1 Introduction. 2 Stochastic differential delay ... - York University

STABILITY OF STOCHASTIC DIFFERENTIAL DELAY ITO’S EQUATIONS WITH POISSON JUMPS AND WITH MARKOVIAN SWITCHINGS. APPLICATION TO FINANCIAL MODELS. Swishchuk A.V., Kazmerchuk Y.I. Key words: stochastic stability, markovian switchings, SDDE with jumps, Liapunov functional, (B, S)-securities market.

1

Introduction.

The processes of financial mathematics in conditions of (B, S)-market is patterned by stochastic differential equations Ito. The complexity of financial relations and reality of market impose diverse, often complex, conditions that have their influence on dynamics of securities. Taking those influences in account is serious and actual problem nowadays. It demands applying of deep and various mathematical methods and approaches. The subject of this work is stability of trivial solution of stochastic differential delay Ito’s equations with markovian switchings and with Poisson biffurcations. Throughout the work stochastic analogue of second Liapunov’s method is used having been described in [3]. The issues [2] and [8] contain general theory of differential delay equations. Some results on stochastic differential equations is located in [4],[9] and [10]. Workship [7] contains general survey of some results on SDDE. Subject of [11] is SDDE with markovian switchings of their parameters.

2

Stochastic differential delay equations with Poisson biffurcations.

Let stochastic processes {x(t), t ∈ [−h, T ]} ∈ Rn , {φ(θ), θ ∈ [−h, T ]} ∈ Rn , scalar process of Brownian motion {W (t), t ∈ [0, T ]} and {ν(du, dt), t ∈ [0, T ], u ∈ [−1, +∞)} -centralized Poisson measure with parameter Π(du)dt [10] be set on the probability space (Ω, Σ, P ) and let {φ(θ)} be independent of Wiener process and of measure. For all t ∈ [−h, T ] define: ( φ(t), t ∈ [−h, 0]; Rt Rt R t R +∞ x(t) = (2.1) φ(0) + o f (s, xs )ds + 0 g(s, xs )dW (s) + 0 −1 ux(s) · ν(ds, du); where: f : [0, T ] × Dn [−h, 0] → Rn , g : [0, T ] × Dn [−h, 0] → Rn -are measurable on the set of variables; xt = {x(t + θ), θ ∈ [−h, 0]}; Dn [−h, 0]-space of functionals {φ(t), t ∈ [−h, 0]} ∈ Rn that are right-continuous and have left-sided limits. Let the norm be kφk = sup{|φ(θ)|, θ ∈ [−h, 0]}, phi(0) = x(0) = x. Then {x(t), t ≥ 0} be called a solution of SDDE with Poisson jumps: Z ∞ dx(t) = f (t, xt )dt + g(t, xt )dW (t) + ux(t)ν(dt, du) (2.2) −1

with initial data: x(t) = φ(t), t ∈ [−h, 0]. Consider theorem of existness and uniqueness of solution of (2.2) [1]: Theorem 2.1: Let {f (t, φ)}, {g(t, φ)} be continuous and assume Lipshytz’s condition: |f (t, φ) − f (t, ψ)|2 + |g(t, φ) − g(t, ψ)|2 ≤ k · kφ − ψk2

(2.3)

for all φ, ψ ∈ Dn . If there have been set on the probability space (Ω, Σ, P ) Brownian motion {W (t), t ∈ [0, T ]}, centralized Poisson measure {ν(du, dt)} and independent of them process {φ(θ), θ ∈ [−h, 0]} with realizations that don’t have any breaks of second type and E{kφk4 } < ∞ holds. Then for all T > 0 there exists unique solution of (2.2) at [0, T ].

1

3

Stability theorems.

