OPTIMAL STOPPING AND EXIT TIMES FOR SOME CLASSES OF RANDOM PROCESSES. Vladyslav Tomashyk

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1 NATIONAL TARAS SHEVCHENKO UNIVERSITY OF KYIV UKRAINE OPTIMAL STOPPING AND EXIT TIMES FOR SOME CLASSES OF RANDOM PROCESSES Vladyslav Tomashyk Mechaics ad Mathematics Faculty Departmet of Probability Theory, Statistics ad Actuarial Mathematics Iasi 22

2 Optimal Stoppig Problem a problem cocered with the choosig a time to take a particular actio, i order to maximise a expected reward or miimise a expected cost. This problem is quite popular owadays because of its applicatio to the modelig of behavior of ivestors o the chagig fiacial markets. The mai goal of the preset work is to study the limit behavior of optimal stoppig ad exit times for some classes of radom processes i particular of Ito s diffusio, radom walk ad diffusio process with o-lipschitz diffusio coefficiets.

3 . Limit behavior of optimal stoppig times for Ito s diffusio 2 () t X dx t rx t dt αx t dw t Let us cosider a asset whose price W( t) r, ( ) ( ) ( ) ( ), α = + ( ) oe dimesioal Browia motio, are give costats. satisfies the followig SDE X = x>,. Ito s lemma gives: α 2 X t x r t W t 2 () = exp + α () (Geometric Browia Motio)

4 . Limit behavior of optimal stoppig times for Ito s diffusio We assume the trasactio cost of a >. 3 The discouted profit of the trasactio at the momet ρ s g ( s, x) e ( x a ) =, ρ > is a discoutig factor. τ* The optimal stoppig time is as follows : ( sx, ) * ρτ argmax ( ( ) ) τ = E e X τ a, τ Γ where Γ is the set of all stoppig times. s is Optimal discouted profit will be of the followig form : ( sx, ) * ( *) τ τ * * g = g ( s, x) = E g, X.

5 . Extreme behavior of optimal stoppig times for a Ito diffusio Lemma * Followig statemets take place: ) 2) 3) if r > ρ the if r = ρ if r < ρ a x max γ = ( ) γ ( ) the the ad * g = τ * = τ* ad does ot exist ; ad ( ) { } t X t x max τ * = mi = * ρ s g s, x xe = ; 2 ( ) γ = α r r 2 2 α + α + ρ α 2 2 where 4 * Oksedal B., Stochastic differetial equatios, Berli: Spriger-Verlag, 5 th editio, (2) 332 p.

6 . Limit behavior of optimal stoppig times for Ito s diffusio The mai goal of this part of the work is : 5 to describe the limit behavior of the optimal stoppig times i the case whe of the ratio whe r < ρ, r r, ρ ρ, α α, a a. The cases we have cosidered accordig to the limit behavior a x max = γ a γ ( ) γ ( ) ( ) the ratio does ot have ay ucertaity i the limit. γ ( ) whe ad but x max ( ) ( ) positive value ad γ. a γ are as follows: x *

7 ) Theorem. Let α >, that α 2 r ϕ >. The: 2 if a a, α α >, r r, ρ ρ while γ. Limit behavior of optimal stoppig times for Ito s diffusio ( ) γ, the x max x max ad * P τ * τ. so 6 2) if a a =, α α >, r r, ρ ρ ( ) while * that γ ad if x \{}, max x R+ the P τ * max τ where * if { : * () } τ = t X t > x. so a γ * = = = max γ Remark. τ mi tx () t x for the asset with the price is the optimal stoppig time 2 X t x r α 2 t W t () = exp + α ()

8 . Limit behavior of optimal stoppig times for Ito s diffusio 7 Theorem 2. the stoppig time Uder assumptios of the case 2) of the previous theorem { * () } t X t x τ * = if : > time for a asset whose price X ( t) is as follows : Theorem 3. ad α Let 2 X () exp t x r α = (). 2 t + α W t γ ( ) is a optimal stoppig, a a, r r, α. * P The τ,.

