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Risk and Uncertainty in the Patent Race: a Probabilistic Model

This article develops a novel probabilistic approach to evaluate -to be intended as provide an approximation formulaa patent-protected R&D project at a fixed date under very general conditions. In a real option framework we introduce Spatial Mixed Poisson Processes to describe the dynamics of the project value. In such a fashion, the model is able to account for competition among firms and several sources of uncertainty such as time-to-completion of the project, exogenous shocks, input cost uncertainty, technical uncertainty and asymmetric information under different cost structures. The proposed evaluation procedure is of the Bayesian type, in that it moves from the knowledge of some information collected before the evaluation

Optimal decision-making for taking out a patent is a complex task, exacerbated by the presence of competitiveness and uncertainty. These two characterizing features are intrinsically related, reciprocally entangled. Competitiveness is a source of uncertainty and the latter feeds into the former making it tougher. Indeed, in a competing context the decision maker has to take into account not only the factors that affect his own strategies, but also the factors affecting other investors' decisions.
Uncertainty about others' actions -on the one side-urges taking out the patent in order to avoid being cut out by preemption. On the other side, there are situations in which it may be convenient to wait for the others' move. An example of the latter situation is provided by the millions of vaccine doses for H1N1 flu unsold, or unused and resold from developed to developing countries in 2010. In this case, it would have been optimal to wait for the others' action, at the risk of losing the first mover's benefit.
This paper deals with the basic step of this decision process, which consists in estimating the project value. More specifically, the main contribution consists in proposing a method to approximate the expected value of the patent at a fixed date. The procedure we implement to obtain such a result is based on the information collected before the evaluation date, and in this sense it may be viewed as of the Bayesian type.
Patent race, namely the issue of valuing an R&D project finalized to take out a patent in a competing and uncertain environment, has extensively been treated in the economic literature (Kamien and Schwartz, 1972;Loury, 1979; Dasgupta and Stiglitz, 1980;Fudenberg et al., 1983;Vickers, 1985, 1987;Beath et al., 1989;Nti, 1997  The model we are going to present in the following sections accounts for all the sources of uncertainty aforementioned and the firms are assumed to operate in a competitive environment. Encompassing all the sources of uncertainty in a unique model is quite difficult, because it greatly complicates the algebra, but the task is worth tackling given the importance that uncertainty plays in R&D management and ultimately in policy intervention. Within the real options framework, the task can be accomplished by abandoning the continuous time framework, typical of patent race models, and by using a particular type of discrete process, the Spatial Mixed Poisson Process (SMPP). This process generalizes the standard Spatial Processes derived by jumps occurring according to a Poisson Law.
The adoption of random point processes in applied works is well-acknowledged in the literature. We In our context, from an economic point of view, the use of SMPP allows to overcome some restrictive hypothesis, typical of R&D valuation literature. In particular, first of all, it allows tackling the problem of determining the value of a project related to a patent in the presence of jumps in the underlying asset with negative and positive sizes. Secondly, thinking of a negative jump due to other firms attainment, it can happen, and indeed it does, that the novelty contained in the others' patent is not so crucial, being, let us say, an invention around attainment. 1 This aspect is particularly relevant, in that it captures the fact that negative events can engender heterogesous impacts on the patent value according to the events themselves, but also to the timing of the event. For instance, the same event will likely impact differently on the value of the patent according to whether it occurs before or after the proclamation of a new patent-protecting law or of a new fiscal regulation. In general, the patent under valuation suffers from a negative jump in its value, but this may however not be large enough to make the patent worthless. An original feature of our approach is that the amount of the jumps is not known beforehand. To sum up, the killing jump is not restricted to the first negative jump that may occur, but only a large enough jump can kill the patenting activity. Thirdly, there is a random delay in the transmission of the jump from the stochastic process representing a given source of uncertainty to the cash flow generated by the project. This plays an important role in modeling random processes with different jump behaviors. It has never been clarified in the existing literature why other firms' attainment should bring about an immediate obsolescence in the 1 By invention around it is commonly meant an invention which achieves the same or similar functions using a different means and not violating the claims of the original patent. patent that is being valued. For instance, in the consumer electronics field it takes time before a new product is marketed and, once it is, it takes time before it is sufficiently widespread to cut to nil the profit deriving from the "old" patent. In such a case, the assumption of a no-delay condition would be a naïve approximation. Fourthly, we accommodate the case of the winner-takes-all as a special case of the winner-does-not-take-all game. Fifthly, exogenous shocks are supposed to impact with different intensities on the competing firms. Finally, the adoption of SMPP allows to take into account the dependence between time and size of the jumps in the underlying security. We will turn to these points in more details below.
