SOME SOLUTION CONCEPTS FOR STOCHASTIC PROGRAMMING PROBLEMS R. Caballero, E. Cerdá, M. M. Muñoz and L. Rey () Deparmen of Applied Economics (Mahemaics), Universiy of Málaga,
More informationTerm Structure of Prices of Asian Options
Term Srucure of Prices of Asian Opions Jirô Akahori, Tsuomu Mikami, Kenji Yasuomi and Teruo Yokoa Dep. of Mahemaical Sciences, Risumeikan Universiy 111 Nojihigashi, Kusasu, Shiga 5258577, Japan Email:
More informationSignal Processing and Linear Systems I
Sanford Universiy Summer 214215 Signal Processing and Linear Sysems I Lecure 5: Time Domain Analysis of Coninuous Time Sysems June 3, 215 EE12A:Signal Processing and Linear Sysems I; Summer 1415, Gibbons
More informationA NOTE ON THE ALMOST EVERYWHERE CONVERGENCE OF ALTERNATING SEQUENCES WITH DUNFORD SCHWARTZ OPERATORS
C O L L O Q U I U M M A T H E M A T I C U M VOL. LXII 1991 FASC. I A OTE O THE ALMOST EVERYWHERE COVERGECE OF ALTERATIG SEQUECES WITH DUFORD SCHWARTZ OPERATORS BY RYOTARO S A T O (OKAYAMA) 1. Inroducion.
More informationA general decomposition formula for derivative prices in stochastic volatility models
A general decomposiion formula for derivaive prices in sochasic volailiy models Elisa Alòs Universia Pompeu Fabra C/ Ramón rias Fargas, 57 85 Barcelona Absrac We see ha he price of an european call opion
More informationInductance and Transient Circuits
Chaper H Inducance and Transien Circuis Blinn College  Physics 2426  Terry Honan As a consequence of Faraday's law a changing curren hrough one coil induces an EMF in anoher coil; his is known as muual
More informationWorking Paper On the timing option in a futures contract. SSE/EFI Working Paper Series in Economics and Finance, No. 619
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Biagini, Francesca;
More informationOn Galerkin Approximations for the Zakai Equation with Diffusive and Point Process Observations
On Galerkin Approximaions for he Zakai Equaion wih Diffusive and Poin Process Observaions An der Fakulä für Mahemaik und Informaik der Universiä Leipzig angenommene DISSERTATION zur Erlangung des akademischen
More informationStochastic Calculus and Option Pricing
Sochasic Calculus and Opion Pricing Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Sochasic Calculus 15.450, Fall 2010 1 / 74 Ouline 1 Sochasic Inegral 2 Iô s Lemma 3 BlackScholes
More informationAND BACKWARD SDE. Nizar Touzi nizar.touzi@polytechnique.edu. Ecole Polytechnique Paris Département de Mathématiques Appliquées
OPIMAL SOCHASIC CONROL, SOCHASIC ARGE PROBLEMS, AND BACKWARD SDE Nizar ouzi nizar.ouzi@polyechnique.edu Ecole Polyechnique Paris Déparemen de Mahémaiques Appliquées Chaper 12 by Agnès OURIN May 21 2 Conens
More informationOptimalCompensationwithHiddenAction and LumpSum Payment in a ContinuousTime Model
Appl Mah Opim (9) 59: 99 46 DOI.7/s45895 OpimalCompensaionwihHiddenAcion and LumpSum Paymen in a ConinuousTime Model Jakša Cvianić Xuhu Wan Jianfeng Zhang Published online: 6 June 8 Springer Science+Business
More informationTime Consistency in Portfolio Management
1 Time Consisency in Porfolio Managemen Traian A Pirvu Deparmen of Mahemaics and Saisics McMaser Universiy Torono, June 2010 The alk is based on join work wih Ivar Ekeland Time Consisency in Porfolio Managemen
More informationDYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS
DYNAMIC MODELS FOR VALUATION OF WRONGFUL DEATH PAYMENTS Hong Mao, Shanghai Second Polyechnic Universiy Krzyszof M. Osaszewski, Illinois Sae Universiy Youyu Zhang, Fudan Universiy ABSTRACT Liigaion, exper
