RISK-SENSITIVE ASSET MANAGEMENT WITH GENERAL FACTOR MODELS. ( ) : dst 0 = r(y t )St 0 dt, S0 0 = s 0 0 i(i = 1,, m) : ds i t = S i t : { µ i (Y t )dt

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1 RISK-SENSITIVE ASSET MANAGEMENT WITH GENERAL FACTOR MODELS. ( ) : dst = r(y t )St dt, S = s i(i =,, m) : ds i t = S i t : { µ i (Y t )dt + n+m k= dy t = g(y t )dt + λ(y t )dw t, σ i k(y t )dw k t }, S i = s i Y () = y R n W t = (W k t ) k=,,(n+m) n + m σ, λ, µ, g, r R m (n+m), R n (n+m), R m, R n, R- (A) λ, g, σ, µ, r, (A) x, η R n, ξ R m, µ ξ ξ σσ (x)ξ µ ξ, µ η η λλ (x)η µ η, µ, µ > ( ) (A3) r πt i i ( ), = (,, ) Xt π dx π t X π t = ( π t ) ds t S t + m i= π i t dst i, X π St i = (FRS) Γ T (γ) := sup π A T γ log E [(Xπ T ) γ ] γ (, ) (, ) A T { γ (, ) ( ) Nagai(3) γ (, ) ( ) () ( )HJB (.) ( HJB (.) sup ] sup ˇπ ) π R m [

2 () HJB (.) ˇπ (.)( ) (3) (.) Verification Theorem ( ˇπ ) (FRS) HJB (.) v t + tr(λ(y)λ(y) D v) + γ (Dv) λ(y)λ(y) Dv + g(y) Dv + r(y) [ + sup γ ] π σ(y)σ(y) π + π {µ(y) r(y) + γσ(y)λ(y) Dv} π R m v(t, y) = v t + tr(λ(y)λ(y) D v) + B(y) Dv + (Dv) Q(y)Dv + U(y) =, (.) v(t, y) = Q, B, U { Q(y) := γλ(y) I + γ } γ σ(y) (σ(y)σ(y) ) σ(y) λ(y), B(y) := g(y) + γ γ λ(y)σ(y) (σ(y)σ(y) ) (µ(y) r(y)), U(y) := ( γ) (µ(y) r(y)) (σ(y)σ(y) ) (µ(y) r(y)) + r(y) Theorem.. (A) (A3) (A4) Dξ (y) C ( y + ), C > tr(λ(y)λ(y) D ξ ) + B(y) Dξ + (Dξ ) Q(y)Dξ + U(y), y ξ (A5) (µ(y) r(y)) (σ(y)σ(y) ) (µ(y) r(y)) + r(y), y. γ D := {γ (, ) : (A4) is satisfied} (.) v v v, t, D i v, D ij v L p (, T ; L p loc (Rn )), < p <.. γ D π(t, Y t ) := γ (σ(y t)σ(y t ) ) {µ(y t ) r(y t ) + γσ(y t )λ(y t ) D v(t, Y t )} (FRS) Γ T (γ) = v(, y) [] H. Hata (5) Risk-sensitive asset management with general factor models, preprint. [] H. Nagai (3) Optimal strategies for risk-sensitive portfolio optimization problems for general factor models, SIAM J. Cont. Optim. 4, 779 8, MR =,

3 Local risk-minimization for Barndorff-Nielsen and Shephard models ( ) stochastic volatility Barndorff-Nielsen and Shephard (BNS) locally risk-minimizing (LRM) BNS T > S t : { t ( ) t } S t = S exp µ + βσs ds + σ s dw s. S >, µ, β R, ρ W dσ t = λσ t dt + dh λt, σ >, volatility σ λ >, H subordinator( Lévy ) LRM X c R ξ T X = c + ξ t ds t c F ξ F selffinancing( ) self-financing L - t η t, ξ t V t = η t + ξ t S t t C t C t = V t t ξ sds s C = V C t F ( ) ( )

