Stochastic Target Games with Controlled Loss
|
|
|
- Ήρα Αλεξανδρίδης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Stochastic Target Games with Controlled Loss B. Bouchard Ceremade - Univ. Paris-Dauphine, and, Crest - Ensae-ParisTech USC, February 2013 Joint work with L. Moreau (ETH-Zürich) and M. Nutz (Columbia)
2 Plan of the talk Problem formulation
3 Plan of the talk Problem formulation Examples
4 Plan of the talk Problem formulation Examples Assumptions for this talk
5 Plan of the talk Problem formulation Examples Assumptions for this talk Geometric dynamic programming
6 Plan of the talk Problem formulation Examples Assumptions for this talk Geometric dynamic programming Application to the monotone case
7 Problem formulation and Motivations
8 Problem formulation Provide a PDE characterization of the viability sets [ ] Λ(t) := {(z, p) : u U s. t. E l(z u[ϑ],ϑ t,z (T )) m ϑ V} In which : V is a set of admissible adverse controls U is a set of admissible strategies Z u[ϑ],ϑ t,z is an adapted R d -valued process s.t. Z u[ϑ],ϑ t,z (t) = z l is a given loss/utility function m a threshold.
9 Application in finance Z u[ϑ],ϑ t,z = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) where t,x,y X u[ϑ],ϑ t,x models financial assets or factors with dynamics depending on ϑ Y u[ϑ],ϑ t,x,y models a wealth process ϑ is the control of the market : parameter uncertainty (e.g. volatility), adverse players, etc... u[ϑ] is the financial strategy given the past observations of ϑ. Flexible enough to embed constraints, transaction costs, market impact, etc...
10 Examples [ ] Λ(t) := {(z, p) : u U s.t. E l(z u[ϑ],ϑ t,z (T )) m ϑ V} Expected loss control for l(z) = [y g(x)] Give sense to problems that would be degenerate under P a.s. constraints : B. and Dang (guaranteed VWAP pricing).
11 Examples Constraint in probability : [ Λ(t) := {(z, p) : u U s.t. P Z u[ϑ],ϑ t,z ] (T ) O m ϑ V} for l(z) = 1 z O, m (0, 1). Quantile-hedging in finance for O := {y g(x)}.
12 Examples Matching a P&L distribution = Multiple constraints in probability : [ ( ) ] { u U s.t. P dist (T ), O γ i m i i I, ϑ V} Z u[ϑ],ϑ t,z (see B. and Thanh Nam)
13 Examples Almost sure constraint : Λ(t) := {z : u U s.t. Z u[ϑ],ϑ t,z (T ) O P a.s. ϑ V} for l(z) = 1 z O, m = 1. Super-hedging in finance for O := {y g(x)}. Unfortunately not covered by our assumptions...
14 Setting for this talk (see the paper for an abstract version)
15 Brownian diffusion setting
16 Brownian diffusion setting State process : Z u[ϑ],ϑ solves (µ and σ continuous, uniformly Lipschitz in space) Z(s) = z + s t µ(z(r), u[ϑ] r, ϑ r ) dr + s t σ(z(r), u[ϑ] r, ϑ r ) dw r.
17 Brownian diffusion setting State process : Z u[ϑ],ϑ solves (µ and σ continuous, uniformly Lipschitz in space) Z(s) = z + s t µ(z(r), u[ϑ] r, ϑ r ) dr + Controls and strategies : s t σ(z(r), u[ϑ] r, ϑ r ) dw r.
18 Brownian diffusion setting State process : Z u[ϑ],ϑ solves (µ and σ continuous, uniformly Lipschitz in space) Z(s) = z + s t µ(z(r), u[ϑ] r, ϑ r ) dr + s t σ(z(r), u[ϑ] r, ϑ r ) dw r. Controls and strategies : V is the set of predictable processes with values in V R d.
19 Brownian diffusion setting State process : Z u[ϑ],ϑ solves (µ and σ continuous, uniformly Lipschitz in space) Z(s) = z + s t µ(z(r), u[ϑ] r, ϑ r ) dr + Controls and strategies : s t σ(z(r), u[ϑ] r, ϑ r ) dw r. V is the set of predictable processes with values in V R d. U is set of non-anticipating maps u : ϑ V U, i.e. {ϑ 1 = (0,s] ϑ 2 } {u[ϑ 1 ] = (0,s] u[ϑ 2 ]} ϑ 1, ϑ 2 V, s T, where U is the set of predictable processes with values in U R d.