The rest of the paper is organized as follows. The next Section outlines the basic problem motivating this research. Section 3 is devoted to the construction of a theoretical framework, taking into account multiple sources of uncertainty. IN addition, this Section contains some preliminary results, along with the formal definition of SMPP. Section 4 concerns the technical conditions stopping the patent race. A Bayesian estimate of the value of the project is derived in Section 5 and its economic implications are discussed in Section 6. Finally, Section 7 concludes and outlines future research.
Some mathematical derivations are provided in the Appendix.

The basic problem
Consider a new technology developed by firm A that can possibly be monopolized by taking out a patent. Throughout the paper we will refer toP as the patent, or projectP , or simply the project.
Let us suppose the economic environment to be populated by other N competitive firms investing in R&D to develop new technologies, including the one related toP . We adopt the point of view of firm A and we seek a theoretical estimate of the expected value of the project. In doing so, a real option type approach is followed 2 . The patent race problem is strictly connected to the study of the optimal time for taking out the patentP . Clearly, the patent is attainable after the full development of the related technology, let us say at the random time τ 1 , representing uncertainty in time-to-completion. It follows that the problem of optimal timing for patent protection is considered only when the firm accomplishes the R&D phase and is ready to register the patentP .
At its very essence, the structure of the model works as follows. In accordance with the real options literature, the cash flow is taken as the underlying security, or state. The changes in the cash flow immediately affect the value of the derivative, namely the project. In similarity with financial models, the real options literature treats the value of the project as a derivative, which means that it is supposed to be a function of its underlying security. The value of the project does not coincide with the value of the cash flow in that the firms have forward looking expectations. Typically, 2 Notice that by the words "theoretical estimate" we refer to the fact, that being the jumps in the underlying security stochastic in time and size and being the transmission time of these jumps stochastic itself, it is necessary to estimate the expected value of the impact of the jumps on patenting activity. R&D investments do not generate cash flow in an early stage, but they unfold their productive effects after a (long) period. In turn, this fact does not imply that the project is worthless in the investment period, as rational economic agents take into account future returns and they know that even if the project does not immediately generate cash flow, it will do in the future. The cash flow is supposed to be composed of two parts: a regular flow deriving from the scheduled outflows and an extraordinary flow, subject to stochastic fluctuations due to the economic context and competition.
More specifically, some events occur during the race at discrete times. These events are captured by the jumps of N + 1 distinct stochastic point processes. One of these represents the shocks occurring in the economy, e.g. changes in regulation, demand shifts, taste variations, etc. The remaining N processes represent the progresses of the other N firms taking part in the race. The random jumps occurring in the N + 1 processes affect A's cash flow with a certain random intensity and at random time. We will come back to the details of the transmission mechanism which determines the sizes of the jumps in Section 3. For the time being, we highlight that the time at which the events occur in the N + 1 processes are not necessarily the same time at which the cash flow is affected. Indeed, the jumps in the N + 1 processes do not immediately propagate to A's cash flow, but only after a certain random delay. Moreover, also the N players different from A are affected by exogenous shocks. It is worth recalling that the jumps in the cash flow propagate immediately to the value of the project, without any delay. Since the cash flow jumps with a certain delay with respect to the occurrence of the event, the jumps in the cash flow that will occur "tomorrow", are partially due to the jumps in the N + 1 stochastic processes occurred "today". Hence, to value the patenting activity in a future date one must give an estimate of the jumps in the N + 1 processes already occurred, the effect of which will be exerted in the future.
Summarizing: the underlying security, A's cash flow, is affected by exogenous shocks and by the events affecting the other N players. In turn, the latter are affected by exogenous shocks as well.
Hence, exogenous shocks feed into the underlying security twice, directly and indirectly via the competitors. The transmission of the shocks is not restricted to be immediate and one to one, but occurs with random delay and intensity. Figure 1 graphically depicts this composite situation.

INSERT FIGURE 1 ABOUT HERE
Caption: The jumps occurring in the economy affect the firm's cash flow with a random delay.
