Modeling and estimation of stochastic transition rates in life insurance with regime switching based on generalized Cox processes

Modeling and Estimation of Stochastic Transition Rates in Life Insurance with Regime Switching Based on Generalized

Cox Processes

David Ba˜nos¹ Erik Bølviken², Sindre Duedahl³ and Frank Proske²

Abstract

In this paper we aim at modeling stochastic transition rates of state processes in life insurance by using generalized Cox processes. A feature of our non-Gaussian model is that it can be used to capture ”regime switching”

effects of data which may be due to regulatory changes in insurance markets or external ”shocks” caused e.g. by an economical crisis, natural disasters or epidemics. We propose a method how to estimate the unknown parameters of our model for stochastic transition rates from insurance data by using non-linear filtering techniques for L´evy processes. As a result we also obtain an explicit formula for the unnormalized density of a filtering problem with singular coefficients.

Key words and phrases: Life insurance, Stochastic Transition Rates, L´evy processes, non-linear filtering

AMS 2000 classification: 60G51; 60G35; 60H15; 60H40; 60H15; 91B70

1 Introduction

An important challenge in the risk analysis and risk management of life in- surance companies worldwide has been the accurate modeling of transition

1Inland Norway University of Applied Sciences. PO Box 400, 2418 Elverum, Norway E-mail address: [email protected]

2Centre of Mathematics for Applications (CMA), Department of Mathematics, Uni- versity of Oslo, P.O. Box 1053 Blindern, N-0316 Oslo, Norway.

E-mail address: [email protected], [email protected], [email protected]

3Institute of Computer Science II, Friedrich-Ebert-Alle 144, University of Bonn, 53113 Bonn, Germany.

E-mail address: [email protected]

rates as e.g. mortality rates or disability transition rates in the calcula- tion of insurance premiums. Compared to financial risk of technical interest rates longevity risk e.g. , which is due to increasing life expectancy of policy holders and pensioners, is a source of insurance risk, which has been system- atically underestimated for many years. A reason for the negligence of this type of risk in the insurance business has also been due to the use of deter- ministic models for mortality rates as e.g. the classical Gompertz-Makeham model. The latter models however, which cannot capture the uncertainty of the future dynamics of mortality rates, have led to a miscalculation of insur- ance premiums with respect to defined-benefit pension plans and annuities, from which many insurance companies have suffered substantial losses.

In order to overcome the deficiencies of deterministic models for transi- tion rates, there have been various attempts in the literature in recent years to describe the dynamics of future transition rates rates by using stochastic models. See e.g.the models of Lee, Carter [?] or Cairns, Blake, Dowd [?] in the case of mortality rates.

In this paper we want to study a non-Gaussian stochastic model for stochastic transition rates, which allows for the modeling of ”regime switch- ing” effects of data or more precisely ”regime switching” effects of the jump behaviour or the tails of the distribution of data which may be due to differ- ent types of influence factors as e.g. regulatory changes in insurance markets or external ”shocks” caused by a financial or political crisis, natural disasters or epidemics.

To be more specific, we consider in the following a c´adl´ag stochastic processZt,0≤t≤T with a finite state space S on some probability space (Ω,F, P), which is used as a model for the state of the insured dynamically in time. Further, we denote byN_ik(t) the process which counts the number of transitions from stateitokof the state processZ_t,0≤t≤T in the time interval (0, t].In a regular insurance model with a Markovian state process it is well known that

N_ik(t)− Z t

0

µ_ik(s)ds,0≤t≤T is a P-martingale with respect to the natural filtration

F_t^Z _0≤t≤T, where µ_ik(s) is the transition rate at timeswith respect to a transition from ito j. See e.g. [?].

One of the deficiencies of such a model as mentioned is that the deter- ministic transition rates may not capture the actual future transition rates.

Therefore it is reasonable to assume a stochastic model for the transition ratesµ_ik(t),0≤t≤T :

In the sequel, let µ_ik(t, x),0 ≤ t ≤ T be the transition rate at time t of an insured aged x years with respect to a transition from state i to state j, i, j ∈ S. In particular, the state space S of the insured in the case of a permanent disability insurance consists of the states∗ (”alive”), (”permanently disabled”) and† (”dead”).

In order to estimate stochastic transition rates from insurance data one may think of µ_ik(t, x),0≤ t ≤T as a result of a ”parametrization” of the deterministic transition rates by means of an unknown ”parametrization process”X_t,0≤t≤T.

More precisely, if S={∗,,†} one could assume that µ_∗(t, x) = Y_t⁽¹⁾+ 10^Yt(2)+Yt(3)x,

µ_∗†(t, x) = µ_†(t, x) =Y_t⁽⁴⁾+ 10^Yt(5)+Yt(6)x,

whereYt= (Y_t⁽¹⁾, ..., Y_t⁽⁶⁾),0≤t≤T is a generalized Cox process given by dYt=h(t, Xt)dt+dB^Y_t +

Z

R⁶

ςNλ(dt, dς)

and X_t,0≤t≤T the unknown ”parametrization process” modeled by the stochastic differential equation (SDE)

dX_t=b(X_t)dt+σ(X_t)dB^X_t (1) for Borel functions h, b and σ, where B_t^Y ∈R⁶, B^X_t ∈ R^d are independent Brownian motions and whereNλ is the jump measure of a ”generalized Cox process” with a predictable compensatorµb given by

µ(dt, dς, ω) =b λ(t, Xt, ς)dtν(dς) (2) for a L´evy measureν and a Borel functionλ.