4 F = V T = C T + T ξ sds s C t C t η ξ φ = (η, ξ) C t (φ) R t (φ) R t (φ) := E [(C T (φ) C t (φ)) ] Ft φ R t (φ) R t ( φ) a.s. for every t [, T], risk-minimization LRM K > X = (S T K) + BNS LRM. β = ([3]). ρ = ([]) Föllmer-Schweizer Lévy Malliavin [4] Clark-Ocone ([5]) Malliavin LRM minimal martingale measure(mmm) equivalent martingale measure density Malliavin S t MMM Malliavin [] References [] Arai, T. (5) Local risk-minimization for Barndorff-Nielsen and Shephard models with volatility risk premium. Available at [] Arai, T., Imai, Y. and Suzuki, R. (5) Numerical analysis on local riskminimization for exponential Lévy models, to appear in International Journal of Theoretical and Applied Finance. [3] Arai, T., Imai, Y. and Suzuki, R. (5) Local risk minimization for Barndorff-Nielsen and Shephard models. Available at [4] Arai, T and Suzuki, R. (5) Local risk-minimization for Lévy markets, International Journal of Financial Engineering, Vol., 555. [5] Suzuki, R. (3) A Clark-Ocone type formula under change of measure for Lévy processes with L -Lévy measure, Commun. Stoch. Anal., Vol.7,

5 On the Euler-Maruyama approximation for one-dimensional SDEs with irregular coefficients Dai Taguchi (Ritsumeikan University) joint work with Hoang-Long Ngo (Hanoi National University of Education) In this talk, we consider the Euler-Maruyama approximation for one-dimensional stochastic differential equations. We provide the strong rate of convergence when the drift coefficient is the sum of a bounded variation function on compact sets and a Hölder continuous function, and the diffusion coefficient is a Hölder continuous function. Let X = (X t ) t T be a one-dimensional stochastic differential equation (SDE) X t = x + t b(x s )ds + t σ(x s )dw s, x R, t [, T ], () where W := (W t ) t T is a standard one-dimensional Brownian motion on a probability space (Ω, F, P). Veretennikov [6] show that if the drift coefficient b is bounded measurable and the diffusion coefficient σ is bounded, uniformly elliptic and / + α-hölder continuous for some α [, /], then the equation () has a unique strong solution. One often approximates X by using the Euler-Maruyama scheme which is defined by X (n) t = x + t ( ) b X (n) η n(s) ds + t ( ) σ X (n) η n(s) dw s, t [, T ], where η n (s) = kt/n if s [kt/n, (k + )T/n). It is well-known that if the coefficients b and σ are Lipschitz continuous functions, then the Euler-Maruyama scheme has strong rate of convergence /, that is for any p >, there exists C > such that E[ sup X t X (n) t p ] C t T n. p/ Recently, Yan [7] and Gyöngy and Rásonyi [] have shown that the strong rate in onedimensional setting with Hölder continuous diffusion coefficient. Ngo and Taguchi [4] extended the results in [, 7] for multi-dimensional SDEs with discontinuous drift coefficient (but it is onesided Lipschitz function). Halidias and Kloeden [] prove that if b is increasing, continuous from bellow and σ is a Lipschitz continuous, then X (n) converges to X in L -norm. Their proof is based on upper and lower solutions of the SDE and its Euler-Maruyama approximation, so it is difficult to get any