9 Theorem 4. Let ad. Limit behavior of optimal stoppig times for Ito s diffusio γ x max. ( ), a, α α The * P τ,. 8 ( ), Theorem 5. Let γ a, α α x max * P ad. The τ,, i.e. istataeous sale is the optimal i this case. Remark. It is o eed to build a ew strategy for a ivestor while coefficiets of Ito s diffusio coverge to some fixed values. The ivestor must use the extreme strategy because this strategy will be optimal for the limit process.

10 . Limit behavior of optimal stoppig times for Ito s diffusio 9 All the above theorems ca be paraphrased i terms of istead of ( ) γ usig the ext lemma: ρ ad r Lemma 2. Let α α, r r,, α. The γ ( ) ( ρ r ),. Remark. I the case whe γ ( ) the market degeerates i some way ad the discout factor ρ ad iterest rate r becomes equal i the limit.

11 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio Let ξ, ξ, ξ, K 2 such that Eξ <. be a sequece of idepedet idetically distributed radom variables o some probability space ( Ω, I,P ) The homogeeous Markov chai X = ( X X X K) associated with these variables is,,, : 2 X = x R, X = x+ S, S = ξ, t, Eξ <. t t t i i k= t ( ) { } σ { ξ ξ } I I = Ω I = K - atural filtratio. k k,,, k,, k

12 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio The mai goal of this part of the work is to solve the Optimal Stoppig Problem for the process where ( ) Xt which cosists of fidig the price fuctio ( ) ( ) { τ } V x = sup E g X I <, x R, τ M g x is a measurable fuctio, I { } ad supremum is take i the class x τ is a idicator fuctio M of all Markov times τ with values i [, ] with respect to the atural filtratio ( I k ) k. The Optimal Stoppig Time is defied as follows τ * ( ) I{ τ } = arg max E g X <. τ M x τ

13 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio We cosider the reward fuctios g( x) of the followig form 2 ( ) ( ) k + g x = C x, C R. k= Defiitio. Let η be a radom variable such that E exp( λη ) < for some λ >. Appel Polyomials* is the followig set of polyomials ( ) ( η ) Q y = Q y,, k =,,2, K, defied through the expasio k k exp uy u = Q k y Eexp u k! ( ) ( η ) k k= k k ( ). *Novikov A.A., Shiryaev A. N., O a effective solutio of the optimal stoppig problem for radom walks, Teor. Veroyatost. i Primee., 49:2 (24),

14 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio We cosider the Appel Polyomials geerated by the radom variable Defiitio 2. Fuctio P( y) if exists umber a > such that ( ) ( ) Q y = Q y, M, k =,,2, K. k M = sup S, S =, k k k is called of the type Aa ( ) 3 ( ) =, ( ) [, ), ( ) [ ) P a P y o a P y icreases mootoically o a,. The followig statemets give ecessary ad sufficiet coditios uder which the liear combiatio of the Appel Polyomials is a fuctio of the type Aa ( ).

15 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio We defie the followig liear combiatio of Appel Polyomials Theorem 6. ( ) ( ) ( ) K ( ) P y is of the type A( a) P y = CQ y + C2Q2 y + + CQ y, C > Polyomial ( ) oe of the followig statemets holds a) The derivative of P ( y) does ot have roots o [ ) b) The derivative of P ( y) has roots o [ ) of local extremums of P ( y) ad P ( ) <. if ad oly if, ad ( ) P <., which are ot the poits c) The derivative of P ( y) has roots o [ ) P y The first of the local extremums o [, ) of local extremums of ( )., which cotai the poits miimum ad i each maximum if exists the fuctio is egative ad P ( ) d) The derivative of P ( y) has roots o [ ) of local extremums of P ( y). The first of the local extremums o [ ) maximum ad i each of them the fuctio P ( y) is egative. is., which cotai the poits, is 4