More generally, we may assume in this paper that stochastic transition ratesµ_ik(t, x),0 ≤t≤ T, i, k∈ S are described by a stochastic Gompertz- Makeham modelGM(r, s) given by

µ_ik(t, x) =h^1,r_ik (t, x) + exp(h^2,s_ik(t, x)), (3) whereh^1,r_ik(t, x), h^2,s_ik (t, x) are time-dependent stochastic polynomials of de- greer ands, respectively, that is

h^1,r_ik (t, x) =

r

X

l=0

Y_t^(l)x^l

and

h^2,s_ik (t, x) =

s

X

l=0

Y_t^(r+1+l)x^l for alli, k ∈S.

In order to estimate the unknown ”parametrization” processX_t,0≤t≤ T from (indirectly) observed insurance data

Y_t= (Y_t⁽⁰⁾, ..., Y_t^(r), Y_t^(r+1), ..., Y_t^(r+s))^∗,0≤t≤T, (4) where∗denotes transposition, one can apply non-linear filtering techniques for L´evy processes as proposed in [?] to thesignal processXt∈Rⁿ,0≤t≤T and theobservation process Y_t∈R^m,0≤t≤T :

dXt=b(Xt)dt+σ(Xt)dB_t^X, (5) dYt=h(t, Xt)dt+dB_t^Y +

Z

R^m

ςNλ(dt, dς), (6) wherem=r+s+ 2.

Using the latter non-Gaussian filtering framework, we want to model stochastic transition rates, which are subject to regime switching effects of insurance data. In modeling this phenomenon one could e.g. assume that the ”parametrization” processX_t,0≤t≤T is described by

dX_t=b(X_t)dt+dB^X_t , (7) where the drift coefficient b:Rⁿ −→Rⁿ is a discontinuous vector field. An example of such a discontinuous vector field is

b(t, x) =

a₁ , if kxk ≥τ

a2 else .

Here the vectors a₁, a₂ ∈Rⁿ stand for the different regime switching states the parametrization processXtwill assume, if it exceeds a certain threshold τ at timet, that is kX_tk ≥τ ,or not.

Another example of such a drift coefficient in the case n = 1, which exhibits the feature of mean-reversion in connection with regime switching effects is

b(x) =

a(b1−x) , ifx≥τ a(b2−x) else .

fora, b1, b2 ≥0. In this case the parametrization process Xt may be inter- preted as a mean-reverting process with a mean reversion coefficientsaand different long-run average levelsb₁, b₂ depending on the threshold τ.

The parameters a, b1, b2 and the threshold τ in the above examples are a priori unknown and will be estimated from insurance data by using non- linear filtering techniques.

The non-linear filtering problem for our model is to find the least square estimate to the (possibly transformed) signal process X_t at time t, given the history of the observation process up to timet, that is to determine the conditional expectation

E[f(X_t) F_t^Y

,

wheref is a given Borel function and whereF_t^Y is theσ−algebra, generated by{Y_s,0≤s≤t}.

One of the objectives of this paper is the derivation of an explicit rep- resentation of the unnormalized conditional density with respect to the op- timal filter of the filter problem (??) and (??), when the drift coefficient b in (??) is merely (bounded and) Borel measurable. In solving this problem, we explicitly construct a (weak) solution to a stochastic partial differential equation given by the Duncan-Mortensen-Zakai or shortly Zakai equation for the conditional unnormalized density, which can be regarded as a weak so- lution to a stochastic Fokker-Planck equation with singular coefficients. See [?] in the deterministic case. Our method relies on a representation formula of the unnormalized conditional density found in [?] in the case of regular coefficients and finite L´evy measures, which we want to invoke in connec- tion with an approximation argument and local time techniques. As a result we give an explicit representation of the unnormalized conditional density associated with the least square estimate of the unknown parametrization processX_tof the generalized Cox process (??) in our model for the dynam- ics of stochastic transition rates. In contrast to [?] we do not require in this paper thatbis regular in the sense of Lipschitz continuity or that the L´evy measureν in (??) is finite.

We remark that non-linear filtering has been intensively studied in the literature since the 1960’s. See e.g. Lipster and Shiryaev [?], Kallianpur [?], Fleming and Rishel [?], Xiong [?] and the references therein. See also the innovation approach for the conditional density of the filter process by Fujisaki, Kallianpur and Kunita (see e.g.[?]). As for solutions of the Zakai equation in the Gaussian case we refer the reader to Zakai [?], Gy¨ongy, Krylov [?], [?], Pardoux [?], [?], Kunita [?]. See also [?], [?] or the works [?], [?], which give a generalization of results in [?], [?] to the non-Gaussian case.

The main objective of this paper is to introduce a model for the dynamics

of stochastic transition rates which is able to describe ”regime switching”

effects of the jump or tail distribution behaviour of e.g. observed mortality rates or transition rates in disability insurance by using the generalized Cox process (??) in the framework of non-linear filtering for L´evy processes, where the signal process, that is the parametrization process X_t of (??) is modeled by a SDE with singular coefficients.

A popular model for stochastic transition rates in the case of mortality rates was proposed by Lee, Carter [?]. In this discrete-time model the error terms are Gaussian distributed. A generalization of the Lee-Carter model is the Gaussian two-factor stochastic mortality model by Cairns, Blake and Dowd [?], which is used to describe the different behaviours of mortality rates at lower and higher ages. Reasons for the success of these models in life insurance is the simplicity of their implementation and their predic- tion reliability in forecasting mortality rates under ”usual” circumstances.

However, a disadvantage of these models is that they cannot capture e.g.

the observed skewness and (semi-) heavy tailed innovation distributions of data coming from cohort effects or short term catastrophic events as e.g. the Tsunami in 2004. In recent years there have been therefore several attempts to tackle this problem in the literature. In order to model heavy-tailed dis- tributions of mortality data Giacometti et al. [?] generalized the Lee-Carter model by modeling the distributional behaviour of the error terms by in- finitely divisible distributions in the case of Normal Inverse Gaussian laws.