6 rate of convergence by using their approach. Leobacher and Szölgyenyi [3] introduce a clever way to transfer equation () with piecewise Lipschitz drift coefficient with finite number of discontinuous points, and Lipschitz continuous diffusion coefficient. Using their transformation technique, the equation () is equivalent to SDE with Lipschitz continuous coefficients. Therefore, the new equation can be approximated by its Euler-Maruyama scheme with the standard strong rate of convergence /. In this talk, we provide the rates of strong convergence of the Euler-Maruyama approximation for SDE () when the coefficients b and σ may have a very low regularity which includes every cases of [,, 3, 4, 7]. More preciously, we suppose that the drift coefficient b = b A + b H L (R) is bounded measurable, b A is a function of bounded variation on compact sets and b H is a Hölder continuous with β (, ], and the diffusion coefficient σ is bounded, uniformly elliptic and /+α- Hölder continuous with α [, /]. Under these assumptions for the coefficients b and σ, we obtain that the L -convergence rate for the Euler-Maruyama scheme is β α if α (, /] and log n if α = (Theorem.3 of [5]). For the case b / L (R), we also get the L -convergence rate, by using localization arguments, (Theorem.4 of [5]). Moerover, we obtain the L p -sup convergence rate for any p (Theorem.6,.7,.8 of [5]). The idea of proof is to use the Yamada-Watanabe approximation technique, the removal drift method and the Gaussian upper bounded for the density of the Euler-Maruyama approximation. References [] Gyöngy, I. and Rásonyi, M.: A note on Euler approximations for SDEs with Hölder continuous diffusion coefficients. Stochastic. Process. Appl., 89 (). [] Halidias, N. and Kloeden, P.E.: A note on the Euler-Maruyama scheme for stochastic differential equations with a discontinuous monotone drift coefficient. BIT 48() 5 59 (8). [3] Leobacher, G. and Szölgyenyi M.: A numerical method for SDEs with discontinuous drift. BIT Numer. Math. (5). [4] Ngo, H-L., and Taguchi, D.: Strong rate of convergence for the Euler-Maruyama approximation of stochastic differential equations with irregular coefficients. To appear in Mathematics of Computation. [5] Ngo, H-L., and Taguchi, D.: On the Euler-Maruyama approximation for one-dimensional stochastic differential equations with irregular coefficients. Preprint, arxiv: (5). [6] Veretennikov, A.Yu.: On strong solution and explicit formulas for solutions of stochastic integral equations. Math. USSR Sb. 39, (98). [7] Yan, B. L.: The Euler scheme with irregular coefficients. Ann. Probab. 3, no. 3, 7 94 ().

7 Martingale problems for diffusions on metric graphs TOMOYUKI ICHIBA We shall extend martingale problems for the Walsh Brownian motion ([], [3], [4]) on a star graph to those for diffusions on a metric graph (G, d) with a finite number of vertices (cf. []). Here the metric graph (G, d) is a collection of finite or semi-infinite edges {e E} with N vertices V := {v k, k =,..., N} for some N < + and metric d. We assume that the graph is imbedded in the Euclidian space R and that any two edges can only meet at a vertex. The metric d is defined in the canonical way as the length of a shortest path between two points in G along the edges and the length along each edge is measured with the usual Euclidian metric. The semi-infinite edges isomorphic to R + = [, ) are called external, while the finite edges isomorphic to a finite open interval are called internal. We assign the initial vertex and the terminal vertex for each edge e E via a map δ. The map δ associates each internal edge e to a pair (δ (e), δ (e)) V V with its initial vertex δ (e) and its terminal vertex δ (e). An external edge e has only the initial vertex δ (e) V and we define δ (e) = + formally. We call δ i (e), i =, the end points of e E. We write v e, if v V is one of the end points of e E, and define the set E(v) := {e E : v e} for each v V. We also assume that for each point x G \ V there exist unique e E, v V and r R +, such that x belongs to the edge e(x) with the initial vertex v(x), the terminal vertex δ (e(x)), and the length between x and the initial vertex is r(x) defined by v(x) := δ (e(x)), r(x) := d(v(x), x) ; x G. () Thus we identify the point x G in terms of such triplet (e, v, r) E V R +, and define each function g : G R with this coordinate, i.e., g(x) g(e, v, r), x G. We consider the class D of BOREL-measurable functions g : G R with the following properties: (i) they are continuous in the topology induced by the metric d ; (ii) for every e, v, the function r g e,v (r) := g(e, v, r) is twice continuously differentiable on (, ) and has finite first and second right-derivatives at the origin; (iii) the resulting functions (e, v, r) g e,v(r) and (e, v, r) g e,v(r) are BOREL measurable; and (iv) sup <r<k,e E(v),v V g e,v(r) + g e,v(r) < + holds for all finite K. For notational simplicity we write G (x) := g e,v(r), G (x) := g e,v(r) for x = (e, v, r) with r > () and for each vertex v V with e E(v) we define G (e, v, +) := lim x v,x E(v) ( )i+ G (x), if v = δ i (e), i =,. (3) Let us consider the canonical space Ω := C([, ); R ) of R -valued continuous functions on [, ) endowed with the σ -algebra F of its BOREL sets. We consider also its coordinate mapping ω( ) and the natural filtration F := {F(t), t < } with F(t) := σ(ω(s), s t), t <. For each vertex v V we shall consider a probability measure ν v (de), e E(v) on the set E(v) of edges. For example, if the total number of edges at v is finite, the measure ν v ( ) is a discrete probability measure (cf. []). In the following we shall construct the diffusion on G with drifts b := {b(x), x G}, dispersions σ := {σ(x), x G} and the vertex measures ν := {ν v ( ), v V} via the martingale problem associated with (b, σ, ν). Department of Statistics and Applied Probability, South Hall, University of California, Santa Barbara, CA 936