16 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio Remark. We established that all polyomials ( ) ( ) ( ) K ( ) P y = CQ y + C2Q2 y + + CQ y, C > with positive coefficiets C are of the type A( a). 5 Statemets of the Theorem 6 ca be simplified i particular cases. χ Remark. The symbol k states for the cumulat of the order of the radom variable M defied above. Theorem 7. Let is of the type χ 2 2 > χ. Polyomial ( ) ( ) ( ) ( ) P y = C Q y + C Q y, C, C > A a if ad oly if the coefficiets C, C2 C χ 2 χ. C χ 2 satisfy k

17 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio Theorem 8. is of the type The polyomial P y = C Q y + C Q y + C Q y, C > ( ) ( ) ( ) ( ) ( ) A a if ad oly if oe of the followig statemets hold 6

18 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio The domais of suitable coefficiets for P3 ( y) to be of the type A( a) Built o the ext data: а) χ χ =, χ 3χ χ + χ =, ( χ =, χ =, χ = 2) б) χ χ >, χ 3χ χ + χ =, ( χ =, χ = 2, χ = 5). 2 3

19 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio The domais of suitable coefficiets for P3 ( y) to be of the type A( a) Built o the ext data: в) χ χ =, χ 3χ χ + χ >, ( χ =, χ =, χ = 3) г) χ χ >, χ 3χ χ + χ >, ( χ =, χ = 2, χ = 6). 2 3

20 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio The followig theorem gives a explicit form of the optimal stoppig time 9 Eξ < ( + E ξ ) k+ Theorem 9. Let, <, k ad coefficiets C, C2, K, C be such that polyomial P ( y) = CQ( y) + C2Q2( y) +K K + C Q ( y), C > is of the type A( a) * { *} The the stoppig time if k Xk a radom walk X ( X, X, X, ) τ = is optimal for the = K ad reward fuctio 2 ( ) ( + ) K ( ) + 2 g x = Cx+ C2 x + + C x, C >. Moreover the followig equality holds ( ) ( ) { } ( ) { * : = sup } τ τ < = * τ < V x E g X I E g X I x x τ τ M V x = EP M + x I M + x a *. ad ( ) ( ) { } with positive root *. a

21 2. Optimal stoppig problem for a radom walk with polyomial reward fuctio Remark. I the previous statemets arises some coditios o the χ, χ, χ of the supremum of the radom walk cumulats M = sup Sk. We have proved that exist such radom walks that these k coditios holds. Let the radom walk X ( X, X, X, ) Theorem. = K be defied by the icremets 2 2, with probability p, ξi =, with probability p. p, p 2 Exists such that for all (, p ] iequalities for the maximum of radom walk holds χ χ >, χ 3χ χ + χ >. the followig

22 3. Asymptotics of exit times for diffusio processes We cosider the process which is the solutio of the SDE () ( ) () t ( ) σ ( ( )) ( ) X t = X + b s, X s ds+ s, X s dw s, () with o-radom iitial coditios ad coefficiets that satisfy the followig assumptios a) coefficiets b b) coefficiets b c) coefficiet b σ d) coefficiet ad ad σ σ ( ) σ ( ) ( ) t are cotiuous i both argumets; are of the liear growth b t, x + t, x L + x, t, x R. satisfy Lipschitz coditio; exists icreasig fuctio :, ( ) ( ) b t, x b t, y L x y, t, x, y R. satisfy the followig Yamada coditio ρ ( ) ( ) ( ) ( ) + ρ u du = σ tx, σ ty, ρ x y, t, xy,. such that 2

23 Defiitio 3. Let T >. 3. Asymptotics of exit times for diffusio processes T Space T L ( F P) 2, [ ] [ ] ( Ω ( ) ) L2, T, F B, T, P λ is the followig space 22 elemets of which satisfy the followig coditio T Ω 2 ( ωλ ) ( ) ( ω) X t, dt P d <. Theorem *. Uder coditios (a)-(d) the stochastic differetial T equatio () has uique solutio X ( t) from ( ) L F P for all T >. 2, *Yamada T., Sur l approximatio des solutios d equatios differetielles stochastiques, Z. Wahrscheilichkeitstheorie. Gebiete 36 (976),