Another model in this direction, which is based on non-Gaussian distribu- tions for error terms in the framework of [?], is the paper of Wang et al.

[?]. See also the approach in [?] based on Markov regime switching models or [?], where the authors employ jump diffusions to describe age-adjusted mortality rates.

Contrary to our model (??) and (??), however, the above mentioned models cannot be used to model the rather complex phenomenon of the occurrence of changing types of jumps or types of heavy-tailedness of dis- tributions of real data as a result of different types of ”external” shocks.

The reason for this is that these models are finite-dimensional models (in discrete time). Our model can be regarded as an infinite dimensional model for stochastic transition rates, since one of the unknown parameters is given by the parametrization process Xt,0 ≤ t ≤ T. In this paper we use the powerful tool of non-linear filtering for L´evy processes to efficiently estimate this process from constantly updated observations. Therefore we may expect that our approach is more flexible than those mentioned and also suitable for the modeling of other types of stochastic transition rates beyond mortality rates.

Our paper is organized as follows:

In Section 2 we introduce the framework of our paper and derive an explicit representation of the unnormalized conditional density associated with the least square estimate of the parametrization processX_t,0≤t≤T by constructing an explicit (weak) solution of a Zakai equation with singular coefficients. Further, we study the regularity of the obtained solution. Using the results of Section 2, we finally want to discuss in Section 3 various specifications of our model and its implementation in life insurance based on Monte-Carlo simulation.

2 Framework and Main Results

In this Section we want to introduce the mathematical framework of our general model for stochastic transition rates and to discuss the estimation of the unknown parameters or parameter processes of the the model from constantly updated observations in connection with a non-linear filtering problem for L´evy processes. In solving this problem we derive an explicit representation of the optimal filter of the filtering problem by constructing a (weak)L^p−solution of the Zakai equation for the unnormalized conditional density of the filter process with initial L´evy noise and singular coefficients.

In what follows we consider a L´evy processL_t∈R^m,0≤t≤T, that is a stochastically continuous process with stationary independent increments starting in zero defined on a filtered complete probability space

(Ω^∗,F^∗, π^∗),{F_t^∗}_0≤t≤T,

where{F_t^∗}_0≤t≤T is aπ^∗−augmented filtration generated byL·.

We may here assume from now on thatLt,0≤t≤T is a c`adl`ag process, that is a process, whose paths are right continuous paths and have existing left limits.

By the L´evy-Itˆo theorem the L´evy processLt= (L⁽¹⁾_t , ..., L^(m)_t ),0≤t≤ T can be uniquely decomposed as

L⁽ⁱ⁾_t =

l

X

k=1

a_ikB_t^(k)+b_it+ Z t+

0

Z

R^m0

z_i1_{kzk≥1}N(ds, dz) +

Z t+

0

Z

R^m₀

zi1{kzk<1}Ne(ds, dz),

for 0 ≤ t ≤ T, i = 1, ..., m, where Bt = (B^(k)_t )1≤k≤l ∈ R^l,0 ≤ t ≤ T is a Brownian motion, (a_ik)1≤i≤m,1≤k≤l ∈ R^m×l,(b_i)1≤i≤m ∈ R^m and Ne(ds, dz) =N(ds, dz)−dsν(dz) the compensated Poisson random measure associated with the L´evy process L·. Here ν is aσ−finite measure on the Borel sets B(R^m₀ ), R^m₀ := R^m \ {0}, referred to as L´evy measure, which satisfies the integrability condition

Z

R^d0

1∧ kzk²ν(dz)<∞

for the Euclidean normk·k. See e.g. [?] or [?] for more information on L´evy processes.

In what follows we want to estimate the unknown ”parametrization”

processXt,0 ≤t ≤T from the observed insurance data (??) by analyzing the non-linear filtering problem

dXt=b(Xt)dt+σ(Xt)dB_t^X, (8) dY_t=h(t, X_t)dt+dB_t^Y +

Z

R^m

ςN_λ(dt, dς), (9) for thesignal process Xt∈Rⁿ and theobservation process Yt∈R^m,0≤t≤ T, n, m∈Non a complete probability space (Ω,F, µ) , where the Brownian motionB_t^Y ∈Rⁿ is independent of the Brownian motionB_t^X ∈R^m and the integer valued random measure Nλ, whose predictable compensatorµb with respect to a augmented filtrationF={F_t}_0≤t≤T (generated byB^X_· , B_·^Y,N_λ) is given by

µ(dt, dς, ω) =b λ(t, Xt, ς)dtν(dς) (10) for the L´evy measure ν of Lt ∈ R^m and a Borel function λ. Further the initial condition X₀ in (??) is a random variable, which is independent of B_t^X, B_t^Y and N_λ.

In order to guarantee a unique strong solution to the system (??) and (??), we require for the time being that the continuous coefficientsb:Rⁿ−→

Rⁿ, σ:Rⁿ−→R^n×n, h: [0, T]×Rⁿ−→Rⁿ and λ: [0, T]×Rⁿ×R^m₀ −→R fulfill a linear growth and Lipschitz condition, that is

kb(x)k+kσ(x)k+kh(t, x)k+ Z

R^m0

|λ(t, x, ς)|ν(dς)≤C(1 +kxk) (11)

and

kb(x)−b(y)k+kσ(x)−σ(y)k+kh(t, x)−h(t, y)k +

Z

R^m0

|λ(t, x, ς)−λ(t, y, ς)|ν(dς)

≤Ckx−yk

(12)

for allx, y, tand a constantC <∞, wherek·kstands for a vector or matrix norm.

For the convenience of the reader we now want to give a derivation of the Zakai equation for the unnormalized filter of the non-linear filtering problem (??), (??). See e.g. [?] or [?] in the case of Wiener noise driven obervation processes.

For this purpose denote by π_t : Ω× B(Rⁿ) −→ [0,∞) the regular con- ditional probability measure of the signal process X_t given the σ−algebra F_t^Y, generated by{Y_s,0≤s≤t} and the null setsN. Then

E[f(Xt) F_t^Y

=hπ_t, fi

for allf ∈C_b(Rⁿ) (space of bounded continuous functions), wherehπ_t, fi:=

R

Rⁿf(x)πt(ω, dx).

Suppose that the functionλ: [0, T]×Rⁿ×R^m0 −→R is strictly positive and consider the density process

Λt:= exp{

m

X

i=1

Z t 0

−h_i(s, Xs)dB^Y,i_s −1 2

Z t 0

kh(s, X_s)k²ds +

Z t 0

Z

R^m₀

−logλ(s, Xs, ς)N_λ(ds, dς) + Z t

0

Z

R^m₀

(λ(s, Xs, ς)−1)dsν(dς)}, (13) for 0≤t≤T, whereB_t^Y = (B_s^Y,1, ..., B_s^Y,m)^∗andh(t, x) = (h₁(t, x), ..., h_m(t, x))^∗ (∗ transposition). Further, assume that

E[Λ_T] = 1. (14)

Remark 1 Using stopping time localization of Doleans-Dade exponentials, one obtains e.g. the following sufficient conditions for (??):

sup

0≤t≤T

E

"

exp(6 Z t

0

kh(s, X_s)k²ds + 4

Z t 0

Z

R^m0

(1−λ⁻¹(s, X_s, ς))λ(s, X_s, ς)dsν(dς)

− Z t

0

Z

R^m0

(1−λ⁻⁴(s, Xs, ς))λ(s, Xs, ς)dsν(dς)

#

<∞

(15)

E

"

Z T 0

Z

R^m0

(λ⁻⁴(s, Xs, ς)−1)λ(s, Xs, ς)

ν(dς)ds

#

+E

"

Z T 0

( Z

R^m0

|(λ(s, X_s, ς)−1)|ν(dς))²ds

#

<∞

(16)

E

"

Z T 0

Z

R^m0

|λ(s, X_s, ς) logλ(s, Xs, ς)|dsν(dς)

#

<∞ (17) An example which satisfies the conditions (??), (??) and (??) in the case m= 1 is given by

ν(dς) =ϕ(ς)dς, (18)

where

ϕ(ς) = ( ₁

|ς|^1+α , if |ς| ≤1

0 else

for α∈(0,1) as well as h is a bounded Borel measurable function and λ(s, x, ς) = exp(Ψ(x)|ς|) (19) for a bounded and continuous function Ψ :R−→R.

Define now the probability measure π with Radon-Nikodym derivative on (Ω,F_t) given by

dπ dµ _F

t

= Λt. and require that

Z

R^d0

kzkν(dz)<∞ (20)

Then by Girsanov’s theorem and the uniqueness of semimartigale charac- teristics (see e.g. [?]), the observation process Y_t,0 ≤ t ≤ T becomes a L´evy process being independent of the signal process under the new prob- ability measure π. More precisely, the system (??), (??) has the following representation underπ :

dXt = b(Xt)dt+σ(Xt)dB_t^X

dYt = dBt+dLt, (21)

whereY·is a L´evy process independent ofX·with B_t:=B_t^Y −

Z t 0

(−h(s, X_s))ds,0≤t≤T the Gaussian part and

Lt= Z t

0

Z

R^m0

ςN(ds, dς)

the jump component with respect the Poisson random measureN(ds, dς) :=

Nλ(ds, dς) with compensatordsν(dς).

SinceY·is a L´evy process underπ, we also observe that the (augmented) filtrationF_t^Y, 0≤t≤T is right-continuous.

The so calledunnormalized filterhΨ_t,·i,0≤t≤T is a stochastic process taking values in the space of finite Borel measures on Rⁿ, and is given by the Kallianpur-Striebel-formula, which is a consequence of Bayes’ rule:

Theorem 2 The optimal filter πt has the representation hπ_t, fi= hΨ_t, fi

hΨ_t,1i with

hΨ_t, fi:=Eπ[Ztf(Xt) F_t^Y

for all f ∈C_b(Rⁿ), where Eπ denotes the espectation with respect to π and

where

Z_t:= Λ⁻¹_t

= exp{

m

X

i=1

Z t 0

hi(s, Xs)dB_sⁱ− 1 2

Z t 0

kh(s, X_s)k²ds +

Z t 0

Z

R^m₀

logλ(s, Xs, ς)N(ds, dς) +

Z t 0

Z

R^m0

(1−λ(s, X_s, ς))dsν(dς)},

(22)

for 0≤t≤T under π.

Remark 3 We mention the fact that Eπ[ξ

F_t^Y

=Eπ[ξ|A]

for allF_t−measurable ξ with E_π[|ξ|]<∞,where A:= _

0≤t≤T

F_t^Y. See Proposition 3.15 in [?].

We also need the following Lemmata for the derivation of the Zakai equation:

Lemma 4 Let f ∈C_b^∞(Rⁿ) (space of smooth functions onRⁿwith bounded partial derivatives). Assume that the coefficients b, σ in (??), (??) are bounded and that

E

"

exp 496 Z T

0

kh(s, X_s)k²ds+ Z T

0

Z

R^m0

(1−λ³²(s, Xs(θ), ς))ν(dς)

ds + 32

Z _T

0

Z

R^m0

(1−λ(s, X_s(θ), ς))ν(dς)

ds

!#

<∞,

(23)

E

"

Z T 0

Z

R^m0

|logλ(r, Xr(θ), ς)|^jν(dς) k

dr

!4#

<∞, (24)

for allj= 1,2,4,8 ,k= 1,2,3 and E[

Z T 0

Z

R^m0

1−λ³²(s, X_s(θ), ς)

ν(dς)ds]

+E[(

Z T 0

( Z

R^m0

(1−λ(r, Xr(θ), ς))ν(dς))²ds)⁴]

<∞.