20 Brownian diffusion setting State process : Z u[ϑ],ϑ solves (µ and σ continuous, uniformly Lipschitz in space) Z(s) = z + s t µ(z(r), u[ϑ] r, ϑ r ) dr + Controls and strategies : s t σ(z(r), u[ϑ] r, ϑ r ) dw r. V is the set of predictable processes with values in V R d. U is set of non-anticipating maps u : ϑ V U, i.e. {ϑ 1 = (0,s] ϑ 2 } {u[ϑ 1 ] = (0,s] u[ϑ 2 ]} ϑ 1, ϑ 2 V, s T, where U is the set of predictable processes with values in U R d. The loss function l has polynomial growth and is continuous.
21 The game problem Worst expected loss for a given strategy : [ ( ) ] J(t, z, u) := ess inf E l Z u[ϑ],ϑ t,z (T ) F t ϑ V
22 The game problem Worst expected loss for a given strategy : [ ( ) ] J(t, z, u) := ess inf E l Z u[ϑ],ϑ t,z (T ) F t ϑ V The viability sets are given by Λ(t) := {(z, p) : u U s.t. J(t, z, u) p P a.s.}. Compare with the formulation of games in Buckdahn and Li (2008).
23 Geometric dynamic programming principle How are the properties (z, m) Λ(t) and (Z u[ϑ],ϑ t,z (θ),?) Λ(θ) related?
24 First direction : Take (z, m) Λ(t) and u U such that [ ( ) ] ess inf E l Z u[ϑ],ϑ t,z (T ) F t m ϑ V
25 First direction : Take (z, m) Λ(t) and u U such that [ ( ) ] ess inf E l Z u[ϑ],ϑ t,z (T ) F t m ϑ V To take care of the evolution of the worst case scenario conditional expectation, we introduce : [ ( Sr ϑ := ess inf E l Z u[ϑ r ϑ],ϑ ) ] r ϑ t,z (T ) F r. ϑ V
26 First direction : Take (z, m) Λ(t) and u U such that [ ( ) ] ess inf E l Z u[ϑ],ϑ t,z (T ) F t m ϑ V To take care of the evolution of the worst case scenario conditional expectation, we introduce : [ ( Sr ϑ := ess inf E l Z u[ϑ r ϑ],ϑ ) ] r ϑ t,z (T ) F r. ϑ V Then S ϑ is a submartingale and S ϑ t m for all ϑ V,
27 First direction : Take (z, m) Λ(t) and u U such that [ ( ) ] ess inf E l Z u[ϑ],ϑ t,z (T ) F t m ϑ V To take care of the evolution of the worst case scenario conditional expectation, we introduce : [ ( Sr ϑ := ess inf E l Z u[ϑ r ϑ],ϑ ) ] r ϑ t,z (T ) F r. ϑ V Then S ϑ is a submartingale and S ϑ t m for all ϑ V, and we can find a martingale M ϑ such that S ϑ M ϑ and M ϑ t = S ϑ t m.
28 We have for all stopping times θ (may depend on u and ϑ) [ ( ess inf E l Z u[ϑ θ ϑ],ϑ ) ] θ ϑ t,z (T ) F θ = Sθ ϑ ϑ V Mϑ θ.
29 We have for all stopping times θ (may depend on u and ϑ) [ ( ess inf E l Z u[ϑ θ ϑ],ϑ ) ] θ ϑ t,z (T ) F θ = Sθ ϑ ϑ V Mϑ θ. If the above is usc in space uniformly in the strategies, θ takes finitely many values, we can find a covering formed of B i (t i, z i ), i 1, such that J(t i, z i ; u) M ϑ θ ε on A i := {(θ, Z u[ϑ],ϑ t,z (θ)) B i }.
30 We have for all stopping times θ (may depend on u and ϑ) [ ( ess inf E l Z u[ϑ θ ϑ],ϑ ) ] θ ϑ t,z (T ) F θ = Sθ ϑ ϑ V Mϑ θ. If the above is usc in space uniformly in the strategies, θ takes finitely many values, we can find a covering formed of B i (t i, z i ), i 1, such that Hence J(t i, z i ; u) M ϑ θ ε on A i := {(θ, Z u[ϑ],ϑ t,z (θ)) B i }. K(t i, z i ) M ϑ θ ε on A i. where [ ( ) ] K(t i, z i ) := esssup ess inf E l Z u[ϑ],ϑ t u U ϑ V i,z i (T ) F ti is deterministic.
31 We have for all stopping times θ (may depend on u and ϑ) [ ( ess inf E l Z u[ϑ θ ϑ],ϑ ) ] θ ϑ t,z (T ) F θ = Sθ ϑ ϑ V Mϑ θ. If the above is usc in space uniformly in the strategies, θ takes finitely many values, we can find a covering formed of B i (t i, z i ), i 1, such that Hence J(t i, z i ; u) M ϑ θ ε on A i := {(θ, Z u[ϑ],ϑ t,z (θ)) B i }. K(t i, z i ) M ϑ θ ε on A i. where [ ( ) ] K(t i, z i ) := esssup ess inf E l Z u[ϑ],ϑ t u U ϑ V i,z i (T ) F ti If K is lsc is deterministic. K(θ(ω), Z u[ϑ],ϑ t,z (θ)(ω)) K(t i, z i ) ε M ϑ θ (ω) 2ε on A i.
32 We have for all stopping times θ (may depend on u and ϑ) [ ( ess inf E l Z u[ϑ θ ϑ],ϑ ) ] θ ϑ t,z (T ) F θ = Sθ ϑ ϑ V Mϑ θ. If the above is usc in space uniformly in the strategies, θ takes finitely many values, we can find a covering formed of B i (t i, z i ), i 1, such that Hence J(t i, z i ; u) M ϑ θ ε on A i := {(θ, Z u[ϑ],ϑ t,z (θ)) B i }. K(t i, z i ) M ϑ θ ε on A i. where [ ( ) ] K(t i, z i ) := esssup ess inf E l Z u[ϑ],ϑ t u U ϑ V i,z i (T ) F ti If K is lsc (Z u[ϑ],ϑ t,z (θ(ω)), Mθ ϑ (ω) 3ε) Λ(θ(ω)) is deterministic.
33 To get rid of ε, and for non-regular cases (in terms of K and J), we work by approximation : One needs to start from (z, m ι) Λ(t) and obtain where Λ(t) := (Z u[ϑ],ϑ t,z (θ), Mθ ϑ ) Λ(θ) P a.s. ϑ V { (z, m) : there exist (tn, z n, m n ) (t, z, m) such that (z n, m n ) Λ(t n ) and t n t for all n 1 }
34 To get rid of ε, and for non-regular cases (in terms of K and J), we work by approximation : One needs to start from (z, m ι) Λ(t) and obtain where Λ(t) := (Z u[ϑ],ϑ t,z (θ), Mθ ϑ ) Λ(θ) P a.s. ϑ V { (z, m) : there exist (tn, z n, m n ) (t, z, m) such that (z n, m n ) Λ(t n ) and t n t for all n 1 Remark : M ϑ = M αϑ t,p := p + t αϑ s dw s, with α ϑ A, the set of predictable processes such that the above is a martingale. }
35 To get rid of ε, and for non-regular cases (in terms of K and J), we work by approximation : One needs to start from (z, m ι) Λ(t) and obtain where Λ(t) := (Z u[ϑ],ϑ t,z (θ), Mθ ϑ ) Λ(θ) P a.s. ϑ V { (z, m) : there exist (tn, z n, m n ) (t, z, m) such that (z n, m n ) Λ(t n ) and t n t for all n 1 Remark : M ϑ = M αϑ t,p := p + t αϑ s dw s, with α ϑ A, the set of predictable processes such that the above is a martingale. Remark : The bounded variation part of S is useless : optimal adverse player control should turn S in a martingale. }
36 Reverse direction : Assume that {θ ϑ, ϑ V} takes finitely many values and that (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ(θ ϑ ) P a.s. ϑ V.
37 Reverse direction : Assume that {θ ϑ, ϑ V} takes finitely many values and that (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ(θ ϑ ) P a.s. ϑ V. Then K(θ ϑ, Z u[ϑ],ϑ t,z (θ ϑ )) Mt,m(θ αϑ ϑ ).
38 Reverse direction : Assume that {θ ϑ, ϑ V} takes finitely many values and that Then (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ(θ ϑ ) P a.s. ϑ V. K(θ ϑ, Z u[ϑ],ϑ t,z (θ ϑ )) Mt,m(θ αϑ ϑ ). Play again with balls + concatenation of strategies (assuming smoothness) to obtain ū U such that [ ( ) ] E l Z (u θ ϑ ū)[ϑ],ϑ t,z (T ) F θ Mt,m(θ αϑ ϑ ) ε P a.s. ϑ V.