The delayed transformations are captured by the arrays in the graph.
The cash flow process is such to account for uncertainty of cash flow itself as well as technical uncertainty. The process describing exogenous shocks accounts also for input cost uncertainty. The random processes, by modeling competition, account for uncertainty coming from a competitive environment and asymmetric information under an asymmetric cost structure. The context is assumed to be one in which the winner does not necessarily take all. To formalize the model, let us introduce a probability space with filtration (Ω, F, {F t } t≥0 , P ), containing all the random variables introduced in the following. We denote as T the set of stopping times in [0, +∞), i.e. (1) To model time-to-completion uncertainty, let us define the random time at which the new technology is fully developed by A as τ 1 ∈ T . The optimal timing problem can be rephrased in searching for the random time τ 2 that is optimal for taking outP and such that τ 2 (ω) ≥ τ 1 (ω), for each ω ∈ Ω.
Fix t > 0 and denote as C t the random value of the projectP for A at time t. The quantity C t is random in that it depends on the future events concerning the life of the firms involved in the patent race and on the economic environment.
The patent is assumed to be optimally taken out when the expected value of C t reaches a certain time-dependent deterministic threshold Γ t . More formally, let us introduce the difference process where E indicates the expected value operator. Thus, the stopping time τ 2 can be written as follows: Of course A must compete with several other firms, which try to obtain the registration of the same or similar patents. Therefore, τ 1 or τ 2 may also be equal to +∞ when the patent race ends in favor of a firm different from A. We will label these situations as game over conditions, and they will be discussed in Section 4.
The optimization problem related to τ 2 in (3) depends on two components: the time-dependent threshold Γ t and the expected value of the project E[C t ]. This paper proposes a Bayesian method for estimating E[C t ] in a very general framework. Since the estimation procedure is quite technical, we have chosen to focus only on this issue, without devoting much efforts neither to the identification of Γ t , which can be exogenously or endogenously determined, nor to the optimization problem.
Nevertheless, the results obtained hereafter will set up the basis for a follow up study coping with the optimization problem. For the time being, tackling the optimization problem exceedingly complicates and lengthens the paper.

The sources of uncertainty and the underlying asset
The theoretical analysis of patent race interactions is given through the introduction of quantitative processes translating qualitative occurrences. This approach has been already used in the literature, 6

Overview of the main elements of the model
This subsection introduces the main components of the patent race problem. Since the argument is rather technical, we prefer to treat separately the conditions granting the existence of competition among firms (the so-called game over conditions, see the next section). For the time being it will be assumed that the patent race never stops.
Let us introduce a stochastic process T describing the evolution of A's cash flow, directly related toP . We also assume that all the sources of uncertainty impact on the cash flow generated by the R&D investment project.It follows that T collects R&D expenses, the inflows and the outflows stemming from the events in the economic environment and the competition with the other N firms populating the market. A formal definition of T will be presented below.
The time-dependent value ofP for A can be explicitly written by collecting information on the process T . For this reason T represents the underlying asset.
To formalize the sources of uncertainty and analyze their interaction rules, a multivariate point process is needed, with N + 1 components given by: an exogenous process S that captures the quantitative translation of the shocks occurring in the economic system, including input cost uncertainty; a stochastic process U k describing the cash flows of the k-th firm's R&D process, with k = 1, . . . , N .
All the sources of uncertainty affect A ′ s cash flow (see Figure 2 for a representation of such reticular relationships).

INSERT FIGURE 2 ABOUT HERE
Caption: the arrows point out that the realizations of T are driven by the evolution of the multivariate process (S, U 1 , . . . , U N ) and, furthermore, U k depends on S, for each k.
From A's point of view, U k represents a proxy for the speed of the technological renewal process and the interest in new technologies of the k-th firm involved in the patent race. U k captures uncertainty arising from a competitive environment and asymmetric information under an asymmetric cost structure. In such a way, besides uncertainty arising from time-to-completion, we manage to capture uncertainty due to exogenous shocks, S, technical uncertainty (or entity of R&D expenses, or cost-to-completion), competitive interactions, asymmetric information and finally investment cost asymmetry. All these components impact differently on U k and are somehow reflected on A's cash flow. Modelling separately the dynamics of each competitor's cash flow, U k , we have the highest possible degree of asymmetry and uncertainty. 3 In his setting a Geometric Brownian Motion is assumed to capture the shocks in the economy, which feed into the underlying state dynamics, the industry's inverse demand curve. The underlying state determines the value of the option to invest. The exogenous shocks in the economy are captured by a process S = {(τ S i , ξ S i )} i∈N , where τ S i and ξ S i represent the random time and size of the i-th shock, for each i ∈ N, respectively. In particular, τ S i ∈ T , ξ S i ∈ F τ S i and ξ S i takes on values in R. The random variables ξ S i are supposed i.i.d. and independent from {τ S i } i∈N . Moreover, the following key assumption on the stochastic structure of S will stand in force hereafter. Assumption 1. The process S is a SMPP with mixing distribution Π and baseline intensity measure For a formal definition of SMPP, see Subsection 3.5.
An economic explanation of Assumption 1 is in order.
The process S captures also the component of uncertainty pertaining to the so called catastrophic events, that are related to negative jumps (McDonald and Siegel, 1986; Miltersten and Schwartz, 2004;Schwartz, 2004). However, hypothesizing only negative jumps to occur would be an unrealistic assumption. Indeed, there are many events which could have positive influence, e.g. a more favorable patent-protection law, more effective or rigorous enforcement policies, a complementary discovery accelerating patent attainment, a reduction in patenting cost, a reduction in the renewal fee schedule, etc. In addition, it is unrealistic to assume that the size of the jump is predetermined (in this respect see Weeds, 2002;Lambrect Perraudin, 2003). Abandoning the assumption that S follows a Poisson process, typical in the literature, and allowing it to follow a SMPP, we easily overcome both assumptions. 4 In this framework, jumps are not restricted to take on only negative values. Being ξ S i random, the size of the jump is unknown a priori. As a consequence, it is possible to generalize the effect of a negative jump and remove the restriction that any negative jump, in particular the first, can kill the patenting activity. Negative jumps "small" in size will induce a drawback, but not necessarily put an end to the activity. This characteristic is particularly important for R&D activity characterized by high entry and exit sunk costs. R&D cannot be costlessly suspended and resumed.
Think of the cost of firing the researchers and the legally required termination payments. In this case the Marshallian rule of negative Net Present Values (NPV) to shut down the activity does not hold (Dixit, 1989). The presence of exit sunk costs requires the NPV to be greatly negative before shutting down the plant. Therefore, in this context negative but not fatal jumps are quite likely to occur and the SMPP accommodates for this feature.
For further explanations on the reasonability of Assumption 1, we refer to Subsection 3.5. 4 The SMPP assumption is not strictly necessary to allow the process to take on positive jumps, but it is essential to introduce random sizes and derive our theoretical estimation result.  The sequence of couples U k = (τ Uk i , ξ Uk i ) formalizes the process describing times and sizes of the cash flow of the k-th firm, with k = 1, . . . , N , as they are perceived by A. The process U k collects the k-th firm's cash flow associated to the development of new technologies having the chance to take out new patents, includingP . It can be considered as a proxy for the R&D progress of the N firms playing the race. The randomness in U k also allows to capture the asymmetric information context, in that A cannot perfectly forecast the realizations of U k . The dependence between S and U k is conveniently modeled as follows.
3k )} i∈N , which are assumed to be i.i.d. and independent from S and take values in W Uk = [0, +∞) 2 × R, and define a transformation 3k are random variables as in (7), and represent a stochastic delay, a stochastic scale factor and a random growth factor, respectively.
By defining we obtain a representation of the process U k .

Cash flow: process T .
T is the process followed by A's cash flow and represents the underlying asset of the proposed real option model. In the Introduction, we have seen that given the long time profile of research projects, revisions to investment plans are very common in patenting activity. Therefore, without loss of generality, we can assume that the R&D process involves a regular flow and an extraordinary flow.
The regular flow aims at capturing the deterministic expenses sustained periodically by A to increase R&D. It is composed of a sequence of annuities occurring at fixed dates. Spector  cost uncertainty. Hence, the sources of the realizations of the extraordinary flow are the processes S and U k , with k = 1, . . . , N . Accordingly, we split the extraordinary flow in N + 1 processes T S , T 1 , . . . , T N , that are related to the jumps of T due to S, U 1 , . . . , U N , respectively.
Formally, the process T is a sum of N + 2 terms: The process which are assumed to be independent from S and take values in By defining we have a representation of the process T S associated to the jumps of the process S.  jump in the exogenous process that induces a jump in the cash flow at a later random time. Think of an environmental regulation forcing the firms to invest in expensive "scrubbers" or to buy tradable "allowances" that allow them to pollute, a sort of Clean Air Act. Most of the time the regulation does not force the firms to comply immediately with the new standard, letting the firm deciding when it is optimal to delay the green investment within a reasonable amount of time. In other words, there is a random delay between the jump in the exogenous process S and the related jumps in T .
The functional form of the size of T , ξ TS i , is such to accommodate the ambiguous relationship between good/bad news (Baudry and Dumont, 2006) and a jump in the cash flow. A straightforward example of good news is one that makes the total cost decrease, e.g. a reduction in input cost for any reason, but also good news that make the firm more eager to invest are conceivable. If a new regulation requiring clean energy adoption is enacted while the firm A is investing in more efficient solar panels, the firm will be spurred to finish the R&D phase as soon as possible, in order to commercialize the new product. For this reason, the effect of the news described by the random jump ξ S i is reduced or amplified by a random factor w In a competitive economic framework different types of managerial strategies can be adopted. One of the most natural ones is for A to establish economic relationships with some of the other firms populating the market. In doing so, the firms cluster in two families: A's partners and A ′ s rivals.
We then suppose the existence of N p partners and N r = N − N p rivals. Now, the process T k has to be defined by taking into account this distinction. Without loss of generality, we can order the N partners such that 1, . . . , N p are A's partners, while N p + 1, . . . , N are rivals.
Partnership or competitiveness are key issues to study the effect produced by a jump in U k on A's cash flow. Moreover, in a complex economic system, several technologies and related patents are under development at the same time. Therefore, a jump in the k-th firm's R&D process may induce different effects on the value ofP for A, depending on the relationship between A and the k-th firm, as well as on the degree of similarity between patents. In order to clarify the latter point we introduce an index α  We assume α when the technology implemented forP is closer to that related to the i 2 -th jump than to the one related to the i 1 -th jump. The limit cases are α (i) k = 0, occurring when the i-th jump has nothing to do withP , and α (i) k = 1, occurring when the k-th firm takes out patentP . Such an index enables also to consider the interconnections between different sectors in the economy.
That is, firms operating in the same sector are more likely to compete on very similar projects. If the k-th firm operates in the same sector of A, then the index α (i) k exhibits a high value for each i. Therefore, it is more likely that if a firm, say the k-th one, operating in the same sector of A takes out a patent, we will have a value of α The relation between U k and T in this case is formalized by defining 1k is a random delay, y 2k is a random scale factor and y (i) 3k is a random growth factor. The definition of φ P,k is in accordance with that of φ TS in (7), with a couple of key differences.
A random delay appears, due to the possible delay in the propagation of information from k to A. A term related to α (i) k is introduced, in order to describe the dependence of ξ Tk i on the type of technology implemented. If a new attainment is made by a partner of A, and such an attainment is complementary toP -in the sense that it helps the attainment ofP -then A benefits from this jump, and the value of the patent will increase. In this respect, we notice that the function is increasing with respect to α  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  is a random growth factor, and it formalizes the time dependence, as explained in the discussion of φ TS .
The relationship between T and U k , with k being A's rival is similar to that between T and U k with k partner, but with an opposite sign.
Define the sequence of trivariate i.i.d. random variables 3k )} i∈N , which are assumed to be independent from S and take values in Z R,k = [0, +∞) 2 × R, and introduce a transformation As usual, we formalize the relationship between U k and T by assuming: 2k is a random scale factor and z (i) 3k is a random growth factor. In the rest of the paper, we refer to T = (τ T i , ξ T i ) when referring to the process T without considering which process drives the jumps.  It is worth noting that the same sort of transmission mechanism for the exogenous shocks on A applies to the other N firms. That is, the value of the N firms' projects are supposed to be random and similar to A's, although not perfectly identical because, in general,

Preliminary comments and results
and (w  (14) and (15)  By a purely mathematical point of view, formulas (7) and (4) show that there is a correlation structure between ξ TS i and (τ S i , ξ S i ) as well as between ξ Uk i and (τ S i , ξ S i ). This fact does not allow to consider S as a stochastic process on the line, and we need to treat S as a Spatial Point Process.
In this respect, the benefits of assuming S to be a SMPP are twofold. First of all, it provides a model for simultaneously estimating the number and the size of the jumps in the economy and, consequently, the number and the size of the cash flows of the N + 1 firms populating the market, including A, and related toP . Secondly, according to some recent results, SMPPs can guarantee the invariance of the stochastic structure between S, T S and U k , for each k (see Theorem 1). The same argument applies to the correlation structure between ξ Tk i and (τ Uk i , ξ Uk i ), shown in (9) and (12).
In order to be self-contained, we introduce the formal definition of SMPP.

with distribution G and independent of S.
Then T = Φ φ (S, W) is a SMPP with the same mixing distribution Π and intensity measure where J ⊆ Y and X ∈ φ −1 w (J) if and only if φ(X, w) ∈ J.
Theorem 1 is a key result in our work, since it explains the invariance of SMPPs with respect to a very general class of transformations. As a consequence, we can write:

The game over conditions
In a patent race context, firm A loses the competition in two cases: (i) when A abandons the project to develop the technology associated toP ; (ii) when the technology has been already developed by A, but someone else registers the patent P before A.
The cases (i) and (ii) correspond to a certain jump in A's cash flow, which has been named in the Introduction as killing jump. In both cases, we achieve a situation where the cash flow process T stops, and the value of the project remains permanently below the threshold Γ in (2).
To model this occurrence, we need to introduce a time dependent threshold: such that Θ kills the value of the patent, e.g.
such that A killing jump may also depend on the registration of the patentP by another firm. The consequences in the long-run of this event differ according to the nature of the relationship between A and the winning firm, i.e. partnership or rivalry. However, the analysis of the future exploitation of the technology related toP is out of the scope of this paper. We can generally say that the registration ofP by the k-th firm is the end of the patent race, for each k = 1, . . . , N . This fact involves the relationship between the processes U k and T , and is associated only to the case of α (i) k = 1. Analogously to γ t defined in (19), there exists a time dependent threshold such that 5 The value of the project As clearly explained by Pakes and Simpson (1989) and Shankerman and Pakes (1986), fixed t ≥ 0, C t can be regarded as the value at time t of the project associated to the patentP . Actually, C t is a random variable in that it sums up also the discounted future random cash flow of A, and the estimate of the patent value will be pursued through the computation of its conditioned expected value.
In details, consistently with the definition of T and formula (6), we write C t as the sum of the contributions due to the ordinary and extraordinary flows or, more formally, as the discounted sum of the processes T O , T S , T 1 , . . . , T N : where F o r P e e r R e v i e w and the superscripts indicate the reference processes. By (24), the terms in the right-hand side of (23) can be written as: where β ∈ (0, 1) is an appropriate discount factor. By inserting the game over conditions and by formula (25), (23) becomes: A Bayesian-type mechanism for the estimation of the expected value of C T E t in (26)  In agreement with a commonly used mathematical notation, we introduce D ⊆   1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  • the number of jumps in the economy occurred over I propagated to A's cash flow in the same time interval. We denote this quantity as T E (IS ) (I T ) 5 , and it is given by the aggregation of the contributions of the processes T S , T 1 , . . . , T N , i.e.: We consider two partitions ∆ and Ψ of H as follows: where H r } is increasing with respect to r; where G A further refined partition of H can be obtained by the intersection of the partitions defined in (27) and (28). We have We fix the four integers r, v, h, j and denote by b r ], conditioned on the previous history in period I. i.e.
Proposition 2 allows to get an upper and a lower approximation of Consider the following sequences:φ For any j, h ∈ N, we haveφ {φ j,h } is non-decreasing and {φ j,h } is non-increasing with respect to j and h.
We now conveniently thicken the decomposition ∆ j ∩ Ψ h in (29). Assume that Moreover, assume that  We now add also C TO t to C T E t of (46), and derive an estimate for C t . By (26) we have: In order to solve (47) we need to compute the probabilities involved in it. Denote by F 1 the probability distribution of the random variable τ 1 . Then: The variable α (i) k may be assumed discrete. We have: Define the time dependent set I Θ (t) ⊆ R 2 as follows: The game over conditions, obtained counting the jumps such that P (ξ TS i > −Θ(τ TS i )) and P (ξ Tk i > −Θ(τ Tk i )) can be expressed through the processes T S and T k , respectively, and the set I Θ (t) defined in (50). Let us explain this point. We develop the theory only for T S , being the case of T k analogous.