(25)

Then there exists a c`adl`ag modification of the unnormalized filter hΨ_·, fi. Proof. See Appendix.

Remark 5 An example satisfying the assumptions (??)-(??) in Lemma??

is given by Remark ??.

Lemma 6 Consider F−predictable processes α_t, β_t, γ_t(·),0 ≤ t ≤ T such that

Eπ[ Z T

0

(|α_s|+|β_s|²)ds] < ∞, E_π[

Z T 0

Z

R^m0

|γ_s(ς)|²dsν(dς)] < ∞.

Then

E_π[ Z t

0

α_sds F_t^Y

= Z t

0

E_π[α_s F_s^Y

ds, Eπ[

Z t 0

β_sdBs

F_t^Y

= Z t

0

Eπ[β_s F_s^Y

dBs

E_π[ Z T

0

Z

R^m0

γ_s(ς)N(ds, dς)e F_t^Y

= Z t

0

Z

R^m0

E_π[γ_s(ς) F_s^Y

Ne(ds, dς) and

Eπ[ Z t

0

β_sdB_s^X F_t^Y

= 0.

Proof. The proof is essentially based on the independence of the incre- ments of the process Y_t,0≤t≤T under π and can be e.g. found in [?] or [?] in the case of Brownian motion.

Using the latter auxiliary result, we obtain the following Zakai equation for the unnormalized filter of the non-linear filtering problem (??), (??):

Theorem 7 Assume the conditions of Lemma ??and require that sup

0≤s≤T

Eπ[|Z_s(λ(s, Xs, ς)−1)|^p]<∞ (26) for allς and somep >1. Then the unnormalized filterhΨ_t,·i,0≤t≤T is a c`adl`ag F_t^Y−adapted solution to the Zakai equation, that is to the SPDE

hΨ_t, fi = hΨ₀, fi+ Z _t

0

hΨ_s,Lfids+ Z _t

0

hΨ_s, f·h^∗(s,·)idB_s (27) +

Z t 0

hΨ_s−, f·(λ(s,·, ς)−1)iNe(ds, dς)

for all f ∈ D ⊂ C_c^∞((Rⁿ) (space of compactly supported infinitely often differentiable functions of Rⁿ), where D is a (countable) dense subset of L²(Rⁿ) and L the generator of the diffusion process X· given by

Lf(x) = 1 2

n

X

i,j=1

σij(x) ∂²

∂x_i∂x_jf(x) +

n

X

i=1

bi(x) ∂

∂x_if(x) (28) withσ(x) = (σij(x))1≤i,j≤nandb(x) = (b1(x), ..., bn(x))^∗and whereNe(ds, dς) is the compensated Poisson random measure associated with the L´evy process Yt,0≤t≤T under π.

Proof. It follows from Itˆo’s Lemma forf ∈C_c^∞((Rⁿ) that f(X_t) =f(X₀) +

Z t 0

Lf(X_s)ds+ Z t

0

∇^∗f(X_s)σ(X_s)dB_s^X,

where∇^∗denotes the transposed gradient. On the other hand we know that the processZ_t,0≤t≤T in Theorem??satisfies the SDE

Zt= 1+

m

X

i=1

Z t 0

Zshi(s, Xs)dB_sⁱ+ Z t

0

Z

R^m0

Zs−(λ(s, Xs, ς)−1)Ne(ds, dς). (29) So using integration by parts, we obtain that

Ztf(Xt) = f(X0) + Z t

0

ZsLf(X_s)ds+ Z t

0

Zs∇^∗f(Xs)σ(Xs)dB_s^X +

m

X

i=1

Z t 0

Zsf(Xs)hi(s, Xs)dB_sⁱ +

Z t 0

Z

R^m0

Zs−f(Xs)(λ(s, Xs, ς)−1)Ne(ds, dς).

The conditional expectation with respect toF_t^Y applied to the latter equa- tion combined with Lemma??then gives

hΨ_t, fi = hΨ₀, fi+Eπ[ Z t

0

ZsLf(Xs)ds F_t^Y

+E_π[ Z t

0

Z_s∇^∗f(X_s)σ(X_s)dB_s^X F_t^Y

+

m

X

i=1

E_π[ Z t

0

Z_sf(X_s)h_i(s, X_s)dB_sⁱ F_t^Y

+Eπ[ Z t

0

Z

R^m0

Zs−f(Xs)(λ(s, Xs, ς)−1)Ne(ds, dς) F_t^Y

= hΨ₀, fi+ Z t

0

hΨ_s,Lfids+

m

X

i=1

Z t 0

hΨ_t, f ·h_i(s,·)idB_sⁱ +

Z t 0

Z

R^m0

hΨs−, f ·(λ(s,·, ς)−1)iNe(ds, dς),

where we used Lemma??, Remark??, the continuity of the paths ofX_t,0≤ t≤T, the continuity ofλand (??) in connection with uniform integrability under the measureπ.

Remark 8 The condition (??) in Theorem ??holds, if e.g.

sup

0≤s≤T

Eπ[Z_s^pr]<∞ and

sup

0≤s≤T

Eπ[|λ(s, X_s, ς)−1|^pq]< C

for all ς, some constant C withpr <2, ¹_r+¹_q = 1, r, q >1 are satisfied.The latter conditions are e.g. covered by the conditions B1−B6 in the paper later on.

In addition to the conditions (??), (??) let us from now on also require that the drift coefficient bis bounded andσ =Id(identity).