24 3. Asymptotics of exit times for diffusio processes The followig poitwise covergece is assumed 23 ( ) ( ) ( ) ( ) σ ( ) σ ( ) X X, b tx, b tx,, tx, tx,, (2). Theorem. Uder coditios (a)-(d) ad covergece (2) the process X () t covergeces uiformly i probability to the limit process X () t o ay iterval [, T ]. This is geeralizatio of result obtaied by Gihma ad Skorohod*. Remark. The process X ( t) is the limit process that is the solutio of SDE () with limit iitial coditios ad parameters X, b, σ. *Gihma I.I., Skorohod A.V., Stochastic differetial equatios ad its applicatios, Kiev, Naukova Dumka (982).

25 3. Asymptotics of exit times for diffusio processes The followig assumptios o coefficiets of SDE () are assumed: 24 ) coefficiets b ( x ) ad σ ( x ) deped oly o space; 2) coefficiets b ad σ are cotiuous i both argumets; 3) coefficiets b ad σ are of the liear growth 4) coefficiet b 5) coefficiet satisfy Lipschitz coditio; σ satisfy the Yamada coditio with ρ ( ) = ; x x τ Let τ ad be the first exit times of the processes from some iterval X < l,. ( ],l where ( ) X ( t) ad X () t Theorem. Uder covergece (2) the exit time probability to τ. τ coverge o

26 Refereces 25 ) Oksedal B., Stochastic differetial equatios, Berli: Spriger-Verlag, 5 th editio, (2) 332 p. 2) Novikov A.A., Shiryaev A. N., O a effective solutio of the optimal stoppig problem for radom walks, Teor. Veroyatost. i Primee., 49:2 (24), ) Ikeda N., Wataabe S., Stochastic differetial equatios ad diffusio processes, North-Hollad Publishig Co. ad Kodasha., (98). 4) Yamada T., Sur l approximatio des solutios d equatios differetielles stochastiques, Z. Wahrscheilichkeitstheorie. Gebiete 36 (976), ) Gihma I.I., Skorohod A.V., Stochastic differetial equatios ad its applicatios, Kiev, Naukova Dumka (982). 6) Gihma I.I., Skorohod A.V., Cotrolled radom processes, Kiev, Naukova Dumka (977). 7) Dyki Е.B., Yushkevich A.A., Theorems ad problems about Markov processes, Moscow.: Nauka, p. 8) Schoutes W., Stochastic Processes ad Orthogoal Polyomials, New York: Spriger-Verlag, (2) 63 p. (Lecture Notes i Statist., V. 46). 9) Mishura Y.S., Tomashyk V.V., Limit behavior of optimal sale times for a asset whose price is described by Ito s diffusio, Appl. Stat. Act ad Fi. Math. 2 (27) ) Tomashyk V.V., Mishura Y.S., Optimal stoppig times for a radom walk with polyomial reward fuctios, Appl. Stat. Act ad Fi. Math. -2 (28) -. ) Mishura Y.S., Tomashyk V.V., Optimal stoppig problem for radom walks with polyomial reward fuctios, Teor. Imovir. ta Matem. Statyst. 86 (22)

29 Явний вигляд поліномів Аппеля через кумулянти в.в: 2 ( ) = ( ) = χ 2( ) = ( χ) χ2 Q y, Q y y, Q y y, 3 3( ) = ( χ) χ2( χ) χ3 Q y y 3 y, де χ = μ, χ = μ + μ, χ = 2μ 3 μ μ + μ, μk = E η. k Формула для відшукання кумулянтів випадкової величини: ( iu) m+ χ = = ( ) =! m= ( iu) m μ!. m

Sequences and Series of Functions

Sequences and Series of Functions Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges