Stochastic (Random) Demand Inventory ModelsStochastic (Random) Demand Inventory Models George...

Stochastic (Random) Demand Inventory Models

George Liberopoulos

1G. Liberopoulos: Production Systems29/11/2018

The Newsvendor model• Assumptions/notation

– Single-period horizon– Uncertain demand in the period: D (parts) assume continuous random variable

Density function and cumulative distribution function of D: f(x) and F(x)

– Infinite production/replenishment rate (instantaneous replenishment)– Zero lead time– Overage cost rate: cost per unit of positive inventory remaining at the end of

the period: co (€ per left-over part)– Underage cost rate: cost per unit of unsatisfied demand (negative ending

inventory) : cu (€ per missing part or unsatisfied demand)– No fixed setup production/order cost

• Decision– Order quantity at the beginning of the period: Q (parts)


0

( )( ) ( ) ( ) ( )a

x ax

dF xF a P D a f x dx f adx ==

= ≤ = =∫

The Newsvendor model• Definitions

– (Positive) inventory remaining at the end of the period: I +

– Unsatisfied demand (negative inventory) at the at the end of the period: I –

– Total overage and underage cost at the end of the period: G(Q, D)

– Expected cost: G(Q)


( ) max( ,0)( ) max( ,0)

I Q D Q DI D Q D Q

+ +

− +

= − ≡ −

= − ≡ −

( , ) ( ) ( )o u o uG Q D c I c I c Q D c D Q+ − + += + = − + −

0

0 0

0

( ) [ ( , )] ( , ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

Dx

o ux x

oQ

u

Q

x x

G Q E G Q D G Q x f x dx

c Q x f x dx c x Q f x dx

c Q x f x dx c x Q f x dx

∞

=

∞ ∞

= =

=

+

=

+

∞

= =

= − + −

= − + −

∫

∫ ∫

∫ ∫

The Newsvendor model


( ) max( ,0)I Q QD D+ += ≡ −−

( ) ( )uQx

f xx Q dxc∞

=

−∫

( ) max( ,0)I D DQ Q− += ≡ −−

0

( ) ( )Q

ox

f xc dxQ x=

−∫

( )f x

Q Dμ = Ε[D]

The Newsvendor model• Problem

• First derivative of cost function


0

0

00

1( ) ( ) 0( ) (

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

1 ( ) ( 1) ( )

(

)

0( ) ( ) 1( ) ( )

) [

Q

o ux x Q

Q

ox

ux

x Q x

x xQ

Q

o ux x Q

o u

Q

dG Q d c Q x f x dx c x Q f x dxdQ dQ

dc Q x f x dxdQ

dc x Q f x dxd

Q x f x Q

Q

c f x dx c f x dx

c F Q

x f x

x Q f x x Q f x

c

∞

=

= =

=∞ =

=

=

∞

=

∞

= =

= − + −

= − + +

− +

+ −

= −

− − −

− − −

=

∫ ∫

∫

∫

∫ ∫1 ( )]F Q−

Minimize ( )Q

G Q

The Newsvendor model• In deriving the previous formula, we used Leibnitz’s rule for taking the

derivative of an integral whole limits are functions of the variable with respect to which the derivative is taken. Leibnitz’s rule is:


( ) ( )

( ) ( )

( , ) ( ) ( )( , ) ( ( ), ) ( ( ), )b y b y

a y a y

d df x y db y da yf x y dx dx f b y y f a y ydy dy dy dy

= + −∫ ∫

The Newsvendor model• Second derivative

• First-order condition for minimization

Note:– F(Q) is the fill rate, i.e. the probability that a demand will be satisfied!

– Recall: EOQ model with backorders:


[ ]2

2

( ) ( ) [1 ( )] ( ) ( ) 0o u o udG Q d c F Q c F Q c c f Q

dQ dQ= − − = + ≥

0

* * * 1

( ) 0 ( ) [1 ( )] 0 ( ) ( )

: ( )

o u o u uQ

u u

o u o u

dG Q c F Q c F Q c c F Q cdQ

c cQ F Q Q Fc c c c

=

−

= ⇒ − − = ⇒ + =

⇒ = ⇒ = + +

* bFh b

=+

The Newsvendor model• Special case: D ∼ Normal(μ, σ)


( )

( )

Normal(0,1) standardized Normal cumulative distribution function

( ) ( )

( ) ,

and hence ( ) can be evaluated from standardized Normal t s(

able

D Q QF Q P D Q P

Q

z

F Q z z

z F Q

µ µ µσ σ σ

µσ

− − − = ≤ = ≤ = Φ

−⇒ =Φ =

⇒ Φ

Φ

0 0.5000 0.00000.85 0.8023 0.30231.19 0.9015 0.40151.65 0.9505 0.45052.33 0.9901 0.49013.09 0.9990 0.49

) ( )

9

0.5

0

z zΦ −

The Newsvendor model• Special case: D ∼ Normal(μ, σ) cont’d

• ExampleD ∼ Normal(120, 45) buying price c = 30selling price S = 110salvage price s = 10


( )

* ** * 1: ( )

c c cu o u

u u u

o u o u o u

z

c c cQ QQ F Qc c c c c c

µ µσ σ

+

− − −= ⇒ Φ = ⇒ =Φ + + +

0.80

*0.80

80 0.80 0.8520 80

120 45 0.85 120 38.25 158.85 159

u

o u

c zc c

Q zµ σ

⇒ = = ⇒ =+ +

⇒ = + = + ⋅ = + = ≈

*( )u o uc c cQ zµ σ +⇒ = +

110 30 8030 10 20

u

o

c S cc c s= − = − =

⇒ = − = − =

The Newsvendor model• Extension: Discrete demand

– Uncertain demand in the period: D (parts) assume discrete random variableProbability mass function and cumulative distribution function of D: p(x) and F(x)

– Expected cost: G(Q)

– Problem:

– First-order difference

– First-order condition for minimization


( ) ( ) ( ) ( ) ( ) ( 1)x a

F a P D a p x p a F a F a≤

= ≤ = = − −∑1

0( ) [ ( , )] ( , ) ( ) ( ) ( ) ( ) ( )

Q

o uD x x x QG Q E G Q D G Q x p x c Q x p x c x Q p x

− ∞

= =

= = = − + −∑ ∑ ∑

*

* *

: smallest such that ( 1) ( ) 0 ( ) [1 ( )] 0

: smallest such that ( )

o u

u

o u

Q Q G Q G Q c F Q c F Q

cQ Q F Qc c

+ − ≥ ⇔ − − ≥

⇒ ≥+

0 1( 1) ( ) ( ) ( ) ( ) [1 ( )]

Q

o u o ux x Q

G Q G Q c p x c p x c F Q c F Q∞

= = +

+ − = − = − −∑ ∑

Minimize ( )Q

G Q

The Newsvendor model• Extension: Staring inventory y > 0

– Still want to be at Q* after ordering, because Q* is the minimizer of G(Q)– Order quantity: U– Optimal policy now depends on starting inventory:

Note: – U* ≡ optimal order quantity– Q* ≡ optimal “order-up-to” point ≡ inventory target level ≡ base stock level


* **

*

, if ( )

0, if Q y y Q

U yy Q

− <=

≥

The Newsvendor model• Interpretation of co and cu for the single-period model

– S = selling price (€ per part)– c = variable cost (€ per part)– h = holding cost (€ per part per period)– p = loss-of-goodwill cost (€ per part short per period)


order cost leftover inventory cost shortage cost

sales reven

sal

e

e

u

s

( , ) ( ) ( min( , ))G Q D cQ h Q D p D Q S Q D+ += + − + − −

0

( ) ( ) ( ) ( ) ( )Q

Q

cQ h Q x f x dx p S x Q f x dx Sµ∞

= + − + + − −∫ ∫

*

( ) 0 ( ) ( )(1 ( )) 0

: ( ) ,u o

dG Q c hF Q p S F QdQ

p S cQ F Q c p S c c h cp S h

= ⇒ + − + − =

+ −⇒ = ⇒ = + − = +

+ +

0 0

(

[( ) ]

)

( ) ( )

( ) [ ( , )] ( ) ( ) ( ) ( ) [ ( ) ( ) ]

Q

Q

Q Q

DQ

E D

Q

xf x dx

x Q Qf x dx

G Q E G Q D cQ h Q x f x dx p x Q f x dx S xf x dx Qf x dx

µ

µ

µ

∞

∞+

∞ ∞

−

− = − −−

= = + − + − − +

∫

∫

∫ ∫ ∫ ∫

The Newsvendor model• Extension: infinite periods (infinite horizon) with backorders

Same assumptions as single-period model except that:– Dt = demand in period t; D1, D2, D3, … are i.i.d. with distribution f(x), F(x)– Ut = amount ordered in period t– Optimal policy in each period is “order-up-to” Q– In the long-run, the inventory can never be higher than Q