Using the independence of the increments of the observation process Y·

under π and the probability density of the signal process Xt, which ex- ists in this case, our assumptions on b, h, λ and ν imply that there is an F_t^Y−adapted process Φ(t,·),0 ≤ t ≤ T, called unnormalized conditional density, such that

hΨ_t, fi= Z

Rⁿ

f(x)Φ(t, x)dx,0≤t≤T

for all f ∈Cb(Rⁿ). Hence we can recast the Zakai equation (??) in terms of the unnormalized density and find that Φ(t,·),0 ≤ t ≤ T satisfies a stochastic Fokker-Planck-equation, that is the SPDE

d_tΦ(t, x) = L^∗Φ(s, x)dt+ (30)

Φ(t, x)h^∗(s, x)dB_t+ Z

R^m0

Φ(t⁻, x)(λ(s, x, ς)−1)Ne(dt, dς) Φ(0, x) = p₀(x),

whereL^∗ is the adjoint operator of the generatorL of Xt and wherep0(x) is the probability density of X₀, in a weak sense, that is Φ ∈ L²_loc([0, T]× Rⁿ;L²(Ω)) is F_t^Y−adapted process, which solves the equation

Z

Rⁿ

Φ(t, x)f(x)dx (31)

= Z

Rⁿ

p0(x)f(x)dx+ Z t

0

Z

Rⁿ

Φ(s, x)Lf(x)dxds +

Z t 0

Z

Rⁿ

Φ(s, x)h^∗(s, x)f(x)dxdBs

+ Z t

0

Z

R^m0

Z

Rⁿ

Φ(s⁻, x)(λ(s, x, ς)−1)f(x)dxNe(ds, dς),0≤t≤T for allf ∈C_c^∞((Rⁿ).

In fact, it was shown in [?] that the Zakai equation for the unnormalized density (??) has a unique strong solution Φ(t, x) to (??) in L^p(µ), p ≥ 1, which is twice continuously differentiable inx, under the following conditions

A1 : The L´evy measure ν is bounded.

A2 : The drift coefficientb is contained in C_b^2+β(Rⁿ).

A3 : The initial conditionp0 in (??) is positive and belongs to C_b^2+β(Rⁿ).

A4 : The intensity rateλis strictly positive and λ(·,·, ς)∈ C_b^1,2(R+×R^m)∩C^2+β(R+×R^m) uniformly in ς.

A5 :

n

X

i=1

∂

∂x_ib_i ∈C_b²(Rⁿ)∩C^2+β(Rⁿ).

A6 : The observation functionh is contained in C_b^1,2(R+×Rⁿ)∩C^2+β(R+×Rⁿ).

A7 : Λ_t,0≤t≤T in (??) is a martingale,

where C_b^l,r(R+×R^d) is the space of l-times in t ∈ (0,∞) and r-times in x ∈ R^d continuously differentiable, whose partial derivatives are bounded and have continuous extensions to R+ ×R^d (R+ := [0,∞)). The space C^r+β(U) denotes the space of functions inC^r(U) with all partial derivatives up to orderr being H¨older continuous of orderβ ∈(0,1).

Moreover, the strong solution Φ to (??) has the following explicit repre- sentation:

Φ(t, x, ω) (32)

= E_ϑ^x[p0(X_t^∗(θ)) exp(−

n

X

i=1

Z t 0

∂

∂x_ibi(X_s^∗(θ))ds) exp{

Z T T−t

h^∗(s, X_s−(T^∗ _−t)(θ))dBs(ω)−1 2

Z T T−t

h(s, X_s−(T^∗ _−t)(θ))

2

ds +

Z T T−t

Z

R^m0

log(λ(s, X_s−(T^∗ _−t)(θ), ς))Ne(ds, dς, ω) +

Z T T−t

Z

R^m₀

(log(λ(s, X_s−(T^∗ _−t)(θ), ς))−(λ(s, X_s−(T^∗ _−t)(θ), ς)−1))dsν(dς)}]

whereX_s^∗(θ) =Xs^∗,x(θ),0≤s≤T,is the solution to the time-homogeneous SDE

dX_t^∗ =−b(X_t^∗)dt+dB_t^∗, X₀^∗ =x∈R^d (33) for a Brownian motionB_·^∗,defined on an auxiliary probability space (Θ,K, ϑ).

In order to capture ”regime switching effects” in the framework of our model for the stochastic transition rates µ_ik(t, x),0 ≤ t ≤ T in (??) and in view of Monte Carlo simulation techniques with respect to such transi- tion rates, we now want to extend the representation of Φ (??) under the conditions A1−A7 to the case, when the drift coefficent b of the signal process is merely bounded and measurable. In addition, we aim at relaxing the condition A1 of compound Poisson L´evy measures ν in (??) to that of finite-variation L´evy measures ν satisfying (??). Furthermore, we will show that such a Φ solves the Zakai equation (??) in the weak sense.

To this end we need to recall the concept of stochastic integration over the plane with respect to Brownian local time. See [?]:

Consider elementary functions f_∆: [0,1]×R−→R given by f∆(s, x) = X

(sj,xi)∈∆

fijχ_(sj,sj+1](s).χ_(xi,xi+1](x), (34)

where (xi)_1≤i≤n,(fij)1≤i≤n,1≤j≤mare finite sequences of real numbers, (sj)_1≤j≤m a partition of [0,1] and ∆ = {(s_j, x_i),1≤i≤n,1≤j≤m}. Denote by {L(t, x)}_{0≤t≤1,x∈}

R the local time of a 1−dimensional Brownian motion B.

Then the integral of integration off∆with respect to Lis defined as Z 1

0

Z

R

f_∆(s, x)L(ds, dx) (35)

= X

(sj,xi)∈∆

f_ij(L(s_j+1, x_i+1)−L(s_j, x_i+1)−L(s_j+1, x_i) +L(s_j, x_i)).