More informationTechnical Appendix to Risk, Return, and Dividends
Technical Appendix o Risk, Reurn, and Dividends Andrew Ang Columbia Universiy and NBER Jun Liu UC San Diego This Version: 28 Augus, 2006 Columbia Business School, 3022 Broadway 805 Uris, New York NY 10027,
More informationOn the degrees of irreducible factors of higher order Bernoulli polynomials
ACTA ARITHMETICA LXII.4 (1992 On he degrees of irreducible facors of higher order Bernoulli polynomials by Arnold Adelberg (Grinnell, Ia. 1. Inroducion. In his paper, we generalize he curren resuls on
More informationPricing FixedIncome Derivaives wih he ForwardRisk Adjused Measure Jesper Lund Deparmen of Finance he Aarhus School of Business DK8 Aarhus V, Denmark Email: jel@hha.dk Homepage: www.hha.dk/~jel/ Firs
More informationDynamic Information. Albina Danilova Department of Mathematical Sciences Carnegie Mellon University. September 16, 2008. Abstract
Sock Marke Insider Trading in Coninuous Time wih Imperfec Dynamic Informaion Albina Danilova Deparmen of Mahemaical Sciences Carnegie Mellon Universiy Sepember 6, 28 Absrac This paper sudies he equilibrium
More informationSEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION. Chavchavadze Ave. 17 a, Tbilisi, Georgia, Email: toronj333@yahoo.
SEMIMARTINGALE STOCHASTIC APPROXIMATION PROCEDURE AND RECURSIVE ESTIMATION N. LAZRIEVA, 2, T. SHARIA 3, 2 AND T. TORONJADZE Georgian American Universiy, Business School, 3, Alleyway II, Chavchavadze Ave.
More informationVerification Theorems for Models of Optimal Consumption and Investment with Retirement and Constrained Borrowing
MATHEMATICS OF OPERATIONS RESEARCH Vol. 36, No. 4, November 2, pp. 62 635 issn 364765X eissn 526547 364 62 hp://dx.doi.org/.287/moor..57 2 INFORMS Verificaion Theorems for Models of Opimal Consumpion
More informationON THE PRICING OF EQUITYLINKED LIFE INSURANCE CONTRACTS IN GAUSSIAN FINANCIAL ENVIRONMENT
Teor Imov r.amaem.sais. Theor. Probabiliy and Mah. Sais. Vip. 7, 24 No. 7, 25, Pages 15 111 S 949(5)6344 Aricle elecronically published on Augus 12, 25 ON THE PRICING OF EQUITYLINKED LIFE INSURANCE
More information17 Laplace transform. Solving linear ODE with piecewise continuous right hand sides
7 Laplace ransform. Solving linear ODE wih piecewise coninuous righ hand sides In his lecure I will show how o apply he Laplace ransform o he ODE Ly = f wih piecewise coninuous f. Definiion. A funcion
More information= r t dt + σ S,t db S t (19.1) with interest rates given by a mean reverting OrnsteinUhlenbeck or Vasicek process,
Chaper 19 The BlackScholesVasicek Model The BlackScholesVasicek model is given by a sandard imedependen BlackScholes model for he sock price process S, wih imedependen bu deerminisic volailiy σ
More informationViscosity Solution of Optimal Stopping Problem for Stochastic Systems with Bounded Memory
Viscosiy Soluion of Opimal Sopping Problem for Sochasic Sysems wih Bounded Memory MouHsiung Chang Tao Pang Mousapha Pemy April 5, 202 Absrac We consider a finie ime horizon opimal sopping problem for
More informationadaptive control; stochastic systems; certainty equivalence principle; longterm
COMMUICATIOS I IFORMATIO AD SYSTEMS c 2006 Inernaional Press Vol. 6, o. 4, pp. 299320, 2006 003 ADAPTIVE COTROL OF LIEAR TIME IVARIAT SYSTEMS: THE BET O THE BEST PRICIPLE S. BITTATI AD M. C. CAMPI Absrac.