⇒ In steady-state (long run): Ut = Dt–1


Inventory/backorders

time t

Q

Transient Ut = Dt–1

Steady state

The Newsvendor model• Extension: infinite periods (infinite horizon) with backorders (cont’d)

– Total cost in a period with demand D

– Expected average cost per period

– First-order condition for minimizing G(Q)

Note: – c and S play no role in determining Q*, because in the long run, all demands are

satisfied regardless of Q; therefore, the expected average ordering cost minus revenue per period is (c – S)μ regardless of Q.


order cost sales revenue inventory holding cost backorder cost

( , ) ( ) ( ) ( )G Q D c S D h Q D p D Q+ +

−

= − + − + −

* *

( ) 0 ( ) (1 ( )) 0

: ( ) Newsventor formula: ,u o

dG Q hF Q p F QdQ

pQ F Q c p c hp h

= ⇒ − − =

⇒ = ⇒ = =+

( ) [ ( , )] ( ) [( ) ] [( ) ]D

G Q E G Q D c S hE Q D pE D Qµ + += = − + − + −

The Newsvendor model• Extension: infinite periods (infinite horizon) with lost sales

Same assumptions as infinite horizon with backorders except that:– Unmet demand is not backordered but is lost– In steady-state (long run): Ut = min(Q, Dt–1)


Inventory/lost sales

time t

Q

Transient Ut = min(Q, Dt–1)

The Newsvendor model• Extension: infinite periods (infinite horizon) with lost sales (cont’d)

– Total cost in a period with demand D

– Expected average cost per period

– First-order condition for minimizing G(Q)

Note: – c and S now play a role in determining Q*, because in the long run, the demand

satisfied and the orders are min(Q, D), so they depend on Q; therefore, the expected average ordering cost minus revenue per period is (c – S)E[min(Q, D)].


* *

( ) 0 ( ) ( )(1 ( )) 0

: ( ) Newsvendor formula: ,u o

dG Q hF Q p S c F QdQ

p S cQ F Q c p S c c hp S c h

= ⇒ − + − − =

+ −⇒ = ⇒ = + − =

+ − +

( , ) ( ) ( ) ( )min( , )Q DG Q D c S h Q D p D Q+ += − + − + −

[( )( ) [ ( , )] ( ) [( ) ] [( ]] )D

G Q E G Q D c S hE Q D pE DE D Q Qµ + + + − = = − + − + − −

Lot size – Reorder point (Q, R) model• Assumptions

– Infinite horizon– Continuous review (as opposed to periodic review)– Dt: random stationary demand per unit time (e.g., daily demand)

mean λ ≡ Ε[Dt], variance σt2 = E[(Dt – λ)2]

– Unmet demand is either backordered or lost– τ: Fixed replenishment order lead time – Costs:

• Variable unit production/order cost: c (€ per part)• Fixed setup production/order cost: K (€ per production run/order)• Inventory holding cost rate: h (€ per part per unit time)• Stock-out (shortage/penalty) cost rate (2 cases / 4 situations: see next)

• Order policy– (Q, R) policy: order Q when inventory position falls below R

• Decision variables– Q: lot size (reorder quantity)– R: reorder point


(Q, R) model• Assumptions on stock-outs and stock-out cost rate

– Case 1: Backordered demand• p1 (€ per stock-out occasion)• p2 (€ per part short )• p3 (€ per part short per unit time)

– Case 2: Lost sales• pL (€ per lost sale)


(Q, R) model: Backordered demand


R

time

R+Q

τ

Q

Inventory position

On-hand inventory

Backorders

(Q, R) model: Backordered demand• Analysis

– D: demand during lead time τ• Density function and cumulative distribution function of D: f(x) and F(x)• D = D1 + D2 + …+ Dτ

• Mean: μ ≡ E[D] = Ε[D1 + D2 + …+ Dτ ] = τ Ε[Dt] = τ λ• Variance: σ2 ≡ Var[D] = Ε[(D – μ)2] = Var[D1 + D2 + …+ Dτ] = τ Var[Dt] = τ σt

2


τ

x

Inventory /backorders

time

f(x)

μR

(Q, R) model: Backordered demand• Inventory holding cost

– Safety stock ss ≡ R – μ = R – λ τ– Expected average inventory approximation Ī ≈ ss + Q/2 = R – λ τ + Q/2