The latter integral can be generalized to integrands of the Banach space (H,k·k) of measurable functions f endowed with the norm

kfk = 2 Z 1

0

Z

R

(f(s, x))²exp(−x² 2s)dsdx

√2πs 1/2

(36) +

Z 1 0

Z

R

|xf(s, x)|exp(−x²

2s) dsdx s√

2πs.

Iff is such thatf(t,·) is locally square integrable andf(t,·) continuous intas a map from [0, T] toL²_loc(R),thenf ∈ H and

Z t 0

Z

R

f(s, x)L(ds, dx), 0≤t≤T exists as well as

E

Z t 0

Z

R

f(s, x)L(ds, dx)

≤ kfk for 0≤t≤T. Further, iff(t, x) is differentiable in x, then

Z _t

0

Z

R

f(s, x)L(ds, dx) =− Z _t

0

f⁰(s, B_s)ds , 0≤t≤T , wheref⁰(s, x) denotes the derivative inx. See [?].

Assume now that ¯Bt=

B¯⁽¹⁾_t , ...,B¯_t⁽ⁿ⁾

,0≤t≤T is a Brownian motion, whose components ¯B⁽ⁱ⁾_t are defined on probability spaces (Ω_i,F_i, µ_i), i = 1, ..., n. In what follows we denote by

Bb_t:= (Bb_t⁽¹⁾, ...,Bb_t^(d)) := ¯B_T−t, 0≤t≤T , (37)

the time-reversed Brownian motion. The processBb_t⁽ⁱ⁾ satisfies for each i= 1, ..., dthe equation

Bb⁽ⁱ⁾_t = ¯B⁽ⁱ⁾₁ +Wf_t⁽ⁱ⁾− Z _t

0

Bbs⁽ⁱ⁾

T −sds , 0≤t≤T , a.e., (38) where fW_t⁽ⁱ⁾, 0 ≤t≤ T are independent µ_i-Brownian motions with respect to the filtrations F_t^Bb(i) generated byBb·⁽ⁱ⁾,i= 1, ..., n. See [?].

Using the relation (??) one obtains the following decomposition of local time-space integrals (see [?]):

Z t 0

Z

R

f_i(s, x)L_i(ds, dx) (39)

= Z t

0

fi(s,B¯_s⁽ⁱ⁾)dB¯_s⁽ⁱ⁾+ Z T

T−t

fi(T−s,Bb_s⁽ⁱ⁾)dWf_s⁽ⁱ⁾

− Z T

T−t

fi(T−s,Bb_s⁽ⁱ⁾) Bbs⁽ⁱ⁾

T −sds,

0≤t≤T, a.e. for f_i∈ H, i= 1, ..., n. HereL_i(t, x) is the local time of ¯B·⁽ⁱ⁾

on (Ωi, µ_i),i= 1, ..., n.

In the sequel we also need the following auxiliary result (see also [?]):

Lemma 9 Let B_t,0≤t≤T be a 1−dimensional Brownian motion. Then E

exp

k

Z T 0

|B_t| t dt

<∞ for allk≥0.

Proof. See Appendix.

Let us now assume that the following conditions are satisfied B1 : The L´evy measure ν fulfills condition (??).

B2 : The drift coefficient bis Borel measurable and bounded.

B3 : The initial condition p₀ in (??) is positive and belongs to C_b^2+β(Rⁿ).

B4 : The intensity rateλis strictly positive and λ(·,·, ς)∈ C_b^1,2(R+×Rⁿ)∩C^2+β(R+×Rⁿ) uniformly inς.

B5 : The observation function h is contained in C_b^1,2(R+×Rⁿ)∩C^2+β(R+×Rⁿ).

B6 : λsatisfies (??)-(??), (??)-(??)

and the following integrability conditions

sup

x∈U

E[exp(200{

Z T 0

Z

R^m0

(λ(s,B¯_s^x, ς)−1

(40)

+max¹⁰⁰

n=1

Z T 0

Z

R^m0

(λ²ⁿ(s,B¯_s^x, ς)−1

dsν(dς)})]

< ∞ and

sup

x∈U

E



 Z T

0

Z

R^m0

logλ(s,B¯_s^x, ς)

idsν(dς)

!8

<∞, (41) for alli= 1,2,4,8 and all boundedU ⊂Rⁿ.

We mention that condition B2 implies the the existence of a unique strong solutionX_·^∗ to the the SDE (??). See e.g. [?].

We obtain the following existence result for weak solutions of a singular stochastic Fokker-Planck equation driven by L´evy noise, that is the SPDE (??) with the adjoint operator L^∗ of the generator L of Xt for merely bounded and measurable drift coefficients b:R^d−→R^d:

Theorem 10 Suppose that the conditionsB1−B6hold. Then there exists a weak solutionΦto the SPDE (??), which is given in law by the unnormalized

density and takes the explicit form

Φ(t, x, ω) (42)

= Eϑ[p0( ¯B_t^x(θ)) exp(

n

X

i=1

{ Z t

0

bi( ¯B^x_s(θ))dB¯_s⁽ⁱ⁾+ Z T

T−t

bi(Bb_s^x(θ))dfW_s⁽ⁱ⁾

− Z T

T−t

bi(Bb_s^x(θ)) Bb_s⁽ⁱ⁾ T−sds}) exp{

Z T T−t

h^∗(s,B¯_s−(T^x _−t)(θ))dB_s(ω)−1 2

Z T T−t

h(s,B¯_s−(T^x _−t)(θ))

2

ds +

Z _T

T−t

Z

R^m0

log(λ(s,B¯^x_s−(T−t)(θ), ς))Ne(ds, dς, ω) +

Z T T−t

Z

R^m0

(log(λ(s,B¯_s−(T^x _−t)(θ), ς))