More informationAlmostsure hedging with permanent price impact
Almossure hedging wih permanen price impac B. Bouchard and G. Loeper and Y. Zou November 3, 215 Absrac We consider a financial model wih permanen price impac. Coninuous ime rading dynamics are derived
More informationWhen to Cross the Spread?  Trading in TwoSided Limit Order Books 
When o Cross he Spread?  rading in wosided Limi Order Books  Ulrich Hors and Felix Naujoka Insiu für Mahemaik HumboldUniversiä zu Berlin Uner den Linden 6, 0099 Berlin Germany email: {hors,naujoka}@mah.huberlin.de
More informationWorking Paper When to cross the spread: Curve following with singular control
econsor www.econsor.eu Der OpenAccessPublikaionsserver der ZBW LeibnizInformaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Naujoka, Felix; Hors,
More informationDependent Interest and Transition Rates in Life Insurance
Dependen Ineres and ransiion Raes in Life Insurance Krisian Buchard Universiy of Copenhagen and PFA Pension January 28, 2013 Absrac In order o find marke consisen bes esimaes of life insurance liabiliies
More informationSAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS
The Annals of Probabiliy 2004, Vol. 32, No. 1A, 1 27 Insiue of Mahemaical Saisics, 2004 SAMPLE PATH PROPERTIES OF THE STOCHASTIC FLOWS BY DMITRY DOLGOPYAT, 1 VADIM KALOSHIN 2 AND LEONID KORALOV 3 Universiyof
More informationEmergence of FokkerPlanck Dynamics within a Closed Finite Spin System
Emergence of FokkerPlanck Dynamics wihin a Closed Finie Spin Sysem H. Niemeyer(*), D. Schmidke(*), J. Gemmer(*), K. Michielsen(**), H. de Raed(**) (*)Universiy of Osnabrück, (**) Supercompuing Cener Juelich
More informationINDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES
Journal of Applied Analysis 1, 1 (1995), pp. 39 45 INDEPENDENT MARGINALS OF OPERATOR LÉVY S PROBABILITY MEASURES ON FINITE DIMENSIONAL VECTOR SPACES A. LUCZAK Absrac. We find exponens of independen marginals
More informationA DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL
A DYNAMIC PROGRAMMING APPROACH TO THE PARISI FUNCTIONAL AUKOSH JAGANNATH AND IAN TOBASCO Absrac. G. Parisi prediced an imporan variaional formula for he hermodynamic limi of he inensive free energy for
More informationLectures # 5 and 6: The Prime Number Theorem.
Lecures # 5 and 6: The Prime Number Theorem Noah Snyder July 8, 22 Riemann s Argumen Riemann used his analyically coninued ζfuncion o skech an argumen which would give an acual formula for π( and sugges
More informationOptimal Time to Sell in Real Estate Portfolio Management
Opimal ime o Sell in Real Esae Porfolio Managemen Fabrice Barhélémy and JeanLuc Prigen hema, Universiy of CergyPonoise, CergyPonoise, France Emails: fabricebarhelemy@ucergyfr; jeanlucprigen@ucergyfr
More informationEconomics Honors Exam 2008 Solutions Question 5
Economics Honors Exam 2008 Soluions Quesion 5 (a) (2 poins) Oupu can be decomposed as Y = C + I + G. And we can solve for i by subsiuing in equaions given in he quesion, Y = C + I + G = c 0 + c Y D + I
More informationT ϕ t ds t + ψ t db t,
16 PRICING II: MARTINGALE PRICING 2. Lecure II: Pricing European Derivaives 2.1. The fundamenal pricing formula for European derivaives. We coninue working wihin he Black and Scholes model inroduced in
More information2.5 Life tables, force of mortality and standard life insurance products
Soluions 5 BS4a Acuarial Science Oford MT 212 33 2.5 Life ables, force of moraliy and sandard life insurance producs 1. (i) n m q represens he probabiliy of deah of a life currenly aged beween ages + n
More informationChapter 7. Response of FirstOrder RL and RC Circuits
Chaper 7. esponse of FirsOrder L and C Circuis 7.1. The Naural esponse of an L Circui 7.2. The Naural esponse of an C Circui 7.3. The ep esponse of L and C Circuis 7.4. A General oluion for ep and Naural
More informationIntroduction to Stochastic Calculus
IEOR E477: Financial Engineering: Coninuousime Models Fall 21 c 21 by Marin Haugh Inroducion o Sochasic Calculus hese noes provide an inroducion o sochasic calculus, he branch of mahemaics ha is mos idenified
More informationDifferential Equations. Solving for Impulse Response. Linear systems are often described using differential equations.