(underestimates true value)– Expected average inventory hold cost = h Ī = h(R – λ τ + Q/2)


R

ss = R – λ τ

ss + Q = R – λ τ + Q

μ = λ τ

Inventory /Backorders

time

τ –λ

(Q, R) model: Backordered demandTo see why expected average inventory approximation underestimates true expected average inventory:


0

0 0

[ ] 2 ( ) 2

[( ) ] 2 ( ) ( ) 2 ( ) ( ) 2

(1 ( )) ( )

( ) ( )

( ) ( )

( ) ( )

[( ) ] 0

appr

R R

true

appr true R

R R

R R

R

appr tr

I R E D Q R xf x dx Q

I E R D Q R x f x dx Q RF R xf x dx Q

I I R F R xf x dx

R f x dx xf x dx

Rf x dx xf x dx

R x f x dx

E D R

I I

∞

+

∞

∞ ∞

∞ ∞

∞

+

≈ − + = − +

= − + = − + = + +

− = − −

= −

= −

= −

= − ≤

⇒ ≤

∫∫ ∫

∫∫ ∫∫ ∫∫

ue

(Q, R) model: Backordered demand• Setup cost

– Expected average order frequency = λ/Q– Expected average setup cost per unit time = K λ/Q

• Stock-out (penalty) cost– Assumption: τ << Q/λ ⇒ stock-out per cycle depends only on R– B(R) : Expected stock-out cost per cycle (depends on definition of stock-

out cost rate)– Expected average stock-out cost per unit time = B(R) λ/Q

• Total expected average cost per unit time


( , ) ( )2

R R Rh K BQQQ Q

G λ λλτ = + − + +

(Q, R) model: Backordered demand• Optimization problem

• Optimality conditions


22 2

( , ) ( ) 2 [ ( )]02

2 [ ( )] (1)

( , ) ( ) 0

( ) (2)

G R h K B R K B RQQ Q h

K B RQh

G Q dBhQ

R RR d

dB hQ

R

Q

dR

R

Q λ λ λ

λ

λ

λ

∂ += − − = ⇒ =

∂

+⇒ =

∂= + =

∂

⇒ = −

,Minimize ( , ) ( )

2RQ

QQQ

R R RG KQ

h Bλ λλτ = + − + +

(Q, R) model: Backordered demand• Optimality condition (2): 3 cases

– Case 1: Stock-out cost p1 € per stock-out occasion

– Case 2: Stock-out cost p2 € per part short


1 1

probabilty of stock-out per cycle

11

( )( ) [1 ( )] ( )

Condition (2) : ( ) ( )

dBB R p F R p f Rd

hQ hQp f R f Rp

RR

λ λ

= − ⇒ = −

− = − ⇒ =

2 2 2

( ) expected numberof stock-outs per cycle ( )

22

( )( ) [( ) ] ( ) ( ) [1 ( )]

Condition (2) : [1 ( )] ( ) 1

x Rn Rn R

dBB R p E D R p x R Rf x dx p F Rd

hQ hQp F R F Rp

R

λ λ

∞+

=≡

= − = − ⇒ = − −

− − = − ⇒ = −

∫

(Q, R) model: Backordered demand– Case 3: Stock-out cost p3 € per part short per unit time


2

2

3

expected total w

Average waiting time per backorder when the demand is : ( )

2Average total waiting time of all backorders when the demand is :

( ) [( ) ]( )2 2

( )( ) ( )2x R

DD R

DD R D RD R

x RB R p f x dx

λ

λ λ

λ

+

+ ++

∞

=

−

− −− =

−= ∫ 23

aiting timeof all backorders per cycle

3 3 3

3

3

( ) ( )2

( ) ( ) ( ) [( ) ] ( )

Condition (2) : ( ) ( )

x R

x R

px R f x dx

p p pdB x R f x dx E D R n Rd

p hQ hQn R

RR

n Rp

λ

λ λ λ

λ λ

∞

=

∞+

=

= −

= − − = − − = −

− = − ⇒ =

∫

∫

(Q, R) model: Backordered demand

• Simultaneous solution of conditions (1) and (2)Solve by fixed-point iteration: 1. Given R, solve (1) to find Q2. Given Q, solve (2) to find R3. Repeat until convergence


(Q, R) model: Backordered demandIllustration for case 2: Stock-out cost p2 € per part short– Optimality conditions for case 2

– Assumption: D~Normal(μ, σ) Use standardized cumulative distribution function (cdf) Φ(z) to compute F(R)