Assume: (Lv)(s, φ) = lim t↓0

1 · [E{v(s + t, xs+t (φ))} − v(s, φ)] t

(3.1)

Then definition area D(L) of operator L consists of elements of space C(R+ × Dn ) and hold: 1) For all (s, φ) ∈ R+ × Dn there exist δ > 0 C > 0 such that: sup 0 0, {φ, ψ} ⊂ Sr ≡ {φ ∈ Dn [−h, 0] | kφk < r} : Z 0 |f (t, φ) − f (t, ψ)|2 + |g(t, φ) − g(t, ψ)|2 ≤ Lr · |φ(θ) − ψ(θ)|2 · dθ

(3.3)

−h

Then for functionals: R0 1) G(s, φ(0)) ∈ D(L); 2) V (s, φ) = −h l(θ) · H(s, φ(θ), φ(0)) · dθ ∈ D(L) : 1) (LG)(s, φ) =

∂G(s, φ(0)) + < (∇G)(s, φ(0)), a(s, φ(0)) > + ∂s 1 + sp((∇2 G)(s, φ(0)) · g(s, φ) · g T (s, φ))+ 2 +∞ Z + [G(s, φ + φ(0) · y) − G(s, φ(0))−

(3.4)

−1

− < (∇G)(s, φ(0)), φ(0) · y >]Π(dy) 2) (LV (s, φ)) =l(0) · H(s, φ(0), φ(0)) − l(−h) · H(s, φ(−h), φ(0))− Z 0 dl(θ) − · H(s, φ(θ), φ(0))dθ+ −h dθ Z 0 + l(θ) · (L2 H)(s, φ(θ), φ(0))dθ

(3.5)

−h

where: L2 acts at H(s, φ(θ), φ(0)) by the first and the third arguments by the rule (3.4) with fixed second argument, < ·, · > is inner product; G(s, φ(0)) has continuous second derivatives w.r.t. 00 φ(0); l has a continuous derivative on some open set containing [−r, 0]; H(s, α, beta), Hs0 , Hβ0 , Hββ are continuous in s, α and β; ∇ stands fro dervative w.r.t. φ(0). Consider definition of stability of solution of (2.2). Assume: f (t, 0) = g(t, 0) = 0 (3.6) Then trivial solution of (2.2) be called stable if for all ε > 0 there exists δ > 0 such that for all s ≥ 0 from φ ∈ Sδ = {φ ∈ Dn |kφk < δ} we impose xt (s, φ) ∈ Sε = {φ ∈ Dn |kφk < ε} for all t ≥ s ≥ 0. 2

Trivial solution be called asymptothycally stable if it is stable and for all s ≥ 0 there exists δ > 0 such that: limt→∞ E{|xt (s, φ)|2 } = 0 for all φ ∈ Sδ . We will investigate the stability of trivial solution of (2.2) by stochastic analogue of second method of Liapunov. Let us consider a set: Z 0 2 W := {V ∈ C(R+ × Dn )| c1 · |φ(0)| ≤ V (s, φ) ≤ c2 · |φ(θ)|2 dθ} −h

for some c1 > 0, c2 > 0 and for all s ∈ R+ , φ ∈ Dn . Theorem 3.2 [2]: Assume coditions (3.3) and (3.6) and existance of functional V ∈ W ∩D(L) such that LV ≤ −f , f ∈ W , Then trivial solution of (2.2) is asymptothycally stable.

4

Stability of stochastic differential delay equations with linear Poisson jumps and with Markovian switchings.

Let {Ω, F, {Ft }t≥0 , P } be a complete probability space with filtration satisfying usual conditions (i.e. it is right continuous and F0 contains all P -null sets). Define stochastic processes {x(t), t ∈ [−h, T ]} ∈ Rn , {φ(θ), θ ∈ [−h, 0]} ∈ Rn , scalar Brownian motion {W (t), t ∈ [0, T ]}, centralized Poisson measure {ν(dy, dt), t ∈ [0, T ], y ∈ [−1, +∞)} with parameter Π(dy)dt [10] and for all t ∈ [−h, T ]:   φ(t), tR∈ [−h, 0]; t (4.1) x(t) = φ(0) + o [a(r(s))x(s) + µ(r(s))x(s − τ )]ds+  R R t R +∞ t  + 0 σ(r(s))x(s − ρ)dW (s) + 0 −1 yx(s) · ν(dy, ds); Here: a(·), µ(·), σ(·) are matrix maps with dimension n × n acting from set S = {1, 2, ..., N }; τ > 0, and ρ > 0; {r(t), t ∈ [0, +∞)} is markovian chain taking values at the set S with generator Γ = (γij )N ×N : ( γij δ + o(δ), i 6= j P (r(t + δ) = j|r(t) = i) = 1 + γii δ + o(δ), i = j P where δ > 0, γij ≥ 0 and γii = − i6=j γij . Assume r(·) is independent of W (·). We call x(t) a solution of stochastic differential delay equation with linear Poisson jumps and with Markovian switchings: dx(t) =[a(r(t))x(t) + µ(r(t))x(t − τ )]dt + σ(r(t))x(t − ρ)dW (t)+ Z ∞ + yx(t)ν(dy, dt)