Differenial Equaions Linear sysems are ofen described using differenial equaions. For example: d 2 y d 2 + 5dy + 6y f() d where f() is he inpu o he sysem and y() is he oupu. We know how o solve for y given
More informationPROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE
Profi Tes Modelling in Life Assurance Using Spreadshees PROFIT TEST MODELLING IN LIFE ASSURANCE USING SPREADSHEETS PART ONE Erik Alm Peer Millingon 2004 Profi Tes Modelling in Life Assurance Using Spreadshees
More information3 RungeKutta Methods
3 RungeKua Mehods In conras o he mulisep mehods of he previous secion, RungeKua mehods are singlesep mehods however, muliple sages per sep. They are moivaed by he dependence of he Taylor mehods on he
More informationCointegration: The Engle and Granger approach
Coinegraion: The Engle and Granger approach Inroducion Generally one would find mos of he economic variables o be nonsaionary I(1) variables. Hence, any equilibrium heories ha involve hese variables require
More informationConditional Default Probability and Density
Condiional Defaul Probabiliy and Densiy N. El Karoui, M. Jeanblanc, Y. Jiao, B. Zargari Absrac This paper proposes differen mehods o consruc condiional survival processes, i.e, families of maringales decreasing
More informationINTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES
INTEREST RATE FUTURES AND THEIR OPTIONS: SOME PRICING APPROACHES OPENGAMMA QUANTITATIVE RESEARCH Absrac. Exchangeraded ineres rae fuures and heir opions are described. The fuure opions include hose paying
More informationcooking trajectory boiling water B (t) microwave 0 2 4 6 8 101214161820 time t (mins)
Alligaor egg wih calculus We have a large alligaor egg jus ou of he fridge (1 ) which we need o hea o 9. Now here are wo accepable mehods for heaing alligaor eggs, one is o immerse hem in boiling waer
More informationThe option pricing framework
Chaper 2 The opion pricing framework The opion markes based on swap raes or he LIBOR have become he larges fixed income markes, and caps (floors) and swapions are he mos imporan derivaives wihin hese markes.