39 Reverse direction : Assume that {θ ϑ, ϑ V} takes finitely many values and that Then (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ(θ ϑ ) P a.s. ϑ V. K(θ ϑ, Z u[ϑ],ϑ t,z (θ ϑ )) Mt,m(θ αϑ ϑ ). Play again with balls + concatenation of strategies (assuming smoothness) to obtain ū U such that [ ( ) ] E l Z (u θ ϑ ū)[ϑ],ϑ t,z (T ) F θ Mt,m(θ αϑ ϑ ) ε P a.s. ϑ V. By taking expectation [ ( ) ] E l Z (u θ ϑ ū)[ϑ],ϑ t,z (T ) F t m ε P a.s. ϑ V
40 Reverse direction : Assume that {θ ϑ, ϑ V} takes finitely many values and that Then (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ(θ ϑ ) P a.s. ϑ V. K(θ ϑ, Z u[ϑ],ϑ t,z (θ ϑ )) Mt,m(θ αϑ ϑ ). Play again with balls + concatenation of strategies (assuming smoothness) to obtain ū U such that [ ( ) ] E l Z (u θ ϑ ū)[ϑ],ϑ t,z (T ) F θ Mt,m(θ αϑ ϑ ) ε P a.s. ϑ V. By taking expectation [ ( ) ] E l Z (u θ ϑ ū)[ϑ],ϑ t,z (T ) F t m ε P a.s. ϑ V so that (z, m ε) Λ(t).
41 To cover the general case by approximations, we need to start with where (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ ι (θ ϑ ) P a.s. ϑ V, Λ ι (t) := { (z, p) : (t, z, p ) B ι (t, z, p) implies (z, p ) Λ(t ) }.
42 To cover the general case by approximations, we need to start with where (Z u[ϑ],ϑ t,z (θ ϑ ), Mt,m(θ αϑ ϑ )) Λ ι (θ ϑ ) P a.s. ϑ V, Λ ι (t) := { (z, p) : (t, z, p ) B ι (t, z, p) implies (z, p ) Λ(t ) }. To concatenate while keeping the non-anticipative feature : Martingale strategies : α ϑ replaced by a : ϑ V a[ϑ] A in a non-anticipating way (corresponding set A) Non-anticipating stopping times : θ[ϑ] (typically first exit time of (Z u[ϑ],ϑ ), M a[ϑ] ) from a ball.
43 The geometric dynamic programming principle (GDP1) : If (z, m ι) Λ(t) for some ι > 0, then u U and {α ϑ, ϑ V} A such that (Z u[ϑ],ϑ t,z (θ), Mt,m(θ)) αϑ Λ(θ) P a.s. ϑ V. (GDP2) : If (u, a) U A and ι > 0 are such that (Z u[ϑ],ϑ t,z (θ[ϑ]), M a[ϑ] t,m (θ[ϑ])) Λ ι (θ[ϑ]) P a.s. ϑ V for some family (θ[ϑ], ϑ V) of non-anticipating stopping times, then (z, m ε) Λ(t), ε > 0. Rem : Relaxed version of Soner and Touzi (2002) and B., Elie and Touzi (2009).
44 Application to the monotone case
45 Monotone case : Z u[ϑ],ϑ t,x,y R d R with X u[ϑ],ϑ t,x = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) with values in t,x,y independent of y and Y u[ϑ],ϑ t,x,y y.
46 Monotone case : Z u[ϑ],ϑ t,x,y R d R with X u[ϑ],ϑ t,x = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) with values in t,x,y independent of y and Y u[ϑ],ϑ t,x,y y. The value function is : ϖ(t, x, m) := inf{y : (x, y, m) Λ(t)}.
47 Monotone case : Z u[ϑ],ϑ t,x,y R d R with X u[ϑ],ϑ t,x = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) with values in t,x,y independent of y and Y u[ϑ],ϑ t,x,y y. The value function is : ϖ(t, x, m) := inf{y : (x, y, m) Λ(t)}. Remark :
48 Monotone case : Z u[ϑ],ϑ t,x,y R d R with X u[ϑ],ϑ t,x = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) with values in t,x,y independent of y and Y u[ϑ],ϑ t,x,y y. The value function is : ϖ(t, x, m) := inf{y : (x, y, m) Λ(t)}. Remark : If ϕ is lsc and ϖ ϕ, then Λ(t) {(x, y, m) : y ϕ(t, x, m)}.
49 Monotone case : Z u[ϑ],ϑ t,x,y R d R with X u[ϑ],ϑ t,x = (X u[ϑ],ϑ t,x, Y u[ϑ],ϑ ) with values in t,x,y independent of y and Y u[ϑ],ϑ t,x,y y. The value function is : ϖ(t, x, m) := inf{y : (x, y, m) Λ(t)}. Remark : If ϕ is lsc and ϖ ϕ, then Λ(t) {(x, y, m) : y ϕ(t, x, m)}. If ϕ is usc and ϕ ϖ then Λ ι (t) {(x, y, m) : y ϕ(t, x, m) + η ι } for some η ι > 0.
50 Thm :
51 Thm : (GDP1) : Let ϕ C 0 be such that arg min(ϖ ϕ) = (t, x, m). Assume that y > ϖ(t, x, m ι) for some ι > 0. Then, there exists (u, a) U A that Y u[ϑ],ϑ t,x,y (θ ϑ ) ϕ(θ ϑ, X u[ϑ],ϑ t,x (θ ϑ ), M a[ϑ] t,m (θ ϑ )) P a.s. ϑ V.
52 Thm : (GDP1) : Let ϕ C 0 be such that arg min(ϖ ϕ) = (t, x, m). Assume that y > ϖ(t, x, m ι) for some ι > 0. Then, there exists (u, a) U A that Y u[ϑ],ϑ t,x,y (θ ϑ ) ϕ(θ ϑ, X u[ϑ],ϑ t,x (θ ϑ ), M a[ϑ] t,m (θ ϑ )) P a.s. ϑ V. (GDP2) : Let ϕ C 0 be such that arg max(ϖ ϕ) = (t, x, m). Assume that (u, a) U A and η > 0 are such that Y u[ϑ],ϑ t,x,y (θ[ϑ]) ϕ(θ[ϑ], X u[ϑ],ϑ t,x (θ[ϑ]), M a[ϑ] t,m (θ[ϑ]))+η P a.s. ϑ V. Then, y ϖ(t, x, m ε), ε > 0.
53 Thm : (GDP1) : Let ϕ C 0 be such that arg min(ϖ ϕ) = (t, x, m). Assume that y > ϖ(t, x, m ι) for some ι > 0. Then, there exists (u, a) U A that Y u[ϑ],ϑ t,x,y (θ ϑ ) ϕ(θ ϑ, X u[ϑ],ϑ t,x (θ ϑ ), M a[ϑ] t,m (θ ϑ )) P a.s. ϑ V. (GDP2) : Let ϕ C 0 be such that arg max(ϖ ϕ) = (t, x, m). Assume that (u, a) U A and η > 0 are such that Y u[ϑ],ϑ t,x,y (θ[ϑ]) ϕ(θ[ϑ], X u[ϑ],ϑ t,x (θ[ϑ]), M a[ϑ] t,m (θ[ϑ]))+η P a.s. ϑ V. Then, y ϖ(t, x, m ε), ε > 0. Remark : In the spirit of the weak dynamic programming principle (B. and Touzi, and, B. and Nutz).
54 PDE characterization - waving hands version Assuming smoothness, existence of optimal strategies... y = ϖ(t, x, m) implies Y u[ϑ],ϑ (t+) ϖ(t+, X u[ϑ],ϑ (t+), M a[ϑ] (t+)) for all ϑ.
55 PDE characterization - waving hands version Assuming smoothness, existence of optimal strategies... y = ϖ(t, x, m) implies Y u[ϑ],ϑ (t+) ϖ(t+, X u[ϑ],ϑ (t+), M a[ϑ] (t+)) for all ϑ. This implies dy u[ϑ],ϑ (t) dϖ(t, X u[ϑ],ϑ (t), M a[ϑ] (t)) for all ϑ
56 PDE characterization - waving hands version Assuming smoothness, existence of optimal strategies... y = ϖ(t, x, m) implies Y u[ϑ],ϑ (t+) ϖ(t+, X u[ϑ],ϑ (t+), M a[ϑ] (t+)) for all ϑ. This implies dy u[ϑ],ϑ (t) dϖ(t, X u[ϑ],ϑ (t), M a[ϑ] (t)) for all ϑ Hence, for all ϑ, µ Y (x, y, u[ϑ] t, ϑ t ) L u[ϑ]t,ϑt,a[ϑ]t X,M ϖ(t, x, m) σ Y (x, y, u[ϑ] t, ϑ t ) = σ X (x, u[ϑ] t, ϑ t )D x ϖ(t, x, p) with y = ϖ(t, x, m) +a[ϑ] t D m ϖ(t, x, m)
57 PDE characterization - waving hands version Supersolution property where inf v V sup (u,a) N v ϖ ( µ Y (, ϖ, u, v) L u,v,a X,M ϖ ) 0 N v ϖ := {(u, a) U R d : σ Y (, ϖ, u, v) = σ X (, u, v)d x ϖ + ad m ϖ}.
58 PDE characterization - waving hands version Supersolution property where inf v V sup (u,a) N v ϖ ( µ Y (, ϖ, u, v) L u,v,a X,M ϖ ) 0 N v ϖ := {(u, a) U R d : σ Y (, ϖ, u, v) = σ X (, u, v)d x ϖ + ad m ϖ}. Subsolution property ( ) sup inf µ Y (, ϖ, u[v], v) L u[v],v,a[v] X,M ϖ 0 (u[ ],a[ ]) N [ ] ϖ v V where N [ ] ϖ := {loc. Lip. (u[ ], a[ ]) s.t. (u[ ], a[ ]) N ϖ( )}.
59 References R. Buckdahn and J. Li (2008). Stochastic differential games and viscosity solutions of Hamilton-Jacobi-Bellman-Isaacs equations. SIAM SICON, 47(1) : H.M. Soner and N. Touzi (2002). Dynamic programming for stochastic target problems and geometric flows, JEMS, 4, B., R. Elie and N. Touzi (2009). Stochastic Target problems with controlled loss, SIAM SICON, 48 (5), B. and M. Nutz (2012). Weak Dynamic Programming for Generalized State Constraints, SIAM SICON, to appear. L. Moreau (2011), Stochastic target problems with controlled loss in a jump diffusion model, SIAM SICON, 49, , B. and N. M. Dang (2010). Generalized stochastic target problems for pricing and partial hedging under loss constraints - Application in optimal book liquidation, F&S, online. B. and V. Thanh Nam (2011). A stochastic target approach for P&L matching problems, MOR, 37(3),
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Other Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Statistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Models for Probabilistic Programs with an Adversary
Models for Probabilistic Programs with an Adversary Robert Rand, Steve Zdancewic University of Pennsylvania Probabilistic Programming Semantics 2016 Interactive Proofs 2/47 Interactive Proofs 2/47 Interactive
Credit Risk. Finance and Insurance - Stochastic Analysis and Practical Methods Spring School Jena, March 2009
Credit Risk. Finance and Insurance - Stochastic Analysis and Practical Methods Spring School Jena, March 2009 1 IV. Hedging of credit derivatives 1. Two default free assets, one defaultable asset 1.1 Two
Econ 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
ST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr (T t N n) Pr (max (X 1,..., X N ) t N n) Pr (max
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Bounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
EE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Jesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
5. Choice under Uncertainty
5. Choice under Uncertainty Daisuke Oyama Microeconomics I May 23, 2018 Formulations von Neumann-Morgenstern (1944/1947) X: Set of prizes Π: Set of probability distributions on X : Preference relation
Uniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Example Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Second Order Partial Differential Equations
Chapter 7 Second Order Partial Differential Equations 7.1 Introduction A second order linear PDE in two independent variables (x, y Ω can be written as A(x, y u x + B(x, y u xy + C(x, y u u u + D(x, y
Phys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
SCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Homework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
C.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Solution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
Every set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Section 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Mean-Variance Hedging on uncertain time horizon in a market with a jump
Mean-Variance Hedging on uncertain time horizon in a market with a jump Thomas LIM 1 ENSIIE and Laboratoire Analyse et Probabilités d Evry Young Researchers Meeting on BSDEs, Numerics and Finance, Oxford
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Lecture 34 Bootstrap confidence intervals
Lecture 34 Bootstrap confidence intervals Confidence Intervals θ: an unknown parameter of interest We want to find limits θ and θ such that Gt = P nˆθ θ t If G 1 1 α is known, then P θ θ = P θ θ = 1 α
derivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Areas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
P AND P. P : actual probability. P : risk neutral probability. Realtionship: mutual absolute continuity P P. For example:
(B t, S (t) t P AND P,..., S (p) t ): securities P : actual probability P : risk neutral probability Realtionship: mutual absolute continuity P P For example: P : ds t = µ t S t dt + σ t S t dw t P : ds
Chapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
The Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Lecture 26: Circular domains
Introductory lecture notes on Partial Differential Equations - c Anthony Peirce. Not to be copied, used, or revised without eplicit written permission from the copyright owner. 1 Lecture 6: Circular domains
( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Homework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Approximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Abstract Storage Devices
Abstract Storage Devices Robert König Ueli Maurer Stefano Tessaro SOFSEM 2009 January 27, 2009 Outline 1. Motivation: Storage Devices 2. Abstract Storage Devices (ASD s) 3. Reducibility 4. Factoring ASD
Reminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Matrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Figure A.2: MPC and MPCP Age Profiles (estimating ρ, ρ = 2, φ = 0.03)..
Supplemental Material (not for publication) Persistent vs. Permanent Income Shocks in the Buffer-Stock Model Jeppe Druedahl Thomas H. Jørgensen May, A Additional Figures and Tables Figure A.: Wealth and
Partial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Exercises to Statistics of Material Fatigue No. 5
Prof. Dr. Christine Müller Dipl.-Math. Christoph Kustosz Eercises to Statistics of Material Fatigue No. 5 E. 9 (5 a Show, that a Fisher information matri for a two dimensional parameter θ (θ,θ 2 R 2, can
An Introduction to Signal Detection and Estimation - Second Edition Chapter II: Selected Solutions
An Introduction to Signal Detection Estimation - Second Edition Chapter II: Selected Solutions H V Poor Princeton University March 16, 5 Exercise : The likelihood ratio is given by L(y) (y +1), y 1 a With
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 7η: Consumer Behavior Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Τέλος Ενότητας Χρηματοδότηση Το παρόν εκπαιδευτικό υλικό έχει αναπτυχθεί
ANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Fractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
CHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Solutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
SCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Exercises 10. Find a fundamental matrix of the given system of equations. Also find the fundamental matrix Φ(t) satisfying Φ(0) = I. 1.
Exercises 0 More exercises are available in Elementary Differential Equations. If you have a problem to solve any of them, feel free to come to office hour. Problem Find a fundamental matrix of the given
HOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Bayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
12. Radon-Nikodym Theorem
Tutorial 12: Radon-Nikodym Theorem 1 12. Radon-Nikodym Theorem In the following, (Ω, F) is an arbitrary measurable space. Definition 96 Let μ and ν be two (possibly complex) measures on (Ω, F). We say
D Alembert s Solution to the Wave Equation
D Alembert s Solution to the Wave Equation MATH 467 Partial Differential Equations J. Robert Buchanan Department of Mathematics Fall 2018 Objectives In this lesson we will learn: a change of variable technique
Problem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Instruction Execution Times
1 C Execution Times InThisAppendix... Introduction DL330 Execution Times DL330P Execution Times DL340 Execution Times C-2 Execution Times Introduction Data Registers This appendix contains several tables
Estimation for ARMA Processes with Stable Noise. Matt Calder & Richard A. Davis Colorado State University
Estimation for ARMA Processes with Stable Noise Matt Calder & Richard A. Davis Colorado State University rdavis@stat.colostate.edu 1 ARMA processes with stable noise Review of M-estimation Examples of
MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)
![MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)](/thumbs/81/84712331.jpg "MATH423 String Theory Solutions 4. = 0 τ = f(s). (1) dτ ds = dxµ dτ f (s) (2) dτ 2 [f (s)] 2 + dxµ. dτ f (s) (3)")
1. MATH43 String Theory Solutions 4 x = 0 τ = fs). 1) = = f s) ) x = x [f s)] + f s) 3) equation of motion is x = 0 if an only if f s) = 0 i.e. fs) = As + B with A, B constants. i.e. allowe reparametrisations
Srednicki Chapter 55
Srednicki Chapter 55 QFT Problems & Solutions A. George August 3, 03 Srednicki 55.. Use equations 55.3-55.0 and A i, A j ] = Π i, Π j ] = 0 (at equal times) to verify equations 55.-55.3. This is our third
Math 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Απόκριση σε Μοναδιαία Ωστική Δύναμη (Unit Impulse) Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο. Απόστολος Σ.
Απόκριση σε Δυνάμεις Αυθαίρετα Μεταβαλλόμενες με το Χρόνο The time integral of a force is referred to as impulse, is determined by and is obtained from: Newton s 2 nd Law of motion states that the action
Exercise 2: The form of the generalized likelihood ratio
Stats 2 Winter 28 Homework 9: Solutions Due Friday, March 6 Exercise 2: The form of the generalized likelihood ratio We want to test H : θ Θ against H : θ Θ, and compare the two following rules of rejection:
3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in
Nowhere-zero flows Let be a digraph, Abelian group. A Γ-circulation in is a mapping : such that, where, and : tail in X, head in : tail in X, head in A nowhere-zero Γ-flow is a Γ-circulation such that
Variational Wavefunction for the Helium Atom
Technische Universität Graz Institut für Festkörperphysik Student project Variational Wavefunction for the Helium Atom Molecular and Solid State Physics 53. submitted on: 3. November 9 by: Markus Krammer
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ. Ψηφιακή Οικονομία. Διάλεξη 9η: Basics of Game Theory Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Ψηφιακή Οικονομία Διάλεξη 9η: Basics of Game Theory Mαρίνα Μπιτσάκη Τμήμα Επιστήμης Υπολογιστών Course Outline Part II: Mathematical Tools Firms - Basics of Industrial
The challenges of non-stable predicates
The challenges of non-stable predicates Consider a non-stable predicate Φ encoding, say, a safety property. We want to determine whether Φ holds for our program. The challenges of non-stable predicates
6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Durbin-Levinson recursive method
Durbin-Levinson recursive method A recursive method for computing ϕ n is useful because it avoids inverting large matrices; when new data are acquired, one can update predictions, instead of starting again
Lecture 21: Properties and robustness of LSE
Lecture 21: Properties and robustness of LSE BLUE: Robustness of LSE against normality We now study properties of l τ β and σ 2 under assumption A2, i.e., without the normality assumption on ε. From Theorem
Inverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
ΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
1) Formulation of the Problem as a Linear Programming Model
1) Formulation of the Problem as a Linear Programming Model Let xi = the amount of money invested in each of the potential investments in, where (i=1,2, ) x1 = the amount of money invested in Savings Account
Concrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +](/thumbs/73/69566903.jpg "k A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +")
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Numerical Analysis FMN011
Numerical Analysis FMN011 Carmen Arévalo Lund University carmen@maths.lth.se Lecture 12 Periodic data A function g has period P if g(x + P ) = g(x) Model: Trigonometric polynomial of order M T M (x) =
forms This gives Remark 1. How to remember the above formulas: Substituting these into the equation we obtain with
Week 03: C lassification of S econd- Order L inear Equations In last week s lectures we have illustrated how to obtain the general solutions of first order PDEs using the method of characteristics. We
DESIGN OF MACHINERY SOLUTION MANUAL h in h 4 0.
DESIGN OF MACHINERY SOLUTION MANUAL -7-1! PROBLEM -7 Statement: Design a double-dwell cam to move a follower from to 25 6, dwell for 12, fall 25 and dwell for the remader The total cycle must take 4 sec
Section 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Finite difference method for 2-D heat equation
Finite difference method for 2-D heat equation Praveen. C praveen@math.tifrbng.res.in Tata Institute of Fundamental Research Center for Applicable Mathematics Bangalore 560065 http://math.tifrbng.res.in/~praveen
Math221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Solution Concepts. Παύλος Στ. Εφραιµίδης. Τοµέας Λογισµικού και Ανάπτυξης Εφαρµογών Τµήµα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών
Παύλος Στ. Εφραιµίδης Τοµέας Λογισµικού και Ανάπτυξης Εφαρµογών Τµήµα Ηλεκτρολόγων Μηχανικών και Μηχανικών Υπολογιστών Ισορροπία Nash αγνές στρατηγικές µικτές στρατηγικές Κυρίαρχες στρατηγικές Rationalizability
Depth versus Rigidity in the Design of International Trade Agreements. Leslie Johns
Depth versus Rigidity in the Design of International Trade Agreements Leslie Johns Supplemental Appendix September 3, 202 Alternative Punishment Mechanisms The one-period utility functions of the home
HW 3 Solutions 1. a) I use the auto.arima R function to search over models using AIC and decide on an ARMA(3,1)
HW 3 Solutions a) I use the autoarima R function to search over models using AIC and decide on an ARMA3,) b) I compare the ARMA3,) to ARMA,0) ARMA3,) does better in all three criteria c) The plot of the
Limit theorems under sublinear expectations and probabilities
Limit theorems under sublinear expectations and probabilities Xinpeng LI Shandong University & Université Paris 1 Young Researchers Meeting on BSDEs, Numerics and Finance 4 July, Oxford 1 / 25 Outline
Optimal Fund Menus. joint work with. Julien Hugonnier (EPFL) 1 / 17
Optimal Fund Menus joint work with Julien Hugonnier (EPFL) 1 / 17 Motivation and overview The same firm offers many mutual funds Question: given the heterogeneity of investors, what is the optimal fund
Practice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER
ORDINAL ARITHMETIC JULIAN J. SCHLÖDER Abstract. We define ordinal arithmetic and show laws of Left- Monotonicity, Associativity, Distributivity, some minor related properties and the Cantor Normal Form.
Test Data Management in Practice
Problems, Concepts, and the Swisscom Test Data Organizer Do you have issues with your legal and compliance department because test environments contain sensitive data outsourcing partners must not see?
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Μηχανική Μάθηση Hypothesis Testing
ΕΛΛΗΝΙΚΗ ΔΗΜΟΚΡΑΤΙΑ ΠΑΝΕΠΙΣΤΗΜΙΟ ΚΡΗΤΗΣ Μηχανική Μάθηση Hypothesis Testing Γιώργος Μπορμπουδάκης Τμήμα Επιστήμης Υπολογιστών Procedure 1. Form the null (H 0 ) and alternative (H 1 ) hypothesis 2. Consider
2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
![2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)](/thumbs/40/21364883.jpg "2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)")
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Fourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)