(4.2)

−1

Theorem 4.1: Equation (4.2) has unique solution x(t) of (4.1) for t ∈ [−h, +∞). Proof : There exists a sequence {τk }k≥0 of stopping times that 0 = τ0 < τ1 < ... < τk → ∞ and r(t) is constant at each interval [τk , τk+1 ), : ∀k ≥ 0 : r(t) = r(τk )

τk ≤ t < τk+1 .

Consider equation (4.2) at t ∈ [0, τ1 ∧ T ], that is: dx(t) =[a(r(0))x(t) + µ(r(0))x(t − τ )]dt+ Z ∞ yx(t)ν(dy, dt) + σ(r(0))x(t − ρ)dW (t) + −1

3

(4.3)

with initial data x(t) = φ(t), t ∈ [−h, 0], where φ is continuous process. Using theorem 2.1 we know that equation (4.3) has unique right-continuous solution at [−h, τ1 ∧ T ] with the property: E[sup−h≤s≤τ1 ∧T |x(s)|2 ] < +∞. At the moment τ1 ∧ T we have: x(τ1 ∧ T ) − x(τ1 ∧ T −) = Y1 x(τ1 ∧ T −), where Y1 is a random variable with distribution Π(dy). Further we consider (4.2) at t ∈ [τ1 ∧ T, τ2 ∧ T ]: dx(t) =[a(r(τ1 ∧ T ))x(t) + µ(r(τ1 ∧ T ))x(t − τ )]dt+ Z ∞ + σ(r(τ1 ∧ T ))x(t − ρ)dW (t) + yx(t)ν(dy, dt)

(4.4)

−1

with initial data assigned by function xτ1 ∧T defined: xτ1 ∧T (θ) = x(θ + τ1 ∧ T ) being a solution of (4.3). At the moment τ2 ∧ T we have: x(τ2 ∧ T ) − x(τ2 ∧ T −) = Y2 x(τ2 ∧ T −), where Y2 has the distribution Π(dy) and independent on Y1 . We know that (4.4) has unique rightcontinuous solution at [τ1 ∧ T − h, τ2 ∧ T ]. Continuing this procedure we obtain unique solution x(t) at [−h, T ]. For T is arbitrary defined x(t) exists and is unique at [−h, +∞]. Q.E.D. Assign C 2,1 (Rn × R+ × S; R+ ) the family of all nonnegative functions V (x, t, i) at Rn × R+ × S which have contionuous second derivatives by x and have continuous derivative by t. For V ∈ C 2,1 (Rn × R+ × S; R+ ) introduce an operator LV : Rn × R+ × S → R by rule: LV (x, t, i) =Vt (x, t, i) + Vx (x, t, i) · [a(i)x(t) + µ(i)x(t − τ )]+ 1 + tr[xT (t − ρ)σ T (i)Vxx (x, t, i)σ(i)x(t − ρ)]+ 2 Z ∞ N X + γij V (x, t, j) + [V (x + yx, t, i) − V (x, t, i)−

(4.5)

−1

j=1

− Vx (x, t, i)yx]Π(dy) where: Vt (x, t, i) =

∂V (x, t, i); ∂t

Vx (x, t, i) = (

Vxx (x, t, i) = (

∂V ∂V (x, t, i), ..., (x, t, i)); ∂x1 ∂xn

∂2V (x, t, i))n×n . ∂xi ∂xj

Let us introduce here generalized Ito’s formula [11]. For V ∈ C 2,1 (Rn × R+ × S; R+ ) and for all stopping moments 0 ≤ ρ1 < ρ2 < ∞: Z ρ2 EV (x(ρ2 ), ρ2 , r(ρ2 )) = EV (x(ρ1 ), ρ1 , r(ρ1 )) + E LV (x(s), s, r(s))ds,

(4.6)

ρ1

whether expectations of all intergrals exist. Consider stability theorem. Theorem 4.2: Assume p, c1 , c2 are positive integers and λ1 > 0 and there exists function V (x, t, i) ∈ C 2,1 (Rn × R+ × S; R+ ) such that: c1 kxkp ≤ V (x, t, i) ≤ c2 kxkp

(4.7)

LV (x, t, i) ≤ −λ1 kxkp

(4.8)

1 lim sup ln(E|x(t)|p ) ≤ −γ, t

(4.9)

and: n

for all (x, t, i) ∈ R × R+ × S. Then: t→∞

4

where x(t) is solution of (4.2) with initial data φ(t), γ > 0 is defined by: γ = λ1 /c2 . In other words, trivial solution of (4.2) is p-exponentially stable and p-exponent of Liapunov no greater then −γ. Proof: Fix φ(t) ∈ C[−h, 0] is continuous process. Define: U (x, t, i) = eγt V (x, t, i). By Ito’s formula: Z t EU (x(t), t, r(t)) =EU (x(0), 0, r(0)) + E LU (x(s), s, r(s))ds = 0 Z t = EU (x(0), 0, r(0)) + E eγt [γV (x(s), s, r(s))+ 0

+ LV (x(s), s, r(s))]ds ≤ c2 · Ekφkp + Z t + (γ · c2 − λ1 ) · E eγs |x(s)|p ds; 0

For γc2 = λ1 we have: EU (x(t), t, r(t)) ≤ c2 Ekφkp . But: EU (x(t), t, r(t)) ≥ c1 · eγt · Ekx(t)kp . So: Ekx(t)kp ≤

Ekφkp c2 e−γt c1

and demanded inequality follows here. Q.E.D.

5

Application in finance.

In financial mathematics equation (2.2) describes process of stock cost St with linear Poisson jumps. In this chapter we will consider an equation: Z ∞ dS(t) = [aS(t) + µS(t − τ )]dt + σS(t − ρ)dW (t) + yS(t)ν(dt, dy) (5.1) −1

Parameters µ σ have a sense of coeficients of growth and volatility. Equation (5.1) without jump component was named after Ito. Solutions of this equation are continuous, but solutions of (5.1) at finite interval have finite quantity of breaks of first type. Besides stocks primary securities also include bonds. Values of bond process are determined and independent of outter factors. It has such a look: dB(t) = [bB(t) + νB(t − β)]dt

(5.2)

Within coditions of (B, S)-market consider new terms: Discounted cost of stock : St∗ = St /Bt , where St and Bt are described by (5.1) and (5.2). Capital of holder of securities: Xt = αt St + βt Bt , where: (αt , βt ) is portfolio of holder, i.e. quantity of stocks and bondes which holder holds at the time t. Further we will use self-financing strategies, i.e. portfolio for which: St dαt +Bt dβt = 0. Hence, capital Xt satisfyes an equation: dXt = αt dSt + βt dBt . Whether trivial solution of (5.1) is stable in sense of Liapunov then in according to the definition stochastic stability it can be treated such as: S = (St ) is taking values close to 0 with probability 1. The latter is undesireable in the sense of finance. That is, obtaining such restrictions at the parameters we obtain conditions of taking advantages for the holder of securities. Theorem 5.1: For the functionals G and V we have: 1) G(s, φ(0)) ∈ D(L); R0 2) V (s, φ(0)) = −h l(θ) · H(s, φ(θ), φ(0))dθ ∈ D(L) where: L is infinitesimal operator.

5

And: 1) (LG)(s, φ) =

∂G ∂G(s, φ(0)) +< (s, φ(0)), aφ(0) + µφ(−τ ) > + ∂s ∂φ 1 ∂2G 2 2 + σ φ (−ρ)+ 2 ∂φ2 Z ∞ + [G(s, φ(0) + φ(0)y) − G(s, φ(0))− −1

∂G (s, φ(0))yφ(0)]Π(dy); − ∂φ 2) (LV (s, φ)) =l(0) · H(s, φ(0), φ(0)) − l(−h) · H(s, φ(−h), φ(0))− Z 0 dl(θ) − · H(s, φ(θ), φ(0))dθ+ −h dθ Z 0 + l(θ) · (L2 H)(s, φ(θ), φ(0))dθ −h

where: L2 acts at H(s, φ(θ), φ(0)) by the first and the third arguments by the rule 1) with fixed second argument. This thoerem is consequence of theorem 3.1.

6

Examples.

1. Consider equations: ( dS(t) = [aS(t) + µS(t − τ )]dt + σS(t − ρ)dW (t) dB(t) = [bB(t) + νB(t − β)]dt Let us find restrictions at a, µ, σ, b, ν that trivial solution of (6.1) is stochastically stable. Let us take the functional of Liapunov: Z 0 Z 0 x2 (0) V (x) = +A x2 (θ)dθ + B x2 (θ)dθ; where : A > 0, B > 0 2 −τ −β

(6.1)

(6.2)

Consider first equation. The infinitesimal operator has the look: σ2 2 x (−ρ) + A[x2 (0) − x2 (−τ )]+ 2 + B[x2 (0) − x2 (−ρ)] = (A + B + a)x2 (0)+

LV (x) =x(0)[ax(0) + µx(−τ )] +

+(

σ2 − B)x2 (−ρ) − Ax2 (−τ ) + µx(0)x(−τ ) 2

Write LV (x) as square form with relations to [x(0), x(−τ ), x(−ρ)].     A + B + a µ/2 0 x(0)  · x(−τ ) −A 0 LV (x) = (x(0), x(−τ ), x(−ρ)) ·  µ/2 0 0 σ 2 /2 − B x(−ρ) For square form is negative defined when all main minors of dimension k have the sign (−1)k . So it is sufficient the fullfilment of ineqalities: µ µ A + B + a < 0, (A + B + a)(−A) − · > 0, 2 2 σ2 −A < 0, (−A)( − B) > 0, 2 σ2 µ µ σ2 − B < 0, [(A + B + a)(−A) − · ]( − B) < 0. 2 2 2 2 6

This system of inequalities is equivalent to: σ2 − B < 0, 2 µ2 (A + B + a)(−A) > . 4 A + B + a < 0,

(6.3)

The set of {A, B} satisfying (6.3) is nonempty if: a+

σ2 + |µ| < 0 2

(6.4)

Then there exist positive A, B such that the functional V is positively defined and LV is negative defined. It is equivalent to conditions of therem 4.2 to be performed. That is, (6.4) is condition of stability of trivial solution of (6.1). For the second equation of (6.1) we have: b + |ν| < 0.

(6.5)

2. Consider the process of discounted cost of stock: St∗ = St /Bt . For the investigation of stability of trivial portfolio we will use a vector: Xt =



 Bt . St∗

It is easy to see that Xt is satisfying an equation:   bBt + νBt−β dt+ dXt = Bt−τ ∗ ∗ Bt−β (a − b)St∗ + µSt−τ Bt − νSt Bt   0 + dWt B ∗ σ Bt−ρ St−ρ t Let us write a functional of Liapunov: Z 2 2 V (x) = x1 (0)x2 (0)/2 + A

0

x21 (θ)x22 (θ)dθ

−τ

+B

Z

0

x21 (θ)x22 (θ)dθ; A, B > 0

−ρ

(LV )(x) =x1 (0)x22 (0)[bx1 (0) + νx1 (−β)] + x21 (0)x2 (0)[(a − b)x2 (0)+ + µx2 (−τ )x1 (−τ )/x1 (0) − νx2 (0)x1 (−β)/x1 (0)]+ 1 + x21 (0)σ 2 x22 (−ρ)x21 (−ρ)/x21 (0)+ 2 + A[x21 (0)x22 (0) − x21 (−τ )x22 (−τ )] + B[x21 (0)x22 (0) − x21 (−ρ)x22 (−ρ)] = =bx21 (0)x22 (0) + (a − b)x21 (0)x22 (0) + µx1 (0)x2 (0)x1 (−τ )x2 (−τ )+ 1 + σ 2 x21 (−ρ)x22 (−ρ) + Ax21 (0)x22 (0) − Ax21 (−τ )x22 (−τ )+ 2 + Bx21 (0)x22 (0) − Bx21 (−ρ)x22 (−ρ) = =(A + B + a)(x1 (0)x2 (0))2 + µ(x1 (0)x2 (0))(x1 (−τ )x2 (−τ ))− 1 − A(x1 (−τ )x2 (−τ ))2 + ( σ 2 − B)(x1 (−ρ)x2 (−ρ))2 ; 2 Let us cosider a substitution: η(θ) := x1 (θ)x2 (θ). Then: LV (x) = (a + A + B)η 2 (0) + µη(0)η(−τ ) − Aη 2 (−τ ) + (σ 2 /2 − B)η 2 (−ρ) Write LV as square form with accordance to [η(0), η(−τ ), η(−ρ)]:   a + A + B µ/2 0  −A 0 Λ =  µ/2 0 0 σ 2 /2 − B 7

(6.6)

Notice that we were also considering the form of such a type in item 1. Thus the condition of negative definition has the look: a + |µ| + σ 2 /2 < 0 (6.7) That is, (6.7) is condition of stability of trivial solution of (6.6). Since St∗ is component of vector of solution so it is also stable within condition (6.7). Notice that this condition is imposed only on the parameters of change of stock’s cost. 3. Let us consider capital process: Xt = βt Bt + γt St Strategy (βt , γt ) be called self-financing if: Bt dβt + St dγt = 0. Thus: dXt = βt dBt + γt dSt . For investigation of stability consider a system: ( Xt = βt Bt + γt St , (6.8) Yt = −βt Bt + γt St ¿From (6.8) we have: ( dXt = βt dBt + γt dSt dYt = −βt dBt + γt dSt or:

( dXt = [bβt Bt + νβt Bt−β + aγt St + µγt St−τ ]dt + σγt St−ρ dWt dYt = [−bβt Bt − νβt Bt−β + aγt St + µγt St−τ ]dt + σγt St−ρ dWt

For simplicity impose: β = ρ = τ = 1. Then: 1 βt γt dXt = [(a + b)Xt + (ν +µ )Xt−1 + 2 βt−1 γt−1 βt γt + (a − b)Yt + (−ν +µ )Yt−1 ]dt+ βt−1 γt−1 1 γt + σ (Xt−1 + Yt−1 )dWt 2 γt−1 1 βt γt dYt = [(a − b)Xt + (−ν +µ )Xt−1 + 2 βt−1 γt−1 βt γt + (a + b)Yt + (ν +µ )Yt−1 ]dt+ βt−1 γt−1 1 γt + σ (Xt−1 + Yt−1 )dWt 2 γt−1 Liapunov’s functional: V (x, y) = x2 (0) + y 2 (0) + A

Z

0

−1

x2 (θ) + B

Z

0

y 2 (θ)

−1

Now we have: βt γt +µ )x(−1)+ βt−1 γt−1 βt γt + (−b + a)y(0) + (−ν +µ )y(−1)]+ βt−1 γt−1 βt γt + y(0)[(−b + a)x(0) + (−ν +µ )x(−1)+ βt−1 γt−1 βt γt + (b + a)y(0) + (ν +µ )y(−1)]+ βt−1 γt−1 σ 2 γt2 2 2 2 2 2 + 2 (x(−1) + y(−1)) + A[x (0) − x (−1)] + B[y (0) − y (−1)] 2 γt−1

LV (x, y) =x(0)[(b + a)x(0) + (ν

8

Considering it as a square form of [x(0), y(0), x(−1), y(−1)] we impose matrix with dimension 4 × 4. Analysing the values of main minors we find conditions on a, b, µ, ν, σ. That is, we reduce stability problem to linear algebra task. 4. In the item 1 it was imposed conditions on the coeficients of growth and volatility. But parameters τ, ρ, β could be arbitrary. So let us find restrictions on them. For equation: dx(t) = −bx(t − r)dt, where b > 0 (6.9) consider an integral: −b