More informationOn the Role of the Growth Optimal Portfolio in Finance
QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 144 January 2005 On he Role of he Growh Opimal Porfolio in Finance Eckhard Plaen ISSN 14418010 www.qfrc.us.edu.au
More informationOption PutCall Parity Relations When the Underlying Security Pays Dividends
Inernaional Journal of Business and conomics, 26, Vol. 5, No. 3, 22523 Opion Puall Pariy Relaions When he Underlying Securiy Pays Dividends Weiyu Guo Deparmen of Finance, Universiy of Nebraska Omaha,
More informationThe Heisenberg group and Pansu s Theorem
The Heisenberg group and Pansu s Theorem July 31, 2009 Absrac The goal of hese noes is o inroduce he reader o he Heisenberg group wih is Carno Carahéodory meric and o Pansu s differeniaion heorem. As
More informationAP Calculus AB 2013 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a missiondriven noforprofi organizaion ha connecs sudens o college success and opporuniy. Founded in 19, he College Board was
More informationAP Calculus BC 2010 Scoring Guidelines
AP Calculus BC Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in, he College Board
More informationDistributing Human Resources among Software Development Projects 1
Disribuing Human Resources among Sofware Developmen Proecs Macario Polo, María Dolores Maeos, Mario Piaini and rancisco Ruiz Summary This paper presens a mehod for esimaing he disribuion of human resources
More informationKeldysh Formalism: Nonequilibrium Green s Function
Keldysh Formalism: Nonequilibrium Green s Funcion Jinshan Wu Deparmen of Physics & Asronomy, Universiy of Briish Columbia, Vancouver, B.C. Canada, V6T 1Z1 (Daed: November 28, 2005) A review of Nonequilibrium
More informationAn Optimal Control Approach to InventoryProduction Systems with Weibull Distributed Deterioration
Journal of Mahemaics and Saisics 5 (3):64, 9 ISSN 5493644 9 Science Publicaions An Opimal Conrol Approach o InvenoryProducion Sysems wih Weibull Disribued Deerioraion Md. Aiul Baen and Anon Abdulbasah
More informationOption Pricing Under Stochastic Interest Rates
I.J. Engineering and Manufacuring, 0,3, 889 ublished Online June 0 in MECS (hp://www.mecspress.ne) DOI: 0.585/ijem.0.03. Available online a hp://www.mecspress.ne/ijem Opion ricing Under Sochasic Ineres
More informationPRICING and STATIC REPLICATION of FX QUANTO OPTIONS
PRICING and STATIC REPLICATION of F QUANTO OPTIONS Fabio Mercurio Financial Models, Banca IMI 1 Inroducion 1.1 Noaion : he evaluaion ime. τ: he running ime. S τ : he price a ime τ in domesic currency of
More informationEndpoint Strichartz estimates and global solutions for the nonlinear Dirac equation 1
Endpoin Sricharz esimaes and global soluions for he nonlinear Dirac equaion 1 Shuji Machihara, Makoo Nakamura, Kenji Nakanishi, and Tohru Ozawa Absrac. We prove endpoin Sricharz esimaes for he KleinGordon
More informationA ProductionInventory System with Markovian Capacity and Outsourcing Option
OPERATIONS RESEARCH Vol. 53, No. 2, March April 2005, pp. 328 349 issn 0030364X eissn 15265463 05 5302 0328 informs doi 10.1287/opre.1040.0165 2005 INFORMS A ProducionInvenory Sysem wih Markovian Capaciy
More informationModeling VIX Futures and Pricing VIX Options in the Jump Diusion Modeling
Modeling VIX Fuures and Pricing VIX Opions in he Jump Diusion Modeling Faemeh Aramian Maseruppsas i maemaisk saisik Maser hesis in Mahemaical Saisics Maseruppsas 2014:2 Maemaisk saisik April 2014 www.mah.su.se
More informationDuration and Convexity ( ) 20 = Bond B has a maturity of 5 years and also has a required rate of return of 10%. Its price is $613.