8 Given the vertex measures ν and BOREL measurable functions b : G R +, σ : G R \ {} we define for every g D the process M g ( ; ω) := g(ω( )) g(ω ()) Lg(ω(t)) {ω(t) E} dt, (4) where with a( ) := σ ( ) and the derivatives in () we define the infinitesimal generator Lg(x) := b(x)g (x) + a(x)g (x) ; x G. Local Martingale Problem: For every fixed x G to find a probability measure P on the canonical space (Ω, F), such that ω() = x holds P -a.e.; (ii) {ω(t) V} dt holds P - a.e.; (iii) for every function g D + (respectively, D ), the process M g ( ) in (4) is a continuous local submartingale (resp., martingale) with respect to the filtration F := {F(t+), t < }, where { } D + (resp., D ) := g D : G (e, v, +)ν v (de) (resp., ), v V. E(v) Proposition. Suppose that the drifts are identically zero and the reciprocal of the dispersion coefficient r σ e,v (r) := σ(e, v, r) = σ(x) is locally square integrable, i.e., K ( / σ e,v(r))dr < + for every compact subset K of e := e {δ (e)} {δ (e)}, e E(v), v V. Then the local martingale problem associated with the triplet (, σ, ν) is well-posed. More generally, under appropriate conditions, by the method of time-change and the scale functions we may construct the diffusion on the graph G associated with the triplet (b, σ, ν). For a fixed initial vertex v V and its neighborhood B (as a subset of E(v ) ), the local behavior of the resulting coordinate process X(, ω) := ω( ) in B is characterized by the distance R( ) := r(x( )) (defined as in ()) from the vertex v which satisfies dr(t) = b(x(t))dt + σ(x(t))dβ(t) + dl R (t) ; t (5) locally for some Brownian motion β( ) and the local time growths dl RA (t) = ν v (A) dl R (t) ; t, (6) where L ξ ( ) is the semimartingale local time at the origin for the generic semimartingale ξ( ) and R A ( ) := R( ) {e(x( )) A} for every BOREL subset A of E(v ). This is an extension of the results in [] from the graph with a finite number of edges to that with possibly uncountable number of edges but still with a finite number N of vertices. We shall also discuss some examples (including the WALSH diffusions with N = ) and further extensions. References [] BARLOW, M., PITMAN, J. & YOR, M. (989) On Walsh s Brownian motions. Séminaire de Probabilités XXIII. Lecture Notes in Mathematics 37, [] FREIDLIN, M. & SHEU, S. () Diffusion processes on graphs: stochastic differential equations, large deviation principle. Probab. Theory and Relat. Fields [3] ICHIBA, T., KARATZAS, I., PROKAJ, V. & YAN, M. (5) Stochastic integral equations for Walsh semimartingales. Preprint available at arxiv: [4] WALSH, J. (978) A diffusion with a discontinuous local time. Astérisque 5-53,

9 ( ), ( ), ( ), Poisson {X(t)} ( Surplus ) : (.) X(t) = x + κ t t dn(s) Z N(s), x., κ >, {N(t), t } γ Poisson, {Z k, k =,, } F i.i.d. r.v. s {N(t)}. {X(t)}, T. Lundberg model,., T X(T ) κ, x, γ. F, Laplace Fourier v(x, α, β) E x [ exp{ α T + i β X T } ], (α >, β R ) ( v ). δ (cf.,.3, D ),., D dense,, F. Surplus (.) Ito,.. ( ). α >, β R, v(x, α, β) = E x [exp{ α T + i β X(T )}] : F (λ) df (y) e λ y [ x α v(x, α, β) = κ x v(x, α, β) + γ df (y) { v(x y, α, β) e i β (x y)} + + e i β x F (i β) v(x, α, β) ], x, v(x, α, β) = e i β x, x <., x Laplace. ρ γ /κ, (.) v(λ, α, β) = v(, α, β) ρ { } F (i β) F (λ) /(λ i β) { }. λ ρ F (λ) α/κ, (.), v(, α, β).

10 {X(t)} random walk, Feller ([], Ch 8, Lemma ), v(, α, β) :.. F Fourier F (ξ) = E[e i ξ Z ], (.) log v(, α, β) = π h(ξ) iξ κ γ + γ F ( ξ). lim R α ds dx e iβx R R dξ e iξ x s h(ξ)..3. F δ, M >, < a < a <... < a M,, p k <, k =,,..., M (p + + p M = ), M (.3) df (x) = p k δ(x, a k ) dx.. δ(x, a) a δ. k= (.) (.),.,.4 ( ). F δ (.3), ρ γ κ, ζ ρ + α κ { v(, α, β) = exp k Z M + v(x, α, β) = E x [ exp{ α T + i β X(T ) } ] = v(, α, β) + M l= k Z M + k Z M +, v(x, α, β) p k k! ρ k e i β k,a k,a dy e (i β ζ) y y k }, p k k! ( ρ ) k ( x k, a ) k e ζ (x k,a ) I {x> k,a } p k k! ( ρ ) k + p l dy e ζ (y k,a )+i β (x y a l) (y k, a ) k I {y> k,a } I {<x y<al }. k Z M + ( ).. [] Cramér, H., Historical review of Fillip Lundberg s work on risk theory, Skand. Aktua. (Suppl.), 5 (969), 6-. [] Feller, W., An Introduction to Probability Theory and its Aplications, Vol. II, John Wiley & Sons, 97. [3] Lundberg, F., Über die Wahrscheinlichkeitsfunktion einer Risikenmasse. Skand. Aktua., 3 (93), -83. [4] Nishioka, K. and Igarashi, T., Non-ruin probability of an insurer under the Lundberg model,,, 93 (4), 4 47.

11 *,, 3 Ryo Oizumi*, Toshikazu Kuniya, Yoichi Enatsu,, 3. A R d a (, α) X a A X = x A Ito { dx j a (t) = g j (X a (t), v a, Γ t,x ) da + N k= σ jk (X a (t), v a, Γ t,x ) db k a, j d X j = xj () dx j a (t) := X j a+ε (t + ε) X j a (t) v = v a := (v (a), v (a),, v l (a))

12 [] t Γ t,x := ( ) Γ t,x, Γ t,x,, Γ m t,x,, Γ M t,x t a y A P t (a, x y) P t+ε (a + ε, x y) = A dξ K ε,t (ξ y) P t (a, x ξ) P t (, x y) = n t (x) δ d (x y) }{{} α. () n t (x) = dady F (y, Γ t,x ) P t (, x y) A }{{} t a ε Ito nonlocal Eq.(). Hamilton Jacobi Bellman a w {[ λ,a (x, Γ) inf Hv x (Γ) + λ ] w λ,a (x, Γ) } = v V w λ,α (x, Γ) = F (x, Γ) d H x v (Γ) := g j (x, v, Γ) x j d c jj (x, v, Γ) Γ R M +, j= ψ λ (x, Γ) = α j,j = x j x j + µ (x, v, Γ) (3) da w λ,a (x, Γ) =, (4) [] R. Oizumi, Unification theory of optimal life histories and linear demographic models in internal stochasticity, PLOS ONE 9 (6) (4) e98746.