−(λ(s,B¯_s−(T^x _−t)(θ), ς)−1))dsν(dς)}E( Z T

0

−b^∗( ¯B_s^x(θ))dB¯s)],

where Eϑ denotes the expectation with respect to the product measure ϑ = µ₁×...×µ_nwithB¯·⁽ⁱ⁾ is a Brownian motion on(Ωi, µ_i),i= 1, ..., n,B¯_t^x(θ) :=

x+ ¯B_t(θ) andBb_t^x(θ) :=x+ Bb_t(θ). Further, E(

Z t 0

−b^∗( ¯B_s^x(θ)dB¯s)

= exp(

Z t 0

−b^∗( ¯B^x_s(θ))dB¯s(θ)−1 2

Z t 0

b( ¯B_s^x(θ)

2ds),0≤t≤T is the Doleans-Dade exponential.

Proof. The proof is based on the explicit representation for the unnor- malized density Φ in (??) and an approximation argument with respect to the functionband the L´evy measureν.

Consider a sequence of Borel sets U_r, r≥ 1 of R^m₀ with U_r % R^m₀ such that ν(Ur) <∞ for all r. Define the compound Poisson L´evy measures νr

by

νr(B) = Z

B

1_Ur(ς)ν(dς),

where 1A is the indicator function of a set A. In the sequel we denote by N_r(ds, dς) the Poisson random measure associated with the L´evy measure ν_r,r≥1.

Let us also choose functions b^r∈C_c^∞(Rⁿ), r≥1 such that kb^r(x)k ≤M <∞

for a constantM and all x, r as well as

b^r(x)−→b(x) a.e.

forr−→ ∞.

In the following let us denote by Φ_r the unique (strong) solution to the SPDE (??) with respect to the drift coefficent b^r and by X_t^∗,r, 0 ≤ t ≤ T the strong solution to the SDE

dX_t^∗,r =−b^r(X_t^∗,r)dt+dB^∗_t, X₀^∗,r =x∈Rⁿ (43) for allr.

In what follows let x∈U for a bounded setU ⊂Rⁿ.

Using Girsanov’s theorem and the explicit representation of Φr in (??) forb=b^r and ν=ν_r based on the conditionB6 we find that

Φ_r(t, x, ω) (44)

= E_ϑ[p₀( ¯B_t^x(θ)) exp(

n

X

i=1

{ Z t

0

∂

∂xi

b^r_i( ¯B_s^x(θ))ds) exp{

Z T T−t

h^∗(s,B¯_s−(T^x _−t)(θ))dB_s(ω)−1 2

Z T T−t

h(s,B¯_s−(T^x _−t)(θ))

2

ds +

Z _T

T−t

Z

R^m0

log(λ(s,B¯^x_s−(T−t)(θ), ς))Ne_r(ds, dς, ω) +

Z T T−t

Z

R^m0

(log(λ(s,B¯_s−(T^x _−t)(θ), ς))

−(λ(s,B¯_s−(T^x _−t)(θ), ς)−1))dsνr(dς)}E(

Z T 0

−(b^r( ¯B_s^x(θ)))^∗dB¯s)].

If we apply (??) to ¯B⁽ⁱ⁾· on (Ω_i, µ_i) for fi(s, z)

= b^r_i(x1+ ¯B_s⁽¹⁾(ω1), ...,

xi−1+ ¯B_s⁽ⁱ⁻¹⁾(ωi−1), z, x_i+ ¯B_s⁽ⁱ⁺¹⁾(ω_i+1), ..., x_n+ ¯B⁽ⁿ⁾_s (ω_n))

then we get that

Φr(t, x, ω) (45)

= E_ϑ[p0( ¯B_t^x(θ)) exp(

n

X

i=1

{ Z t

0

b^r_i( ¯B_s^x(θ))dB¯_s⁽ⁱ⁾+ Z T

T−t

b^r_i(Bb_s^x(θ))dWf_s⁽ⁱ⁾

− Z T

T−t

b^r_i(Bb_s^x(θ)) Bb⁽ⁱ⁾s

T −sds}) exp{

Z T T−t

h^∗(s,B¯_s−(T^x _−t)(θ))dBs(ω)−1 2

Z T T−t

h(s,B¯_s−(T^x _−t)(θ))

2

ds +

Z T T−t

Z

R^m0

log(λ(s,B¯^x_s−(T−t)(θ), ς))Ne_r(ds, dς, ω) +

Z T T−t

Z

R^m0

(log(λ(s,B¯_s−(T^x _−t)(θ), ς))

−(λ(s,B¯_s−(T^x _−t)(θ), ς)−1))dsνr(dς)}E(

Z T 0

−(b^r( ¯B_s^x(θ)))^∗dB¯s)].

Then it follows from the mean value theorem that E[(Φr(t, x, ω)−Φ(t, x, ω))²]

=E[(p₀( ¯B_t^x(θ)))²(I₁^r+I₂^r+I₃^r)²Z 1 0

exp(I₀^r+τ(I₁^r+I₂^r+I₃^r))dτ2

], whereEis an expectation with respect to a probability measure under which Y·associated with the bounded and measurable drift coefficientbis the L´evy process of the type in (??) and where

I₀^r:=I_0,1^r +I_0,2^r +I_0,3^r with

I_0,1^r :=

n

X

i=1

{ Z t

0

b^r_i( ¯B_s^x(θ))dB¯_s⁽ⁱ⁾+ Z T

T−t

b^r_i(Bb^x_s(θ))dWf_s⁽ⁱ⁾

− Z T

T−t

b^r_i(Bb_s^x(θ)) Bb_s⁽ⁱ⁾ T−sds},