Graduae School of Business Adminisraion Universiy of Virginia UVAF38 Duraion and Convexiy he price of a bond is a funcion of he promised paymens and he marke required rae of reurn. Since he promised
More informationMultiprocessor SystemsonChips
Par of: Muliprocessor SysemsonChips Edied by: Ahmed Amine Jerraya and Wayne Wolf Morgan Kaufmann Publishers, 2005 2 Modeling Shared Resources Conex swiching implies overhead. On a processing elemen,
More informationTWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE
Annals of he Academy of Romanian Scieniss Series on Mahemaics and is Applicaions ISSN 2666594 Volume 3, Number 2 / 211 TWO OPTIMAL CONTROL PROBLEMS IN CANCER CHEMOTHERAPY WITH DRUG RESISTANCE Werner Krabs
More informationCommunication Networks II Contents
3 / 1  Communicaion Neworks II (Görg)  www.comnes.unibremen.de Communicaion Neworks II Conens 1 Fundamenals of probabiliy heory 2 Traffic in communicaion neworks 3 Sochasic & Markovian Processes (SP
More information1. Introduction. We consider a ddimensional stochastic differential equation (SDE) defined by
SIAM J. CONROL OPIM. Vol. 43, No. 5, pp. 1676 1713 c 5 Sociey for Indusrial and Applied Mahemaics SENSIIVIY ANALYSIS USING IÔ MALLIAVIN CALCULUS AND MARINGALES, AND APPLICAION O SOCHASIC OPIMAL CONROL
More informationDifferential Equations and Linear Superposition
Differenial Equaions and Linear Superposiion Basic Idea: Provide soluion in closed form Like Inegraion, no general soluions in closed form Order of equaion: highes derivaive in equaion e.g. dy d dy 2 y
More informationOptimal Reinsurance/Investment Problems for General Insurance Models
Opimal Reinsurance/Invesmen Problems for General Insurance Models Yuping Liu and Jin Ma Absrac. In his paper he uiliy opimizaion problem for a general insurance model is sudied. he reserve process of he
More informationChapter 8: Regression with Lagged Explanatory Variables
Chaper 8: Regression wih Lagged Explanaory Variables Time series daa: Y for =1,..,T End goal: Regression model relaing a dependen variable o explanaory variables. Wih ime series new issues arise: 1. One
More information4 Convolution. Recommended Problems. x2[n] 1 2[n]
4 Convoluion Recommended Problems P4.1 This problem is a simple example of he use of superposiion. Suppose ha a discreeime linear sysem has oupus y[n] for he given inpus x[n] as shown in Figure P4.11.
More informationLECTURE 7 Interest Rate Models I: Short Rate Models
LECTURE 7 Ineres Rae Models I: Shor Rae Models Spring Term 212 MSc Financial Engineering School of Economics, Mahemaics and Saisics Birkbeck College Lecurer: Adriana Breccia email: abreccia@emsbbkacuk
More informationAppendix A: Area. 1 Find the radius of a circle that has circumference 12 inches.
Appendi A: Area workedou s o OddNumbered Eercises Do no read hese workedou s before aemping o do he eercises ourself. Oherwise ou ma mimic he echniques shown here wihou undersanding he ideas. Bes wa
More informationPATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM
PATHWISE PROPERTIES AND PERFORMANCE BOUNDS FOR A PERISHABLE INVENTORY SYSTEM WILLIAM L. COOPER Deparmen of Mechanical Engineering, Universiy of Minnesoa, 111 Church Sree S.E., Minneapolis, MN 55455 billcoop@me.umn.edu
More informationGraduate Macro Theory II: Notes on Neoclassical Growth Model
Graduae Macro Theory II: Noes on Neoclassical Growh Model Eric Sims Universiy of Nore Dame Spring 2011 1 Basic Neoclassical Growh Model The economy is populaed by a large number of infiniely lived agens.