13 (cf. [, ]) sharp Varadhan Z Z d Bernoulli Ginzburg-Landau [] C. Kipnis and C. Landim, Scaling Limits of Interacting Particle Systems, 999, Springer. [] S.R.S. Varadhan, Nonlinear diffusion limit for a system with nearest neighbor interactions II, in Asymptotic Problems in Probability Theory, Stochastic Models and Diffusions on Fractals, Pitman Res. Notes Math. Ser., 83 (994), 75-8.,

14 On fluctuations of eigenvalues of Gaussian beta ensembles at high temperature Trinh Khanh Duy Institute of Mathematics for Industry, Kyushu University Introduction Gaussian beta ensembles, as generalizations of Gaussian orthogonal ensembles, Gaussian unitary ensembles and Gaussian symplectic ensembles, were initially defined as eigenvalues ensembles with the joint density, (λ,..., λ n ) (λ) β e β ( n n )) i= V (λi) dλ = exp β ln λ j λ i V (λ i dλ, where V (λ) = λ /4 and (λ) = i<j (λ j λ i ) denotes the Vandermonde determinant. They can be also viewed as the equilibrium measure of a one dimensional Coulomb log-gas at the inverse temperature β. Let L n,β = n δ n λj / n j= be the empirical distributions of the scaled Gaussian beta ensembles. Then for fixed β, it is well known that the empirical measures L n,β converge weakly to the semicircle distribution, where the semicircle distribution, denoted by sc, is a probability measure supported on [, ] with the density 4 x /(π). This means that for any bounded continuous function f, L n,β, f = n f(λ j / n) sc, f = f(x) 4 x n π dx a.s. as n. j= The fluctuations around the semicircle distribution were also investigated. More precisely, it was shown in [7] that for a sufficiently nice function f with sc, f =, n j= i<j f(λ j / n) N d (, σf ) as n. d Here denotes the convergence in distribution. Note that beta ensembles with general potential V were considered in [7]. Matrix models for Gaussian beta ensembles were introduced by Dumitriu and Edelman [4]. They are symmetric tridiagonal matrices, also called Jacobi matrices, whose components are independent and are distributed as N (, ) χ (n )β χ (n )β N (, ) χ (n )β T n,β = β χ β N (, ) i=

15 Here χ k, for k >, denotes the (/ )-chi distribution with k degrees of freedom or the square root of the gamma distribution Γ(k, ). By those matrix models, Dumitriu and Edelman [5] established the above central limit theorem for polynomials f by different methods. The limiting behaviour of Gaussian beta ensembles in the regime where n with β = α/n (α being a positive constant) has been considered recently. It was shown first in [, 6] that the mean empirical distribution (also the mean spectral measure) converges weakly to a deterministic distribution µ α, and then in [3] that the empirical distributions L n,β themselves converge weakly to the same limit. Namely, for any polynomial p, as n with nβ = α, L n,β, p µ α, p in probability. The probability measure µ α here is the (scaled) measure of associated Hermite polynomial []. Note that the support µ α is the whole real line and the measure µ α is determined by its moments. It is the purpose of this talk to investigate the fluctuations of the empirical distributions around µ α. The main result is as follows. Theorem.. For a polynomial p of positive degree, as n with nβ = α, L n, p E[ L n, p ] n d N (, ˆσ p). The main tool used here is the martingale difference central limit theorem. It is expected that the central limit theorem should hold for a larger class of nice function f. References [] R. Allez, J-P. Bouchaud, and A. Guionnet: Invariant beta ensembles and the Gauss- Wigner crossover, Physical review letters 9 (), no. 9, 94. [] R. Askey and J. Wimp: Associated Laguerre and Hermite polynomials, Proc. Roy. Soc. Edinburgh Sect. A 96 (984), no. -, [3] F. Benaych-Georges, S. Péché: Poisson statistics for matrix ensembles at large temperature, J. Stat. Phys. 6 (5), no. 3, [4] I. Dumitriu and A. Edelman: Matrix models for beta ensembles, J. Math. Phys. 43 (), no., [5] I. Dumitriu and A. Edelman: Global spectrum fluctuations for the β-hermite and β-laguerre ensembles via matrix models, J. Math. Phys. 47 (6), 633. [6] T.K. Duy and T. Shirai: The mean spectral measures of random Jacobi matrices related to Gaussian beta ensembles, Electron. Commun. Probab. (5), no. 68, 3. [7] K. Johansson: On fluctuations of eigenvalues of random Hermitian matrices, Duke Math. J. 9 (998), no., 5 4.

16 CONCENTRATION FOR FIRST PASSAGE PERCOLATION. First passage percolation [] First passage percolation. ({η(j, x)} (j,x) N Z d, P); P(η(j, x) = ) = p (, ]. ω p (.) ω p = (k,x) N Z d η(k, x)δ (k,p /d x). (P, ω ) intensity counting measure Lebesgue measure N R d Poisson point process (.) p /d ω p p ω ω p, n { n } (.) T n (ω p ) = min x k x k α : x = and {(k, x k )} n k= ω p k= subadditive ergodic theorem (.3) µ p = lim n n T n(ω p ) P-a.s. µ p time constant 3. Theorem (concentration around the mean). For any χ < /, there exist C, C, λ > independent of p such that for any n N, P( n T n (ω p ) n E[T n (ω p )] > n χ ) < C exp{ C n λ }. Theorem (rate of convergence). For any γ (, ), there exists C > independent of p such that for any n N, µ p n E[T n (ω p )] < Cn γ. Theorem 3 (continuity). µ p is continuous in p [, ]. Theorem Concentration First Passage Percolation T n α > jump References [] F. Comets, R. Fukushima, S. Nakajima, N. Yoshida. Limiting results for the free energy of directed polymers in random environment with unbounded jumps Journal of Statistical Physics.

17 , [, ] Dirichlet Varadhan,. ([, Theorem.7]). (E, B, µ) σ-, (E, F) L (E, µ) Dirichlet, (T t ) t> L (E, µ) Markov. A, B, P t (A, B) = T t B dµ. lim t log P t(a, B) = d(a, B). t d(a, B) A, B, (E, F). A ( ). (E, B, µ) Dirichlet (E, F), Hilbert (H, (, ) H ) : F L (E H; µ) E (f, g) = ( f, g) H dµ E. E H- b, c, ϵ >, A ϵ > { } max b Hf dµ, c Hf dµ ϵe (f, f) + A ϵ f dµ, f F., E(f, g) = ( f, g) H dµ + E E E E (b, f) H g dµ + (c, g) H f dµ, E E f, g F (T t ) t>, (E, F) d Varadhan., A, B B, P t (A, B) = T t B dµ, A ( d(a, B) = sup essinf f F x B lim t log P t(a, B) = d(a, B) t {., F = f F b : h F b, ( f, f)h dµ. E f(x) esssup E x A ) f(x) } h dµ, essinf, esssup µ [] Ariyoshi, T. and Hino, M., Small-time asymptotic estimate in local Dirichlet spaces, Electron. J. Probab. (5), [] Hino, M. and Ramírez, J. A., Small-time Gaussian behavior of symmetric diffusion semigroups, Ann. Probab. 4 (3),

18 Parametrix method for simulation of stochastic differential equations Tomooki Yuasa Abstract In this talk, we introduce an exact simulation method for stochastic differential equations (SDE s) via Parametrix method with localization. This method can be expressed as two methods: the Forward method and the Backward method. In particular, the Backward method does not require the regularity of the coefficients of the SDE. Although the method has no bias, the error will be big because the variance is not generally finite. Accordingly, we consider an important sampling method together with and a new idea called the localization technique. Preparation Here, we simply explain the -dimensional SDE with the Forward method only. The general case can be obtained similarly. Let X t X t (x) be the weak solution of -dimensional SDE { dxt = σ(x t )dw t + b(x t )dt, t [, T ], () X = x, where W t is a -dimensional Wiener process. Let ξ i, i Z > be a sequence of i.i.d. random variables with common density f ξ which satisfies f ξ (,) (x) = dx a.e. and f ξ [, ) (x) > dx a.e. Also, let τ, τ i = i j= ξ j, i Z > be a point process and let N t = inf{n Z ; τ n t < τ n+ }, t [, T ] be its associated renewal process. Xτ π i, i Z denotes the Euler-Maruyama scheme of () with X π = x and the random partition π = { = τ < τ <... < τ NT T }. We define q n, n Z, for every n Z > and s = < s < < s n < T, q n (s, s,..., s n ) = T s n dxf ξ (x) n i= f ξ(s i+ s i ) and q (s ). Basic form of Parametrix method Theorem. (P. Andersson, V. Bally and A. Kohatsu-Higa (5)). Suppose that the coefficients of () satisfy σ Cb (R), b C b (R) and a := σ is uniformly elliptic. Then, for every f Cc (R), [ ] f(xt π E[f(X T )] = E ) N T θ τi+ τ q NT (τ, τ,..., τ NT ) i (Xτ π i, Xτ π i+ ). i= Dept. of Mathematics, Ritsumeikan University, -- Nojihigashi, Kusatsu, Shiga , Japan,

19 where for every t [, T ] and x, y R, θ t (x, y) = ( ) ( ) a (y) + (a(y) a(x))h () t (x, y) + a (y)h () t (x, y) b (y) + (b(y) b(x))h () t (x, y), h () t (x, y) = y x b(x)t a(x)t and h () t (x, y) = ( y x b(x)t a(x)t ) a(x)t. Also, if there exists ω such that N T (ω) =, then we understand N T (ω) i= θ τi+ τ i (Xτ π i (ω), Xτ π i+ (ω)) =. 3 Samplings of the sequence ξ i Once one defines f ξ, we can obtain the concrete Parametrix method. 3. Exponential sampling We arbitrarily choose λ >. Let ξ i, i Z > be a sequence of i.i.d. random variables with common density f ξ (x) = λe λx [, ) (x), x R. Namely, the common distribution of ξ i is the Exponential distribution with parameter λ and N t is a Poisson process. Also, for every n Z > and = s < s < < s n < T, q n (s, s,..., s n ) = e λt λ N T. 3. Beta sampling We arbitrarily choose τ > T and γ (, ). Let ξ i, i Z > be a sequence of i.i.d. random variables with common density f ξ (x) = γ x γ τ γ (,τ) (x), x R. Then, the common distribution F ξ (x) = ( x τ ) γ (,τ) (x) + [τ, ) (x), x R. Also, for every n Z > and = s < s < < s n < T, q n (s, s,..., s n ) = ( ( T sn τ ) τ )( γ ) n n τ γ i= (s i+ s i). γ 3.3 Gamma sampling We arbitrarily choose γ ( α, ) and ϑ >. Now, we need to choose α (, ) in Forward method, too. Let ξ i, i Z > be a sequence of i.i.d. random variables with common density f ξ (x) = Γ( γ)ϑ γ x e x γ ϑ (, ) (x), x R. Namely, the common distribution of ξ i is the Gamma distribution with parameter γ and ϑ. Also, for every n Z > and = s < s < < s n < T, q n (s, s,..., s n ) = Γ( γ, T s n ϑ ) Γ( γ) ( Γ( γ)ϑ ) n e sn n γ ϑ i= (s i+ s i ). γ Here, Beta sampling and Gamma sampling are two example of importance samplings which lead to methods with finite variance. Note that we must choose parameters for smaller variance and shorter trial time. However, the variance is still big even if we use importance sampling and its application although successful has some limitations. Thus, we suggest a new localization technique and give their theoretical properties in this talk. This technique can theoretically improve the importance sampling methods proposed above. In some sense, it becomes a space importance sampling method while the ones described above are time importance sampling methods.

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