– Φ(z) and hence F(R) can be evaluated from standardized Normal cdf tables


2

2

2 [ ] (1( ) ) 1 (2)( )F RK p hQQh p

n Rλλ

+= = −

( )

Normal(0,1)

( ) ( )

( ) ,

D R RF R P D R P

RF R z z

µ µ µσ σ σ

µσ

− − − = ≤ = ≤ = Φ

−⇒ = Φ =

(Q, R) model: Backordered demandUse standardized loss function L(z) to compute n(R)

L(z) and hence n(R) can be evaluated from standardized loss function tablesIt can be shown that


Normal(0,1)density function

~ Normal(0,1) ( ) [( ) ] ( ) ( )y z

Y L z E Y z y z y dyϕ∞

+

=

⇒ ≡ − = −∫

( ) ( ) [1 ( )]

( ) ( ) ( ) ( )[1 ( )],

L z z z zRn R L z z R z z

ϕµσ σϕ µ

σ

= − −Φ−

⇒ = = + − −Φ =

Normal(0,1)

( ) [( ) ]

( ) ( ),

D R Rn R E D R E L

Rn R L z z

µ µ µσ σσ σ σ

µσσ

+

+

− − − = − = − =

−⇒ = =

(Q, R) model: Backordered demandUnder the assumption D~Normal(μ, σ) , the optimality conditions become


2

2

2 [ ( )] (1)

( ) 1 (2)

(3)

K p L zQh

Qhzp

Rz

λ σ

λ

µσ

+=

Φ = −

−=

(Q, R) model: Backordered demandFixed point iteration algorithm for case 2 under the assumption D~Normal(μ, σ)


1 00 0 0 0

2 1

1

1 1

Step 1

Step 2

Step 3Step

2 , 1 , , 1

2 [ ( )]:

: 1

:: OR 1, GOTO Ste4 p 1

nn

nn

n n

n n n n

Q hKQ z R z nh p

K p L zQh

Q hzp

R zQ Q R R n n

λ µ σλ

λ σ

λµ σ

ε ε

−

−

−

− −

= = Φ − = + =

+=

= Φ −

= +

− ≥ − ≥ ⇒ ← +

(Q, R) model: Lost sales


R

time

R+Q

τ

Q

Inventory position

On-hand inventory

Lost sales

(Q, R) model: Lost salesModify total expected average cost per unit time function

Optimality conditions


expected lostsales per cycle

expected lostsales per unit time

( , ) (2

( ) )LnQG Q R h R K p n RQ Q

R λ λλτ = + − + + +

( ) 1

2 [ ( )] (1) (same as case 2 of backordered demands)

( , ) ( ) ( )1 0 1 0

1 [1 ( )] 0

( ) 1 (2) (Newsvendor formula:

1

L

L L

L

Lo

L L

K p n RQh

p pG Q R dn R dn RhR Q

dn RdR dR Qh dR

p F RQh

pQhF R cpQh Qh p

λ

λ λ

λ

λλ λ

+=

∂ = + + = ⇒ + + = ∂

⇒ − + − =

⇒ = − =+ +

, )

( same as case 2 of backordered demands)

u L

L

Qh c p

Qh p

λ

λ

= =

<< ⇒

(Q, R) model: Lost salesNote

– G(Q, R) is an approximation, because it overestimates the order frequency

– In the lost sales model, not all demand is met; on average, Q demands are met and n(R) demands are not met

⇒Average number of demands per cycle = Q + n(R) parts per cycle– Number of demands per unit time = λ parts per unit time⇒ Order frequency = λ/[Q + n(R)] instead of λ/Q⇒A more accurate expression for G(Q, R) is:

– Usually, n(R) << Q, so the order frequency can still be approximated by λ/Q


( ) (( , ) ( ) ( )

2 )LQG

nQ R h R n R K p n R

Q R n RQλ λλτ = + − + + + + +

(Q, R) model: Summary


Situation D~Normal(μ,σ) Eq. (2) Eq. (2) D~Normal(μ,σ)

p1 (€ per stock-out occasion)

p2 (€ per part short)p3 (€ per part short per unit time)

pL (€ per lost sale)