More informationDETERMINISTIC INVENTORY MODEL FOR ITEMS WITH TIME VARYING DEMAND, WEIBULL DISTRIBUTION DETERIORATION AND SHORTAGES KUNSHAN WU
Yugoslav Journal of Operaions Research 2 (22), Number, 67 DEERMINISIC INVENORY MODEL FOR IEMS WIH IME VARYING DEMAND, WEIBULL DISRIBUION DEERIORAION AND SHORAGES KUNSHAN WU Deparmen of Bussines Adminisraion
More informationAnswer, Key Homework 2 David McIntyre 45123 Mar 25, 2004 1
Answer, Key Homework 2 Daid McInyre 4123 Mar 2, 2004 1 This prinou should hae 1 quesions. Muliplechoice quesions may coninue on he ne column or page find all choices before making your selecion. The
More informationAP Calculus AB 2010 Scoring Guidelines
AP Calculus AB 1 Scoring Guidelines The College Board The College Board is a noforprofi membership associaion whose mission is o connec sudens o college success and opporuniy. Founded in 1, he College
More informationThe Torsion of Thin, Open Sections
EM 424: Torsion of hin secions 26 The Torsion of Thin, Open Secions The resuls we obained for he orsion of a hin recangle can also be used be used, wih some qualificaions, for oher hin open secions such
More informationDistributed and Secure Computation of Convex Programs over a Network of Connected Processors
DCDIS CONFERENCE GUELPH, ONTARIO, CANADA, JULY 2005 1 Disribued and Secure Compuaion of Convex Programs over a Newor of Conneced Processors Michael J. Neely Universiy of Souhern California hp://wwwrcf.usc.edu/
More information1. y 5y + 6y = 2e t Solution: Characteristic equation is r 2 5r +6 = 0, therefore r 1 = 2, r 2 = 3, and y 1 (t) = e 2t,
Homework6 Soluions.7 In Problem hrough 4 use he mehod of variaion of parameers o find a paricular soluion of he given differenial equaion. Then check your answer by using he mehod of undeermined coeffiens..
More informationTail Distortion Risk and Its Asymptotic Analysis
Tail Disorion Risk and Is Asympoic Analysis Li Zhu Haijun Li May 2 Revision: March 22 Absrac A disorion risk measure used in finance and insurance is defined as he expeced value of poenial loss under a
More informationNiche Market or Mass Market?
Niche Marke or Mass Marke? Maxim Ivanov y McMaser Universiy July 2009 Absrac The de niion of a niche or a mass marke is based on he ranking of wo variables: he monopoly price and he produc mean value.
More informationAnalysis of Tailored BaseSurge Policies in Dual Sourcing Inventory Systems
Analysis of Tailored BaseSurge Policies in Dual Sourcing Invenory Sysems Ganesh Janakiraman, 1 Sridhar Seshadri, 2, Anshul Sheopuri. 3 Absrac We sudy a model of a firm managing is invenory of a single
More informationSensitivity Analysis for Averaged Asset Price Dynamics with Gamma Processes
Sensiiviy Analysis for Averaged Asse Price Dynamics wih Gamma Processes REIICHIRO KAWAI AND ASUSHI AKEUCHI Absrac he main purpose of his paper is o derive unbiased Mone Carlo esimaors of various sensiiviy
More informationValuation of Life Insurance Contracts with Simulated Guaranteed Interest Rate
Valuaion of Life Insurance Conracs wih Simulaed uaraneed Ineres Rae Xia uo and ao Wang Deparmen of Mahemaics Royal Insiue of echnology 100 44 Sockholm Acknowledgmens During he progress of he work on his
More information1 HALFLIFE EQUATIONS
R.L. Hanna Page HALFLIFE EQUATIONS The basic equaion ; he saring poin ; : wrien for ime: x / where fracion of original maerial and / number of halflives, and / log / o calculae he age (# ears): age (halflife)
More informationJournal Of Business & Economics Research September 2005 Volume 3, Number 9
Opion Pricing And Mone Carlo Simulaions George M. Jabbour, (Email: jabbour@gwu.edu), George Washingon Universiy YiKang Liu, (yikang@gwu.edu), George Washingon Universiy ABSTRACT The advanage of Mone Carlo
More informationCoupling Online and Offline Analyses for Random Power Law Graphs
Inerne Mahemaics Vol, No 4: 40946 Coupling Online and Offline Analyses for Random Power Law Graphs FanChungandLinyuanLu Absrac We develop a coupling echnique for analyzing online models by using offline
More informationUNIVERSITY OF CALGARY. Modeling of Currency Trading Markets and Pricing Their Derivatives in a Markov. Modulated Environment.
UNIVERSITY OF CALGARY Modeling of Currency Trading Markes and Pricing Their Derivaives in a Markov Modulaed Environmen by Maksym Terychnyi A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL
More information