Z

t−r

x(θ − r)dθ =

Z

t−2r

t

x0(t)dθ = x(t) − x(t − r)

t−r

Then (6.9) is equivalent to: x0(t) = −bx(t) − b2

Z

t−r

x(θ)dθ

(6.10)

t−2r

The initial data is function at [−2r, 0]. Let us consider (6.10) according to stability problem. Impose: Z 0 Z −r V (x) = x2 (0)/2 + A x2 (θ)dθ + B (θ + 2r)x2 (θ)dθ −r

−2r

The infinitesimal operator: LV (x) =x(0)[−bx(0) − b

2

Z

−r

x(θ)dθ] + A[x2 (0) − x2 (−r)]+

−2r

+ B[(−r + 2r)x2 (−r) − (−2r + 2r)x2 (−2r) −

−r

Z

x2 (θ)dθ] =

−2r

=(A − b)x2 (0) − b2 x(0)rx(θ) + (Br − A)x2 (−r) − Brx2 (θ), θ ∈ (−2r, −r). Considering it as a square form impose: 0 < br < 1

(6.11)

5. Consider an equation with jump component: dSt = (aSt + µSt−τ )dt + σSt−ρ dWt +

+∞

Z

ySt ν(dt, dy)

(6.12)

−1

Similar to preceding: V (x) = x2 (0)/2 + A

Z

0

x2 (θ) + B

−τ

1 1 LV (x) =x(0)(ax(0) + µx(−τ )) + σ 2 x2 (−ρ) + 2 2

Z

Z

0

x2 (θ)

−ρ +∞

[(x(0) + yx(0))2 −

−1 2

− x2 (0) − 2x(0)yx(0)]Π(dy) + A[x2 (0) − x (−τ )] + B[x2 (0) − x2 (−ρ)]   R +∞ 0 a + A + B + 12 −1 y 2 Π(dy) µ/2  Λ= µ/2 −A 0 2 0 0 σ /2 − B Condition of stability has the look: a+

1 2

Z

+∞

y 2 Π(dy) + |µ| + σ 2 /2 < 0

(6.13)

−1

Remark. Stability of stochastic differential delay equations (sdde) Ito is sdudied in [3]. Stability of sdde Ito with jumps is studied in [1]. Stability of sdde with Markovian switchings is investigated in [11]. Novelty of this work is to prove the theorems of existence, uniqueness and stability for sdde with jumps and Markovian switchings. The results connected with applications of stability theorems with delay to mathematical finance are also new. 9

References. [1] Tsarkov M.L., Sverdan M.L., Yasynsky V.K. ”Stability in stochatic modelling of the complex dynamical systems” Kyiv, 1996 [2] Hale J. ”Theory of functional-differential equations” Moscow, 1986 [3] Kushner H. ”On the stability of processes defined by stochastic differnce-differential equations” J.Diff.Eq., 4(1968), 424-443 [4] Gihman I.I., Skorohod A.V. ”Stochastic differential equations” Moscow,1968 [5] Khasminsky R.”Stochastic stability of differential equations” Moscow,1969 [6] Kolmanovsky V., Nosov V. ”Stability of functional-differential equations” Moscow,1972 [7] Ivanov A.F.,Swishchuk A.V. ”Stochastic differential delay equations and stochastic stability: a survey of some results” SITMS Research Report 2/99, University of Ballarat, Australia, Jan. 1999 [8] Bellman R., Cook K. ”Differential delay equations” Moscow, 1978 [9] Vatanabe, Ikeda ”Stochastic differential equations and diffusion processes” [10] Skorohod A.V. ”Asymtothycal methods in the theory of stochastic differential equations”, Naukova Dumka, Kyiv, 1989 [11] Mao X.,Matasov A.,Piunovski A. ”Stochastic differntial delay equations with Markovian switchings” Bernoulli,6(1),2000,pp.73-90

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