( )B R

1[1 ( )]p F R−1

( ) hQf Rp λ

=

2

( ) 1 hQF Rp λ

= −2 ( )p n R

23 ( ) ( )2 x R

p x R f x dxλ

∞

=

−∫3

( ) hQn Rp

=

( ) L

L

pF RQh p

λλ

=+

( )Lp n R

1

( ) hQzpσϕ

λ=

2

( ) 1 hQzp λ

Φ = −

3

( ) hQL zpσ

=

( ) L

L

pzQh p

λλ

Φ =+

2 [ ( )]Eq. (1): K B R RQ zh

λ µσ

+ −= =

1[1 ( )]p z−Φ

2 ( )p L zσ

( )Lp L zσ

( )B R

(Q, R) model: Service Levels• Service levels in (Q, R) systems

– Type 1 Service (replaces stock-out cost p1 € per stock-out occasion)S1 ≡ Probability of not stocking out during the lead timeS1 = P(D ≤ R) = F(R)

– Optimization problem

– Solution


,

1

Minimize ( , )2

subject to ( ) (i.e., subject to )

Q RG hR QQ

QF R

R K

S

λλτ

α α

= + − +

≥ ≥

*

*

* 1

2 EOQ

= minimum such that ( )

continuous r.v. ( )

KQh

R R F R

D R F

λ

α

α−

= =

≥

⇒ =

(Q, R) model: Service Levels– Type 2 Service (replaces stock-out cost p2 € per part short)

S2 ≡ Proportion of demands met from stockS2 = 1 – n(R)/Q

– Optimization problem

– Note: Now the constraint depends on both R and Q


,

2

Minimize ( , )2

( )subject to 1 (i.e., subject to )

RQG h K

n

QQQ

R

SQ

R

R

λλτ

β β

= + − +

− ≥ ≥

(Q, R) model: Service Levels– Type 2 Service (cont’d)

Approximate solution


*

* *

* *

* * *

** * * 1

2 EOQ

= minimum such that ( ) (1 )

continuous r.v. ( ) (1 )

~ Normal( , ) ( ) ( ) (1 )

(1 ),

KQh

R R n R Q

D n R Q

D n R L z Q

QR z z L

λ

β

β

µ σ σ β

βµ σσ

−

≈ =

≤ −

⇒ = −

⇒ ≡ = −

−⇒ = + =

(Q, R) model: Service Levels– Type 2 Service (cont’d)

More accurate solutionConsider first-order conditions (1) and (2) for case 2


2

2 [ ( )] (1), ( ) 1 (2)

(2) imputed stock-out cost[1 ( )]

2 { ( ) [1 ( )] }(1) quandratic function in

( ) 2 ( )positive root: (3)1 ( ) 1 ( )

(1 )( ) (4)

K pn R QhQ F Rh p

QhpF R

K hn R F R Qh

n R K n RQF R h F

R

Q

n

Q

R

Q

λλ

λ

λ λ

λ

βσ

+= = −

⇒ = ≡−

+ −⇒ = ≡

= + + − −

−=

(Q, R) model: Random Lead Time• Extension: Random lead-time

– L: random lead time– Mean: τ ≡ Ε[L], variance σL

2 ≡ Ε[(L– τ)2]– D: demand during lead time L

• Density function and cumulative distribution function of D: f(x) and F(x)• D = D1 + D2 + …+ DL, where L is a random variable• It can be shown (see next page) that:

Mean: μ ≡ E[D] = τ λVariance: σ2 ≡ Var[D] = Ε[(D – μ)2] = τ σt

2 + λ2σL2

Everything else holds!!


(Q, R) model: Random Lead Time• Derivation of μ and σ2


2

2 2 2

|

2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

|

2

| |

Mean: [ ] [ [ | ]] [ ]

Variance: [ ] [( ) ] [ 2 ] [ ] 2 [ ] [ ][ [ | ]] 2

where we used:[ | ] [

L D L L

t L t LL D L

D L D

t L

L

E D E E D L E L

Var D E D E D D E D E D EE E D L

E D L E Va

λ

µ

µ τλ

σ µ µ µ µ

µ µ τσ λ σ λ τ µ τσ λ σ λ

τ

λ

λ

τ τ

σ σ

≡ = = =

≡ = − = − + = − +

= − + = + + − = + + −

=

=

+

2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

|

[ | ] [ | ] ]

[ [ | ]] [ ] ( [ ] ) ( )

t

t t t L t LL D L L

r D L E D L L L

E E D L E L L Var L

σ λ

σ λ τσ λ τ τσ λ σ τ τσ λ σ λ τ

+ = +

= + = + + = + + = + +

Date post:	10-May-2020
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