Start with i = 1 and suppose that ξ TS 1 > −Θ(τ TS 1 ). Then go to i = 2 and check if ξ TS 2 > −Θ(τ TS 2 ). If this is the case, go to i = 3 and so on. The process stops at jumpī, where ξ TS ı ≤ −Θ(τ TS ı ). We then argue that the number of realizations of T S falling in the set I Θ (t) becomes equals to 1 for i =ī.

Consider now the set {t
Since T S is a simple process and by condition (43) we can assume that there exists an index dependent v(i) , for each i ∈ N. For the arguments developed above, we have a critical threshold associated toī, sayv(ī), such that the number of realizations of the process T S falling in I v Θ (t) is 0, for v = 1, . . . ,v(ī) − 1, while it is 1 when v =v(ī). Therefore, the indexv(ī) can be defined as: We put a subscript S and k to v, to distinguish the cases of T S and T k . The process T S and T k are not independent, since they both depend on the realizations of S. Therefore: Hence, by (53), the game over conditions in (47) become: By applying Definition 1, condition (54) becomes: Denote the right-hand side of (55) as Υ.
By substituting (48) and (55) into (47), we obtain: Equation (56) claims that the value of the project is given by the sum of two components. The first one, p(t), represents the discounted expected value of the jumps in the underlying security, while the second one concerns the discounted expected value of the ordinary expenses. In this quantity,  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  Looking at the time profile of the estimate we can claim that, p(t) comes from the algebraic sum of two components: where p where p In other words, p(t) =p H (t) is a special case of (56) and its occurrence is purely accidental.
Therefore, we argue that paying a little price in terms of algebraic effort, the model better fits the real world, providing us with a theoretical estimate of the value of the project. So far, the comparison between our model and an alternative naive one has been kept as simple as possible, but the divergence can be worsened if one considers also that computation of p(t) entails taking into account some remarkable features of the race that a naive approach completely neglects. Such as: the possibility of both positive and negative jumps, the possibility of non deadly jumps which are usually omitted from the models, the possibility of either the winner-taks-all or the winner-does not-take-all, the different sensitivity to exogenous shock of different firms. Put another way, the higher the uncertainty in the race, the greater the divergence between p(t) andp H (t).
A quantitative appraisal of the difference is far beyond the scope of this paper. We limit ourselves to bring to notice how neglecting some source of uncertainty can heavily bias the estimation of the project.  1  2  3  4  5  6  7  8  9  10  11  12  13  14  15  16  17  18  19  20  21  22  23  24  25  26  27  28  29  30  31  32  33  34  35  36  37  38  39  40  41  42  43  44  45  46  47  48  49  50  51  52  53  54  55  56  57  58  59  The real option theory is based on the definition of an underlying asset which evolves stochastically over time and uncertainty feeds back dynamically in the value of the derivative. In this paper we have shown that by exploiting this basic idea it is possible to model patent race taking into account the multiple sources of uncertainty it implies, without requiring unrealistic or abrupt assumptions, such as winner-takes-all or winner-does-not-take-all. At the same time, we depart from the standard real option framework by considering an underlying asset evolving in discrete time as a Spatial Mixed Poisson Process. To our best knowledge, this is the first paper that models the entire set of uncertainty sources in a unique framework, providing a theoretical estimate of the value of the project. Moreover, the introduction of a distance measure between competing projects enables to account for both complementarity between innovations and winner-takes-all as possible outcomes of the race. Finally, our model allows to handle the case in which players of the race form alliances.
The processes U k are generated by the exogenous process S, for each k = 1, . . . , N . Therefore, fixed k = 1, . . . , N , there exists a family of sets of indexes {j k,1 , . . . , j k,Kk } ⊂ {i 1 , . . . , i K }, such that: {(τ Uk jk,1 , ξ Uk jk,1 ), . . . , (τ Uk andK k ≡ U k(I * S ) (I k ), where the subscript (I * S ) means that the cash flows of the processes U k stem from events captured by jumps of S occurring in I S under (20).